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Predicting high school completion using student performance in high school algebra

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Predicting high school completion using student performance in high school algebra a mixed methods research study
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Chiado, Wendy S
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Algebra ( lcsh )
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Too many of our nation's youth have failed to complete high school. Determining why so many of our nation's students fail to graduate is a complex, multi-faceted problem and beyond the scope of any one study. The study presented herein utilized a thirteen-step mixed methods model developed by Leech and Onwuegbuzie (2007) to demonstrate within a large urban Rocky Mountain school district, the existence of a significant relationship between high school algebra performance and high school completion and to provide one suggested framework by which to explain the nature of this relationship. Quantitative analysis showed that a failing grade in a student's first high school algebra course is not only a significant predictor of failure to graduate high school but was more liable, than several other predictors, to forecast which students are more likely to leave high school before earning a diploma within four years. However, quantitative examination revealed only the existence of a significant relationship between these two variables. It fell short of providing any elucidation as to why performance in high school algebra can be considered as a predictor of failure to graduate high school. Qualitative analysis, using the tradition of phenomenology (Moustaskas, 1994), was employed to expand on the quantitative analysis and offer a hypothesis from a student-centered perspective as to why this relationship may exist. Several students, who have returned to school after dropping out or who have been in high school for more than four years, were interviewed about their individual experiences in high school algebra classes and the greater high school environment. The qualitative analysis results did not conclusively agree with the quantitative results, and failed to conclusively point to high school algebra as the overwhelming reason for high school dropout. Nonetheless the interviews revealed that the students perceived numerous negative sources of self-efficacy and expectancies that, when combined with the uniqueness of the high school algebra experience, increased the likelihood of failing to graduate high school.
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Includes bibliographical references.
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by Wendy S. Chiado.

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Full Text
PREDICTING HIGH SCHOOL COMPLETION
USING STUDENT PERFORMANCE
IN HIGH SCHOOL ALGEBRA:
A MIXED METHODS RESEARCH STUDY
by
Wendy S. Chi ado
M. S., Naval Postgraduate School, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Educational Leadership and Innovation
2012


This thesis for the Doctor of Philosophy degree by
Wendy S. Chi ado
has been approved for the
Educational Leadership and Innovation Program
by
Dr. Michael Marlow, Chair and Advisor
Dr. Alan Davis
Dr. Honorine Nocon
Dr. Mary Thurman
Date April 10. 2012
n


Chiado, Wendy S., Ph.D., Educational Leadership and Innovation
Predicting High School Completion Using Student Performance In High School Algebra:
A Mixed Methods Research Study
Thesis directed by Dr. Michael Marlow
ABSTRACT
Too many of our nations youth have failed to complete high school. Determining why
so many of our nations students fail to graduate is a complex, multi-faceted problem and
beyond the scope of any one study. The study presented herein utilized a thirteen-step
mixed methods model developed by Leech and Onwuegbuzie (2007) to demonstrate
within a large urban Rocky Mountain school district, the existence of a significant
relationship between high school algebra performance and high school completion and to
provide one suggested framework by which to explain the nature of this relationship.
Quantitative analysis showed that a failing grade in a students first high school algebra
course is not only a significant predictor of failure to graduate high school but was more
liable, than several other predictors, to forecast which students are more likely to leave
high school before earning a diploma within four years. However, quantitative
examination revealed only the existence of a significant relationship between these two
variables. It fell short of providing any elucidation as to why performance in high school
algebra can be considered as a predictor of failure to graduate high school. Qualitative
analysis, using the tradition of phenomenology (Moustaskas, 1994), was employed to
expand on the quantitative analysis and offer a hypothesis from a student-centered
perspective as to why this relationship may exist. Several students, who have returned to
school after dropping out or who have been in high school for more than four years, were


interviewed about their individual experiences in high school algebra classes and the
greater high school environment. The qualitative analysis results did not conclusively
agree with the quantitative results, and failed to conclusively point to high school algebra
as the overwhelming reason for high school dropout. Nonetheless the interviews revealed
that the students perceived numerous negative sources of self-efficacy and expectancies
that, when combined with the uniqueness of the high school algebra experience,
increased the likelihood of failing to graduate high school.
The form and content of this abstract are approved. I recommend its publication.
Approved: Dr. Michael Marlow
IV


DEDICATION
I dedicate this work to my wonderful family. For my father Martin, my mother
Rhoda, my mother-in-law Eunice, and my sister-in-laws, Carole and Peggy, you always
believed in my abilities and supported me in all of my endeavors. You are no longer with
me, but I feel you are forever watching over me, and I will carry my memories of you
forever. To my sisters, Judy and Rachel, I am thankful we have each other and am
grateful for the many years of friendship. To my nephew Matthew, and my niece
Amanda, always know how much I love being your aunt and enjoy your company! For
Lewis, my husband of over 28 years, it has been quite a journey. You have accompanied
me along every stop, and I could have not achieved this Ph.D. without your support. I
love you and will always prize our time together. For my son Seth, you have grown into
an amazing young man, husband, and father. You serve our country as a dedicated and
loyal sailor and, following in your parents footsteps, have brought us great joy and pride.
You are everything to me, and no mother could be prouder of her child than I am of you.
For my daughter-in-law Sarah, you are such a precious addition to our family; I will love
you forever. And for my dear, dear grandchildren, Calista, Carter, and the grandson on
the way, my little Oliver Riley Martin Chiado, there is no greater love than what a
Grandmother has for her grandchildren. You are, and will always be, in my heart and
soul; and I look forward to our many years of cherished memories.
v


ACKNOWLEDGMENTS
I would like to thank my wonderful and dedicated dissertation committee,
Dr. Michael Marlow, Dr. Alan Davis, Dr. Honorine Nocon, and Dr. Mary Thurman.
Each and every one of you has patiently provided me invaluable advice, kind words, and
never-ending assistance throughout my doctoral education and dissertation process. I
would have never achieved my doctorate without your support. Although I had many
wonderful teachers throughout my doctoral program, I would especially like to thank
Dr. Nancy Leech, a once-in-a-lifetime teacher, and my many kind colleagues of the
Mixed Methods SIG, especially Dr. Tony Onwuegbuzie. Everyone was so patient at
answering my many questions; and because of that, you instilled a love for learning new
research methods and for understanding the value of mixed methods research.
I would like to especially extend a heartfelt thanks to my Ph.D. advisor,
Dr. Michael Marlow. Your friendship, dedication, and faith in your doctoral students are
unequaled; and we all feel honored and lucky to be part of your classes and educational
endeavors. You have taught me so much over the years, and your unfailing and patient
belief in my abilities was pivotal to my reaching this huge milestone. I know I have the
best committee and advisor in the university!
vi


TABLE OF CONTENTS
I. INTRODUCTION....................................................................1
Research Questions.............................................................8
Research Hypotheses...........................................................10
Limitations of the Dissertation Study.........................................10
Researchers Perspective......................................................13
Significance of This Research.................................................21
Conceptual Framework..........................................................26
II. LITERATURE REVIEW AM) SYNTHESIS...............................................28
Goal and Objective of the Literature Review...................................29
Rationale and Purpose for the Literature Review...............................30
Literature Review Questions...................................................30
Literature Review Data Collection Process.....................................32
Literature Review Legitimacy and Validity.....................................33
Literature Review Synthesis Methodology.......................................34
Interpreting the Results of the Literature Review: Steps One through Three...40
Step One: The high school dropout environment and algebras unique role...43
Step Two: Self-efficacy, the high school dropout, and the high school algebra
experience.................................................................54
Self-efficacy and the high school experience...........................55
Self-efficacy and the high school algebra experience...................64
Step three: The relationship between social learning theory and self-efficacy and
the behaviors leading to high school dropout...............................70
The relationship between self-efficacy and social learning theory......73
Behaviors, expectancies, and self-efficacy sources of the high school dropout
.......................................................................77
vii


Step 4: Self-efficacy, reinforcements, and expectancies in high school algebra
exacerbating the potential to engage in high school dropout behaviors...84
Moving Forward: Why the Research is Needed and Potential Themes of Self-
Efficacy, Expectancies, and Behaviors to Explore in the Qualitative Aspect of
the Research..........................................................86
III. METHODS....................................................................92
Demographics and Context of the School District Under Study................94
The high school dropout sample classification............................94
Research Study Design.......................................................96
Overall Research Sample Design and Data Collection Procedures...............97
Quantitative Methods........................................................98
Research Question One: Does poor performance in a students first high school
algebra course serve as a significant predictor of failure to complete high school?
...............................................................98
Research Question Two: Will poor performance in high school algebra be more
likely than the other predictor variables of reading comprehension ability, writing
performance ability, low socioeconomic status, English grade, English language
learners, ethnicity, sex, and special education status to lead to subsequent failure
to complete high school?...................................................100
Effect size and power......................................................102
Quantitative sample and data collection....................................104
Quantitative variables.....................................................105
Missing data...............................................................108
Qualitative Methods............................................................109
Research Questions Three and Four: How do student-held expectancies and
perceived reinforcements from their experience in first high school algebra class
lead to the potential of the student to engage in behaviors that are symptomatic of
a student who will probably fail to finish high school? How do a students self-
efficacy beliefs, resulting from ones experience in his/her first high school
algebra class, influence the potential of a student to engage in behaviors that are
symptomatic of a student who will probably fail to finish high school?..110
Qualitative sample and data collection..................................Ill
vm


Qualitative measures.......................................................114
Qualitative analysis.......................................................115
Validity of qualitative analysis...........................................116
Mixed Methods Data Analysis...................................................116
Limitations...................................................................116
IV. RESULTS......................................................................118
Quantitative Results..........................................................119
Does poor performance in the first high school algebra course serve as a
significant predictor of subsequent failure to complete high school?.122
Does the high school grade in which algebra is first taken serve as a predictor of
high school graduation success?......................................122
Is poor performance in high school algebra more likely than the other predictor
variables of reading comprehension ability, writing performance ability, low
socioeconomic status, English language learners, ethnicity, sex, and special
education status to lead to subsequent failure to complete high school?...123
Qualitative Results...........................................................127
How do student-held expectancies and perceived reinforcements from their
experience in first high school algebra class lead to the potential of the student to
engage in behaviors that are symptomatic of a student who will probably fail to
finish high school? How do a students self-efficacy beliefs, resulting from ones
experience in his/her first high school algebra class, influence the potential of a
student to engage in behaviors that are symptomatic of a student who will
probably fail to finish high school?.................................129
Perceived sources of self-efficacy and expectancies in a students first high
school algebra class..........................................130
Misdirected social persuasion in a first high school algebra class......130
Intense affective states in a first high school algebra class...........132
Insufficient mastery experiences in a first high school algebra class...135
Ill-suited vicarious experiences in a first high school algebra class...136
Expectancies in a first high school algebra class.......................137
IX


Perceived sources of self-efficacy and expectancies in a students greater high
school experience..................................................140
Misdirected social persuasion in the greater high school environment.140
Intense affective states in the greater high school environment......142
Expectancies in the greater high school environment................143
Behaviors leading to high school dropout or not graduating on time...144
Summary of Qualitative Findings..........................................145
V. DISCUSSION................................................................148
Bottom Line up Front: Although Mixed Methods Research does not Prove
Conclusively that Algebra Performance is the Singular Cause of Failure to Complete
High School, the Findings from Analysis of this Studys Quantitative and Qualitative
Data Samples Points toward Algebra Performance as a Noteworthy Contributor to the
School District Dropout Problem.....................................................150
The quantitative results showed that a students failure to pass the first high school
algebra class is significantly associated with failure to graduate high on time and after
four years. The quantitative results also showed that the high school grade in which
algebra is taken is significantly related to failure to graduate high school........151
Within the qualitative study sample, all four sources of self-efficacy were present
in the students high school algebra class; it was largely the self-efficacy sources
of misdirected social persuasion and intense affective state and negative
expectancies that were experienced by students in high school algebra classes. 154
Within the qualitative study sample, the self-efficacy sources of misdirected
social persuasion and affective states combined with negative expectancies in the
greater high school environment intensified those which were experienced in a
first high school algebra course................................................164
The result of the interplay between sources of self-efficacy and expectancies in
high school algebra and the greater high school experience: A potential to engage
in behaviors characteristic of the high school dropout..........................167
Taking algebra in the ninth grade transition year can be a risky endeavor for
students, as evident by the studys quantitative and qualitative results........170
The integrated quantitative and qualitative results showed that the relationship
between the failure to pass algebra and high school dropout crossed ethnic
boundaries......................................................................171
x


The Value of Mixed Methods Analysis in Investigating the Relationship between
High School Algebra Performance and High School Graduation................172
Mixed methods research provided insight into the depth and breadth of the
problem and the results of the qualitative analysis significantly enhanced the
results of the quantitative analysis by showing practical and economic
significance...........................................................172
Results from the quantitative method of study were not totally congruent with the
results from the qualitative method of study...........................174
Concluding Thoughts.......................................................175
We need to pay attention to the teachers of ninth grade algebra........175
Alternative schools are onto something. Economically and logically we cannot
create a district of all alternative school, but why cant we replicate some of the
delivery methods at least in the ninth grade and especially in core classes such as
algebra................................................................176
Future Research and Recommended Prescription for Success..................177
REFERENCES....................................................................180
APPENDIX......................................................................189
xi


LIST OF TABLES
Table II. 1 Literature review synthesis questions, as mapped to the study research
questions, goal and rationale...............................................31
Table II.2 A summary of the sources of self-efficacy and expectancies developed from
the review of the literature common to experiences in both high school
algebra and high school experiences to examine in the qualitative analysis,
that can lead to the potential to engage in the three behaviors characteristic of
the high school dropout..........................................90
Table III. 1 The demographical AFGR breakdown for the school district under study
at the end of the 2010 academic school year......................96
Table III.2 Independent and dependent variables and their corresponding levels of
measurement......................................................108
Table III.3 Demographical and academic characteristics of the student interviewees. ..113
Table IV. 1 Grade in which a student first takes algebra for the 2004 to 2008 student
cohort.....................................................................120
Table IV.2 Numbers of algebra pass/fail and English pass/fail students in the same year
as their first algebra course for the 2004 to 2008 student cohort...................121
Table IV.3 Descriptive statistics for the student cohort of school years 2004 to 2008.. 121
Table IV.4 Chi-Square analysis of high school algebra performance association to high
school graduation...................................................................122
Table IV. 5
Chi-Square analysis of the high school grade in which algebra is first taken
and its association to high school graduation.............................123
Table IV.6
Logistic regression predicting graduation success from the combination of
independent variables.............................................126
Table IV.7 Summary of themes of student-perceived sources of self-efficacy,
expectancies, and behaviors, as derived from analysis of the interview data
and as related to the conceptual framework. Bold type indicates answers
related to high school experience only, italics indicate answers from algebra
only, and all capitals indicate answers from both high school
algebra classes and the greater high school experience...................146
Xll


LIST OF FIGURES
Figure II. 1 Path Through the Literature Review...................................35
Figure II.2 Step One: The High School Dropout Environment and Algebras Unique
Role...................................................................36
Figure II.3 Step Two: Self-Efficacy, the High School Dropout, and the High School
Algebra Experience.....................................................37
Figure II.4 Step Three: The Relationship Between Social Learning Theory and Self-
Efficacy and the Behaviors Leading to High School Dropout..............38
Figure II. 5 Step Four: Self-Efficacy, Reinforcements, and Expectancies in High School
Algebra Exacerbating the Potential to Engage in High School Dropout
Behaviors..............................................................39
xiii


CHAPTER
I. INTRODUCTION
There is a disconcerting national trend of United States students who fail to
graduate high school on time or who never receive a high school diploma. Recent
statistics show that on average only 70% of our students earn a diploma each year,
leaving almost one-third of our nations students without a high school diploma. Every
year an estimated 1.2 million students do not graduate on time. At any one time, there
are 3.5 million to 6 million individuals between the ages of 16 and 24 who have
completely dropped out of high school, meaning they have failed to earn a high school
diploma and are not enrolled in high school. The statistics are more severe for minority
populations. Nearly one-half of the annual dropout population is comprised of Black,
Hispanic, and Native American youth (Bloom, 2010; Bridgeland, Dilulio, & Morrison,
2006; Stillwell, 2010; Sweeten, Bushway, & Paternoster, 2009). Some go so far as
calling the high school dropout problem an epidemic and label certain United States high
schools as dropout factories (Bridgeland et al., 2006; Tyler & Lofstrom, 2009). For many
students, high schools have gone from the launching point to prosperity and being part of
the American dream to becoming the stalling point in a lifes progression.
Literature spanning many years and scores of investigative approaches reveal that
the failure to graduate high school is a complex and multi-faceted issue. This has yielded
discussion and research into a wide and diverse range of variables and factors, all
presenting a relationship to the failure to graduate high school. The relationship has
been viewed through an equally wide assortment of theoretical frameworks and analysis
methods.
1


Representative studies have explored the relationship between high school
graduation and predictor variables such as academic performance, student attitudes,
classroom behaviors, adolescent self-esteem, and school dissatisfaction. Other inquiries
have focused on failure to graduate high school and its relationship to failing in the ninth
and tenth grades or to a shortage of credits at the end of ninth and tenth grades. The role
of the ninth grade and its standing as a transition year is also of critical concern to
investigators of the high school dropout matter (Allensworth & Easton, 2005; Ekstrom,
Goertz, Pollack, & Rock, 1986). Allensworth and Easton (2005) additionally established
that the organizational structure of the high school can predict which students is on-track
to graduate. They and other researchers did additional research to show that signs of poor
performance as early as pre-school and elementary school can also predict high school
graduation success (Allensworth & Easton, 2005; Hickman, Bartholomew, Mathwig, &
Heinrich, 2008).
Another direction of research has inspected the dropout issue for its relationship
to demographics such as race, ethnicity, and socioeconomic position, or as tied to family
structure such as birth order, number of parents in the household, and education level of
the mother. Other studies claim that although a number of issues are involved in the high
school dropout matter, some are more significant (e.g., grades and student attitudes) than
others (Allensworth & Easton, 2005; Ekstrom et al., 1986; Rumberger, 1987).
Since the causes of failing to complete high school have been shown to be
complex and many, it is a challenge to know which set, if there is such a concept as a
right set, of predictors can be used to identify those students who are in the greatest
danger of dropping out. Identifying a right set of predictors can lead to efficiently and
2


effectively structuring focused interventions. No one predictive model is comprehensive,
and not all risk factors are equally effective at identifying students who are in danger of
dropping out of high school before earning a diploma (Gleason & Dynarski, 2002).
Indeed, Dynarski and Gleason (1998) are emphatic in their plea for the academic
community to identify superior predictors of high school dropouts. They state that often
used predictors such as high absenteeism, low grades, over-age attendance, and
teen parenthood, while important in identifying potential dropouts, if they are the only
predictors used, can lead to our missing the dropout prediction of two out of every three
students (Dynarski & Gleason, 1998).
Given the complexity and number of factors discussed in the literature, it is
unlikely that any future study will offer an absolute solution to the high school dropout
problem. It is doubtful that any singular intervention will be 100% effective in stopping
students from dropping out of high school.
Yet, it is this researchers firm belief and educational philosophy that the
discovery and understanding of a previously unknown risk factor, shown to have a
significant relationship on the proclivity to drop out of high school, is invaluable and
adds worth to the existing body of research. A study that reveals the existence and nature
of an additional predictor of high school completion, such as performance in high school
algebra, offers a return in the ability to identify more students early enough in their
secondary school tenure to intervene and promote high school success. Such work
enlarges the toolbox of identifiers and interventions used by a school district in the
quest to ensure that all students graduate with a high school diploma in hand.
3


The school district at the center of this dissertation study is not unlike the rest of
the United States and the state in which it is located. It has experienced a distressing
failure to graduate high school rate of about 30% during each of the past few years. In
addition to this less than acceptable high school dropout rate, the district has expressed a
long-standing concern that students are not achieving success in high school
mathematics, particularly in their first high school algebra course, which is initially
experienced by a majority of students in the ninth grade. Almost 30% of the districts
high school students fail algebra each year. Several administrators, including the school
districts recently retired superintendent, are bewildered about the co-existence of nearly
equal high school students who fail to graduate and high school students who fail algebra.
These administrators have openly speculated their suspicions that the two variables are
related (personal communication, February, 2008).
The former governor of Colorado, Roy Romer, when serving as the
superintendent of the Los Angeles school district, also articulated this same supposition
and voiced that of all the obstacles to graduation, algebra was the most concerning to
him. He personally believes that algebra, more than any other course, has resulted in
high school students dropping out (Helfand, 2006).
The research problem driving this mixed methods study centered on this
coexistence of nearly identical unsatisfactory student high school algebra performance
and failure to graduate rates and the concurrent lack of methodical research and
explanation as to the existence and nature of this concurrence. The school district is
striving to design and implement interventions calculated to help more students succeed
in high school. Administrators are determined to identify more students early enough in
4


the educational process in order to implement means that prevent students from leaving
high school before earning their diploma. Despite thoughts that high school algebra
performance may have a role in the failure to graduate, the school district has not
investigated if and why such a relationship exists between these two variables. By not
using high school algebra performance as an indicator of a students propensity to drop
out of high school, an opportunity to help a large portion of students may be missed.
Additionally, interventions designed to increase success in high school algebra, while
helping students in this academic area, may be limited in its total worth, if it is not
simultaneously addressed with the impact that high school algebra performance has on
resultant graduation success.
A corollary research problem influencing the direction of this dissertation study
and defining the direction for the qualitative aspect emanated from my own personal
experience. Over the past 17 years, I have worked with multiple volunteer committees
and on studies on behalf of the school district. I have been privy to several discussions
about student interventions, including those intended to bolster algebra achievement and
others aspiring to minimize the high school dropout rate. I have participated in task force
proceedings centering on how to help students overcome the difficulties in ninth grade
mathematics and have been appointed to committees focused on programs for students in
danger of dropping out of high school. Along with others, I have heard about
implementation of various software programs, mentoring efforts, and teacher training
ventures and have participated in dialogue about employing community assets, tutoring
programs, and tailoring interventions. We have each offered thoughts and expertise
toward outlining what is postulated to be the causes of each problem: poor algebra
5


performance and incomplete high school education, which have all been discussed and
worked on in parallel paths.
However, as I reflected back on my participation with the school district, it struck
me that there was something missing from these discussions and deliberations, especially
as I theorized about the existence and nature of a relationship between high school
algebra performance and high school graduation. Not only have I observed the
discussions about high school algebra performance and high school dropout rates
proceeding independently, without any vocalized suspicion of a linkage, but rarely did I
hear any talk attempting to understand the issues from the students perspective. Have
we missed a major aspect of understanding the problem by failing to appreciate the dual
problems of poor high school algebra performance and unacceptable high school
graduation rates from the students perceptions of abilities, values, and expectations?
Have we tried to correct both poor algebra results and high school dropout rates without
acknowledging a key element necessary to craft and field more effective interventions?
Did we lack an understanding of students beliefs and perceptions of the causes of
difficulties in high school algebra and the resultant potential to engage in behavior
leading to the likelihood of dropping out of high school? Therefore, a second research
problem warranting study was the school districts lack of focus on the students
perceived aspect of academic difficulty in high school algebra and its impact on the high
school dropout rate.
This dissertation used a multi-step mixed methods research model designed by
Leech and Onwuegbuzie (2007) to structure and to execute the study into both research
problems. The study was conducted within a large, urban school district located in the
6


Rocky Mountains portion of the United States. The goal of the mixed methods study was
to determine if a significant relationship existed between student performance in high
school algebra and high school graduation. If such a relationship existed, the subsequent
goal was to then understand the perceptions, from a student-centered perspective, of the
relationship between poor performance in high school algebra and failure to graduate
high school. The use of both quantitative and qualitative methodologies was sought to
understand the depth and breadth of the hypothesized relationship between high school
algebra performance and high school graduation and to address the districts premise that
high school algebra performance is an indicator of subsequent failure to graduate high
school.
Quantitative methods were used first to determine if there is a significant
relationship between student performance in high school algebra and high school
graduation and to determine the relative significance of high school algebra performance
as compared to other predictor variables on high school graduation. Subsequently,
qualitative methods were used to examine the nature of this hypothesized relationship
from the students perspective and, as guided by the Theory of Social Learning of Rotter
and the Concept of Self-Efficacy by Bandura, to provide insight into their perceptions of
reinforcements, expectancies, and self-efficacy beliefs and the resultant potential to
engage in behaviors that led to the likelihood of dropping out of high school.
Poor algebra performance was defined as receiving a failing algebra grade (which
for the school district under study is a grade below a D). Dropping out was defined to
be any student who did not graduate high school in four years and who does not yet hold
7


a high school diploma. It does not include the population of students who earned a
General Education Development (GED) certificate.
The study had three main objectives. The research objectives were to (a) improve
prediction abilities and, thus, identify a greater number of students who are in danger of
failing to complete high school; (b) explore student beliefs and expectations for poor
performance in high school algebra and identification of those factors that may also
influence the propensity to drop out of high school; and (c) influence development and
implementation of research-based interventions that are designed to concurrently reduce
the occurrence of poor high school algebra performance and failure to complete high
school. The mixed methods research objectives derived from the stance that any partial
or full solution to the research problems underlying this study dictates a continuum of
research that neither quantitative nor qualitative methods can address alone (Collins,
Onwuegbuzie, & Sutton, 2006; Leech & Onwuegbuzie, 2007; Onwuegbuzie et al., 2007).
In addition, as in other school districts, several millions of school district dollars are
being invested in computer and other intervention programs, to include extra periods of
algebra instruction, instruction to parents to better help their students, and steps to
prevent problems in earlier grades (Helfand, 2006). The third objective sought to
communicate the need to consider other factors, including student perspectives, prior to
developing and implementing costly interventions.
Research Questions
The studys research questions were predetermined to maintain consistency with
the research problem areas and were specifically constructed to dictate the methods,
boundaries, and direction of the research (Plano Clark & Badiee, 2010). Additionally,
8


Teddlie and Tashakkori (2009), in their latest mixed methods research treatise, further
advised in a discussion about the pivotal role of research questions that the questions
should be focused on what is not known about a phenomenon. Therefore, the research
questions for the study were framed accordingly in order to narrow the scope of study
and its accompanying literature search to the problem areas previously discussed and to
scrutinize the missing student perspective on the phenomenon of algebra performance
and its impact on high school completion. The questions were as follows.
1. Does poor performance in a students first high school algebra course serve as
a significant predictor of failure to complete high school?
2. Will poor performance in high school algebra be more likely than other
predictor variables of reading comprehension ability, writing performance
ability, low socioeconomic status, English grade, English language learners,
ethnicity, sex, and special education status to lead to subsequent failure to
complete high school?
3. How do student-held expectancies and perceived reinforcements from their
experience in first high school algebra class lead to the potential of the student
to engage in behaviors that are symptomatic of a student who will probably
fail to finish high school?
4. How do a students self-efficacy beliefs, resulting from ones experience in
his/her first high school algebra class, influence the potential of a student to
engage in behaviors that are symptomatic of a student who will probably fail
to finish high school?
9


Research Hypotheses
There were four research hypotheses that drove the need for this effort:
1. Performance in ones first high school algebra class is related to high school
graduation.
2. Ethnicity, socioeconomic status, and gender are not significant factors in the
relationship of performance in ones first high school algebra class to high
school graduation.
3. Performance in ones first high school algebra class is one of the largest
significant factors, when compared to others in a model that includes several
predictor variables of high school graduation.
4. Student participation in high school algebra classes result in lowered self-
efficacy and negative expectancies, which lead to the potential to engage in
behaviors that lead to dropping out of high school.
Limitations of the Dissertation Study
As previously cited, the research has touted the existence of numerous factors
associated with the failure of students to graduate high school. In both the quantitative
and qualitative efforts to address the research problems, there were potentially an
unlimited number of variables that could have been scrutinized for singular or relative
significance on high school graduation success. The overall scope of this study was
deliberately limited to the significance and nature of the relationship between high school
algebra performance and high school graduation success, primarily as a result of hearing
and discussing the particular concerns of school district leadership about the existence of
this potential relationship.
10


Boundaries delimiting the direction for the study were also set down within both
the quantitative and qualitative portions of the dissertation. The first portion of the
quantitative segment of the research (research question one) attempted to determine only
if there was a significant correlation between high school algebra course performance and
high school graduation using a defined quantitative sample. There are other potential
mediating factors shown by research to be significant predictors of failure to graduate
high school, such as the number of credits earned at the end of freshman year and poor
overall grade point average (Allensworth & Easton, 2005). These factors, which can be
postulated to simultaneously occur when one fails a required high school course such as
algebra, were not considered.
Despite the desire to limit this study to the relationship between high school
algebra performance and high school graduation, it was intriguing to obtain some insight
into the relative importance of high school algebra performance with respect to other
selected variables on high school graduation. After all, a student does not take algebra in
a void; and thus, quantitative research question two examined this relationship. The
comparison predictor variables were limited to those for which the school district
routinely collects data in order to evaluate student performance and to report results, in
accordance with school district, state, and federal policies. Although previous research,
along with this studys results, may reveal other predictor variables warranting future
comparison study, the desire was to not place an unnecessary burden on the school
district by requesting additional data. The intent was to stay synergistic with school
district assessment practices by showing results from using the same variables that are
used to analyze school district data.
11


The qualitative analysis was limited to a focus on the nature of a hypothesized
relationship between high school algebra performance and high school graduation from
the students perspective, looking to describe the relationship in terms of student
expectations, reinforcements, behavior potentials, and self-efficacy beliefs, a limitation
consciously placed due to personal interest in Social Learning Theory and self-efficacy
concepts, especially as advocated by Rotter and Bandura respectively. This study did not
account for other potential factors such as teacher pedagogy methods or student learning
differences.
Interviews were conducted with students who have returned to school after
dropping out of high school at least once or who have not yet graduated after attending
high school for more than four years. Ideally interviews would have been held with
students who have failed and passed high school algebra and who have both graduated
and failed to graduate from high school so as to highlight differences in perceptions, but
it was felt that students who have graduated were not readily available.
The school district used in this study has an extensive alternative school program
with a pool of students available for interviews: dropouts who have returned to school or
students who are still working on their diplomas after more than four years of high school
attendance. The qualitative research was meant to provide a window into student
expectations, reinforcements, behavior potentials, and self-efficacy beliefs, as related to
high school algebra performance and its relationship to high school graduation.
Therefore, it was felt that limiting the interviews to those students who have failed to
graduate high school in four years was sufficient to accomplish this purpose. Future
study can be recommended to examine a comparison of these qualitative variables with
12


those held by students who have passed algebra and have successfully completed high
school in four years.
Two other study limitations concerned the samples used in the study. The data
sample used to select students for interviews in the qualitative analysis was
chronologically different than the sample used for statistical testing in the quantitative
analysis. This difference was overcome by the demographic similarities of the sample
groups. The target interviewee sample group was similar enough to those in the
statistical analysis sample and, therefore, was representative of the greater group of
students who fail to graduate high school in four years.
Finally, quantitative and qualitative analysis was limited to samples of those
students who were in the school district at the start of the ninth grade and does not
include those who entered into the school district in grades ten through twelve. Although
one cannot completely eliminate the influence of extraneous variables, this delimitation
was done to help minimize this influence by removing variability caused by students
coming from other school districts.
Researchers Perspective
I am not without a history of individualized experiences and biases that may have
consciously or unconsciously influenced the demeanor of this study. Researchers enter
into study with experiences that can explicitly and implicitly sway their research
(Creswell, 1998; Creswell & Plano Clark, 2007). My individualized experiences and
biases were formulated by a number of attributes to include (a) education and
volunteerism as a student and parent in a public school system, (b) experience and
involvement as a parent and advocate of a high school dropout, and (c) investigation and
13


learning as a participant and believer in several research methods courses. Although
purposeful effort was made to sift out the influence of lifetime experiences and biases in
gathering and analyzing data, it was impossible to do so with 100% certainty and was
especially difficult during the qualitative aspect of the research when I attempted to draw
close to my research subjects. I had to consciously think of my own preconceived notions
about why students drop out in the school district in which the study took place and not
let these opinions sway the conduct of or answers from the student interviews.
Teddlie and Tashakkori (2009) discuss the requirement for a researcher to
acknowledge and understand the role that an individuals worldview can have in
research. They have characterized worldview as a system of beliefs and practices that
influence how researchers select both the questions they study and methods they use to
study them.
Creswell further describes five foundational philosophical assumptions that can
be used to characterize ones individual system of beliefs and practices, and I used the
same five assumptions as the pillars forming my worldview and influencing my practice
of research. These five pillars are indicative of my primary belief in the practice of
pragmatism, or in placing emphasis on the answers to the research questions over all
other matters, such as the actual methods used in a study. Openly discussing these five
pillars helped me to admit explicit bias and allows both the reader and myself an
understanding of the historical context and life influences from which I entered, directed,
and, perhaps unconsciously, swayed this study (Creswell, 1998; Creswell & Plano Clark,
2011).
14


The first assumption is the ontological assumption, whereby one asks what is the
nature of reality? Is there a singular, concrete reality; or are there multiple, elastic
realities from which one can examine a particular research problem? Does the researcher
believe in the singular reality or multiple realities? I assert that the question of an
existence and nature of a relationship between designated variables does not need to be
an either/or proposition. There co-exists both the singular and the multiple, and a
researcher can approach a particular research problem claiming the existence of both
states of reality. This claim to a co-existence begs a personal preference for and
proclivity toward using mixed methods research, relying on quantitative methods to
disclose the singular reality and qualitative methods to confirm one of several potential or
multiple realities.
In this case, the singular reality exists in the presence of a relationship, a sole
tangible concept, which is the hypothesized relationship of high school algebra
performance to high school graduation. The desire to prove this hypothesis, the singular
reality, lends itself to the use of quantitative methods to expose its existence. Once the
existence of the relationship, or the singular reality, is demonstrated, the reason for and
the nature of its being can be accredited to multiple realities. Not only can it be attributed
to multiple realities but, as will be presented later in the discussion about the rationale
and purpose for using mixed methods, it is essential to use multiple realities to
understand the true composition and significance of this relationship. In addition,
proving the existence of the singular reality is not good enough in todays educational
environment, at least from this researchers perspective. One needs to understand the
nature of the relationship, which is actually comprised of multiple realities.
15


These multiple realities are formed from the numerous ways in which we can
describe the relationship (the singular reality). The multiple realities can emanate from
differing group perspectives (i.e., students, teachers, parents), differing research methods
(i.e., grounded theory, phenomenology, bibliography), and differing theoretical
frameworks (i.e., Social Learning Theory, behaviorism, symbolic interactionism). All of
these accounts form realities that can simultaneously exist, and all can be shown to be
equally valid by using qualitative methods.
The second assumption underlying my worldview centers on the epistemological
aspect and focuses on the association between researchers and the researched subjects.
Creswell is an advocate for narrowing the gulf between researchers and the researchers
subjects, which is more characteristic of constructivism than pragmatism. A pragmatic
approach to epistemology, however, calls for the researcher to collect data according to
what works when attempting to answer a research question (Creswell & Plano Clark,
2011). How I have structured the qualitative aspects of the research design shows my
agreement with and favor for the constructivist goal of becoming close to the research
subjects, with the researcher as an integral part of the collection methodology. Yet this
may also be a somewhat pragmatic approach, since I firmly believe what works best is to
collect data from the eye of the matter, which is from the students themselves. Moreover
I came into this study as one who has experienced the high school dropout occurrence,
albeit as a parent of a high school dropout. I had hoped that my genuine interest in
gaining an appreciation for why students drop out of high school and a never-ending
desire to come to terms with why this even occurred in my own family demonstrated a
commonality and a natural empathy with my interviewees. I believed that this allowed
16


me to draw close to my research subjects, encouraging honesty and complete openness in
their responses to my questions.
Continuing in a discussion of my worldview, there is a third assumption: the role
of values within the research. Another term for this is the axiological nature of the study.
Pragmatists take multiple stances. They believe in both the biased and unbiased
perspective. There are two personal biases that are admittedly driving the purpose and
course of this research study. These same values are two that I firmly believe should be
foundational to the nature of public school education within the United States. These are,
first, the strong belief that there should be equity in the attention paid to students
problems in the public school system and, second, that educators and parents must listen
to students viewpoints about their educational experiences.
Growing up in an urban area of the country as a product of an ethnically diverse
and socially mixed public school system, I cherish diversity in the school system and
appreciate the need for the public school system to equally educate and provide
assistance to students of all backgrounds, especially to those who are lagging behind.
There is an achievement gap, one which I do not deny, between certain groups of
minorities and majority populations in the school district at the focus of this study.
Correspondingly there is a pressing need for public schools to take action and for
academia to research methods by which to narrow this gap, another requirement that I do
not refute.
Despite this acknowledgement, I cannot help but feel there is an ignored and cast
aside segment of the student population whose problems and academic failures get
subjugated by the intense attention paid to the achievement gap. This suppresses true
17


achievement of equality in the public school system under study. This group is one to
which my son belongs: the forgotten, and, from my biased viewpoint, an often neglected
group, the majority sector of the population. There appears to be, especially from my
vantage point as a parent and volunteer, little or not enough attention paid to the
educational problems of students from the white, male, middle-class segment of the
population; and this minimizes the attainment of true equality in public school education.
Who is trying to determine why students, such as my son from a middle class,
educated, two-parent home with a good family support system, drop out of high school?
Since I value diversity and equality in the public school system, I entered into this study,
in no small part, for a rather selfish reason: to search for some influence or predictor
variable that turns a blind eye to racial and socioeconomic considerations and that may be
a predictor that cuts across all groups of children to include the white, middle-class
portion of the population. To counter this entering bias, I used quantitative techniques to
determine whether or not factors such as race, socioeconomic status, and gender are
significant when looking at the relationship between high school algebra performance
and high school graduation.
The second value biasing the research is another one that emerged from my
vantage point as a parent and volunteer. This is the criticality of listening to the students
perceptions of their educational experience. As a mother, I watched, and was part of the
process, as many educators, counselors, and social health professionals tried to identify
the reasons and causes for my sons struggles and eventual act of dropping out of high
school. Yet, very rarely did I observe any of these professionals ask my son about his
perspective on troubles in school. Repeatedly this led an educator or professional to
18


make a quick and mistaken judgment that my sons problems were largely due to a
discipline issue or a lazy nature. My son's greatest academic success came in the
classrooms of teachers who would take the time to talk and listen to him and adjust his
educational experience accordingly.
In addition, as I reflected back on the numerous times in which I had been part of
some committee or task force examining a particular school districts educational
problem, it occurred to me that I rarely heard anyone speak of the issue from the
perspective of or the focus of the students themselves. These dual experiences with my
sons educational path and with my volunteer work have nurtured my bias that todays
educational environment has sadly neglected to pay attention to a voice that may provide
valuable insight to academic struggles: the voices of our students. This particular bias
resulted in my choice of using phenomenology as the tradition underlying the qualitative
portion of the study. Phenomenology allows one to see lived experiences from the eyes
of those who have lived the experiences and to examine these experiences from a
diversity of viewpoints (Moustaskas, 1994). It permitted an investigation into the first
persons (the student) perceptions of experiences in high school algebra and its influence
on subsequent high school graduation.
The fourth assumption addresses the literary style of the researcher and inquires
into the language in which the researcher will write. One of the largest challenges to the
legitimacy of a mixed methods study is the manner in which the research is written up
and in the resultant meanings that are communicated to the reader. Often this yields
difficulty in showing the value of using both quantitative and qualitative methods. There
has been significant debate in what comprises a quantitative versus qualitative style of
19


writing and presentation. For example, many quantitative researchers use tables and
charts to visually display data; yet the organization of this data is characteristic of the
subjectivity of qualitative research. Many qualitative researchers use tables and charts to
provide the rigor of organization of the quantitative approach (Sandelowski, 2008).
A pragmatist uses both styles of expression. Quantitative methodology
traditionally calls for third person and emotionally bland, detached language. Qualitative
methodology usually uses the first person and takes on a more emotional, passionate
tone. In a mixed methods treatise, one can use the traditional language of either research
genre as appropriate in the specific section or use one or the other throughout (Creswell
& Plano Clark, 2007).
Since I am emotional and passionate about public school education and am
working to discover research-based methods to improve the experience and results for all
children, I have frequently used the first person style of expression typical of qualitative
reporting. This will also include discussion about the quantitative aspects. However,
since these emotions and passions can often lead to an unstructured flow of thoughts, I
have also relied on expressive aspects typical of quantitative methods to include a visual
organization of data and results. This was done in order to provide a bounded structure
and framework for the analysis and discussion of the qualitative portions of the study.
Finally, the fifth assumption, the methodology, involves the process used in the
research (Creswell, 1998). I am most definitely in keeping with the pragmatic camp,
believing in using the best research method that can yield results and understanding. The
literature shows a diverse range of multi-faceted issues related to high school completion.
Therefore, we need to deduce and break down the factors that result in a less than desired
20


high school graduation rate into potential causes such as ethnicity, performance in math
and English courses, and perceptions of negative influences in the high school algebra
and greater school environments. This will subsequently induce possible theories or
factors from the involved parties perspectives as to why. A mixed method approach was
the most suitable manner in which to confirm some of the potential factors that
demonstrate an association between high school algebra performance and high school
graduation and to further provide some insight as to why this association may exist.
One concluding thought as to my rationalization for the use of mixed methods.
This research approach was a logical fit with my academic and professional experience. I
have a love for numbers and science but a passion for words and stories. My experience
and training, primarily in the military and in working with space systems, are chockfull of
forays into the black and white definitiveness and cause and effect world of quantitative
methods. Much of my undergraduate and graduate education is in the sciences where
objectivity in the form of statistics and experimentation reign. On the other hand,
additional graduate schooling is in the field of education; and recent professional and
volunteer work required me to make subjective sense of happenings and problems.
Because of this experience, I have resorted to the expressiveness and grayish
explanations typical of qualitative methods.
Significance of This Research
The United States high school dropout issue has been described as a Silent
Epidemic (Bridgeland et al., 2006). Seeking out and understanding the reasons why so
many of our youth drop out of high school is an economic and social necessity for
America to retain its place as a leader in the 21st century. There are numerous
21


consequences for both individuals and society that result from failure to earn a high
school diploma; and a disproportionate part of the impact is felt by minorities, who drop
out at a greater level than others (Chapman, Laird, & KewalRamani, 2010; Stillwell,
2010; Sum, Khatiwada, & McLaughlin, 2009).
Individually, high school dropouts are more likely to be unemployed and earn less
income annually and over a lifetime. Looking at the populace of, ages 16 through 24, on
average during the year 2008, high school dropouts were employed 22% less than high
school graduates, 31% less than those with one to three years of college, and 41% less
than those who graduated from college. According to Sum, Khatiwada, & McLaughlin
(2009), this same high school dropout age group was also more likely to be unemployed
longer. In 2007, the 16 through 24-year-old high school dropouts had a 40%
unemployment rate that lasted the entire year (Sum et al., 2009).
Earning potential decreases for the high school dropout. In 2009, high school
dropouts earned 31% less than high school graduates and 46% less than those with some
college education (Bridgeland et al., 2006). As opposed to earlier generations when high
school dropouts could easily enter the workforce, they now compete with an unskilled
labor pool of immigrants from other countries, which has a further negative effect on
their lifetime income earning potential (Bridgeland et al., 2006; Sum et al., 2009).
Accordingly, between 1974 and 2004 the median earnings of families headed by a high
school dropout declined by one-third (American Psychological Association, 2010).
High school dropouts also exhibit additional consequences of their failure to
graduate. They have riskier health behaviors and are less likely to have health insurance
plans. They exhibit more illegal drug usage, experience poorer mental health, and have
22


reduced home ownership rates. Dropouts are also more likely than high school and
college graduates to have marital problems (Bridgeland et al., 2006; Rouse & Kemple,
2009; Sum et al., 2009; Sweeten et al., 2009; Tyler & Lofstrom, 2009).
High school dropouts collectively encumber our society as well. They frequently
receive financial and health assistance from local government. In return, these high
school dropouts contribute less tax revenue to the public coffers, providing on average
42% less state and federal tax income. Crime rates and the accompanying cost of jail,
probation, and parole are affected because high school dropouts are more prone to
commit crimes, including violent crimes, and are more likely than those who earn a high
school diploma to end up on death row (Campolieti, Fang, & Gunderson, 2010; Sum et
al., 2009; Tyler & Lofstrom, 2009). One state legislator went so far as to say, Dropping
out of high school is an apprenticeship for prison (Sum et. al., 2009, p. 10).
When high school dropouts become parents themselves, the result is a cycle that
often reflects continual poverty and negative educational results. Unfortunately this cycle
is difficult to break. Sixteen percent of children under the age of 18 live in households
where the head of the household is a high school dropout (Ziomek-Daigle, 2010).
Children of less educated parents tend to do poorer in their own educational
achievements. Many dropouts become single parents (primarily females) and have
children earlier in life than do high school graduates. The dropout parents are struggling
financially and, as a result, become dependent on public and family assistance. Nearly 37
of every 100 dropouts were living in poor families in 2006-2007. A poor family is
classified as having an income 25% below the federal poverty income threshold. A poor
family is four times more likely to have a parent who is a high school dropout as opposed
23


to a family with a parent who has a high school diploma or higher (Sum et al., 2009).
The cycle persists as the more impoverished the family is, the more likely the children
will have future academic difficulty (Archambault, Janosz, Fallu, & Pagani, 2009; Sum et
al., 2009; Tyler & Lofstrom, 2009).
Nowadays, gaining proficiency in mathematics, specifically algebra, may be
almost as important as earning a high school diploma. There is much more depth needed
to education in our present information-driven society than in the generations before us.
Education needs to go beyond the old saying: reading, writing, and arithmetic. The
basics are not good enough anymore; and todays 21st century citizen needs to gain the
critical thinking, problem solving, and deliberate analysis skills that are characteristic of
mathematical disciplines such as algebra. The National Mathematics Advisory Panel,
which was convened in 2006 by President Bush, espoused that the current American
student should strive to understand the intricacies and patterns that comprise the
disciplines of mathematics, including algebra, in order for the United States to keep pace
in our changing world by producing more college graduates, including engineers and
scientists (Vogel, 2008).
The Mathematical Association of America further proclaims that algebra literacy
is essential to understand much of science, statistics, business, or todays technology
disciplines (Katz, 2007). Algebra, especially, is often viewed as a gateway course,
providing individuals a greater ability to gain access to a college education and technical
careers (Stein, Kaufman, Sherman, & Hillen, 2011). Even if one does not aspire to a
college education or to enter a technical career, algebra literacy can be beneficial to
having a successful life. During our recent economic crisis, one can note the
24


miscomprehension of many Americans who did not understand the ramifications of
variable rate mortgages and its resulting negative impact on the overall nations economy
(Rand Corporation, 2008; Schornick, 2010).
Of special interest is the work by Rose and Betts (2004) that, when using the High
School and Beyond (1980-1992) data set, significantly showed a relationship between
successfully passing algebra and increased incomes at a point ten years after completing
high school. The increased income was significantly higher for females who had some
college education. Although the study did not show if this was either the result of
completing algebra in itself or the fact that algebra opens the door to additional career
and training opportunities, it is still a noteworthy finding.
In particular, and discouragingly, the United States ranks 26 out of 39 countries in
problem solving, which is a large component of the algebra curriculum, according to the
Program for International Assessment (Schornick, 2010). The American College Testing
(ACT) has repeatedly emphasized that preparedness to complete college, to include
mathematical readiness, is vital for Americans to be economically competitive.
Unfortunately, many of our students are just not keeping pace with the rest of the world
students (ACT, 2004, 2005).
Disciplined research that examines the dual issues of students high school algebra
performance and dropping out of high school, with an ultimate goal of showing the
relationship between the two, has the potential to add value to the existing body of
scholarship. This is not only needed but is in great demand so our nation can make a dent
in the dismal trends of poor performance in algebra and students dropping out of high
school. Failure to pay attention and to conscientiously seek every answer and tool
25


available that can prevent students from dropping out of high school is a collective
societal failure to help our country maintain its status as a world leader and a failure to
help our youth benefit from the lifetime rewards of a successful high school education.
Finally, although some argue we should focus more attention and resources on ways in
which to prevent students from dropping out of high school at earlier ages and school
grades (Rouse & Kemple, 2009), this should not be our only focus. In my firm belief,
especially from the vantage point as a parent of a high school dropout, it is criminal to
ignore any means possible to intervene and reverse the propensity of a student to drop out
of high school.
Conceptual Framework
The conceptual framework guiding the qualitative portion of the research will
integrate social learning theory and self-efficacy concepts. In his social learning theory,
Rotter (1982) represents the relationship between behavior and a persons particular
environment in an equation:
Behavior Potential (BP) = f (Reinforcement Value (RV) + Expectancy (E))
Bandura in his many treatises on the concept of self-efficacy offers that self-
efficacy originates from one or a combination of four sources. These sources are (a)
Mastery Experiences (ME), (b) Vicarious Experiences (VE), (c) Social Persuasion (SP),
and (d) Affective States (AS). I will discuss in the literature review how social learning
theory and the concept of self-efficacy are closely related and because of this, one can
substitute the sources of self-efficacy for reinforcement values in Rotters original
formula. The formula thus becomes:
BP = RV= ME + VE + SP + AS + E, where
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RV = ME + VE + SP + AS
The literature review will disclose that students perceive several sources of
unfavorable self-efficacy beliefs and negative expectancies in both their high school
algebra classes and greater high school environment. These perceptions lead to the
potential to engage in behaviors that are typical of high school dropout.
The literature review will further highlight that passing algebra is instrumental in
a students ability to earn a high school diploma. Qualitative analysis will suppose a
framework such that the sources of self-efficacy beliefs and expectancies in algebra
exacerbate those that originate in the greater high school experience and that this
increases the potential to engage in behaviors leading to high school dropout. Therefore,
the conceptual framework for this research is represented by the formula:
BP dropout ((ME + VE + SP +AS)algebra + Ealgebra) ((ME + VE + SP + AS)high school
Ebigh school)
This formula should not be interpreted literally as a mathematical formula but as a
representation of a proposed concept. Whereas the plus symbol, (+), does symbolize the
additive power of four perceived sources of self-efficacy and expectancies in a students
algebra and greater high school experiences, the multiplication symbol (*) does not
connote a straight multiplicative process. It signifies the capacity of the sources of self-
efficacy beliefs and expectancies in algebra to intensify that of those from the greater
high school experience which eventually results in the potential to engage in behaviors
leading to high school dropout.
27


CHAPTER
II. LITERATURE REVIEW AND SYNTHESIS
The literature review for the research study into the relationship between high
school algebra performance and high school graduation was put into practice in a manner
consistent with the overall approach of this study. The review is primarily organized in
step with a literature review and synthesis framework advocated for use in mixed
methods research studies (Onwuegbuzie, Collins, Leech, Dellinger, & Jiao, 2007). These
authors offer a literature review framework that is similar to the 13 steps recommended
for the general conduct of mixed methods research. I used many of the recommended
steps to provide rigor and structure for this studys literature review. In addition to using
a framework to provide a deliberate structure, the mechanics of the literature review
progression, along with the synthesis results, are presented in a detailed and explicit
manner. Frels and Onwuegbuzie (2011) campaign for translucency in this foundational
aspect of research, especially within a dissertation discourse, since it provides the reader
an unambiguous understanding of the writers intentions and thought processes behind
the course of the literature review (Frels & Onwuegbuzie, 2011; Onwuegbuzie, Collins,
et al., 2007).
The initial steps of the literature review, the synthesis formulation stage,
involved delineating the goal, objective, rationale, and purpose for the examination. This
stage culminated with the formulation of literature review synthesis questions, which are
similar to research study questions; these literature review questions were then used to
guide the ensuing course of the literature review.
28


Goal and Objective of the Literature Review
The primary goal for the literature review was to provide justification and
structure for the conduct of the research into the hypothesized existence and nature of the
relationship between high school algebra performance and high school graduation. This
goal had two parts. The first part set the stage for why the examination into the
hypothesized relationship was a creditable and crucial research effort. Why was studying
the matter of high school algebra so imperative to the understanding of high school
graduation rates? This involved probing the literature for evidence that illustrated
algebras fundamental role within a triumphant high school career, both in its successful
completion and its timing within a students course sequence. This piece of the goal
would provide the substantiation needed for conducting quantitative and qualitative work
on the proposed relationship.
The second part of the goal was to formulate a conceptual framework to guide the
qualitative research portion of the study. I entered into the investigation knowing that the
tradition of phenomenology would guide the qualitative aspect with a predisposition to
somehow weave in the Social Learning Theory of Rotter and the self-efficacy concepts of
Bandura. However, I was unclear about how to structure the thinking and subsequent
analysis of the qualitative data in a relationship that would make use of these concepts.
The formation of a conceptual framework is often pivotal to demonstrating how a study
answers a research question(s). Therefore, using the literature review to help define a
blueprint for construction of a conceptual framework served as the second component of
the goal (Bordage, 2009; Miles & Huberman, 1994).
29


The overarching objective for the literature review was to evaluate the legitimacy
of the premise which instigated the dissertation research effort: that previous study about
the existence and nature of a relationship between high school algebra performance and
high school graduation was relatively nonexistent. I had based this supposition on the
limited results yielded from precursory searches of the literature conducted after
discussions with administrators of the school district under study and before formally
proposing the study to the university and school district.
Rationale and Purpose for the Literature Review
The rationale for the overall research study, and for the literature review, is
significance enhancement. Significance enhancement advocates the combining of
research from a multiple methods in order to solidify and strengthen the overall findings
of a study (Collins et al., 2006). In this case, the rationale was to demonstrate a need to
augment what is present in the current body of research about predictors of high school
graduation failure with new research data and research findings through an investigation
into the particular role of high school algebra.
The purpose of the literature review was to demonstrate the need to follow up on
results of previous research into the separate enclaves of student high school algebra
performance and high school dropout rate and the lack of research into a possible
association. This was to provide further evidence of the need to explore algebras
potentially pivotal role in a students high school education success.
Literature Review Questions
Developing thoughtful and pointed literature review questions are analogous in
utility to generating research study questions. The literature review questions are critical
30


to the accomplishment of a research study by helping the investigator structure and
organize the direction of the literature assessment in order to yield maximum
effectiveness to the overarching study (Onwuegbuzie, Collins, et al., 2007). My literature
review was designed to answer six questions, four of which were composed to directly
correlate with the studys research questions. The last two questions were fashioned to
be reflective of the goal and rationale of the literature review, respectively. The literature
review questions, as mapped to the studys research questions, goal, and rationale are
displayed in Table II. 1.
Table II. 1 Literature review synthesis questions,
as mapped to the study research questions, goal and rationale.
Literature Review Questions Study Research Questions
1. Given the existing body of literature about the high school dropout problem in the United States, what factors substantiate the need to examine the distinctive role of high school algebra performance, as compared to other factors, in predicting high school graduation? 1. Does poor performance in a students first high school algebra course serve as a significant predictor of failure to complete high school?
2. What research has been conducted that would justify the use of the predictor variables of algebra performance, reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special education status as a model for examining subsequent failure to complete high school? 2. Will poor performance in high school algebra be more likely than other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special education status to lead to subsequent failure to complete high school?
31


Table II.l (cont.)
Literature Review Questions Study Research Questions
3. What is the conjectured role of self- efficacy, with respect to high school graduation and high school algebra performance? 3. How do a students self-efficacy beliefs, resulting from ones experience in his/her first high school algebra class, influence the potential of a student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school?
4. What does the literature suggest is the relationship between social learning theory and self-efficacy and how does this lend itself to students engaging in behaviors that lead them to dropping out of school? 4. How do student-held expectancies and perceived reinforcements from their experience in the first high school algebra class lead to the potential to engage in behaviors that are symptomatic of failure to finish high school?
Literature Review Goal
5. What conceptual framework can be developed offering a proposed connection between perceived self- efficacy beliefs, reinforcements, and expectancies that result from students experiences in their first high school algebra course leading to the potential to engage in behaviors typical of high school dropouts? Develop a conceptual framework for the qualitative analysis.
Literature Review Rationale
6. What quantitative, qualitative, and/or mixed methods research has been conducted that examines the hypothesized relationship between high school algebra performance and high school graduation? Significance enhancement: The need to augment what is present in the current body of research about predictors of high school graduation failure with new research data and research findings through an investigation into the particular role of high school algebra.
Literature Review Data Collection Process
A search was made of several web and library subscription databases, both within
and outside the education and social sciences disciplines. This included Google Scholar,
Wilson Web, Academic Search Premier, Education Full-Text, Educational Resources
Information Center (ERIC), Academic Search Premier, and the University of Colorado
library databases. I was also able to take advantage of my military background and the
32


online reference databases available through the Air Force Portal and Naval Postgraduate
School. I additionally searched a website called Virtual Learning Resource Center, an
education-based, electronic center. While this site did not provide peer-reviewed articles,
it did provide some interesting information from newspaper publications and popular
media websites, such as Los Angeles Times, New York Times, and PBS Parents.
In addition to the computer-based searches, a hand search was conducted through
many of the professional journals I receive, which includes Educational Researcher,
Review of Education Research, Journal of Education Research, and American Education
Research Journal. I have also gleaned information from past annual conferences of and
papers of the American Education Research Association, Northern Rocky Mountain
Education Research Association, and Hawaii International Conference on Education.
Search terms were numerous and varied with the intent of trying several
combinations in order to uncover what pre-existed in the literature. Keywordssuch as
high school mathematics performance, student perceptions, self-efficacy, social learning
theory, high school algebra proficiency, high school algebra difficulties, high school
algebra requirements, high school obstacles, and mixed methodswere all used in
Boolean combination with other terms, such as high school graduation, high school
dropout, failure to graduate, and high school algebra performance.
Literature Review Legitimacy and Validity
I sought references that reflected a review of multiple research methodologies and
looked for quantitative, qualitative, and mixed methods research in order to increase the
legitimacy and quality of the review synthesis. Establishing legitimacy ensures there is
33


enough information about a concept that can be surmised from a thorough search of the
literature (Onwuegbuzie, Collins, et al., 2007).
Literature review validity was sought by using peer reviewed articles (or by
noticing a lack of such peer reviewed publications) for consideration when determining
the findings from the literature synthesis. In addition, articles were also reviewed with a
critical eye. I sought articles that presented both quantitative and qualitative data. I also
reviewed articles that presented rigorous study, which included explanations of justifiable
research techniques and clear presentation of findings and discussion (Frels &
Onwuegbuzie, 2011; Onwuegbuzie, Collins, et al., 2007).
Literature Review Synthesis Methodology
Figure IF 1 depicts the four-step sequential path taken though the literature review.
Step one is the launching point, where I embarked on the literature review by looking into
the greater realm of the high school dropout problem and some of the many predictors of
high school dropout. In step one I deliberately sought to narrow the search to a smaller,
inclusive sphere within the high school dropout boundary that contained arguments
pointing to the criticality and timing of student high school algebra performance and its
relationship to high school success. In this beginning stage of the research, I also
inquired about what research has been conducted that would allow me to justify the use
of the predictor variables of algebra performances, reading comprehension ability,
writing performance ability, low socioeconomic status, English grade, English language
learners, ethnicity, sex, and special education status as a model for examining subsequent
failure to complete high school.
34


In steps two and three, I further constricted the boundaries of the literature review
search sphere to inspect the topics of self-efficacy and social learning theory, as they
related first to a students high school experiences and then specifically to high school
algebra experiences. Finally the innermost sphere represents the concluding portion of
the literature review, step four. This is where I use the findings from the previous steps
in the search to develop a conceptual framework (the goal of the literature review) that
proposes a lens for demonstrating the relationship between high school algebra
performance and high school dropout. In this step, I also provided justification for
continuing on with my research, the rationale for the literature review.
Figures II.2 through II. 5 provide more detail about each of the four steps
comprising the trail through the literature review, with each step intending to provide
answers to specific literature review questions.
Path through the Literature Review
Figure II. 1 Path Through the Literature Review
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The sequential path through the literature starts with an examination of the issues
surrounding the U.S. high school dropout issue and narrows to those that can highlight
the importance of acceptable high school algebra performance to graduation. The path
then narrows to an examination of self-efficacy and social learning theory matters as they
relate to high school dropout and high school algebra experiences. The path ends at a
proposed conceptual framework depicting a unique lens through which to view the
relationship between the two and a discussion about why further study is needed.
Step One: The High School Dropout
Environment and Algebra's Unique Role
Figure IL2 Step One: The High School Dropout Environment and
Algebras Unique Role
Step one of the literature review will address the first two literature review
questions. First, given the existing body of literature about the high school dropout
problem in the United States, what factors substantiate the need to examine the distinctive
role of high school algebra performance, as compared to other factors, in predicting high
school graduation? This diagram depicts the funnel of predictors to be identified, with
36


particular attention being focused on core course difficulties, the ninth grade transition
year, and failing a course in the early high school years.
Second, what research has been conducted that would justify the use of the
predictor variables of algebra performance, reading comprehension ability, writing
performance ability, low socioeconomic status, English grade, English language learners,
ethnicity, sex, and special education status as a model for examining subsequent failure to
complete high school?
Step Two: Self-Efficacy, the High School Dropout
and the High School Algebra Experience
Figure II.3 Step Two: Self-Efficacy, the High School Dropout, and the
High School Algebra Experience
Step two of the literature review addresses the third literature review question:
What is the conjectured role of self-efficacy, with respect to high school graduation and
high school algebra performance?
37


Step Three: The Relationship between Social Learning
Theory and Self-Efficacy and the Behaviors Leading to
High School Dropout
Figure H.4 Step Three: The Relationship Between Social
Learning Theory and Self-Efficacy and the Behaviors
Leading to High School Dropout.
Step three of the literature review addresses the fourth literature review question:
What does the literature suggest is the relationship between social learning theory and
self-efficacy and how does this lend itself to students engaging in behaviors that lead
them to dropping out of school?
38


Step 4: Self-Efficacy, Reinforcements, and Expectancies
in High School Algebra Exacerbating the Potential to
Engage in High School Dropout Behaviors
Figure II.5 Step Four: Self-Efficacy, Reinforcements,
and Expectancies in High School Algebra Exacerbating
the Potential to Engage in High School Dropout Behaviors
Step four of the literature review addresses the fifth literature review question:
What conceptual framework can be developed offering a proposed connection between
perceived self-efficacy beliefs, reinforcements, and expectancies that result from
students experiences in their first high school algebra course leading to the potential to
engage in behaviors typical of high school dropouts?
I synthesized the literature results and developed a conceptual framework using
the concepts of self-efficacy and Theory of Social Learning. This suggests a proposed
connection between student-perceived self-efficacy beliefs, reinforcements, and
expectancies that have resulted from experiences in high school algebra and high school
dropout. The inner sphere shows that student-perceived self-efficacy beliefs,
reinforcements, and expectancies developed in their high school algebra experiences leak
39


through a porous boundary between high school algebra and the greater high school
experience, which exacerbates the potential to engage in high school dropout behaviors.
This figure does not illustrate the answer to literature question six: What quantitative,
qualitative, and/or mixed methods research has been conducted that examines the
hypothesized relationship between high school algebra performance and high school
graduation?
Interpreting the Results of the Literature Review: Steps One through Three
Research over the years has examined many topics and theories related to the
alarming national high school dropout rate, and numerous reasons have been offered to
explain the failure to complete high school. The factors associated with high school
dropout are expansive and cover a broad range of categories. They range from personal
and family traits to school performance and academic characteristics to behavioral and
emotional patterns. Some of these proposed causes support the criticality of algebra to
success in a high school education; others do not.
Personal attributes of sex and ethnicity are correlated with high school graduation.
By most recognized definitions of high school graduation rates, a disparity to be
addressed further in the discussion of methods, females demonstrate a better high school
completion rate than do males. For example, examining 2008 data compiled by the
National Center for Educational Statistic (NCES), females (90.5%) had a higher status
completion rate than males (89.3%). Higher status completion means that one holds
either a high school diploma or general equivalency diploma (GED) credential. Using
another measure, the status dropout rate, which addresses the percentage of students who
are not enrolled in high school and who have not earned any high school credential
40


(diploma or GED), males have a higher status dropout rate (8.5%) than females (7.5%)
(Chapman et al., 2010).
Status completion and status dropout measures reflect that ethnicity also matters
when examining high school success. Status completion rates were higher for
Asian/Pacific Islanders (95.5%) and White (94.2%) than for Blacks (82.5%) and
Hispanics (75.5%). Status dropout rates showed a similar pattern with less Asian/Pacific
Islanders (4.4%) and Whites (4.8%) going without a high school credential than Blacks
(9.9%) and Hispanics (18.3%) (Chapman et al., 2010).
Unfavorable familial circumstances have long been associated to graduation
failure. Coming from a single family home, a large family home, an urban family home
or a home with other dropouts (parents and older siblings) has been linked with lower
graduation rates. Families headed by dropouts struggle to provide parental involvement
and support in the form of study aids and out-of-school learning opportunities. These
families are often headed by adults who have had children very early in life or include
older siblings who themselves have dropped out of high school (Bridgeland et al., 2006;
Hickman et al., 2008; Swanson, 2006).
Family socioeconomic status is a circumstance that plays a major role in
predicting graduation in more than one way. Students from lower socioeconomic status
and below the poverty line are more likely to drop out of school than students who are
not (Ekstrom et al., 1986). In 2008, the dropout rate for low income students was about
four and one-half times that of students from high income families (Chapman et al.,
2010). These students are subject to secondary effects from being part of a lower
socioeconomic stratum. One is that many low socioeconomic families are prone to
41


increased mobility; and their children are placed in multiple, changing school
environments. Students of high mobility families have trouble adjusting academically,
socially, and emotionally to repeated new school environments, resulting in higher
incidence of disengagement in academic settings and increased dropout rates (Hickman et
al., 2008; Rumberger, 1987).
Another secondary effect of low family socioeconomic status is that many of their
high school-aged students need to work in order to supplement the family income. While
holding some outside job responsibility can increase student achievement, there is a
negative correlation with graduation for students who work over 20 hours a week
(Ekstrom et al., 1986; Gleason & Dynarski, 2002; Natriello, 1987; Orfield, 2006;
Rumberger, 1987; Tyler & Lofstrom, 2009).
School performance and academic determinants, such as poor reading, writing,
and math abilities, are manifested by low and failing course grades, which in turn impacts
eventual high school graduation success. Low standardized test scores in the school
years prior to entering high school have been quantitatively correlated to greater
incidence of high school dropout. Some of these academic predictors show up as early as
elementary school; reading comprehension in third grade is an example of an early
predictor. Looking at the year prior to high school entry, dropouts exhibit significantly
lower eighth grade mathematics and English performance records (Allensworth &
Easton, 2005; Allensworth, Nimi, Montgomery, & Lee, 2009; Bridgeland et al., 2006;
Ekstrom et al., 1986; Gewertz, 2006; Gleason & Dynarski, 2002; Hickman et al., 2008;
Neild, 2009; Tyler & Lofstrom, 2009).
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Certain behavioral and emotional patterns are representative of high school
dropouts. These include nonconforming behaviors, such as excessive absences,
incomplete schoolwork, and disciplinary infractions. A 50-year longitudinal study of
adolescent boys who have displayed delinquent and conduct-disorder behavior has shown
these youth to be 17 times more likely to drop out of school (Hickman et al., 2008).
Absences are significant indicators of future dropout occurrence. During high
school, for example, data from a survey of over 700 students revealed that graduates were
suspended for 1.51 days, while dropouts were suspended for 6.2 days (Hickman et al.,
2008). Much like the set of academic predictors, patterns of behavior often show up
early in a students life, e.g., student kindergarten attendance records can predict
attendance in high school. The more a student is absent in kindergarten the greater
likelihood of dropping out in high school (Ekstrom et al., 1986; Hickman et al., 2008).
Another problem behavior, and one which will be subsequently examined in
depth, is the detachment from school that dropouts develop because they find school
unrewarding, both academically and socially. There are several negative feelings and
experiences reported by the dropouts themselves. They detail that their high school
experience has yielded feelings of low self-esteem about their academic abilities,
alienation from school and from other students, and a general disinterest in the school
environment (Bridgeland et al., 2006; Ekstrom et al., 1986; Gleason & Dynarski, 2002;
Natriello, 1987; Orfield, 2006; Rumberger, 1987; Stillwell, 2010; Sweeten et al., 2009;
Ziomek-Daigle, 2010).
Step One: The high school dropout environment and algebras unique role.
Given the massive discussion about the diverse causes of high school dropout,
there is ample basis for determining algebras comparative influence and role within a
43


model of predictors of high school graduation. However, algebras singular role with
respect to high school graduation success is also worthy of study. A combination of
factors elevates the need to investigate algebras influence, with the first being the very
nature of algebraic content and configuration.
Algebra is a unique course and is often the first in which students are asked to
employ abstract reasoning and problem solving. Researchers have demonstrated that the
abstract nature of algebra increases its difficulty over arithmetic. Students often have
trouble with using abstractness for the required construction of meaning, generalized
relationships, and management of multiple representations of algebraic objects (Rakes,
Valentine, McGatha, & Ronau, 2010; Vogel, 2008).
A second characteristic, the structure of algebraic material compounds its
exclusivity and unfamiliarity to students. For example, students often fail to recognize
the differences between expressions and equations. An algebra equation requires
consideration as a single object rather than a collection of objects, another concept that is
difficult for students to conceptualize. Additionally, the meaning of equality is often
confused within algebras context. Taken together, these typical content structural
challenges often prevent students from recognizing the utility of algebra for generalizing
numerical relationships.
A third distinctive characteristic of algebra is its language. Algebra has an
obscure mathematical lexicon of its own that students must quickly master. It is a
foreign language filled with symbols, expressions, linear equations, quadratic
equations, functions, and polynomials. Each of these three foundational aspects of
algebra (abstract reasoning, mathematical structure, and language usage) in itself causes
44


challenges for students. The confluence of all three potentially forms an impermeable
barrier to mastering algebra for many students (Rakes et al., 2010).
The distinctiveness of algebra leads to a domino effect for many high school
students; and failing to catch onto its many characteristics prohibits success in future,
other required mathematical coursework. An extra barrier to algebraic mastery is the
usual circumstance of limited availability and utility of support from outside the school.
According to Weiss, Carolan, and Baker-Smith (2010), mathematics learning, including
algebra, is more school dependent than other topics, such as reading or science. Algebra,
therefore, may be a course where one not only struggles but the levels of frustration are
so high that one walks away from math and other coursework, never to come back
(Vogel, 2008).
Students are increasingly required to take algebra, even though many struggle
with the uniqueness of algebra and may not be adequately prepared to undertake this
course. The topic of mandatory algebra instruction for all high school students has
garnered much attention. Recent trends in national, state, and local education policies
point to three particular aspects of algebra enrollment that, in addition to its unique
nature, thus elevate algebras criticality to the high school dropout issue. These are (a)
algebra enrollment in the transition or ninth year of high school, (b) algebras inclusion as
a core or required high school course, and (c) algebra failure in the early high school
years.
Algebra is a required mathematics course in many schools and is increasingly part
of todays high school landscape. High stakes testing, such as state assessment tests
brought on by the No Child Left Behind act, is one reason for the addition of more
45


rigorous and mandated classes, including algebra to the high school curriculum. Because
of this, many states must ensure they enroll all students, even the unprepared, in rigorous
and academically challenging core courses, such as algebra (Medina, 2010).
Two other causal factors have induced the addition of algebra as a standard in
high school programs across the nation. The first is the movement to ensure United
States students are prepared to enter the 21st century workforce. Since algebraic skills
have long been thought to be foundational for entry into technical professions needed to
compete in todays global economy, there is a growing fear that America is losing ground
with other industrialized nations in nurturing such a skilled workforce (Allensworth et
al., 2009; Gamoran & Hannigan, 2000). Listening to PBS radio the other day and
following a story on President Obamas visit to an Intel microchip plant, I heard him
quoted as saying: Our high school graduates had better know algebra (PBS Radio
Broadcasting, 2012; Stein et al., 2011).
The second issue is the reported unpreparedness of our students to complete
college courses. The remediation rate of students when they enter college, especially in
mathematic skills, e.g., algebra, is of considerable concern. In response, a number of
states have responded to the unpreparedness of college students with dictating that their
school districts include algebra into the high school curriculum (Stein et al., 2011). For
example, in New York State, 28% of incoming college freshmen in the year 2005 were
enrolled in a remedial course in reading, writing, or mathematics. Because of this, New
York State has started to grade their high schools on how many of their students are in
remediation while in college (Medina, 2010).
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Many states and school districts now require the passing of algebra as a core
course to fulfill the requirements for a high school diploma. By 2007, a total of 13 states,
with 16 more states expected to follow suit, were mandating the inclusion of algebra as a
required course for all students (Schornick, 2010). Some states and school districts are
not only requiring the successful completion of algebra but are directing it at the earliest
opportunity in a high school career, i.e., the ninth grade (Allensworth et al., 2009;
Medina, 2010; Neild, 2009; Neild, Stoner-Eby, & Furstenberg, 2008; Schornick, 2010;
Stein et al., 2011). Unfortunately, the experiences in, and the results of, courses such as
algebra which students take in the ninth grade often shape the success of the rest of their
academic careers (Allensworth et al., 2009; Gamoran & Hannigan, 2000; Neild, 2009;
Neild et al., 2008).
The first two years of a students high school experience is make or break time
in an educational trajectory, especially for teenagers identified as at risk (Natriello, 1987).
Ninth grade is an especially crucial, transition year for many adolescents. For some
students, it represents the final stages in the prolonged exiting off the educational track.
Research has shown that a positive experience in ninth grade is significantly correlated to
later success in high school, and academic difficulties in the ninth year have a negative
correlation to later high school graduation (Allensworth & Easton, 2005; American
Psychological Association, 2010; Archambault, Janosz, Fallu et al., 2009; Archambault,
Janosz, Morizot, & Pagani, 2009; Hickman et al., 2008; Neild, 2009; Neild et al., 2008;
Roderick & Camburn, 1999).
Academic difficulty is quickly evident after transitioning from middle school to
high school. Dropouts show lower grades than their peers as early as the first semester of
47


ninth grade (Alspaugh, 1998; Hickman et al., 2008; Neild, 2009; Neild et al., 2008).
Hickman et al. (2008) demonstrated that the ninth grade first semester overall grade point
average (GPA) was significantly lower for dropouts than for graduates, and this disparity
is maintained into the second semestera mean of 2.75 to 1.27, respectively.
Ninth grade also represents the second transitional juncture for a majority of
American students, including most of the students in the district under study. The first
transition is from elementary school to middle school. Alspaughs (1998) research
showed that students who undergo both these transitions (i.e., who do not go to a K-8
school before entering high school) tend to show a statistically significant greater
achievement loss in high school and a statistically greater significant likelihood of
dropping out of high school (Alspaugh, 1998).
Several dynamics converge to amplify the swirl of a students ninth year of
school. Curricula demands finally catch up with many students in their freshman year.
Insufficiently prepared students and students holding less than average grade equivalent
skill sets, especially in mathematics and reading, especially struggle and flounder in an
environment where they must pass and earn credits to progress. In Philadelphia, for
example, ninth graders who scored at grade level on standardized math tests were 42%
more likely to get promoted to the tenth grade than those who scored two years or below
grade level (Neild, 2009; Neild & Balfanz, 2006).
There is a sudden reduction in adult scaffolding and a corresponding need for
ninth graders to assume personal responsibility for their own education and behavior, to
include speaking up when things go wrong. Adult scaffolding can be thought of as the
protective framework, or hand-holding, that has surrounded students and which has
48


provided explicit directions on most aspects of their elementary and middle school
education. This includes actions such as deliberately telling students when to take notes,
asking them to hand in their homework and ensuring they come to class.
Due to the blanketing of structure and organization imposed on students in their
elementary and middle school education years, many are unlikely to have thus learned to
assume self-responsibility for accomplishing tasks necessary for success in education
prior to entering high school. Adolescents entering the ninth grade are also simply not
mature enough to understand they must possess self-initiative to complete work, ask the
teacher for help when needed, and earn passing grades to gain the credits needed to
advance and eventually graduate. The reduction in adult supervision often leads to non-
productive patterns of behavior, such as incomplete or late work, inattentiveness to class
proceedings, tardiness and unexcused absences, all of which have been found to be
negative influences on achievement and the eventual earning of a diploma (Neild et al.,
2008; Roderick & Cambum, 1999). By the time adolescents gain the maturity to realize
they alone are primarily responsible for their educational accomplishments, it is often
too late.
When a student enters ninth grade, in addition to a reduction in adult scaffolding,
they must relate to a new set of educators. In the overwhelming majority of cases, they
have had scarce personal contact or a previous relationship with the new teachers or other
school adults. Students quickly feel lost and anonymous, lacking ready access to a
trusting adult to whom they can talk, especially in this pivotal transition year, when it is
so easy to fall off track and get into academic difficulty (Bridgeland et al., 2006; Neild,
2009; Neild et al., 2008).
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Corollary to the changes in the adult influences is a changing of the peer cohort
influence of previous educational years. Peer cohorts and the participation with new
activities and new social networks can yield either positive or negative effects on student
achievement. As an example, in the case of ninth graders who had poor eighth grade
academic performancegrades of mostly C or lowergoing to high school with mostly
new peers who were not their colleagues in the eighth grade showed a positive correlation
to improved grades (Neild, 2009).
The newness of the fresh high school environment matters but so does the actual
academic composition of the ninth year, such as the number of core courses that a
freshman takes. Hickman et al. (2008) did a study of a high school cohort of 119
students enrolled in high school from 2002 to 2005. The cohort included 60 graduating
students and 59 dropout students. High school dropouts took significantly more core
courses, a mean of 6.02, than graduates, a mean of 4.36. It is likely that because they
took more core courses in the ninth year their GPAs were lower immediately than their
peers and persisted in this lowered state throughout the year.
The type of courses a student takes has another relationship to high school
graduation. Because dropouts take significantly more core courses in the ninth grade
than those who graduate, their individual schedules leave little room for electives or
personal interest courses, such as the arts, that have a better likelihood of resulting in
academic success typical of other high school students (Hickman et al., 2008).
Requiring students to take more mathematics, to include algebra, does not
necessarily equate to increased achievement in high school or to high school graduation.
Hoffer (1998) used data from the National Education Longitudinal Study of 1988
50


(NELS:88) and Logistic Regression analysis to show that dropout rates between schools
requiring two years and schools requiring three years of mathematics are about equal
without adjusting for the effects of student background differences. In other words, there
was neither an increase nor a decrease in high school achievement, when the number of
mathematics courses required for graduation was raised; and there was no negative or
positive effect on high school dropout rates (Hoffer, 1997). This lack of an increase in
dropout rates can be attributed in part to schools that allow students to take watered down
math courses or to not take difficult math courses such as algebra.
The Chicago Public School System requires that all students take algebra in the
ninth grade. Allensworth et al. (2009), in a longitudinal study of the consequences of
mandating Algebra 1 for all Chicago high school freshmen in the years 1994 to 2004,
also found no increase in dropout rates in an urban setting with an almost 50% dropout
rate. While more students earned math credits (probably attributed to more students
taking Algebra I), they did, however, find that algebra failure rates increased, grades
slightly decreased, and test scores did not improve (Allensworth et al., 2009). They also
discovered that absenteeism increased for students of previously measured average and
high math abilities, perhaps as a result of boredom and unchallenging curriculum that is
designed for students of all skill levels. Algebra for all may increase the number of
students who take algebra, including minority students, and may increase the number of
students passing algebra; but it also increases the number of students who fail and get
behind in academic credits as well (Gamoran & Hannigan, 2000).
The majority of students in the school district under study takes algebra and does
so in the ninth grade. In addition, all students are now required to pass three years (six
51


semesters) of math. While algebra is not required as one of the math classes the state high
stakes testing in the ninth year includes many algebraic topics. Thus, many of the
students are forced to take algebra in preparation for the state tests, with scarce regard for
whether or not students are ready to take these required courses. And because of this,
more unprepared and low achieving students in the district are taking algebra than ever
before.
Highly related to the difficulty of adjusting to high school in the ninth year is the
frequent occurrence of a student ending up with a shortage of credits at the end of this
academic year. If a student fails algebra, it can lead to the student getting behind in the
number of credits needed to graduate high school in four years. Falling behind in credits
due to failing a class and not earning credit for the failed course during the early high
school years significantly increases the risk of failing to graduate high school, especially
graduating on time after four years of attendance.
Students who are falling behind are noticeable almost immediately as they begin
their high school careers and are often failing some or all of their core courses. In the
Chicago public school system, for example, more than 40% of freshmen fail a major or
core course in their first semester of freshmen year (Allensworth & Easton, 2005; Neild,
2009; Roderick & Cambum, 1999). In Philadelphia, it was found that the percentage of
courses failed in the ninth grade was a strong predictor of eventual dropout. Using a
representative five-course load, an increase of 20 percentage points, or one extra
failed course, increases the odds of dropping out by approximately one-third (Neild et al.,
2008).
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The Chicago Public School System uses an on-track indicator that is comprised of
two components. Those who are on track at the conclusion of the ninth grade have failed
no more than one semester of a core course (defined to be English, math, social studies,
and science) and have completed enough credits to be promoted to the tenth grade. In a
study conducted with cohorts of their students in the years between the 1993-1994 school
year and the 1999-2000 school year, the Chicago found the on-track indicator to be more
accurate for prediction of eventual graduation than eighth grade test scores or student
demographical characteristics, such as race/ethnicity and economic status. In 1999, for
example, the school system found that a student who was on track at the end of the ninth
year was more than three and one-half times likely to graduate than one who had failed a
semester of a core course and was behind on credits. Students who were off track in the
ninth grade had a 22% on-time graduation rate as compared to 81% for students who
were on track with credits and passed courses at the end of the ninth grade (Allensworth
& Easton, 2005; Neild, 2009).
Other, more recent, studies in Texas and Pennsylvania provide additional
evidence of the negative correlation between getting behind in credits in the ninth grade
and high school dropout rates. The studies go so far as to claim that 16 out of every 100
studentsand in some school systems as high as almost one-third of all dropoutsnever
get beyond the typical required ninth grade credit totals (Neild, 2009; Neild & Balfanz,
2006; Swanson, 2006).
School districts vary in standards for what is needed for promotion to the next
high school grade. Some require the passing of enough credits and the passing of certain
core courses. Others simply require a student to earn a specified number of credits, in
53


total and by content area, in order to earn a high school diploma. Regardless, a student
who does not earn enough credits in the first year will need to pick up the pace and make
up the credits in order to graduate on time. They must also graduate with enough credits
in the required content areas. Further, the longer it takes a student to complete the
requirements for high school graduation, the higher the risk of failing to earn a high
school diploma (Chapman et al., 2010; Roderick & Camburn, 1999).
A typical student enters school in the ninth grade and must stay on track, passing
enough courses, and passing certain types of courses, in order to maintain academic and
core course credit progress toward earning a diploma in the standard four years. The
literature clearly showed that failing core courses and getting behind in credits in ninth
grade was more predictive of high school dropout, with more students, including this
districts students taking algebra in the ninth grade (Allensworth & Easton, 2005; Neild,
2009; Neild & Balfanz, 2006; Roderick & Camburn, 1999), it provide reason for
examining this in my study as well.
Step Two: Self-efficacy, the high school dropout, and the high school algebra
experience.
Individual student self-efficacy beliefs and perceptions of the reinforcements and
expectancies originating from both the greater high school environment and, specifically,
high school algebra classes all figure in the potential to manifest behaviors characteristic
of the high school dropout. The typical high school atmosphere and high school algebra
class is fraught with sources of negative self-efficacy beliefs, reinforcements, and
expectancies. Examining two closely related schools of thought, Banduras concept of
self-efficacy and Rotters Social Learning Theory, will provide insight as to why a
54


typical high school environment and high school algebra classes are full of these
negative effects.
Self-efficacy and the high school experience.
Self-efficacy is the belief in ones abilities to affect change or impact a situation.
It is a self-judgment of personal capability to organize and execute courses of actions and
behaviors required to attain designated types of performances. Self-efficacy impacts how
well one uses their skills and plays a major role in human functioning, affecting how one
thinks, feels, and acts. Self-efficacy also affects how one motivates oneself in a given
situation and provides individuals the capacity to alter the environment and influence
their own actions or behavior. It has a very powerful influence on peoples actions and
beliefs about what to expect from their actions. Self-efficacious individuals set higher
standards for themselves and use more efficient strategies to gain the results needed in
certain environments. They are also more resilient in their actions, set progressively
higher goals for themselves, and are more satisfied with their performance (Bandura,
2012).
Bandura hypothesized, and has shown in numerous studies, that self-efficacy
mediates past experience and performance and is a key driver in predicting future
performance. Low self-efficacy can lead people to think that things are tougher than they
really are. This can foster stress and a narrow outlook on how to accomplish tasks in a
certain domain. Low self-efficacy can also be a strong determinant on future levels of
accomplishment (Pajeres, 1996b). Persons with high self-efficacy can view hard tasks,
such as getting through the rigors of the freshman year of high school, as a challenge to
overcome. They often show a sense of commitment to this quest for mastery. These
persons will strive to work through obstacles and view failures as something that can be
55


worked through by resorting to their own devices. Most importantly, they believe they
can learn needed skills or knowledge.
Those with low self-efficacy have a weak commitment to overcoming challenges
and obstacles. They do not think they have the ability to gain the needed skills or
knowledge and do not display persistence in the mastery of hard tasks, e.g., gaining the
proficiency needed to pass course examinations. They would rather take the easy path
and resign themselves to failure (Bandura, 1986, 1994, 1995, 2012; Bandura, Regalia,
Caprara, Barbaranelli, & Pastorelli, 2001).
Peoples self-efficacy beliefs are so powerful that this often overpowers the
concrete results of past performance, forming a filter through which to interpret new
phenomena. This new filter, in turn, influences subsequent actions and behavior
outcomes (Pajeres, 1996b). Bandura, in his 1977 seminal work on self-efficacy,
postulated that self-efficacy beliefs and filters formed from self-efficacy beliefs, are a
major determinant of whether a person will attempt a given task and how much effort
will be expended in the accomplishment of a given task when obstacles are present
(Bandura, 1977).
High school dropouts are more likely to report lower feelings of self-efficacy and
encounter multiple sources of negative self-efficacy beliefs than those who stay in school
(Dynarski & Gleason, 1998). General characteristics of high self-efficacious adolescents,
those who stay in school, are greater persistence, effort, and an intrinsic interest in
individual learning and results.
Those with lower self-efficacy reduce their goals and slacken their efforts in the
face of adversity. High self-efficacious students will not be easily deterred by an
56


occasional failure and generally work harder and persist longer to accomplish a task than
low self-efficacious students (Bandura, 2012; Pajeres, 1996a; Schunk, 1984;
Zimmerman, Bandura, & Martinez-Pons, 1992).
Bandura has promoted four sources of efficacy beliefs. These sources contribute
to the conviction in ones ability to perform a task or function successfully within a
specific domain. Together or alone, these sources shape the self-efficacy beliefs
foundational to ones success in their educational pursuits. These sources stem from and
are unique to an individuals interplay with a given environment (Bandura, 1977). The
sources are (a) Mastery Experiences (ME), (b) Vicarious Experiences (VE), (c) Social
Persuasion (SP), and (d) Affective States (AS).
A mastery experience derives from the actual doing of a task. Bandura (1986)
states that mastery experiences have the strongest, although not the only, influence on
self-efficacy, since these experiences offer genuine evidence about the resultant quality of
the completed task. Self-efficacy increases if one believes they are successful at mastery
of a task; if not, there is a decrease in self-efficacy.
Repeated failures not attributed to perceived individual effort and the level of
difficulty of mastery experiences may be a strong detriment on self-efficacy. Success
with hard tasks in challenging situations raises self-efficacy. On the other hand, if a
student has experienced success primarily in easy tasks and unchallenging environments,
they come to expect success routinely and do not learn to persevere. When faced with a
difficult task, they are easily discouraged, give up, and fail, thereby lowering beliefs in
self-efficacy (Bandura, 1995, 2012; Oettingen, 1995).
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Many future dropouts enter high school molded by the mastery experiences of
earlier schooling, when they did not need to apply much effort to succeed and there were
no challenges that would help them gain a sense of perseverance. They developed a false
sense of self-efficacy that was based on getting by in mastery experiences that required
very little expenditure of energy. The new ninth grader quickly faces a novel
environment that contains much harder and more numerous, specific mastery experiences
in which they must put forth extended effort and succeed in order to make the grade and
nurture the growing of self-efficacy. They need to succeed with some degree of self-
reliance and less adult scaffolding in mastery experiences, e.g., doing class work,
completing homework, and passing exams in order to gain academic credit and progress
toward graduation. These mastery experiences become a prominent part of high school,
and failure quickly lowers expectations of success and feelings of self-efficacy
(Allensworth & Easton, 2005; Neild, 2009; Neild et al., 2008).
There are many ninth graders who enter high school without previous success in
mastery experiences, much less challenging ones in which they would have learned how
to persevere. Neild (2009) pointed out there are too many students who have previously
passed coursework but who start to fail in the ninth grade, when inadequate preparation
of less than sufficient reading and math skills catch up with them.
Another problem related to mastery experiences is that ninth graders generally do
not see these experiences as yielding immediate gratification and rewards. End of course
grades, gaining academic credit, and earning a diploma are distal goals, especially for
relatively immature ninth graders. Dropouts report scarce experience with proximal
academic tasks, let alone with increasingly challenging tasks. There is little opportunity
58


to gain a taste of success and the resulting self-efficacy that is generated with appropriate
and rewarding mastery experiences in high school (Schunk, 1984).
The second source of efficacy is vicarious experience. Mastery sources are strong
sources of self-efficacy, and so are vicarious sources, which sometimes act independently
of mastery sources. Vicarious sources become especially prominent when mastery
experiences are limited or absent.
Vicarious experience is knowledge that is gained indirectly from oneself and from
the behaviors of others. With respect to oneself, vicarious experiences allow for
reflection on a new situation. It gives the learner the chance to mediate what they can do
based on what they have observed themselves do in the past.
Modeling by others can take on a substantive role in forming self-efficacy beliefs
and in the subsequent affect on behaviors (Bandura, 1986, 1995; Pajeres, 1996a, 2002;
Schunk, 1984). Bandura (2012) has shown the importance of vicarious sources in
shaping self-efficacy through the findings of two meta-analysis studies that demonstrated
vicarious sources of self-efficacy had lowered or raised self-efficacy independent of
performance outcomes (Bandura, 2012).
One of the strongest vicarious experiences originates from the modeling of
behaviors by others who are similar, with the strength of this source of self-efficacy
increasing accordingly to the degree of closeness of the model to oneself. The successes
of models similar to oneself raise self-efficacy, while their failures lower it (Bandura,
1986, 1995). In a study of sources of self-efficacy beliefs, shaped by social modeling and
its relationship to mathematical outcomes, Schunk and Hanson (1989) found that
elementary students self-efficacy beliefs were stronger when influenced by observing
59


peers than when influenced through observing teachers, even though the same
mathematical operations were modeled by both groups. The power of peer models is
already strong in elementary students, as discussed within this study. This influence
grows stronger through the school years until a student enters high school (Schunk &
Hanson, 1989).
The ninth grade is a point at which students elevate the significance of social
interactions and relationships, and the influence of peer groups grow accordingly (Neild,
2009). Adolescents tend to deliberately choose peers who share similar interests and
values. Peer groups start taking on increasingly important roles and become
progressively more influential on the high school student (Bandura, 1994; Nichols &
White, 2001). These peer models may or may not facilitate academic achievement and
subsequent graduation. Academic achievement may suppress identification with certain
peer groups and success in school; 42% of the over 400 dropouts interviewed in
Baltimore and Philadelphia stated they spent time with people who were not interested in
school (Bridgeland et al., 2006).
On the other hand, the lack of belonging to a peer group has been also positively
correlated to high school graduation for some students. Struggling middle school
students who enter high school with a different peer group actually do better once in high
school. Those with lower ability, who continue to identify with similar students, may be
the victim of interactions that oppose academic achievement (Neild, 2009; Nichols &
White, 2001).
It is possible to gain self-efficacy via vicarious experience without similar role
models, but this is only when it is gained from competent models who adequately convey
60


knowledge, skills, and strategies of competency (Oettingen, 1995). Unfortunately, many
dropouts do not benefit from such role models in their high school educators. Students
often reported they believed their teachers were unqualified and of low quality
(Bridgeland et al., 2006).
Social persuasion is the third source of self-efficacy. Social persuasion is the
response one receives from others, such as teacher feedback and encouragement; and it is
usually displayed in the form of spoken judgments from others. Verbal advice often
provides students needed reinforcement of positive remarks on their efforts and work, but
social persuasion can be a two-edged sword. Withholding feedback is harmful, and it is
as equally destructive to self-efficacy to provide negative and unrealistic feedback
(Bandura, 1986, 1995; Pajeres, 1996a). Constantly berating a ninth grade algebra student
for not being able to solve a problem on the board is just as negative as praising a student
for finishing class work without first checking the students comprehension.
Social persuasion as a self-efficacy booster requires precision in use and origin.
Social persuasion, which is tied to individual accomplishments and not just task
completion, is most effective in bolstering self-efficacy (Schunk, 1984). It is of use only
when the source is believed to be trustworthy and sincere (Oettingen, 1995); and as
reported by many dropouts, there seems to be a lack of such sources in the high school
environment.
Poor attitudes, little caring, modest help, and shallow expectations are examples
of the lack of social persuasion self-efficacy sources in the high school experience of
many dropouts. Dropouts have expressed that teachers themselves appear disengaged
with the class and clearly emit signals showing their dislike of teaching. Students further
61


perceive that their teachers, for the most part, care only about presenting the material and
just standing in front of the board. The teachers talk past the students instead of
communicating with them, rarely involving the students in their education (Bridgeland et
al., 2006; Gewertz, 2006; Gleason & Dynarski, 2002; Orfield, 2006; Schornick, 2010).
Interviewees from Baltimore and Philadelphia focus groups examining the high school
dropout problem pointed out, Some of their [students] best days were when their
teachers noticed them, got them involved in class, and told them they were doing well
(Bridgeland et al., 2006).
The fourth source of efficacy is the affective states, which are the emotions,
attitudes, and physical and emotional responses from items in an environment. These
sources impact ones judgments of own self-efficacy. Negative moods and attitudes,
stress, and frustrations generally yield lower self-efficacy. When someone is in a positive
mood and holds positive attitudes, they tend to be more self-efficacious. What is
important is how one perceives their physical and emotional responses to their
environment (Bandura, 1986, 1993, 1995).
There are many origins of negative physical and emotional responses for the
typical high school dropout, and it starts immediately upon entering their freshman year.
Ninth grade has long been associated with feelings of anxiety and confusion; and many
students quickly believe they do not belong in school, due in large measure to lack of
individual caring on the part of school adults. In addition, ninth graders are faced with
frustrating school policies that trigger mandatory grade failure, such as too many
absences. This may discourage students from persisting in school for the remainder of
62


the quarter, semester, or year, eroding their commitment to school success (Neild, 2009;
Wheelock & Miao, 2005).
As they continue on in school, growing boredom and irrelevancy causes many
feelings of negativity and lack of motivation in some high school students. As a
collective group, high school dropouts are unmotivated, not challenged enough, and
bored by classes. As reported in a survey of 467 dropouts, these negative responses to
school were cited as more central to a decision to leave school than actual grades. Only
35% of students who took the survey cited academic failure as the reason for dropping
out of high school (Bridgeland et al., 2006; Gewertz, 2006). Course material was pointed
out as irrelevant to the real world, and 70% to 80% of students stated that they wanted
more authentic world learning opportunities (Bridgeland et al., 2006; Gewertz, 2006).
Self-efficacy is a fluid construct holding individualistic, domain, and time-altering
attributes. Peoples self-efficacy beliefs are a product of what they personally believe to
be true in a given environment and at any one time. This involves the weighing and
integration of selective information from multiple sources in ones surroundings during a
particular epoch of time (Oettingen, 1995).
Individuals perceive their beliefs of self-efficacy differently throughout their
lifespan and across diverse domains. The older a student gets, the more pessimistic they
grow in their self-perception of individual abilities and expectancies of performance.
Older students tend to expect much less from their behaviors with far less positive
outcomes; and sometimes, to them, it accordingly does not make sense expend much of
an effort in school (Archambault, Janosz, Morizot, et al., 2009; Kloostermann & Cougan,
1994). Younger students, especially in the earlier elementary school years, report a much
63


greater sense of belief in their academic abilities than do older students (Pajeres, 1996b).
Additionally, research has shown that by the fifth grade students develop separate verbal
and math self-efficacy measures, due to their increasing ability to distinguish their
competency in different academic areas (Zimmerman & Martinez-Pons, 1990).
Specific student self-efficacy beliefs in high school are likely to not be the same
as they were in earlier school years, with a noticeable change from the eighth to ninth
grade. Finally, since the typical high school student is old enough to distinguish
competencies in different academic areas, self-efficacy beliefs that are formed through
experience in algebra are likely to be different from those formed in other subjects, such
as English or other math courses (e.g., geometry).
Self-efficacy and the high school algebra experience.
Algebra, as mentioned earlier, is an academic discipline encompassing distinctive
characteristics that affords students a school experience that is unlike that provided by
other classes. Because of this, self-efficacy beliefs, and the sources of these self-efficacy
beliefs, developed from algebra involvement are distinctive from the self-efficacy beliefs
that originate in other areas of high school. With the increasing emphasis on algebra as a
core mathematics course that must be mastered in order to successfully earn a high school
diploma, the concept of algebra and mathematics self-efficacy, the sources of algebra and
mathematics self-efficacy, and their resultant influence on successful algebra outcomes
with the subsequent high school performance are appealing areas for exploration.
Mathematics self-efficacy is strongly related to and is predictive of mathematics
performance outcomes (Hackett & Betz, 1989). Unfortunately, many students lack the
necessary beliefs in their ability to succeed in mathematics, particularly in algebra. For
instance, students were asked to rate their perceptions of competency in nine academic
64


areas on a self-efficacy questionnaire that was given to over 100 ninth and tenth graders
in a large Eastern city. Algebra was rated at almost the lowest self-efficacy. Only the
ability to learn foreign language was judged to be lower (Zimmerman et al., 1992).
Mastery experiences and social persuasion sources have been found to be
especially critical to the formation of algebra self-efficacy. Lopez et al. (1997) used path
analysis to demonstrate that mastery experiences and social persuasion were each
significant predictors of algebra self-efficacy. Algebra self-efficacy was then predictive
of algebra outcomes, in this case passing grades. Although vicarious experiences and
affective states were not significant predictors in themselves, the path from self-efficacy,
which was a summation of paths from all four sources, continued to demonstrate that the
overall self-efficacy concept was significant in predicting algebraic results. This group of
researchers also demonstrated the difference between students perceived mastery
experiences and actual test scores in predicting algebra self-efficacy. They showed that
while perceived mastery experiences significantly predicted self-efficacy, actual
standardized math scores did not. Clearly, ones self-efficacy beliefs from their mastery
experiences were more important on future algebra outcomes than actual performance.
The ease at which students transition into algebra is dependent on prior mastery
experiences in mathematics. However, due to the uniqueness of algebra, earlier
mathematics mastery experiences may be inadequate toward advancing the new mastery
experiences required to promote algebra self-efficacy. Sufficiency in past mathematics
mastery experiences and getting good grades in earlier math courses may not be enough
to maintain mathematics self-efficacy. Students who previously coasted by and were not
challenged enough are prone to not persevere when it gets harder to maintain good grades
65


in a high school mathematics course such as algebra (Kloostermann & Cougan, 1994;
Stipek, 1981). Additionally, gaining mastery experiences in algebra can be very different
from gaining other mathematics mastery experiences, again, because of algebras
abstractness, structure, and language (Helfand, 2006; Kollars, 2008; Pajeres, 1996a; Rand
Corporation, 2008).
Students who have experienced mathematics only at a concrete or procedural
level, characteristic of many U.S. classrooms, must negotiate the complex gap from
concrete to abstract reasoning without the benefit of prior mastery experiences in this
new way of thinking (Rakes et al., 2010). Teaching for content and not mathematical
reasoning or meaning is a practice that starts in elementary school education and
continues into the high school years of mathematics instruction. In the U.S., as compared
to Russia, China, and Japan, our students rarely write arithmetic sentences to represent
situations. Consequently, when our students move to algebra, they have developed the
predisposition to understand algebraic formulas as something they need to calculate
rather than as expressions that represent something (Thompson, 2008).
Too much instructional emphasis is placed on this mechanical and rote learning of
the steps of algebra. Teaching methods that focus on the rigid, procedural aspects, in
reality, interfere with the student gaining the requisite conceptual understanding skills
(Lobato, 2008) and falls short of ensuring that students understand the meanings behind
the algebraic concept (Rakes et al., 2010; Thompson, 2008). Once they waver from a
scripted path as defined by a mechanical method of instruction, students are not given the
tools necessary to find their way and gain successful mastery experiences (Sfard, 1991).
66


As one example, when studying how to use functions and graphs in algebra,
students must begin to understand and interpret one set of algebraic objects in terms of
another (e.g., a function equation with its graph, data set by its equation, or data set by its
graph). Leinhardt, Zaslavsky, & Stein (1990) found that students generally have no
problem with gaining procedural fluency at plotting points and equations but do have
problems with the ability to extract meaning from graphical representations. The
difficulty lies in the connection of a graph to the construct being represented.
Specifically, students are readily capable of demonstrating procedural fluency; but this
procedural understanding is unable to guide students through problems involving
interpretation (Leinhardt, Zaslavsky, & Stein, 1990).
Further complicating the issue, teaching methods used to convey content often
exacerbate the trouble with gaining mastery algebra experiences. Both students and
teachers expect immediate rewards for teaching and learning efforts. However, the
understanding of abstract mathematical ideas often requires a lengthy, iterative process,
and innovative instructional methods, something that is rarely addressed in our country.
In reading through the 2008 Final Report from the National Mathematics Advisory Panel
about instructional changes needed in our algebra curriculum, it was dismaying to see a
lack of information about what is required with respect to instructional pedagogy to
ensure that our high schools do a better job of conveying abstract reasoning and meaning
skills (Rand Corporation, 2008). Teachers of high school algebra, especially, must be
adaptive in instructional pedagogy so as to nurture students mathematical learning and
success.
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It stands to reason, given the a general lack of belief in algebraic self-efficacy,
that students need sources of strong social persuasion in addition to credible mastery
experiences. Teachers must effectively model, illustrate, and explain with multiple
instructional methods. They then need to provide for effective feedback mechanisms,
making certain students understand the content of the algebraic material (Borko &
Whitcomb, 2008). And to be most efficient for increasing algebraic self-efficacy, this
feedback needs to be task and situation-specific (Kloostermann & Cougan, 1994).
Disappointingly, and similarly to the shortage of mastery experiences, there is a
deficiency of strong social persuasion sources in our nations mathematics classrooms.
Mathematics teachers are using boring and uninspiring teaching methods. They simply
lecture, provide notes, and work problems on the board, rarely engaging their students.
Former high school mathematics students in one study pointed out the lack of social
persuasion in their high school algebra courses. They could not ask teachers for help, and
teachers failed to make the classes relevant or applicable to real life. Students also
alleged that their teachers had low expectations of them, which negatively affected
students perceptions of their high school math class experience and course outcomes
(Schomick, 2010).
Vicarious experiences are a third form of self-efficacy at play in algebra classes.
These vicarious algebra sources can be teachers or other youth and can add to or decrease
from existing levels of algebraic self-efficacy. Children who have observed verbal
modeling of division operations by proficient teachers were noted to have gained more
confidence and increased skill in performing division operations than those who did not
observe this modeling (Zimmerman et al., 1992). Students who were part of like-student,
68


clique groupings experienced less variance (good and bad) in Algebra I achievement
results in a study of 230 high school students in two high schools in the mid-south
(Nichols & White, 2001). In any case, an encouraging and participatory set of vicarious
sources in mathematics classes can lead to increased student engagement and, thus, more
positive outcomes (Weiss, Carolan, & Baker-Smith, 2010).
On the other hand, poor vicarious sources, such as math teachers who simply lack
their own mastery knowledge of or a demonstrative interest in the algebra content area,
can be damaging to mathematics self-efficacy, another issue highlighted by the National
Mathematics Advisory Panel (Rand Corporation, 2008). Many algebra teachers do not
model or show a real interest in gaining algebra knowledge for the sake of learning and
preparation for the future. What is largely demonstrated is that teaching algebra is
equated to merely earning a credit or passing the standardized state tests at the end of the
year (Roschelle, Singleton, Sabelli, Pea, & Bransford, 2008).
Algebra is replete with the fourth source of self-efficacy, the affective state
sources. Many of these sources lower ones sense of algebraic self-efficacy. Two such
affective state sources are mathematics anxiety and mathematics frustration. Taking part
in algebra instruction holds a good potential for intensifying previously rooted feelings in
these two areas (Hackett & Betz, 1989). The feelings of anxiety do not diminish upon
high school entry as students grow increasingly frustrated and less confident about their
ability to grapple with the unique learning challenges of algebra, especially in lieu of the
emphasis on procedural learning. One critic of the 2008 Final Report from the National
Mathematics Advisory Panel states, The single-minded focus on proficiency is likely
69


one of the major sources of mathematics anxiety that is rampart in our population
(Thompson, 2008).
Student perceptions of relationships with mathematics teachers also do not
facilitate a reduction in anxiety and frustration from algebra participation. Schornick
(2010) discussed several negative perceptions, as revealed from phenomenological, first-
person accounts of the high school math experience of several students. They described
their high school math classes as something they just had to take in order to get by. Math
learning and class efforts were not valued by their teachers. High school math teachers
were perceived as uncaring, and they did not demonstrate the relevancy of algebra to
other academic courses and real world applications. The lack of teacher demonstrated
relevancy to the real world especially stood out in the memories of these interviewees.
One student remembered that the only class where he saw relevancy was in trigonometry,
when the teacher showed students how it was used in construction (Schornick, 2010).
These negative student perceptions allow for students to dislike mathematics, lose
interest in learning and performing in math classes (e.g., algebra), and become jaded in
class. Algebra then grows beyond the desire of some students to work hard to master the
contents (Allensworth et al., 2009; Gamoran & Hannigan, 2000; Kaput, 2000). Students
who believe algebra is useful in their lives and who enjoy mathematics are more
confident of their mathematical prowess. Students who are confident in their abilities
will do better in their algebra courses (Kloostermann & Cougan, 1994).
Step three: The relationship between social learning theory and self-efficacy
and the behaviors leading to high school dropout.
Perceptions play a major role in the development of a persons individual self-
efficacy beliefs. Perceptions also influence peoples potential to engage in certain
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behaviors in their differing environments, such as the environments of a high school
algebra class and or greater high school atmosphere. Social learning theory proposes that
to understand human behavior one needs to exam the interplay, or interaction, that takes
place between an individual and the environment. The environment is comprised of both
the physical setting and other individuals who are part of the setting. Peoples potential
to behave in a particular fashion can be attributed to perceived reinforcements and
expectancies from their individual interactions with the environment (Mearns, 2004;
Rotter, 1982). Rotter (1982) represents the relationship of a persons interaction within a
particular environment with this equation:
Behavior Potential = f (Reinforcement Value + Expectancy)
This formula illustrates that the likelihood of a person engaging in a particular
behavior, the Behavior Potential (BP), is a function of the relationship between the
environments Reinforcement Values (RV) and Expectancies (E). Reinforcement value
refers to the personal desirability of different sources of activities, and expectancy is the
subjective probability that a given behavior will lead to a particular outcome. The
formula does not literally equate to a one-way relationship but is reciprocating. Engaging
in certain behaviors will influence how one perceives the reinforcements and
expectancies in their environments. Oppositely, peoples perceptions about the
reinforcements and expectancies in their respective environments will shape the potential
to employ specific behaviors (Pajeres, 1996a).
People frequently generalize their reinforcement values and expectancies and
their potential to engage in associated behaviors from one situation to another and from
past experiences to new experiences. People take notice of which reinforcements lead to
71


certain outcomes and then form expectancies about other, similar situations (Schunk,
1984). Individuals are especially prone to generalize when a new experience occurs in a
similar environment as before, when there is no comparable basis from which to form
new reinforcement values and expectancies, and when there is nothing in the similar
environment that facilitates the construction of a new set of reinforcement values and
expectancies (Mearns, 2004; Rotter, 1982; Rotter, Geller, & Hamsher, 1968).
Consider a students educational environment in the years prior to entering high
school. If this earlier educational environment has yielded the expectancy that employing
only minimal effort and nominal investment in time is necessary to receive the
reinforcement that enough content was mastered to pass a core class, such as
mathematics, and automatically advance in school, what will stop them from anticipating
the same reinforcements and expectancies in a new but similar situation? The students
will continue to employ the same minimal effort and nominal investment in time
behavior, thinking they are in a comparable school environment once they reach the ninth
grade. Sorry to say, algebra students, especially in the ninth grade, are quickly faced
instead with a dissimilar environment with opposing reinforcement values, among them
an inability to easily master content and unpleasant expectancies, such as the requirement
to work harder without automatic progression. They can either change their behavior of
putting forth minimal effort and nominal investment in time or they can choose to engage
in a new class of behaviors, such as avoiding the effort and time needed to pass the
course. This in turn can lead to solidification of the new reinforcement value of an
inability to easily master content and the expectation of needing to work harder to
progress, values and expectancies that might generalize and carry forth to other high
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school subject classes, repeating, until one gets hopelessly behind and drops out of high
school altogether.
Conversely, if one has previously come to expect feelings of anxiety and
irrelevancy from math classes and has been rewarded by negative feedbackand because
of this has been engaged in behaviors of schoolwork avoidancethen these patterns may
also carry forth to the new environment of ninth grade and into an algebra class. This
will continue, unless the environment offers a new set of reinforcements and expectancies
and the resultant development of desired behaviors.
Individual perceptions are central to the relationship between behavior potential,
reinforcement values, and expectancies and an absolute classification of these concepts is
impossible. However, this does not preclude the identification of several themes of
perceived reinforcement values and expectancies associated with the potential to engage
in behaviors characteristic of a high school dropout.
Before examining the role of social learning theory and themes of reinforcement
values and expectancies leading to dropout in high school, it is desirable to highlight the
close relationship between Banduras concepts of self-efficacy and Rotters Social
Learning Theory. Since these two theories are so tightly coupled, I assert it is possible to
equate and substitute the four sources of self-efficacymastery experiences, vicarious
experiences, social persuasion, and affective statesfor the reinforcement values within
Rotters predictive formula of behavior and to combine the theories into one formulaic
representation.
The relationship between self-efficacy and social learning theory.
Self-efficacy beliefs are often associated with both behaviors and expectancies.
When faced with task demands, most people behave in a way to make desired things
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happen rather than let things merely happen to them. They act in ways to either please
others or themselves. People with high self-efficacy expect to succeed and see
reinforcement value in their surroundings; they maintain or increase their efforts. People
with low self-efficacy expect failure and see negative reinforcement values in their
environment; they withdraw or give up (Bandura, 2001).
Self-efficacy shows a demonstrated relationship between the potential to engage
in certain academic behaviors and academic success. People avoid activities and
behaviors they believe exceed their abilities or coping skills, and low self-efficacy leads
to evasion of behaviors that are perceived as threatening and emotionally draining.
Student beliefs about what they have accomplished and what they expect they can
accomplish affect their academic performance, as individuals have a tendency to
participate in actions that they deem desirable. They often avoid behaviors and groups of
actions where they receive negative reinforcement and, thus, expect little results in areas
such as algebra tasks, no matter what the benefit even if they persevered (Bandura, 2012;
Kloostermann & Cougan, 1994). Tasks, such as algebra school work, often supply
unpleasant feelings and negative teacher feedback and actions and are, thus, to be ignored
and avoided. There is little motivation to change a behavior, if one believes they have no
personal power over the outcome or have scant expectations that they can manipulate
outcomes with their actions (Bandura, 1995). Students may know they will benefit from
passing classes, such as algebra; but they will avoid the actions needed to reach this
academic milestone, if they really do not believe in the value of this benefit.
Strong self-efficacy beliefs can change this avoidance behavior and bring about
desired outcomes as it can catalyze a student to be more engrossed in the subject matter,
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which, in turn, prompts the exertion needed to succeed (Bandura et al., 2001; Pajeres,
1996a). Archambault et al. (2009) found a relationship between student engagement and
low grades in high school mathematics courses, although the research did not specify if
this was in algebra coursework. A decrease in student engagement significantly
correlated to low grades in math. We will see shortly that the particular behavior of
disengagement from high school academic work is a one that is typical of, and a strong
predictor of, the eventual high school dropout (Archambault, Janosz, Fallu, et al., 2009).
Bandura (1985, 2005, 2012) maintains that self-efficacy beliefs are not only
connected with behaviors but are also tightly linked with expectancy outcomes. People
anticipate likely outcomes of their actions, and act upon what they believe they can do
well at, in addition to the outcomes they expect from their actions. Low self-efficacy
beliefs leads to low expectancies and low expectancies reciprocate by resulting in low
self-efficacy beliefs; these two concepts also jointly affect behavior. One must have faith
in their ability to and expect they can manipulate outcomes with their behaviors. Again,
those with high self-efficacy envision doing well in many situations; those with low self-
efficacy predict failure (Bandura, 1986, 1995, 2012).
If students come to expect low rewards from their environment, then they may not
engage in behaviors appropriate for the environment. Students disengage from and avoid
high school work not only because they have been reinforced with signs that they cannot
do well but because they come to expect the lack of immediate gratification and rewards.
They may come to expect the need to exert too much effort and experience too much of a
struggle to get a passing grade in core courses in order to earn credits toward graduation.
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Lopez et al. (1997) demonstrated how self-efficacy produced negative
expectancies, which, in turn, affected interests and lowered outcomes in algebra
performance at a Midwestern high school. Self-efficacy beliefs directly affected interests
and outcomes; but these interests and outcomes were also mediated by, and strongly
correlated to, student expectancies (Lopez, Lent, Brown, & Gore, 1997).
As a general rule, therefore, many parts of a persons self come together to
orchestrate the potential to take on certain behaviors, including individually perceived
self-efficacy beliefs and expectancies. Together these parts influence a students
potential to engage in specific academic behaviors. Individuals make judgments about
the effects of their actions. These judgments are regulated heavily by their self-efficacy
beliefs, as framed within a given particular environment. They also need to anticipate the
probable effects of different courses of actions and events (expectancies). In the end, this
set of judgments and anticipation allows one to adjust behavior accordingly to the
situation at hand (Bandura, 1977; Bandura et al., 2001; Hackett & Betz, 1989;
Kloostermann & Cougan, 1994).
Since the literature shows that self-efficacy is related to the potential to engage in
certain behaviors, we can modify Rotters predictive formula, as follows, to a new
formula which adds the factor of self-efficacy to the previously mentioned relationship:
BP = RV + E + SE, where SE = Self-efficacy
However, self-efficacy beliefs are developed from the four sources of self-
efficacy, also previously discussed. These four sources can be perceived differently by
individuals, based on the persons interplay with a particular environment (Bandura,
1977). Since these four sources can be perceived differently, they are basically the same
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as Rotters definition of reinforcement valuethe personal desirability of different
sources of activities in ones respective environments. Therefore, I assert, we can further
modify Rotters predictive formula from:
BP = RV + E +SE, to
BP = ME + VE + SP + AS + E, where
RV = ME + VE + SP + AS, and
ME = Mastery Experiences
VE = Vicarious Experiences
SP = Social Persuasion
AS = Affective States
This resultant formula depicts my proposal to use the sources of self-efficacy to
replace reinforcement values in Rotters original formula and to represent the relationship
of both sources of self-efficacy and expectancies to the potential to engage in behavior
within a given environment. I also find it unnecessary to use the concept of self-efficacy
twice in this formula, since it is judged to be adequately represented by its four sources.
This formula will form the foundation for the conceptual framework that will guide the
qualitative phase of the study. Before presenting this conceptual framework, it is
preferred to detour for a bit to discuss some of the major findings pertaining to
representative behaviors of high school dropouts and general categories of sources of
self-efficacy and expectancies associated with these groupings of behaviors.
Behaviors, expectancies, and self-efficacy sources of the high school dropout.
The literature makes the case for three major groupings of behavior which are
exhibited by high school dropouts. These three categories are academic disengagement,
rule non-compliance, and self-regulatory deficiency. All three categories have a
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significant role in predicting high school dropout, and all three are associated themselves
with themes of sources of self-efficacy and student expectancies (American
Psychological Association, 2010; Archambault, Janosz, Fallu, et al., 2009; Bridgeland et
al., 2006; Hand, 2010). I will discuss the three categories of behavior and briefly touch
upon some of the sources of self-efficacy and expectancies that can be coupled with these
categories. A more comprehensive and resultant showing of these findings will be
synthesized and subsequently presented in the final step of the literature review, along
with its accompanying table, Table II.2.
Academic disengagement can be thought of as a lack of involvement and
withdrawal from school activities and demands. Students who are disengaged report
feelings of disconnectedness with their classes, other individuals, and the school
environment as a whole. Disengagement is a strong predictor of a students success and
eventual educational outcomes. Students who are engaged in school perform better
academically and avoid problem behaviors, such as dropping out (Archambault, Janosz,
Fallu, et al., 2009; Archambault, Janosz, Morizot, et al., 2009; Weiss et al., 2010). In a
survey of over 13,000 Canadian high school dropouts, Archambault, Janosz, Fallu, and
Pagani (2009) found that low student engagement significantly predicted high school
dropout.
Academic disengagement is a multi-factored concept possessing three
dimensions: behavioral, affective, and cognitive. The behavioral aspect includes aspects
such as participating in and preparedness for class, willingness to engage in effortful
learning, and to what extent a student demonstrates these behaviors in classroom work,
discussions, and activities. The affective component refers to notions to include students
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feelings, interests, perceptions, and attitudes toward school. This embraces liking school
and the subject matter, feeling involved in school, perceiving one can do the work, and a
general enthusiasm for learning. The cognitive aspect encompasses the students thinking
they want to learn more about a topic and their use of self-regulation strategies
(Archambault, Janosz, Fallu, et al., 2009; Archambault, Janosz, Morizot, et al., 2009;
Weiss et al., 2010).
There are a number of student-perceived sources of self-efficacy and expectancies
that are associated with the behavior of disengagement. Affective states, for example,
that students perceive as unrewarding, unchallenging, and uninteresting become
associated with expectancies that active participation in school produces little benefit.
Several researchers have discussed how dropouts report restricted opportunity for
rewards from school participation, realize limited association between high school
attendance and life goals, and have, generally, lowered expectancies about the value of a
high school diploma. Young adults, in general, have a hard time with long-term planning
and delayed satisfaction. More so, dropouts, as a group, expect to realize immediate
rewards for their efforts, lack the ability to consider any distant goals, and are able to see
only what is right ahead of them (Bridgeland et al., 2006).
Because of these negative sources of self-efficacy and expectancies, dropouts
report they tended avoid participating in class activities or homework (Archambault,
Janosz, Fallu, et al., 2009; Archambault, Janosz, Morizot, et al., 2009; Eckstein &
Wolpin, 1999; Gewertz, 2006; Hickman et al., 2008; Tyler & Lofstrom, 2009). In
interviews with 467 students from a diverse racial/ethnic group, concerning the reasons
for dropping out of school, 47% stated that the classes were not interesting, and 69%
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stated that they were not motivated or inspired (Bridgeland et al., 2006). Others reported
they withdrew from class participation in large part due to another perceived self-efficacy
sourcesocial persuasion. Feedback from teachers frequently demonstrated a lack of
positive expectation of a students capacity to perform the work (Roderick & Cambum,
1999; Weiss et al., 2010). In the same study, 68% of the students interviewed stated that
they would have worked harder if they had been subject to higher expectations for
meeting academic standards (Bridgeland et al., 2006). On the other hand, while they
would have worked harder, they also needed, and expected, more support to fulfill these
higher standards (Bridgeland et al., 2006).
Similar to their general high school experience, students confront sources of self-
efficacy and expectancies that result in academic disengagement from high school
algebra classes. In addition to negative self-efficacy sourcessuch as difficult content, a
mechanical learning process, and uncomfortable feelings (i.e., frustration, confusion,
inadequacy, and anxiety)students have described that algebra classes were simply
boring. They were taught directly from the textbook, and there was no opportunity to
reflect on how and why algebra was done in an exacting fashion.
If technology was used, a method more commonly used to capture the interest of
todays youth; it was not the kind of technology that enticed students to actively
participate in class. Technology usage typically consisted of merely using overhead
projectors and computers to do textbook work, and was no student interaction with the
computers themselves (Kollars, 2008; Roderick & Camburn, 1999; Schornick, 2010).
Students have come to expect an algebra experience that is bland and irrelevant and that
does not require them to pay attention, especially since there is scarce opportunity to
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actively participate. In addition, just throwing work at them without extra help from out-
of-touch teachers has made it difficult for students to expect to succeed. This all
facilitates the tendency for students to disengage from algebra.
Another general category of behavior displayed by dropouts is a lack of rule
compliance. This is typified by actions, such as skipping school and classes, not
accomplishing homework and class work, disrupting classroom activities, and acting rude
toward teachers (Archambault, Janosz, Fallu, et al., 2009; Tyler & Lofstrom, 2009). All
of these behaviors impact school performance and ultimately graduation success.
Research shows that attendance and completing homework is eight times more predictive
of failure to graduate than test scores (Allensworth & Easton, 2007). Attendance, in
particular, is an early indicator of dropping out of high school, since dropouts display a
greater incident of absenteeism beginning with their first semester of ninth grade
(Hickman et al., 2008).
The Bridgeland study of the nations high school dropout problem confirmed the
magnitude of these behaviors26% of the dropouts interviewed did no homework, and
80% did one hour or less of homework each day (Bridgeland et al., 2006). Several
students in an interview protocol concerning the high school math experiences of college
students reported similar behavior with respect to their high school math classes and
declared that they often did no math homework (Schornick, 2010).
Sources of self-efficacy, such as a lack of belief in ability to complete homework
or a lack of quick mastery of the subject material, especially when combined with a
teachers inability to recognize students who do not comprehend the material, fuel a
students potential to engage in rule non-compliance. This is associated with the
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expectancy of an inability to ever catch up with class work and its corollary of severe
penalties for failure to do so. Students quickly expect that if they fall behind they will not
be able to catch up. Therefore, they report it is easier to just give up and not try. As an
example, in one survey, 43% of dropouts stated that they missed too many days of school
and could not hope to catch up (Azzam, 2007; Bridgeland, et al., 2006).
The third category of behavior that is representative of the high school dropout
population is a theme I have branded as self-regulatory deficiency. Self-regulating in a
given school environment means that a student assumes a degree of individual
responsibility for their own learning. This includes motivating oneself to get school work
done, resisting distractions in the surroundings, planning and organizing school work, and
self-initiating the taking and reviewing of class notes, all of which assist in the learning
process.
Students who self-regulate set near-term goals for themselves and are personally
committed to initiating actions consistent with learning. They also start to rely less on
some adults (e.g., parents) but are aware of personal learning roadblocks and know when
they need to seek help from other adults (e.g., teachers). Students who employ self-
regulation strategies have a higher sense of self-efficacy and personal control over their
high school surroundings (Zimmerman et al., 1992; Zimmerman & Martinez-Pons,
1990). In one study of high school students, use of self-regulated learning strategies was
positively correlated with mathematical self-efficacy, while their reliance on adults to
facilitate learning was negatively correlated with mathematical self-efficacy (Zimmerman
& Martinez-Pons, 1990).
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Unfortunately, use of self-regulatory strategies is somewhat counter to the
nature of an adolescent, especially in the immature and younger student of early high
school years. Teenagers have a hard time taking on personal responsibility for their
environment. This is particularly true in the case of dropouts (Bridgeland et al., 2006).
Zimmerman and Martinez-Pons (1990) did research that showed students in grades 8
through 11 declined in their efforts to seek adult assistance (e.g., parents) but increased in
seeking teachers assistance. Dropouts are not entirely consistent with this trend. They
decline in their efforts to seek assistance from both adults and teachers (Zimmerman &
Martinez-Pons, 1990).
Besides the immaturity of younger high school students, the sudden appearance of
freedom counters the interest in self-regulatory strategies. Dropouts have stated that they
had too much freedom and not enough rules in their lives (38% of the dropouts in one
study). Additionally, 51% of this same dropout group reflected back and stated that they
should have accepted some personal responsibility for their educational successes.
Unfortunately, they recognized this well after they had left school, when they were older
and likely more mature (Bridgeland et al., 2006).
Fueling the new state of freedom is the lure of other things in an adolescents
lifesocial activities, being free on the streets, and hanging out with friendsas reported
by high school students taking part in the Philadelphia and Baltimore focus group of
dropouts (Bridgeland et al., 2006). This lack of self-regulation capacity, combined with
too much freedom and not enough rules, leads to dropout expectancies that there is no
need for them to control their own destiny (Gleason & Dynarski, 2002).
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Step 4: Self-efficacy, reinforcements, and expectancies in high school algebra
exacerbating the potential to engage in high school dropout behaviors.
Review of the literature has provided a flavor for the types of themes of student-
perceived sources of self-efficacy and expectancies that originate from both high school
algebra classes and the greater high school experience. These themes of sources of self-
efficacy and expectancies, which are common or unique to both domains, can then be
associated to the three themes of behavior which the literature shows are characteristic of
the high school dropout population. Because of this, student-perceived sources of self-
efficacy, expectancies, and behaviors are the pivotal factors in the conceptual framework
I propose as a lens through which to qualitatively study the hypothesized relationship
between algebra and graduation. The proposed conceptual framework takes advantage of
the initial structure of Rotters Social Learning Theory, Banduras concept of self-
efficacy, and the relationship between the two, and is depicted as follows:
BP dropout (RV algebra Ealgebra) (RVhigh school Ehigh school), where
RV = ME + VE + SP + AS, ultimately finalizing the conceptual framework as
BP dropout = ((ME + VE + SP + AS)algebra + Ealgebra) ((ME + VE + SP + AS)high
school Ehigh school)_________________________________________________________________
This equation represents the cumulating supposition reached from the literature
review synthesis. It is a supposition that demands further analysis from the gathering of
first-person perceptions of high school dropouts: the sources of self-efficacy and
expectancies resulting from high school algebra experiences increase the severity of those
gained from the greater high school experience. In other words, the multiplicative power
of high school algebra sources of self efficacy and expectancies with the greater high
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school environment sources of self efficacy and expectancies aggravate the potential to
engage in behaviors characteristic of a high school dropout.
To re-emphasize, this formula should not be interpreted literally as a
mathematical formula but as a representation of a proposed concept. Whereas the plus
symbol, (+), does symbolize the additive power of four perceived sources of self-efficacy
and expectancies in a students algebra and greater high school experiences, the
multiplication symbol (*) does not connote a straight multiplicative process. It signifies
the capacity of the sources of self-efficacy beliefs and expectancies deriving from high
school algebra experiences to intensify those from the greater high school experience
which eventually results in the potential to engage in behaviors leading to high school
dropout. Do not make the same mistake that many of our beginning algebra students
make: reading the above formula with a predisposition to understand it as something you
need to calculate rather than as an expression one needs to understand in representing a
specified concept.
Bandura (1995, 2012) claims that global viewpoints of self-efficacy can be useful
when considering its relationship to individual behaviors in broad areas, such as the
expansive high school arena. However, he also cautions that one needs to use judgment
of self-efficacy in the correct contextual settings to avoid generalized personality trait
conclusions and to provide predictive accuracy about a certain situation. Domain-
specific assessments of self-efficacy are more predictive than general predictions. It is
even suggested that within a domain one needs to constrict further and look at task-
specific contexts. Algebra is one such task-specific setting within the content of both
mathematics and high school (Bandura, 1995, 2012). Pajeres (1992b) reinforced
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Banduras position, stating it best: If the purpose of a study is to achieve explanatory
power, self-efficacy judgments should be consistent and tailored to the domain of
functioning and/or task under investigation (p. 550).
On the other hand, Bandura and Pajeres discuss how self-efficacy can generalize
or cut across different tasks and domains. A general sense of self-efficacy allows one to
transfer beliefs of competency and behaviors from one domain to the next. This occurs
when the environment offers expectancies for similar judgments of competency or skill
sets across domains (Bandura, 1986; Pajeres, 1996b). Therefore, if a student is
encouraged to develop persistence and self-regulatory skills in algebra, it should
generalize to behaviors in other academic areas in a high school career. However, the
reverse is true as well. Sources of self-efficacy and expectancies developed in algebra
may very well transfer to general high school behaviors (Oettingen, 1995).
The proposed conceptual framework strives to use a domain-specific assessment
of student-perceived sources of self-efficacy (along with expectations) to drive a level of
specificity when studying the relationship between high school algebra and high school
graduation. It also provides for student-perceived sources of self-efficacy and
expectancies gained in high school algebra to bleed over and generalize negatively (or
positively) to the greater high school experience. Finally the framework additionally
allows me to explore the use of self-efficacy beliefs in the qualitative portion of the study
to explain findings from the quantitative component.
Moving Forward: Why the Research is Needed and Potential Themes of Self-
Efficacy, Expectancies, and Behaviors to Explore in the Qualitative Aspect of the
Research.
As discussed and presented, the literature offers arguments for factors that
singularly support, explain and predict poor high school algebra performance and poor
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high school graduation rates. There is scant research that spotlights the issues jointly or
looks at the hypothesized relationship between these two issues, much less provide a
persuasive argument about the reason for its existence.
The few sources that directly referred to the influence of high school algebra
performance on high school graduation were either non-research based or were in one
case, a superficial quantitative exam of algebras impact on graduation success, as
compared with other predictors. Two sources were non-research based documentsone
from a large city newspaper article and the other from an online school administrator
journal website. As previously cited, former Colorado Governor Roy Romer expressed
his opinion that algebra was the largest contributor to the high school dropout problem.
He had observed this correlation while serving as superintendent of the Los Angeles
school district, which was subsequent to the Los Angeles school districts mandate that
successful completion of algebra serve as a requirement for high school graduation
(Helfand, 2006). The other article from the online journal District Administration
presents a non-research based supposition that the initial high school algebra class was
likely to have a large impact on a students decision to drop out, because of the causal
effects of frustration and disengagement (Vogel, 2008).
The one piece of quantitative factor located in the research is a 2006 doctoral
dissertation that used logistic regression to determine that Algebra I was an important
predictor of high school dropout and that students who failed Algebra I were 4.1 times
more likely to drop out than those who passed the course. Unfortunately, this study was
extremely narrow in focus and did not include any qualitative aspect. The sole research
question and focus of this research was to only examine the academic and demographic
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Full Text

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PREDICTING HIGH SCHOOL COMPLETION USING STUDENT PERFORMANCE IN HIGH SCHOOL ALGEBRA: A MIXED METHODS RESEARCH STUDY by Wendy S. Chiado M. S. Naval Postgraduate School, 1988 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Educational Leadership and Innovation 2012

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ii This thesis for the Doctor of Philosophy degree by Wendy S. Chiado has been approved for the Educational Leadership and Innovation Program by Dr. Michael Marlow, Chair and Advisor Dr. Alan Davis Dr. Honorine Nocon Dr. Mary Thurman Date April 10 2012

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iii Chiado, Wendy S., Ph.D., Education al Leadership and Innovation Predicting High School Completion Using Student Performance In High School Algebra: A Mixed Methods Research Study Thesis directed by Dr. Michael Marlow ABSTRACT T outh have failed to complete high school. Determining why faceted problem and beyond the scope of any one study. The study presented herein utilize d a thirteen step mixed methods model developed by Leech and Onwuegbuzie (2007) to demonstrate within a large urban Rocky Mountain school district, the existence of a significant relationship between high school algebra performance and high s chool completion and to provide one suggested frame work by which to explain the nature of this relationship. Q uantitative analysis show ed course is not only a significant predictor of failure to graduate high school but wa s more liable than sev eral other predictors, to forecast which students are more likely to leave high school before earning a diploma within four years. However, quantitative examination reveal ed only the existence of a significant relationship between these two variables. It fell short of providing any elucidation as to why performance in high school algebra can be considered as a predictor of failure to graduate high school. Qualitative analysis, using the tradition of phenomenology (Moustaskas, 1994), was employed to expand on the quantitative analysis and offer a hypothesis from a student centered perspective as to why this relationship may exist. Several students, who have returned to school after dropping out or who have been in high school for more than four years, were

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iv interviewed about their individual experiences in high school algebra classes and the greater high school environment. T he qualitative analysis results did not conclusively agree with the quantitative results, and failed to conclusively point to hig h school algebra as the overwhelming reason for high school dropout Nonetheless the interviews revealed that the students perceive d numerous negative sources of self efficacy and expectancies that, when combined with the uniqueness of the high school alge bra experience, increase d the likelihood of failing to graduate high school. The form and content of this abstract are approved. I recommend its publication. Approved: Dr. Michael Marlow

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v DEDICATION I dedicate this work to my wonderful family. For my father Martin, my mother Rhoda, my mother in law Eunice, and my sister in laws, Carole and Peggy, you always believed in my abilities and supported me in all of my endeavors. You are no longer with me, but I feel you are forever watching over me, and I will carry my memories of you forever. To my sisters, Judy and Rachel, I am thankful we have each other and am grateful for the many years of friendship. To my nephew Matthew, and my niece Amanda, always know how much I love being your aunt and enjoy yo ur company! For Lewis, my husband of over 28 years, it has been quite a journey. You have accompanied me along every stop, and I could have not achieved this Ph.D. without your support. I love you and will always prize our time together. For my son Set h, you have grown into an amazing young man, husband, and father. You serve our country as a dedicated and You are everything to me, and no mother could be proud er of her child than I am of you. For my daughter in law Sarah, you are such a precious addition to our family; I will love you forever. And for my dear, dear grandchildren, Calista, Carter, and the grandson on the way, my little Oliver Riley Martin Chia do, there is no greater love than what a Grandmother has for her grandchildren. You are, and will always be, in my heart and soul; and I look forward to our many years of cherished memories.

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vi ACKNOWLEDGMENTS I would like to thank my wonderful and dedicat ed dissertation committee, Dr. Michael Marlow, Dr. Alan Davis, Dr. Honorine Nocon, and Dr. Mary Thurman. Each and every one of you has patiently provided me invaluable advice, kind words, and never ending assistance throughout my doctoral education and di ssertation process. I would have never achieved my doctorate without your support. Although I had many wonderful teachers throughout my doctoral program, I would especially like to thank Dr. Nancy Leech, a once in a lifetime teacher, and my many kind col leagues of the Mixed Methods SIG, especially Dr. Tony Onwuegbuzie. Everyone was so patient at answering my many questions; and because of that, you instilled a love for learning new research methods and for understanding the value of mixed methods researc h. I would like to especially extend a heartfelt thanks to my Ph.D. advisor, Dr. Michael Marlow. Your friendship, dedication, and faith in your doctoral students are unequaled; and we all feel honored and lucky to be part of your classes and educational e ndeavors. You have taught me so much over the years, and your unfailing and patient belief in my abilities was pivotal to my reaching this huge milestone. I know I have the best committee and advisor in the university!

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vii TABLE OF CONTENTS I. INTRODUCTION ................................ ................................ ................................ ......... 1 Research Questions ................................ ................................ ................................ ....... 8 Research Hypot heses ................................ ................................ ................................ .. 10 Limitations of the Dissertation Study ................................ ................................ ......... 10 ................................ ................................ ............................ 13 Significance of This Research ................................ ................................ .................... 21 Conceptual Framework ................................ ................................ ............................... 26 II. LITER ATURE REVIEW AND SYNTHESIS ................................ ........................... 28 Goal and Objective of the Literature Review ................................ ............................. 29 Rationale and Purpose for the Literature Review ................................ ....................... 30 Literature Review Questions ................................ ................................ ....................... 30 Literature Review Data Collection Process ................................ ................................ 32 Literature Review Legitimacy and Validity ................................ ............................... 33 Literature Review Synthesi s Methodology ................................ ................................ 34 Interpreting the Results of the Literature Review: Steps One through Three ............. 40 ....... 43 Step Two: Self efficacy, the high school dropout, and the high school algebra experience. ................................ ................................ ................................ ............ 54 Self efficacy and the high school experience. ................................ ................ 55 Self efficacy and the hi gh school algebra experience. ................................ .... 64 Step three: The relationship between social learning theory and self efficacy and the behaviors le ading to high school dropout. ................................ ...................... 70 The relationship between self efficacy and social learning theory. ................ 73 Behaviors, expectancies, and self efficacy sources of the high school dropout ................................ ................................ ................................ ......................... 77

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viii Step 4: Self e fficacy, reinforcements, and expectancies in high school algebra exacerbating the potential to engage in high school dropout behaviors. .............. 84 Moving Forward: Why the Research is Needed and Potential Themes of Self Efficacy, Expectancies, and Behaviors to Explore in the Qualitative Aspect of the Research. ................................ ................................ ................................ ... 86 III. METHODS ................................ ................................ ................................ ................. 92 Demographics and Context of the School District Under Study ................................ 94 The high school dropout sample classification. ................................ .................... 94 Research Study Design ................................ ................................ ............................... 96 Overall Re search Sample Design and Data Collection Procedures ............................ 97 Quantitative Methods ................................ ................................ ................................ .. 98 algebra course serve as a significant predictor of failure to complete high school? ................................ ................................ ................................ ............................... 98 Research Question Two: Will poor performance in high school algebra be more likely than the other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special education status to lead to subsequent failure to complete high school? ................................ ................................ .................... 100 Effect size and power. ................................ ................................ ......................... 102 Quantitati ve sample and data collection. ................................ ............................ 104 Quantitative variables ................................ ................................ ........................ 105 Missing data. ................................ ................................ ................................ ....... 108 Qualitative Method s ................................ ................................ ................................ .. 109 Research Questions Three and Four: How do student held expectancies and perceived reinforcements from their experience in firs t high school algebra class lead to the potential of the student to engage in behaviors that are symptomatic of rst high school algebra class, influence the potential of a student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school? ............ 110 Qualitative sample and data collection. ................................ .............................. 111

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ix Qualitative measures. ................................ ................................ .......................... 114 Qualitative analysis. ................................ ................................ ............................ 115 Validity of qualitative analysis. ................................ ................................ .......... 116 Mixed Methods Data Analysis ................................ ................................ .................. 116 Limitations ................................ ................................ ................................ ................ 116 IV. RESULTS ................................ ................................ ................................ ................. 118 Quantitative Results ................................ ................................ ................................ .. 119 Does poor performance in the first high school algebra course serve as a significant predictor of subsequent failure to complete high school? ................. 122 Does the high school grade in which algebra is first taken serve as a predictor of high school graduation success? ................................ ................................ ......... 122 Is poor performance in high school algebra more likely than the other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English language learners, ethnicity, sex, and specia l education status to lead to subsequent failure to complete high school? ............ 12 3 Qualitative Results ................................ ................................ ................................ .... 127 How do student held expectancies and perceived reinforcements from their experience in first high school algebra class lead to the potential of the student to e ngage in behaviors that are symptomatic of a student who will probably fail to experience in his/her first high school algebra class, influence the potential of a student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school? ................................ ................................ .... 129 Perceived sources of self school algebra class. ................................ ................................ ...................... 130 Misdirected social persuasion in a first high school algebra class. ............... 130 Intense affective states in a first high school algebra class. .......................... 132 Insufficient mastery experiences in a first high school algebra class. .......... 135 Ill suited vicarious experiences in a first high school algebra class. ............ 136 Expectancies in a first high school algebra class. ................................ ......... 137

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x Perceived sources of self efficacy a school experience. ................................ ................................ ......................... 140 Misdirected social persuasion in the greater high school environ ment. ....... 140 Intense affective states in the greater high school environment. .................. 142 Expectancies in the greater high school environment. ................................ .. 143 Behaviors leading to high school dropout or not graduating on time. .......... 144 Summary of Qualitative Findings ................................ ................................ ............. 145 V. DISCUSSION ................................ ................................ ................................ ........... 148 Bottom Line up Front: Although Mixed Methods Research does not Prove Conclusively that Algebra Performance is the Singular Cause of Failure to Complete alitative Data Samples Points toward Algebra Performance as a Noteworthy Contributor to the School District Dropout Problem ................................ ................................ .............. 150 The algebra class is significantly associated with failure to graduate high on time and after four years. The quantitative results also showed that the high school grade in whi ch algebra is taken is significantly related to failure to graduate high school. .............. 151 Within the qualitative study sample, all four sources of self efficacy were present efficacy sources of misdirected social persuasion and intense affective state and negative expectancies that were experienced by students in high scho ol algebra classes. 154 Within the qualitative study sample, the self efficacy sources of misdirected social persuasion and affectiv e states combined with negative expectancies in the greater high school environment intensified those which were experienced in a first high school algebra course. ................................ ................................ ......... 164 The result of the interplay between sources of self efficacy and expectancies in high school algebra and the greater high school experience: A potential to engage in behaviors characteristic of the high school dropout. ................................ ...... 167 Taking algebra in the ninth grade transition year can be a risky endeavor for and qualitative results. ............. 170 The integrated quantitative and qualitative results showed that the relationship between the failure t o pass algebra and high school dropout crossed ethnic boundaries. ................................ ................................ ................................ .......... 171

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xi The Value of Mixed Methods Analysis in Investigating the Relat ionship between High School Algebra Performance and High School Graduation ............................ 172 Mixed methods research provided insight into the dep th and breadth of the problem and the results of the qualitative analysis significantly enhanced the results of the quantitative analysis by showing practical and economic significance. ................................ ................................ ................................ ........ 172 Results from the quantitative method of study were not totally congruent with the results from the qualitative method of study. ................................ ...................... 174 Concluding Thoughts ................................ ................................ ................................ 175 We need to pay attention to the teachers of ninth grade algebra. ....................... 175 Alternative schools are onto something. Economically and logically we cannot delivery methods at least in the ninth grade and especially in core classes such as algebra. ................................ ................................ ................................ ................ 176 Future Research and Recommended Prescription for Success ................................ 177 REFERENCES ................................ ................................ ................................ ............... 180 APPENDIX ................................ ................................ ................................ ..................... 189

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xii LIST OF TABLES Table II.1 Literature review synthesis questions, as mapped to the study research questions, goal and rationale. ................................ ................................ .......... 31 Table II.2 A summary of the sources of self efficacy and expectancies developed from the review of the literatu re common to experiences in both high school algebra and high school experiences to examine in the qualitative analysis, that can lead to the potential to engage in the three behaviors characteristic of the high school dropout. ................................ ................................ ................. 90 Table III.1 The demographical AFGR breakdown for the school district under study at the end of the 2010 academic school year. ................................ ................. 96 Table III.2 Independent and dependent variables and their corresponding levels of measurement. ................................ ................................ ................................ 108 Table III.3 Demographical and academic characteristics of the student interviewees. .. 113 Table IV.1 Grade in which a student first takes algebra for the 2004 to 2008 student cohort. ................................ ................................ ................................ ........... 120 Table IV.2 Numbers of algebra pass/fail and English pass/fail students in the same year as their first algebra course for the 2004 to 2008 student cohort. ................. 121 Table IV.3 Descriptive statistics for the student cohort of school years 2004 to 2008. 121 Table IV.4 Chi Square analysis of high school algebra performance association to high school graduation. ................................ ................................ ......................... 122 Table IV.5 Chi Square analysis of the high school grade in which algebra is first taken and its association to high school graduation. ................................ .............. 123 Table IV.6 Logistic reg ression predicting graduation success from the combination of independent variables. ................................ ................................ .................. 126 Table IV.7 Summary of themes of student p erceived sources of self efficacy, expectancies, and behaviors, as derived from analysis of the interview data and as related to the conceptual framework. Bold type indicates answers related to high school experience only, italics indicate answers from al gebra only, and all capitals indicate answers from both high school algebra classes and the greater high school experience. ............................... 146

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xiii LIST OF FIGURES Figure II.1 Path Through the Literature Review ................................ ............................... 35 Figure II.2 S Role ................................ ................................ ................................ ................. 36 Figure II.3 Step Two: Self Efficacy, the High School Drop out, and the High School Algebra Experience ................................ ................................ ........................ 37 Figure II.4 Step Three: The Relationship Between Social Learning Theory and Self Ef ficacy and the Behaviors Leading to High School Dropout. ...................... 38 Figure II.5 Step Four: Self Efficacy, Reinforcements, and Expectancies in High School Algebra Exacerbating the Potential to Engage in High School Dropout Behaviors ................................ ................................ ................................ ........ 39

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1 CHAPTER I. INTRODUCTION There is a graduate high school on time or who never receive a high school diploma. Recent statistics show that on average only 70% of our students earn a diploma each year, leaving almost one third year an estimated 1.2 million students do not graduate on time. At any one time, there are 3.5 million to 6 million individuals between the ages of 16 and 24 who have completely dropped out o f high school, meaning they have failed to earn a high school diploma and are not enrolled in high school. The statistics are more severe for minority populations. Nearly one half of the annual dropout population is comprised of Black, Hispanic, and Nati ve American youth ( Bloom, 2010 ; Bridgeland, Dilulio, & Morrison, 2006 ; Stillwell, 2010 ; Sweeten, Bushway, & Paternoster, 2009 ) Some go so far as calling the high school dropout problem an epidemic and label certain United States high schools as dropout factories ( Bridgeland et al., 2006 ; Tyler & Lofstrom, 2009 ) For many students, high schools have gone from the launching point to prosperity and being part of the American dream to be coming Literature spanning many years and scores of investigative approaches reveal that the failure to graduate high school is a complex and multi faceted issue. This has yielded discussion and research into a wide and diverse range of variables and factors, all presenting a relationship to the failure to graduate high school. The relationship has been viewed through an equally wide assortment of theoreti cal frameworks and analysis methods.

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2 Representative studies have explored the relationship between high school graduation and predictor variables such as academic performance, student attitudes, classroom behaviors, adolescent self esteem, and school dissa tisfaction. Other inquiries have focused on failure to graduate high school and its relationship to failing in the ninth and tenth grades or to a shortage of credits at the end of ninth and tenth grades. The role of the ninth grade and its standing as a transition year is also of critical concern to investigators of the high school dropout matter ( Allensworth & Easton, 2005 ; Ekstrom, Goertz, Pollack, & Rock, 1986 ) Allensworth and Easton (2005) additionally established that the organizational structure of the high school can predict which students is on track to graduate. They and other researchers did additional research to show that signs of poor performance as early as pre school and elementary school can also predict high school graduation success ( Allensworth & Easton, 2005 ; Hickman, Bartholomew, Mathwig, & Heinrich, 2008 ) Another direction of research has inspected the dropout issue for its relationship to demographics such as race, ethnicit y, and socioeconomic position, or as tied to family structure such as birth order, number of parents in the household, and education level of the mother. Other studies claim that although a number of issues are involved in the high school dropout matter, some are more significant (e.g., grades and student attitudes) than others ( Allensworth & Easton, 2005 ; Ekstrom et al., 1986 ; Rumberger, 1987 ) Since the causes of failing to complete high school have been shown to be complex and many, it is a challenge to know which set, if there is such a concept as a danger of dropping out. Identifying a right set of predictors can lead to efficiently and

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3 effectively structuring focused interventions. No one predictive model is comprehensive, and not all risk factors are equally effective at identifying students who are in danger of dropping out of high school before earning a dipl oma ( Gleason & Dynarski, 2002 ) Indeed, Dynarski and Gleason (1998) are emphatic in their plea for the academic community to identify superior predictors of high school dropouts. They state that often used predictors such as high absenteeism, low grades, over age attendance, and teen parenthood, while important in identifying potential dropouts, if they are the only predictors used, can lead to our missing the dropout prediction of two out of every three students ( Dynarski & Gleason, 1998 ) Given the complexity and number of factors discussed in the literature, it is unlikely that any future study will offer an absolute solution to the high school dropout problem. It is doubtful that any singula r intervention will be 100% e ffective in stopping students from dropping out of high school. discovery and understanding of a previously unknown risk factor, shown to have a signi ficant relationship on the proclivity to drop out of high school, is invaluable and adds worth to the existing body of research. A study that reveals the existence and nature of an additional predictor of high school completion, such as performance in hig h school algebra, offers a return in the ability to identify more students early enough in their secondary school tenure to intervene and promote high school success. Such work ict in the quest to ensure that all students graduate with a high school diploma in hand.

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4 The school district at the center of this dissertation study is not unlike the rest of the United States and the state in which it is located. It has experienced a d istressing failure to graduate high school rate of about 30% during each of the past few years. In addition to this less than acceptable high school dropout rate, the district has expressed a long standing concern that students are not achieving success i n high school mathematics, particularly in their first high school algebra course, which is initially experienced by a majority of students in the ninth grade high school students fail algebra each year. Several administrato rs, including the school existence of nearly equal high school students who fail to graduate and high school students who fail algebra. These administrators have openly speculated the ir suspicions that the two variables are related (personal communication, February, 2008). The former governor of Colorado, Roy Romer, when serving as the superintendent of the Los Angeles school district, also articulated this same supposition and voiced that of all the obstacles to graduation, algebra was the most concerning to him. He personally believes that algebra, more than any other course, has resulted in high school students dropping out ( Helfand, 2006 ) The research problem driving this mixed methods study center ed on this coexistence of nearly identical unsatisfactory student high school algebra performance and failure to graduate rates and the concurrent lack of methodical research a nd explanation as to the existence and nature of this concurrence. The school district is striving to design and implement interventions calculated to help more students succeed in high school. Administrators are determined to identify more students earl y enough in

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5 the educational process in order to implement means that prevent students from leaving high school before earning their diploma. Despite thoughts that high school algebra performance may have a role in the failure to graduate, the school distr ict has not investigated if and why such a relationship exists between these two variables. By not out of high school, an opportunity to help a large portion of studen ts may be missed. Additionally, interventions designed to increase success in high school algebra, while helping students in this academic area, may be limited in its total worth, if it is not simultaneously addressed with the impact that high school alge bra performance has on resultant graduation success. A corollary research problem influencing the direction of this dissertation study and defining the direction for the qualitative aspect emanate d from my own personal experience. Over the past 17 years, I have worked with multiple volunteer committees and on studies on behalf of the school district. I have been privy to several discussions about student interventions, including those intended to bolster algebra achievement and others aspiring to minimize the high school dropout rate. I have participated in task force proceedings centering on how to help students overcome the difficulties in ninth grade mathematics and have been appointed to committees focused on programs for students in danger of droppin g out of high school. Along with others, I have heard about implementation of various software programs, mentoring efforts, and teacher training ventures and have participated in dialogue about employing community assets, tutoring programs, and tailoring interventions. We have each offered thoughts and expertise toward outlining what is postulated to be the causes of each problem: poor algebra

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6 performance and incomplete high school education, which have all been discussed and worked on in parallel paths. However, as I reflect ed back on my participation with the school district, it str uck me that there was something missing from these discussions and deliberations, especially as I theorize d about the existence and nature of a relationship between high scho ol algebra performance and high school graduation. Not only have I observed the discussions about high school algebra performance and high school dropout rates proceeding independently, without any vocalized suspicion of a linkage, but rarely did I hear we missed a major aspect of understanding the problem by failing to appreciate the dual problems of poor high school algebra performance and unacceptable high school gradua Have we tr ied to correct both poor algebra results and high school dropout rates without acknowledging a key element necessary to craft and field more effective intervention s? Did difficulties in high school algebra and the resultant potential to engage in behavior leading to the likelihood of dropping out of high school? Therefore, a second rese arch problem warranting study was perceived aspect of academic difficulty in high school algebra and its impact on the high school dropout rate. This dissertation used a multi step mixed methods research model designed by Leech and Onwuegbuzie (2007) to structure and to execute the study into both research problems. The study was conducted within a large, urban school district located in the

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7 Rocky Mountains portion of the United States. The goal of th e mixed methods study was to determine if a significant relationship exist ed between student performance in high school algebra and high school graduation. If such a relationship exist ed the subsequent goal was to then understand the perceptions, from a st udent centered perspective, of the relationship between poor performance in high school algebra and failure to graduate high school. The use of both quantitative and qualitative methodologies was sought to understand the depth and breadth of the hypothesi zed relationship between high school high school algebra performance is an indicator of subsequent failure to graduate high school. Quantitative methods were used fi rst to determine if there is a significant relationship between student performance in high school algebra and high school graduation and to determine the relative significance of high school algebra performance as compared to other predictor variables on high school graduation. Subsequently, qualitative methods were used to examine the nature of this hypothesized relationship Theory of Social Learning of Rotter and the Concept of Self Efficacy by Bandur a, to provide insight into their perceptions of reinforcements, expectancies, and self efficacy beliefs and the resultant potential to engage in behaviors that le d to the likelihood of dropping out of high school. Poor algebra performance was defined as r eceiving a failing algebra grade (which was defined to be any student who did not graduate high school in four years and who does not yet hold

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8 a high school diploma. It does not in clude the population of students who earned a General Education Development (GED) certificate. The study ha d three main objectives. The research objectives we re to (a) improve prediction abilities and, thus, identify a greater number of students who are in danger of failing to complete high school; (b) explore student beliefs and expectations for poor performance in high school algebra and identification of those factors that may also influence the propensity to drop out of high school; and (c) influence development and implementation of research based interventions that are designed to concurrently reduce the occurrence of poor high school algebra performance and failure to complete high school. The mixed methods research objectives derive d from the stan ce that any partial or full solution to the research problems underlying this study dictates a continuum of research that neither quantitative nor qualitative methods can address alone ( Collins, Onwuegbuzie, & Sutton, 2006 ; Leech & Onwuegbuzie, 2007 ; Onwuegbuzie et al., 2007 ) In addition, as in other school districts, several millions of school district dollars are being invested in computer and other intervention programs, to include extra p eriods of algebra instruction, instruction to parents to better help their students, and steps to prevent problems in earlier grades ( Helfand, 2006 ) The third objective s ought to communicate t he need to consider other factors, including student perspectives, prior to developing and implementing costly interventions. Research Questions the research problem areas and were specifically constructed to dictate the methods, boundaries, and direction of the research ( Plano Clark & Badiee, 2010 ) Additionally,

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9 Teddlie and Tashakkori (2009), in their latest mixed methods research treatise, further advised in a discussion about the pivotal role of research questions that the questions should be focused on what is not known about a phenomenon. Therefore, the research questions for the study we re framed accordingly in order to narrow the scope of study and its accompanying literature search to the problem areas previously discussed and to scrut inize the missing student perspective on the phenomenon of algebra performance and its impact on high school completion. The questions we re as follows. 1. a significant predict or of failure to complete high school? 2. Will poor performance in high school algebra be more likely than other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language le arners, ethnicity, sex, and special education status to lead to subsequent failure to complete high school? 3. How do student held expectancies and perceived reinforcements from their experience in first high school algebra class lead to the potential of t he student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school? 4. his/her first high school algebra class, influence the potenti al of a student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school?

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10 Research Hypotheses There we re four research hypotheses that drove the need for this effort : 1. algebra class is related to high school graduation. 2. Ethnicity, socioeconomic status, and gender are not significant factors in the school graduation. 3. rst high school algebra class is one of the largest significant factors, when compared to others in a model that includes several predictor variables of high school graduation. 4. Student participation in high school algebra classes result in lowered self efficacy and negative expectancies, which lead to the potential to engage in behaviors that lead to dropping out of high school. Limitations of the Dissertation Study As previously cited, the research has touted the existence of numerous factors associated with the failure of students to graduate high school. In both the quantitative and qualitative efforts to address the research problems, there were potentially an unlimited number of variables that could have been scrutinized for singular or relative sig nificance on high school graduation success. The overall scope of this study wa s deliberately limited to the significance and nature of the relationship between high school algebra performance and high school graduation success, primarily as a result of h earing and discussing the particular concerns of school district leadership about the existence of this potential relationship.

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11 Boundaries delimiting the direction for the study were also set down within both the quantitative and qualitative portions of th e dissertation. The first portion of the quantitative segment of the research (research question one) attempt ed to determine only if there wa s a significant correlation between high school algebra course performance and high school graduation using a defi ned quantitative sample There are other potential mediating factors shown by research to be significant predictors of failure to graduate high school, such as the number of credits earned at the end of freshman year and poor overall grade point average ( Allensworth & Easton, 2005 ) These factors, which can be postulated to simultaneously occ ur when one fails a required high school course such as algebra, were not considered. Despite the desire to limit this study to the relationship between high school algebra performance and high school graduation, it wa s intriguing to obtain some insight in to the relative importance of high school algebra performance with respect to other selected variables on high school graduation. After all, a student does not take algebra in a void; and thus, quantitative research question two examine d this relationship The comparison predictor variables were limited to those for which the school district routinely collects data in order to evaluate student performance and to report results, in accordance with school district, state, and federal policies. Although pre vious research, comparison study, the desire was to not place an unnecessary burden on the school district by requesting additional data. The intent was to stay synerg istic with school district assessment practices by showing results from using the same variables that are used to analyze school district data.

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12 The qualitative analysis was limited to a focus on the nature of a hypothesized relationship between high school algebra performance and high school graduation from expectations, reinforcements, behavior potentials, and self efficacy beliefs, a limitation consciously placed due to pe rsonal interest in Social Learning Theory and self efficacy concepts especially as advocated by Rotter and Bandura respectively This study did not account for other potential factors such as teacher pedagogy methods or student learning differences. Interviews were conducted with students who have returned to school after dropping out of high school at least once or who have not yet graduated after attending high school for more than four years. Ideally interviews would have been held with students w ho have failed and passed high school algebra and who have both graduated and failed to graduate from high school so as to highlight differences in perceptions but it wa s felt that students who have graduated were not readily available. The school distric t used in this study has an extensive alternative school program with a pool of students available for interviews: dropouts who have returned to school or students who are still working on their diplomas after more than four years of high school attendance The qualitative research was meant to provide a window into student expectations, reinforcements, behavior potentials, and self efficacy beliefs, as related to high school algebra performance and its relationship to high school graduation. Therefore, i t was felt that limiting the interviews to those students who have failed to graduate high school in four years wa s sufficient to accomplish this purpose. Future study can be recommended to examine a comparison of these qualitative variables with

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13 those he ld by students who have passed algebra and have successfully completed high school in four years. Two other study limitations concern ed the samples used in the study. The data sample used to select students for interviews in the qualitative analysis wa s c hronologically different than the sample used for statistical testing in the quantitative analysis. This difference was overcome by the demographic similarities of the sample groups. The target interviewee sample group wa s similar enough to those in the statistical analysis sample and, therefore, wa s representative of the greater group of students who fail to graduate high school in four years. Finally, quantitative and qualitative analysis was limited to samples of those students who were in the school d istrict at the start of the ninth grade and does not include those who entered into the school district in grades ten through twelve. Although one cannot completely eliminate the influence of extraneous variables, this delimitation was done to help minimi ze this influence by removing variability caused by students coming from other school districts. I am not without a history of individualized experiences and biases that may have consciously or unconsciously influence d the demeano r of this study. Researchers enter into study with experiences that can explicitly and implicitly sway their research ( Creswell, 1998 ; Creswell & Plano Clark, 2007 ) My individualized experiences and biases we re formulated by a number of attributes to include (a) education and volunteerism as a student and parent in a public school system, (b) experience and involvement as a parent and advocate of a high school drop out, and (c) investigation and

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14 learning as a participant and believer in several research methods courses. Although purposeful effort was made to sift out the influence of lifetime experiences and biases in gathering and analyzing data, it was impossible to do so with 100% certainty and was especially difficult during the qualitative aspect of the research when I attempted to draw close to my research subjects. I had to consciously think of my own preconceived notions about why students drop out in the sch ool district in which the study took place and not let these opinions sway the conduct of or answers from the student interviews. Teddlie and Tashakkori (2009) discuss the requirement for a researcher to acknowledge and understand the role that an individu influence how researchers select both the questions they study and methods they use to Creswell further describes five foundatio nal philosophical assumptions that can d the same five assumptions as the pillars forming my worldview and influencing my practice of research. These five pillars are indic ative of my primary belief in the practice of pragmatism, or in placing emphasis on the answers to the research questions over all other matters, such as the actual methods used in a study. Openly discussing these five pillars help ed me to admit explicit bias and allow s both the reader and myself an understanding of the historical context and life influences from which I entered, directed, and, perhaps unconsciously, swayed this study ( Creswell, 1998 ; Creswell & Plano Clark, 2011 )

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15 The first assumption is the ontological assumption, whereby one asks what is the nature of reality? Is there a singular, concrete reality; or are there multiple, elastic realities from which one can examine a particular research problem? Does the researcher believe in the singular reality or multiple realities? I assert that the question of an existence and nature of a relationship between designated variables does not need to be an either/or p roposition. There co exists both the singular and the multiple, and a researcher can approach a particular research problem claiming the existence of both states of reality. This claim to a co existence begs a personal preference for and proclivity towar d using mixed methods research, relying on quantitative methods to disclose the singular reality and qualitative methods to confirm one of several potential or multiple realities. In this case, the singular reality exists in the presence of a relationship, a sole tangible concept, which is the hypothesized relationship of high school algebra performance to high school graduation. The desire to prove this hypothesis, the singular reality, len ds itself to the use of quantitative methods to expose its existen ce. Once the existence of the relationship, or the singular reality, is demonstrated, the reason for and the nature of its being can be accredited to multiple realities. Not only can it be attributed to multiple realities but, as will be presented later in the discussion about the rationale and purpose for using mixed methods, it i s essential to use multiple realities to understand the true composition and significance of this relationship. In addition, proving the existence of the singular reality is no nature of the relationship, which is actually comprised of multiple realities.

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16 These multiple realities are formed from the numerous ways in which we can describe the relationship (the singular reality). The multiple realities can emanate from differing group perspectives (i.e., students, teachers, parents), differing research methods (i.e., grounded theory, phenomenology, bibliograph y), and differing theoretical frameworks (i.e., Social Learning Theory, behaviorism, symbolic interactionism). All of these accounts form realities that can simultaneously exist, and all can be shown to be equally valid by using qualitative methods. The s econd assumption underlying my worldview centers on the epistemological aspect and focuses on the association between researchers and the researched subjects. subjects which is more characteristic of constructivism than pragmatism. A pragmatic approach to epistemology, however, calls for the researcher to collect data according to what works when attempting to answer a research question ( Creswell & Plano Clark, 2011 ) How I have structured the qualita tive aspects of the research design show s my agreement with and favor for the constructivist goal of becoming close to the research subjects, with the researcher as an integral part of the collection methodology. Yet this may also be a somewhat pragmatic approach, since I firmly believe what works best is to collect data from the eye of the matter, which is from the students themselves. Moreover I came into this study as one who has experienced the high school dropout occurrence, albeit as a parent of a h igh school dropout. I had hope d that my genuine interest in gaining an appreciation for why students drop out of high school and a never ending desire to come to terms with why this even occurred in my own family demonstrate d a commonality and a natural e mpathy with my interviewees. I believe d that this allow ed

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17 me to draw close to my research subjects, encouraging honesty and complete openness in their responses to my questions. Continuing in a discussion of my worldview, there is a third assumption: the role of values within the research. Another term for this is the axiological nature of the study. Pragmatists take multiple stances. They believe in both the biased and unbiased perspective. There are two personal biases that are admittedly driving the purpose and course of this research study. These same values are two that I firmly believe should be foundational to the nature of public school education within the United States. These are, first, the strong belief that there should be equity in the a problems in the public school system and, second, that educators and parents must listen Growing up in an urban area of the country as a product of an ethnically divers e and socially mixed public school system, I cherish diversity in the school system and appreciate the need for the public school system to equally educate and provide assistance to students of all backgrounds, especially to those who are lagging behind. There is an achievement gap, one which I do not deny, between certain groups of minorities and majority populations in the school district at the focus of this study. Correspondingly there is a pressing need for public schools to take action and for acade mia to research methods by which to narrow this gap, a nother requirement that I do not refute. Despite this acknowledgement, I cannot help but feel there is an ignored and cast aside segment of the student population whose problems and academic failures ge t subjugated by the intense attention paid to the achievement gap. This suppresses true

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18 achievement of equality in the public school system under study. This group is one to which my son belongs: the forgotten, and, from my biased viewpoint, an often neg lected vantage point as a parent and volunteer, little or not enough attention paid to the educational problems of students from the white, male, middle class segment of the population; and this minimizes the attainment of true equality in public school education. Who is trying to determine why students, such as my son from a middle class, educated, two parent home with a good family support system, drop out of high sch ool? Since I value diversity and equality in the public school system, I entered into this study, in no small part, for a rather selfish reason: to search for some influence or predictor variable that turns a blind eye to racial and socioeconomic consider ations and that may be a predictor that cuts across all groups of children to include the white, middle class portion of the population. To counter this entering bias, I used quantitative techniques to determine whether or not factors such as race, socioe conomic status, and gender are significant when looking at the relationship between high school algebra performance and high school graduation. The second value biasing the research is another one that emerge d from my perceptions of their educational experience. As a mother, I watched, and was part of the process, as many educators, counselors, and social health pro fessionals tried to identify school. Yet, very rarely did I observe any of these professionals ask my son about his perspective on troubles in school. Repeatedly this led an educator or professional to

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19 discipline issue or a lazy nature. My son's greatest academic success came in the classroom s of teachers who would take the time to talk and listen to him and adjust his educational experience accordingly. In addition, as I reflected back on the numerous times in which I had been part of problem, it occurred to m e that I rarely heard anyone speak of the issue from the perspective of or the focus of the students themselves. These dual experiences with my educational environment ha s sadly neglected to pay attention to a voice that may provide valuable insight to academic struggles: the voices of our students. This particular bias resulted in my choice of using phenomenology as the tradition underlying the qualitative portion of the study. Phenomenology allows one to see lived experiences from the eyes of those who have lived the experiences and to examine these experiences from a diversity of viewpoints ( Moustaskas, 1994 ) It permitted an investigation into the first on subsequent high school g raduation. The fourth assumption addresses the literary style of the researcher and inquires into the language in which the researcher will write. One of the largest challenges to the legitimacy of a mixed methods study is the manner in which the research is written up and in the resultant meanings that are communicated to the reader. Often this yields difficulty in showing the value of using both quantitative and qualitative methods. There has been significant debate in what comprises a quantitative ver sus qualitative style of

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20 writing and presentation. For example, many quantitative researchers use tables and charts to visually display data; yet the organization of this data is characteristic of the subjectivity of qualitative research. Many qualitativ e researchers use tables and charts to provide the rigor of organization of the quantitative approach ( Sandelowski, 2008 ) A pragmatist uses both styles of expression. Quantitative methodology traditionally calls for third person and emotionally bland, detached language. Qualitative methodology usually uses the first person and takes on a more emotional, passionate tone. In a mixed methods treatise, on e can use the traditional language of either research genre as appropriate in the specific section or use one or the other throughout ( Creswell & Plano Clark, 2007 ) Since I am emotional and passionate about public school education and am working to discover research based methods to improve the experience and results for all children, I have frequently used the first person style of expression typical of qualitative reporting. This will also include discussion about the quantitative aspects. However, since these emotions and passions can often lead to an unst ructured flow of thoughts, I have also relied on expressive aspects typical of quantitative methods to include a visual organization of data and results. This wa s done in order to provide a bounded structure and framework for the analysis and discussion o f the qualitative portions of the study. Finally, the fifth assumption, the methodology, involves the process used in the research ( Creswell, 1998 ) I am most definitely in keeping with the pragmatic camp, believing in using the best research method that can yield results and understanding. The literature shows a diverse ran ge of multi faceted issues related to high school completion. Therefore, we need to deduce and break down the factors that result in a less than desired

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21 high school graduation rate into potential ca u s es such as ethnicity performance in math and English courses and perceptions of negative influences in the high school algebra and greater school environments This will subsequently induce possible theories or approach wa s the most suitable manner in which to confirm some of the potential factors that demonstrate an association between high school algebra performance and high school graduation and to further provide some insight as to why this association may ex ist. One concluding thought as to my rationalization for the use of mixed methods. This research approach was a logical fit with my academic and professional experience. I have a love for numbers and science but a passion for words and stories. My expe rience and training, primarily in the military and in working with space systems, are chockfull of forays into the black and white definitiveness and cause and effect world of quantitative methods. Much of my undergraduate and graduate education is in the sciences where objectivity in the form of statistics and experimentation reign. On the other hand, additional graduate schooling is in the field of education; and recent professional and volunteer work required me to make subjective sense of happenings and problems. explanations typical of qualitative methods. Significance of This Research Epidemic ( Bridgeland et al., 2006 ) Seeking out and understanding the reasons why so many of our youth drop out of high sch ool is an economic and social necessity for America to retain its place as a leader in the 21 st century. There are numerous

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22 consequences for both individuals and society that result from failure to earn a high school diploma; and a disproportionate part o f the impact is felt by minorities, who drop out at a greater level than others ( Chapman, Laird, & KewalRamani, 2010 ; Stillwell, 2010 ; Sum, Khatiwada, & McLaughlin, 2009 ) Individually, high school dropouts are more likely to be unemployed and earn less income annually and over a lifeti me. Looking at th e populace of ages 16 through 24, on average during the year 2008, high school dropouts were employed 22% less than high school graduates, 31% less than those with one to three years of college, and 41% less than those who graduated from college. According to Sum, Khatiwada, & McLaughlin (2009), this same high school dropout age group was also more likely to be unemployed longer. In 2007, the 16 through 24 year old high school dropouts had a 40% unemployment rate that lasted the entire year ( Sum et al., 2009 ) Earning potential decreases for the high school dropout. In 2009 high school dropouts earned 31% less than high school graduates and 46% less than those with some college education ( Bridgeland et al., 2006 ) As opposed to earlier generations when h igh school dropouts could easily enter the workforce, they now compete with an unskilled labor pool of immigrants from other countries, which has a further negative effect on their lifetime income earning potential ( Bridgeland et al., 2006 ; Sum et al., 2009 ) Accordingly, between 1974 and 2004 the median earnings of families headed by a high school dropout declined by one third ( American Psychological Association, 2010 ) High school dropouts also exhibit additional consequences of their failure to graduate. They have riskier health behaviors and are les s likely to have health insurance plans. They exhibit more illegal drug usage, experience poorer mental health, and have

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23 reduced home ownership rates. Dropouts are also more likely than high school and college graduates to have marital problems ( Bridgeland et al., 2006 ; Rouse & Kemple, 2009 ; Sum et al., 2009 ; Sweeten et al ., 2009 ; Tyler & Lofstrom, 2009 ) High school dropouts collectively encumber our society as well. They frequently receive financial and health assistance from local government. In return, these high school dropouts contribute less tax revenue to the publ ic coffers, providing on average 42% less state and federal tax income. Crime rates and the accompanying cost of jail, probation, and parole are affected because high school dropouts are more prone to commit crimes, including violent crimes, and are more likely than those who earn a high school diploma to end up on death row ( Campolieti, Fang, & Gunderson, 2010 ; Sum et al., 2009 ; Tyler & Lofstr om, 2009 ) out of high school et al., 2009, p. 10). When high school dropouts become parents themselves, the result is a cycle that often reflects continual poverty and negative educational results. Unfortunately this cycle is difficult to break Sixteen percent of children under the age of 18 live in households where the head of the household is a high school dropout ( Ziomek Daigle, 2010 ) Children of less educated parents tend to do poorer in their own educational achievements. Many dropouts become single parents (primarily females) and have children earlier in life than do high school graduates. The dropout parents are struggling financially and, as a result, become dependent on public and family assistance. Nearly 37 of every 100 dropouts were living in poor families in 2006 2007. A poor family is classified as having an income 25% below the federal poverty income threshold. A poor family is four times more likely to have a parent who is a high school dropout as opposed

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24 to a family with a parent who has a high school diploma or higher ( Sum et al., 2009 ) The cycle persists as the more impoverished the family is, the more likely the children will have future academic difficulty ( Archambault, Janosz, Fallu, & Pagani 2009 ; Sum et al., 2009 ; Tyler & Lofstrom, 2009 ) Nowadays, gaining proficiency in mathematics, specifically algebra, may be almost as important as earning a high school diploma. There is much more depth needed to education in our present information driven society than in the generations before us. basics are not good enough anymore st century citizen needs to gain the critical thinking, problem solving, and deliberate analysis skills that are characteristic of mathematical disciplines such as algebra. The National Mathematics Advisory Panel, which was convened in 2006 by President Bus h, espoused that the current American student s hould strive to understand the intricacies and patterns that comprise the disciplines of mathematics, including algebra, in order for the United States to keep pace in our changing world by producing more coll ege graduates, including engineers and scientists ( Vogel, 2008 ) The Mathematical Association of America further proclaims that algebra literacy is essential disciplines ( Ka tz, 2007 ) Algebra, especially, is often viewed as a gateway course, providing individuals a greater ability to gain access to a college education and technical careers ( Stein, Kaufman, Sherman, & Hillen, 2011 ) Even if one does not aspire to a college education or to enter a technical career, algebra literacy can be beneficial to havi ng a successful life. During our recent economic crisis, one can note the

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25 miscomprehension of many Americans who did not understand the ramifications of variable rate mortgages ( Rand Corporation, 2008 ; Sc hornick, 2010 ) Of special interest is the work by Rose and Betts (2004) that, when using the High School and Beyond (1980 1992) data set, significantly showed a relationship between successfully passing algebra and increased incomes at a point ten years after completing high school. The increased income was significantly higher for females who had some college education. Although the study did not show if this was either the result of completing algebra in itself or the fact that algebra opens the door to additional car eer and training opportunities, it is still a noteworthy finding In particular, and discouragingly, the United States ranks 26 out of 39 countries in problem solving, which is a large component of the algebra curriculum, according to the Program for Inter national Assessment ( Schornick, 2010 ) The American College Testing (ACT) has repeatedly emphasized that preparedness to complete college, to include mathematical readiness, is vital for Americans to be economically competitive. Unf ortunately, many of our students are just not keeping pace with the rest of the world students ( ACT, 2004 2005 ) Disciplined research that examines the dual issues of students high school algebra performance and dropping out of high school, with an ultimate goal of showing the relationship between the two, has the potential to add value to the existing body of scho larship. This is not only needed but is in great demand so our nation can make a dent in the dismal trend s of poor performance in algebra and students dropping out of high school. Failure to pay attention and to conscientiously seek every answer and tool

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26 available that can prevent students from dropping out of high school is a collective societal failure to help our country maintain its status as a world leader and a failure to help our youth benefit from the lifetime rewards of a successful high school e ducation Finally, although some argue we should focus more attention and resources on ways in which to prevent students from dropping out of high school at earlier ages and school grades ( Rouse & Kemple, 2009 ) this should not be our only focus. In my firm belief, especially from the vantage point as a parent of a high school dropout, it is criminal to ignore any means possible to intervene and reverse the propensity of a student to drop out of high school. Conceptual Framework The conceptual framework guiding the qualitative portion of the research will integrate social learning theory and self eff icacy concepts. In his social learning theory, environment in an equation: Behavior Potential (BP) = f (Reinforcement Value (RV) + Expectancy (E)) Bandura in his many tre atises on the concept of self efficacy offers that self efficacy originates from one or a combination of four sources. These sources are (a) Mastery Experiences (ME), (b) Vicarious Experiences (VE), (c) Social Persuasion (SP), and (d) Affective States (AS ). I will discuss in the literature review how social learning theory and the concept of self efficacy are closely related and because of this, one can substitute the sources of self formula. The for mula thus becomes: BP = RV= ME + VE + SP + AS + E, where

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27 RV = ME + VE + SP + AS The literature review will disclose that students perceive several sources of unfavorable self efficacy beliefs and negative expectancies in both their high school algebra classes and greater high school environment. These perceptions lead to the potential to engage in behaviors that are t ypical of high school dropout. The literature review will further highlight that passing algebra is instrumental in a framework such that the sources of self efficacy bel iefs and expectancies in algebra exacerbate those that originate in the greater high school experience and that this increases the potential to engage in behaviors leading to high school dropout. Therefore, the conceptual framework for this research is rep resented by the formula : BP dropout = ((ME + VE + SP +AS) algebra + E algebra ) ((ME + VE + SP + AS) high school + E high school ) This formula should not be interpreted literally as a mathematical formula but as a representation of a proposed concept. Whereas the plus symbol, (+), does symbolize the additive power of four perceived sources of self efficacy and expectancies in a stude algebra and greater high school experiences, the multiplication symbol (*) does not connote a straight multiplicative process. It signifies the capacity of the sources of self efficacy beliefs and expectancies in algebra to intensify that of those fro m the greater high school experience which eventually result s in the potential to engage in behaviors leading to high school dropout.

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28 CHAPTER II. LITERATURE REVIEW AND SYNTHESIS The literature review for the research study into the relationship between high school algebra performance and high school graduation was put into practice in a manner consistent with the overall approach of this study. The review is primarily organized in step with a literature review and synthesis framework advocated for use in mixed methods research studies ( Onwuegbuzie, Collins, Leech, Dellinger, & Jiao, 2007 ) These authors offer a literature review framework that is similar to the 13 steps recommended for the general conduct of mixed methods re search. I used many of the recommended In addition to using a framework to provide a deliberate structure, t he mechanics of the literature review progression, along with the synthes is results, are presented in a detailed and explicit manner. Frels and Onwuegbuzie (2011) campaign for translucency in this foundational aspect of research, especially within a dissertation discourse, since it provides the reader an unambiguous understand the course of the literature review ( Frels & Onwuegbuzie, 2011 ; Onwuegbuzie, Collins, et al., 2007 ) The initial steps of the literature review, the synthesis formulation stage, involved delineating the goal, objective, rationale, and purpose for the examination. This stage culminated with the formulation of literature review synthesis questions, which a re similar to research study questions ; the se literature review questions were then used to guide the ensuing course of the literature review.

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29 Goal and Objective of the Literature Review The primary goal for the literature review was to provide justificat ion and structure for the conduct of the research into the hypothesized existence and nature of the relationship between high school algebra performance and high school graduation. This goal had two parts. The first part set the stage for why the examina tion into the hypothesized relationship was a creditable and crucial research effort. Why was studying the matter of high school algebra so imperative to the understanding of high school graduation rates? This involved probing the literature for evidence that illustrated completion and its timing with of the goal would provide the substantiation needed for conducting quantit ative and qualitative work on the proposed relationship. The second part of the goal was to formulate a conceptual framework to guide the qualitative research portion of the study. I entered into the investigation knowing that the tradition of phenomenolo gy would guide the qualitative aspect with a predisposition to somehow weave in the Social Learning Theory of Rotter and the self efficacy concepts of Bandura. However, I was unclear about how to structure the thinking and subsequent analysis of the quali tative data in a relationship that would make use of these concepts. The formation of a conceptual framework is often pivotal to demonstrating how a study answers a research question(s). Therefore, using the literature review to help define a blueprint f or construction of a conceptual framework served as th e second component of the goal ( Bordage, 2009 ; Miles & Huberman, 1994 )

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3 0 The overarching objective for the literature review was to evaluate the legitimacy of the premise which instigated the dissertation research effort: that previous study about the existence and nature of a relationship between high schoo l algebra performance and high school graduation was relatively nonexistent. I had based this supposition on the limited results yielded from precursory searches of the literature conducted after discussions with administrators of the school district unde r study and before formally proposing the study to the university and school district. Rationale and Purpose for the Literature Review The rationale for the overall research study, and for the literature review, is significance enhancement. Significance enhancement advocates the combining of research from a multiple methods in order to solidify and strengthen the overall findings of a st udy ( Collins et al., 2006 ) In this case, the rationale was to demonstrate a need to augment what is present in the current body of research ab out predictors of high school graduation failure with new research data and research findings through an investigation into the particular role of high school algebra. The purpose of the literature review was to demonstrate the need to follow up on results of previous research into the separate enclaves of student high school algebra performance and high school dropout rate and the lack of research into a possible association This was to provide further evidence of the need to explore potentiall y success. Literature Review Questions Developing thoughtful and pointed literature review questions are analogous in utility to generating research study questions. The literature review questions are c ritical

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31 to the accomplishment of a research study by helping the investigator structure and organize the direction of the literature assessment in order to yield maximum effectiveness to the overarching study ( Onwuegbuzie, Collins, et al., 2007 ) My literature review was designed to answer six questions, four of which were composed to directly be reflective of the goal and rationale of the literature review, resp ectively. The literature displayed in Table II.1. Table II.1 Literature review synthesis questions, as mapped to the study research questions goal and rationale Literature Review Questions Study Research Questions 1. Given the existing body of literature about the high school dropout problem in the United States, what factors substantiate the need to examine the distinctive role of high school algebra performance, as compared to other factors, in predicting high school graduation? 1. first high school algebra course serve as a significant predictor of failure to complete high school? 2. What research has been cond ucted that would justify the use of the predictor variables of algebra performance, reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special education status as a model for examining subsequent failure to complete high school? 2. Will poor performance in high school algebra be more likely than other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, Eng lish grade, English language learners, ethnicity, sex, and special education status to lead to subsequent failure to complete high school?

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32 Table II.1 (cont.) Literature Review Questions Study Research Questions 3. What is the conjectured role of self efficacy, with respect to high school graduation and high school algebra performance? 3. efficacy in his/her first high school algebra class, influence the potential of a student to eng age in behaviors that are symptomatic of a student who will probably fail to finish high school? 4. What does the literature suggest is the relationship between social learning theory and self efficacy and how does this lend itself to students engaging in behaviors that lead them to dropping out of school? 4. How do student held expectancies and perceived reinforcements from their experience in the first high school algebra class lead to the potential to engage in behaviors that are symptomatic of failure to finish high school? Literature Review Goal 5. What conceptual framework can be developed offering a proposed connection between perceived self efficacy beliefs, reinforcements, and experiences in their first hi gh school algebra course leading to the potential to engage in behaviors typical of high school dropouts? Develop a conceptual framework for the qualitative analysis. Literature Review Rationale 6. What quantitative, qualitative, and/or mixed methods r esearch has been conducted that examines the hypothesized relationship between high school algebra performance and high school graduation? Significance enhancement: The need to augment what is present in the current body of research about predictors of hi gh school graduation failure with new research data and research findings through an investigation into the particular role of high school algebra. Literature Review Data Collection Process A search was made of several web and library subscription databases, both within and outside the education and social sciences disciplines. This included Google Scholar, Wilson Web, Academic Search Premier, Education Full Text, Educational Resources Information Center (ERIC), Academic Search Premier, and the Uni versity of Colorado library databases. I was also able to take advantage of my military background and the

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33 online reference databases available through the Air Force Portal and Naval Postgraduate School. I additionally searched a website called Virtual L earning Resource Center, an education based, electronic center. While this site did not provide peer reviewed articles, it did provide some interesting information from newspaper publications and popular media websites, such as Los Angeles Times, New York Times, and PBS Parents. In addition to the computer based searches, a hand search was conducted through many of the professional journals I receive, which includes Educational Researcher, Review of Education Research, Journal of Education Research, and Am erican Education Research Journal. I have also gleaned information from past annual conferences of and papers of the American Education Research Association, Northern Rocky Mountain Education Research Association, and Hawaii International Conference on Ed ucation. Search terms were numerous and varied with the intent of trying several combinations in order to uncover what pre existed in the literature. Keywords such as high school mathematics performance, student perceptions, self efficacy, social learning theory, high school algebra proficiency, high school algebra difficulties, high school algebra requirements, high school obstacles, and mixed methods were all used in Boolean combination with other terms, such as high school graduation, high school dropou t, failure to graduate, and hi gh school algebra performance. Literature Review Legitimacy and Validity I sought references that reflected a review of multiple research methodologies and looked for quantitative, qualitative, and mixed methods research in o rder to increase the legitimacy and quality of the review synthesis. Establishing legitimacy ensures there is

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34 enough information about a concept that can be surmised from a thorough search of the literature ( Onwuegbuzie, Collins, et al., 2007 ) Literature review validity was sought by using peer reviewed articles (or by noticing a lack of such peer reviewed publications) for consideration when determining the findings from the literature synthesis. In addition, articles were also reviewed with a critical eye I sought articles that presented both quantitative and qualitative data. I also reviewed articles that presented rigorous study, which included explanations of justifiable research techniques and clear presentation of findings and discussion ( Frels & Onwuegbuzie, 2011 ; Onwuegbuzie, Collins, et al., 2007 ) Literature Review Synthesis Methodology Figure II.1 depicts the four step sequential path taken though the literature review. Step one is the launching point, where I embarked on the literature review by looking into the greater realm of the high school dropout problem and some of the many predictors of high school dropout. I n step one I deliberately sought to narrow the search to a smaller, inclusive sphere within the high school dro pout boundary that contained arguments pointing to the criticality and timing of student high school algebra performance and its relationship to high school success. In this beginning stage of the research, I also inquired about what research has been con ducted that would allow me to justify the use of the predictor variables of algebra performances, reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special ed ucation status as a model for examining subsequent failure to complete high school.

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35 In steps two and three, I further constricted the boundaries of the literature review search sphere to inspect the topics of self efficacy and social learning theory, as th ey algebra experiences. Finally the innermost sphere represents the concluding portion of the literature review, step four. This is where I use the findings from th e previous steps in the search to develop a conceptual framework (the goal of the literature review) that proposes a lens for demonstrating the relationship between high school algebra performance and high school dropout. In this step, I also provided jus tification for continuing on with my research the rationale for the literature review Figures II.2 through II.5 provide more detail about each of the four steps comprising the trail through the literature review, with each step intending to provide answe rs to specific literature review questions. Figure II.1 Path Through the Literature Review

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36 The sequential path through the literature starts with an examination of the issues surrounding the U.S. high school dropout issue and narrows to those that can highlight the importance of acceptable high school algebra performance to graduation. The path then narrows to an examination of self efficacy and social learning theory matters as they relate to high school dropout and high school algebra experiences. The path ends at a proposed conceptual framework depicting a unique lens through which to view the relationship between the two and a discussion about why further study is needed. Figure II.2 Step One: The High School Dropout Environment and Step one of the literature review will address the first two literature review questions. First, given the existing body of literature about the high school dropout problem in the Un ited States, what factors substantiate the need to examine the distinctive role of high school algebra performance, as compared to other factors, in predicting high school graduation? This diagram depicts the funnel of predictors to be identified, with

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37 pa rticular attention being focused on core course difficulties, the ninth grade transition year, and failing a course in the early high school years. Second, w hat research has been conducted that would justify the use of the predictor variables of algebra pe rformance, reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special education status as a model for examining subsequent failure to complete high school? Figure II.3 Step Two: Self Efficacy, the High School Dropout, and the High School Algebra Experience Step two of the literature review addresses the third literature review question: What is the conjectured role of self efficacy, with respect to high school graduation and high school algebra performance?

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38 Figure II.4 Step Three: The R elationship B etween S ocial L earning T heory and S elf E fficacy and the B ehaviors L eading to H igh S chool D ropout Step three of the literature review addresses the fourth literature review question: What does the literature suggest is the relationship between s ocial l earning t heory and self efficacy and how does this lend itself to students engaging in behaviors that lead them to dropping out of school?

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39 Figure II.5 Step Four: Self Efficacy, Reinforcements, and Expectancies in High School Algebra Exacerbating the Potential to Engage in High School Dropout Behaviors Step four of the literature review addresses the fifth literature review question: What conceptual framework can be developed offering a proposed connection between perceived self efficacy beliefs, reinforcements, and expectancies that result from eriences in their first high school algebra course leading to the potential to engage in behaviors typical of high school dropouts? I synthesized the literature results and developed a conceptual framework using the concepts of self efficacy and Theory of Social Learning. This suggests a proposed connection between student perceived self efficacy beliefs, reinforcements, and expectancies that have resulted from experiences in high school algebra and high school dropout. The inner sphere shows that student perceived self efficacy beliefs, reinforcements, and expectancies developed in their high school algebra experiences leak

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40 through a porous boundary between high school algebra and the greater high school experience, which exacerbates the potential to enga ge in high school dropout behaviors. This figure does not illustrate the answer to literature question six: What quantitative, qualitative, and/or mixed methods research has been conducted that examines the hypothesized relationship between high school a lgebra performance and high school graduation? Interpreting the Results of the Literature Review: Steps One through Three Research over the years has examined many topics and theories related to the alarming national high school dropout rate, and numerous reasons have been offered to explain the failure to complete high school. The factors associated with high school dropout are expansive and cover a broad range of categories. They range from personal and family traits to school performance and academic characteristics to behavioral and emotional patterns. Some of these proposed causes support the criticality of algebra to success in a high school education; others do not. Personal attributes of sex and ethnicity are correlated with high school graduatio n. By most recognized definitions of high school graduation rates, a disparity to be addressed further in the discussion of methods, females demonstrate a better high school completion rate than do males. For example, examining 2008 data compiled by the National Center for Educational Statistic (NCES) females (90.5%) had a higher status completion rate than males (89.3%). Higher status completion means that one holds either a high school diploma or general equivalency diploma (GED) credential. Using an other measure, the status dropout rate, which addresses the percentage of students who are not enrolled in high school and who have not earned any high school credential

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41 (diploma or GED), males have a higher status dropout rate (8.5%) than females (7.5%) ( Chapman et al., 2010 ) Status completion and status dropout measures reflect that ethnicity also matters when examining high school success. Status completion rates were higher for Asian/Pacific Islanders (95.5%) and White (94.2%) than for Blacks (82.5%) and Hispanics (75.5%). Status dropout rates showed a similar pattern with l ess Asian/Pacific Islanders (4.4%) and White s (4.8%) going without a high school credential than Blacks (9.9%) and Hispanics (18.3%) ( Chapman et al., 2010 ) Unfavorable familial circumstances have long been associated to graduation failure. Coming from a single family home, a large family home, an urban family home or a home with other dropouts (paren ts and older siblings) has been linked with lower graduation rates. Families headed by dropouts struggle to provide parental involvement and support in the form of study aids and out of school learning opportunities. These families are often headed by ad ults who have had children very early in life or include older siblings who themselves have dropped out of high school ( Bridgeland et al., 2006 ; Hickman et al., 2008 ; Swanson, 2006 ) Family socioeconomic status is a circums tance that plays a major role in predicting graduation in more than one way. Students from lower socioeconomic status and below the poverty line are more likely to drop out of school than students who are not ( Ekstrom et al., 1986 ) In 2008, the dropout rate for low income students was about four and one half times that of students from high income families ( Chapman et al., 2010 ) These students are subject to secondary effects from being part of a lower socioeconomic stratum. One is that m any low socioeconomic families are prone to

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42 increased mobility; and their children are placed in multiple, changing school environments. Stu dents of high mobility families have trouble adjusting academically, socially, and emotionally to repeated new school environments, resulting in higher incidence of disengagement in academic settings and increased dropout rates ( Hickman et al., 2008 ; Rumberger, 1987 ) Another secondary effect of low family socioeconomic status is that many of the i r high school aged students need to work in order to supplement the family income. While holding some outside job responsibility can increase student achievement, there is a negative correlation with graduation for students who work over 20 hours a week ( Ekstrom et al., 1986 ; Gleason & Dynarski, 2002 ; Natriello, 1987 ; Orfield, 2006 ; Rumberger, 1987 ; Tyler & Lofstrom, 2009 ) School performance and academic determinants, such as poor reading, writing, and math abilities, are manifested by low and failing course grades which in turn impacts eventual high school graduation success Low standardized test scores in the school years prior to entering high school have been quantitatively correlated to greater incidence of high sch ool dropout. Some of these academic predictors show up as early as elementary school; reading comprehension in third grade is an example of an early predictor. Looking at the year prior to high school entry, dropouts exhibit significantly lower eighth gr ade mathematics and English performance records ( Allensworth & Easton, 2005 ; Allensworth, Nimi, Montgomery, & Lee, 2009 ; Bridgeland et al., 2006 ; Ekstrom et al., 1986 ; Gewertz, 2006 ; Gleason & Dynarski, 2002 ; Hickman et al., 2008 ; N eild, 2009 ; Tyler & Lofstrom, 2009 )

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43 Certain behavioral and emotional patterns are representative of high school dropouts. The se include nonconforming behaviors, such as excessive absences, incomplete schoolwork, and disciplinary infractions. A 50 year longitudinal study of adolescent boys who have display ed delinquent and conduct disorder behavior has shown the se youth to be 17 times more likely to drop out of school ( Hickman et al., 2008 ) Absences are significant indicators of future dropout occurrence. During high school, for example, data from a survey of over 700 students revealed that graduates were suspended for 1.51 days, while dropouts were suspended for 6.2 days ( Hickman et al., 2008 ) Much like the set of academic predictors, patterns of behavior ofte n show up attendance in high school. The more a student is absent in kindergarten the greater likelihood of dropping out in high school ( Ekstrom et al., 1986 ; Hickman et al., 2008 ) Another problem behavior, and one which will be subsequently examined in depth, is the detachment from school that dropouts develop because they find school unrewarding, both academically and socially. There are several negative feelings and experie nces reported by the dropouts themselves. They detail that their high school experience has yielded feelings of low self esteem about their academic abilities, alienation from school and from other students, and a general disinterest in the school environ ment ( Bridgeland et al., 2006 ; Ekstrom et al., 1986 ; Gleason & Dynarski, 2002 ; Natriello, 1987 ; Orfield, 2006 ; Rumberger, 1987 ; Stillwell, 2010 ; Sweeten et al., 2009 ; Ziomek Daigle, 2010 ) Step One: The high school dropout environment and Given the massive discussion about the diverse causes of high school dropout, within a

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44 respect to high school graduation success is also worthy of study. A combination of t being the very nature of algebraic content and configuration. Algebra is a unique course and is often the first in which students are asked to employ abstract reasoning and problem solving. Researchers have demonstrated that the abstract nature of algeb ra increases its difficulty over arithmetic. Students often have trouble with using abstractness for the required construction of meaning, generalized relationships, and management of multiple representations of algebraic objects ( Rakes, Valentine, McGatha, & Ronau, 2010 ; Vogel, 2008 ) A second characteristic, the structure of algebraic material compounds its exclusivity and unfamiliarity to students. For example, students often fail to recognize the differences between expressions and equations. An algebra equation requires consideration as a single object rather than a collection of objects, another concept that is difficult for student s to conceptualize. Additionally, the meaning of equality is often typical content structural challenges often prevent students from recognizing the utility of algebra for generalizing numerical re lationships. A third distinctive characteristic of algebra is its language. Algebra has an obscure mathematical lexicon of its own that students must quickly master. It is a expressions, linear equations, quadratic equations, functions, and polynomials. Each of these three foundational aspects of algebra (abstract reasoning, mathematical structure, and language usage) in itself causes

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45 challenges for students. The confluence of all three potentially forms an impermeable barrier to mastering algebra for many students ( Rakes et al., 2010 ) The distinctiveness of algebra leads to a domino effect for many high school students; and failing to catch onto its many characteristics prohibits success in future, other required mathematical coursework. An extra barrier to algebraic mastery is the usual circumstance of limited availability and utility of support from outside the school. According to Weiss, Carolan, and Baker Smith (2010), mathematics learning, including algebra, is more school dependent than other topics, such as reading or science. Algebra, therefore, may be a course where one not only struggles but the levels of frustration are so high that one walks away from math and other coursework, never to come back ( Vogel, 2008 ) Students are increasingly required to take algebra, even though many struggle with the uniqueness of algebra and may not be adequate ly prepared to undertake this course. The topic of mandatory algebra instruction for all high school students has garnered much attention. Recent trends in national, state, and local education policies point to three particular aspects of algebra enrollm ent that, in addition to its unique a core or required high school cou rse, and (c) algebra failure in the early high school years. Algebra is a required mathematics course in many schools and is increasingly part brought on by the No Child Left Behind act, is one reason for the addition of more

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46 rigorous and mandated classes, including algebra to the high school curriculum. Because of this, many states must ensure they enroll all students, even the unprepared, in rigorous and academically c hallenging core courses, such as algebra ( M edina, 2010 ) Two other causal factors have induced the addition of algebra as a standard in high school programs across the nation. The first is the movement to ensure United st century workforce. Since al gebraic skills have long been thought to be foundational for entry into technical professions needed to with other industrialized nations in nurturing such a skilled w orkforce ( Allensworth et al., 2009 ; Gamoran & Hannigan, 2000 ) Listening to PBS radio the other day and following a story on President Ob ( PBS Radio Broadcasting, 2012 ; Stein et al., 2011 ) The second issue is the reported unpreparedness of our students to complete college c ourses. The remediation rate of students when they enter college, especially in mathematic skills, e.g., algebra, is of considerable concern. In response, a number of states have responded to the unpreparedness of college students with dictating that the ir school districts include algebra into the high school curriculum ( Stein et al., 2011 ) For example, in New York State, 28% of incoming college freshmen in the year 2005 were enrolled in a remedial course in reading, writing, or mathematics. Because of this, New York State has started to grade their high schools on how many of their students are in remediation while in college ( Medina, 2010 )

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47 Many s tates and school districts now require the passing of algebra as a core course to fulfill the requirements for a high school diploma. By 2007, a total of 13 states, with 16 more states expected to follow suit, were mandating the inclusion of algebra as a required course for all students ( Schornick, 2010 ). Some states and school districts are not only requiring the successful completion of algebra but are directing it at the earliest opportunity in a high school career, i.e., the ninth grade ( Allensworth et al., 2009 ; Medina, 2010 ; Neild, 2009 ; Neild, Stoner Eby, & Furstenberg, 2008 ; Schornick, 2010 ; Stein et al., 2011 ) Unfortunately, the experiences in, and the results of, courses such as algebra w hich students take in the ninth grade often shape the success of the rest of their academic careers ( Allensworth et al., 2009 ; Gamoran & Hannigan, 2000 ; Neild, 2009 ; Neild et al., 2008 ) in an educational trajectory, especially for teenagers identified as at risk ( Natriello, 1987 ) Ninth grade is an especially crucial, transition year for many adolescents. For some students, it represents the final stages in the prolonged exiting off the educational track. Research has shown that a positive experience in ninth grade is significantly correlated to later success in high school, and academic difficulties in the ninth year have a negative correlation to later high school graduation ( Allensworth & Easton, 2005 ; American Psychological Association, 2010 ; Archambault, Janosz, Fallu et al., 2009 ; Archambault, Janosz, Morizot, & Pagani, 2009 ; Hickman et al., 2008 ; Neild, 2009 ; Neild et al., 2008 ; Roderick & Camburn, 1999 ) Academic difficulty is quickly evident after transition ing from middle school to high school Dropouts show lower grades than their peers as early as the first semester of

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48 ninth grade ( Alspaugh, 1998 ; Hickman et al., 2008 ; Neild, 2009 ; Neild et al., 2008 ) Hickman et al. (2008) demonstrated that the ninth grade first semester overall grade point average (GPA) was significantly lower for d ropouts than for graduates, and this disparity is maintained into the second semester a mean of 2.75 to 1.27, respectively. Ninth grade also represents the second transitional juncture for a majority of American students, including most of the students in the district under study. The first showed that students who undergo both these transitions (i.e., who do not go to a K 8 school before entering high school) tend to show a statistically significant greater achievement loss in high school and a statistically greater significant likelihood of dropping out of high school ( Alspaugh, 1998 ) Several dynamics school. Curricula demands finally catch up with many students in their freshman year. Insufficiently prepared students and students holding less than average grade equivalent skill sets, especial ly in mathematics and reading, especially struggle and flounder in an environment where they must pass and earn credits to progress. In Philadelphia, for example, ninth graders who scored at grade level on standardized math tests were 42% more likel y to g et promoted to the tenth grade than those who scored two years or below grade level ( Neild, 2009 ; Neild & Balfanz, 2006 ) There is a sudden reduction in adult scaffolding and a corresponding need for ninth graders to assume personal responsib ility for their own education and behavior, to include speaking up when things go wrong. Adult scaffolding can be thought of a s the protective framework has surrounded students and which has

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49 provided explicit directions on most as pects of their elementary and middle school education. This includes actions such as deliberately telling students when to take notes, asking them to hand in their homework and ensuring they come to class Due to the blanketing of structure and organization imposed on students in their elementary and middle school education years many are unlikely to have thus learned to assume self responsibility for accomplishing tasks necessary for success in education prior to entering high school. Adolesce nts entering the ninth grade are also simply not mature enough to understand they must possess self initiative to complete work, ask the teacher for help when needed, and earn passing grades to gain the credits needed to advance and eventually graduate. T he reduction in adult supervision often leads to non productive patterns of behavior, such as incomplete or late work, inattentiveness to class proceedings, tardiness and unexcused absences, all of which have been found to be negative influences on achiev ement and the eventual earning of a diploma ( Neild et al., 2008 ; Roderick & Camburn, 1999 ) By the time adolescents gain the maturity to realize they alone are primarily responsible for their educationa l accomplishments, it is often too late. When a student enters ninth grade, in addition to a reduction in adult scaffolding, they must relate to a new set of educators. In the overwhelming majority of cases, they have had scarce personal contact or a prev ious relationship with the new teachers or other school adults. Students quickly feel lost and anonymous, lacking ready access to a trusting adult to whom they can talk, especially in this pivotal transition year, when it is so easy to fall off track and get into academic difficulty ( Bridgeland et al., 2006 ; Neild, 2009 ; Neild et al., 2008 )

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50 Corollary to the changes in the adult influences is a changing of the peer cohort influence of previous educational years. Peer c ohorts and the participation with new activities and new social networks can yield either positive or negative effects on student achievement. As an example, in the case of ninth graders who had poor eighth grade academic performance grades of mostly C or lower going to high school with mostly new peers who were not their colleagues in the eighth grade showed a positive correlation to improved grades ( Neild, 2009 ) The newness of the fresh high school environment matters but so does the actual acad emic composition of the ninth year, such as the number of core courses that a freshman takes. Hickman et al. (2008) did a study of a high school cohort of 119 students enrolled in high school from 2002 to 2005. The cohort included 60 graduating students and 59 dropout students. High school dropouts took significantly more core courses, a mean of 6.02, than graduates, a mean of 4.36. It is likely that because they took more core courses in the ninth year the ir GPAs were lower immediately than their peers and persisted in this lowered state throughout the year. The type of courses a student takes has another relationship to high school graduation. Because dropouts take significantly more core courses in the ninth grade than those who graduate, their indiv idual schedules leave little room for electives or personal interest courses, such as the arts, that have a better likelihood of resulting in academic success typical of other high school students ( Hickman et al., 2008 ) Requiring students to take more mathematics, to include algebra, does not necessarily equate to increased ac hievement in high school or to high school graduation. Hoffer (1998) used data from the National Education Longitudinal Study of 1988

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51 (NELS:88) and Logistic Regression analysis to show that dropout rates between schools requiring two years and schools req uiring three years of mathematics are about equal without adjusting for the effects of student background differences. In other words, there was neither an increase nor a decrease in high school achievement, when the number of mathematics courses required for graduation was raised; and there was no negative or positive effect on high school dropout rates ( Hoffer, 1997 ) This lack of an increase in dropout rates can be attributed in part to schools that allow students to take watered down math courses or to not take difficult math courses such as algebra. The Chicago Public School System requires that all students take algebra in the ninth grade. Allensworth et al. (2009), in a longitudinal study of the consequences of mandat ing Algebra 1 for all Chicago high school freshmen in the years 1994 to 2004, also found no increase in dropout rates in an urban setting with an almost 50% dropout rate. While more students earned math credits (probably attributed to more students taking Algebra I), they did, however, find that algebra failure rates increased, grades slightly decreased, and test scores did not improve ( Allensworth et al., 2009 ) They also discovered that absenteeism increased for students of previously measure d average and high math abilities, perhaps as a result of boredom and unchallenging curriculum that is designed for students of all skill levels. Algebra for all may increase the number of students who take algebra, including minority students, and may increase the num ber of students passing algebra; but it also increases the number of students who fail and get behind in academic credits as well ( Gamoran & Hannigan, 2000 ) The majority of students in the school district under study takes algebra and does so in the ninth grade. In addition, all students are now required to pass three years (six

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52 semesters) of math. While algebra is not required as one of the math classes the state high stakes testing in the ninth year includes many algebrai c topics. Thus, many of the students are forced to take algebra in preparation for the state tests with scarce regard for whether or not students are ready to take these required courses. And b ecause of this, more unprepared and low achieving students i n the district are taking algebra than ever before. Highly related to the difficulty of adjusting to high school in the ninth year is the frequent occurrence of a student ending up with a shortage of credits at the end of this academic year. If a student fails algebra, it can lead to the student getting behind in the number of credits needed to graduate high school in four years. Falling behind in credits due to failing a class and not earning credit for the failed course during the early high school year s significantly increases the risk of failing to graduate high school, especially graduating on time after four years of attendance. Students who are falling behind are noticeable almost immediately as they begin their high school careers and are often fai ling some or all of their core courses. In the Chicago public school system, for example, more than 40% of freshmen fail a major or core course in their first semester of freshmen year ( Allensworth & Easton, 2005 ; Neild, 2009 ; Roderick & Camburn, 1999 ) In Philadelphia, it was found that the percentage of courses failed in the ninth grade was a strong predictor of eventual dropout. Using a representative five course load, an increase of 20 percentage points, or one extra failed course, increases the odds of dropping out by approximately one third ( Neild et al., 2008 )

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53 The Chicag o Public School System uses an on track indicator that is comprised of two components. Those who are on track at the conclusion of the ninth grade have failed no more than one semester of a core course (defined to be English, math, social studies, and sci ence) and have completed enough credits to be promoted to the tenth grade. In a study conducted with cohorts of their students in the years between the 1993 1994 school year and the 1999 2000 school year, the Chicago found the on track indicator to be mor e accurate for prediction of eventual graduation than eighth grade test scores or student demographical characteristics, such as race/ethnicity and economic status. In 1999, for example, the school system found that a student who was on track at the end o f the ninth year was more than three and one half times likely to graduate than one who had failed a semester of a core course and was behind on credits. Students who were off track in the ninth grade had a 22% on time graduation rate as compared to 81% f or students who were on track with credits and passed courses at the end of the ninth grade ( Allensworth & Easton, 2005 ; Neild, 2009 ) Other, more recent, studies in Texas and Pennsylvania provide additional evidence of the negat ive correlation between getting behind in credits in the ninth grade and high school dropout rates. The studies go so far as to claim that 16 out of every 100 students and in some school systems as high as almost one third of all dropouts never get beyond the typical required ninth grade credit totals ( Neild, 2009 ; Neild & Balfanz, 2006 ; Swanson, 2006 ) School districts vary in standards for what is needed for promotion to the next high school grade. Some require the passing of enough credits and the passing of certain core courses. Others simply require a student to earn a specified number of credits, in

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54 total and by content area, in order to earn a high school diploma. Regardless, a student who does not earn enough credits in the first year will need to pick up the pace and make up the c redits in order to graduate on time. They must also graduate with enough credits in the required content areas. Further, the longer it takes a student to complete the requirements for high school graduation, the higher the risk of failing to earn a high school diploma ( Chapman et al., 2010 ; Roderick & Cam burn, 1999 ) A typical student enters school in the ninth grade and must stay on track, passing enough courses, and passing certain types of courses, in order to maintain academic and core course credit progress toward earning a diploma in the standard four years. The literature clearly showed that failing core courses and getting behind in credits in ninth grade was more predictive of high school dropout, with more students, including this ( Allensworth & Easton, 2005 ; Neild, 2009 ; Neild & Balfanz, 2006 ; Roderick & Camburn, 1999 ) it provide reason for examining this in my study as well. Step Two: Self efficacy, the high school dropou t, and the high school algebra experience. Individual student self efficacy beliefs and perceptions of the reinforcements and expectancies originating from both the greater high school environment and, specifically, high school algebra classes all figure in the pote ntial to manifest behaviors characteristic of the high school dropout. The typical high school atmosphere and high school algebra class is fraught with sources of negative self efficacy beliefs, reinforcements, and expectancies. Examining two closely rel self

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55 typical high school environment and high school algebra classes are full of these negative effects. Self efficacy and the high school experience. Self It is a self judgment of personal capability to organize and execute courses of actions and behaviors required to attain designated types of perform ances. Self efficacy impacts how well one uses their skills and plays a major role in human functioning, affecting how one thinks, feels, and acts. Self efficacy also affects how one motivates oneself in a given situation and provides individuals the cap acity to alter the environment and influence beliefs about what to expect from their actions Self efficacious individuals set higher standards for themselves and use more efficient strategies to gain the results needed in certain environments. They are also more resilient in their actions, set progressively higher goals for themselves, and are more satisfied with their performance ( Bandura, 2012 ) Bandura hypothesized, and has shown in numerous studies, that self efficacy mediates past experience and performance and is a key driver in predicting future performance. Low self efficacy can lead people to think that things are tougher than th ey really are. This can foster stress and a narrow outlook on how to accomplish tasks in a certain domain. Low self efficacy can also be a strong determinant on future levels of accomplishment ( Pajeres, 1996b ) Persons with high self efficacy can view hard tasks, such as getting through the rigors of the freshman year of high school, as a challenge to overcome. They often show a sense of commitment to this quest for mastery. These persons will strive to work through obstacles an d view failures as something that can be

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56 worked through by resorting to their own devices. Most importantly, t hey believe they can learn needed skills or knowledge. Those with low self efficacy have a weak commitment to overcoming challenges and obstacles They do not think they have the ability to gain the needed skills or knowledge and do not display persistence in the mastery of hard tasks, e.g., gaining the proficiency needed to pass course examinations. They would rather take the easy path and resig n themselves to failure ( Bandura, 1986 1994 1995 2012 ; Bandura, Regalia, Caprara, Barbaranelli, & Pastorelli, 2001 ) efficacy beliefs are so powerful that this often overpowers the concrete results of past performance, forming a filter through which to interpret new phenomena. This new filter, in turn, influences sub sequent actions and behavior outcomes ( Pajeres, 1996b ) Bandura, in his 1977 seminal work on self efficacy, postulated that self efficacy beliefs and filters formed from self efficacy beliefs, are a major determinant of whether a p erson will attempt a given task and how much effort will be expended in the accomplishment of a given task when obstacles are present ( Bandura, 1977 ) High school dropouts are more likely to report lower feelings of self efficacy and encounter multi ple sources of negative self efficacy beliefs than those who stay in school ( Dynarski & Gleason, 1998 ) General characteristics of high self efficacious adolesce nts, those who stay in school are greater persistence, effort, and an intrinsic interest in individual learning and results. Those with lower self efficacy reduce their goals and slacken their efforts in the face of adversity. High self efficacious s tudents will not be easily deterre d by an

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57 occasional failure and generally work harder and persist longer to accomplish a task than low self efficacious students ( Bandura, 2012 ; Pajeres, 1996a ; Schunk, 1984 ; Zimmerman, Bandura, & Martinez Pons, 1992 ) Bandura has promoted four sources of efficacy beliefs. These sources contribute specific domain. Togethe r or alone, these sources shape the self efficacy beliefs from and are unique to an interplay with a given environment ( Bandura, 1977 ) The sources are (a ) Mastery Experiences (ME), (b) Vicarious Experiences (VE), (c) Social Persuasion (SP), and (d) Affective States (AS). A mastery experience derives from the actual doing of a task. Bandura (1986) states that mastery experiences have the strongest, althoug h not the only, influence on self efficacy, since these experiences offer genuine evidence about the resultant quality of the completed task. Self efficacy increases if one believes they are successful at mastery of a task; if not, there is a decrease in self efficacy. Repeated failures not attributed to perceived individual effort and the level of difficulty of mastery experiences may be a strong detriment on self efficacy. Success with hard tasks in challenging situations raises self efficacy. On the o ther hand, if a student has experienced success primarily in easy tasks and unchallenging environments, they come to expect success routinely and do not learn to persevere. When faced with a difficult task, they are easily discouraged, give up, and fail, thereby lowering beliefs in self efficacy ( Bandura, 1995 2012 ; Oettingen, 1995 )

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58 Many future dropouts enter high school molded by the mastery experiences of earlier schooling, when they did not need to apply much effort to succeed and there were no challenges that would help them gain a sense of perseverance. They developed a f alse sense of self efficacy that was based on getting by i n mastery experiences that required very little expenditure of energy. The new ninth grader quickly faces a novel environment that contains much harder and more numerous, specific mastery experienc es in which they must put forth extended effort and succeed in order to make the grade and nurture the growing of self efficacy. They need to succeed with some degree of self reliance and less adult scaffolding in mastery experiences, e.g., doing class wo rk, completing homework, and passing exams in order to gain academic credit and progress toward graduation. These mastery experiences become a prominent part of high school, and failure quickly lowers expectations of success and feelings of self efficacy ( Allensworth & Easton, 2005 ; Neild, 2009 ; Neild et al., 2008 ) There are many ninth graders who enter high school without previous success in mastery experiences, much less challenging ones in w hich they would have learned how to persevere. Neild (2009) pointed out there are too many students who have previously passed coursework but who start to fail in the ninth grade, when inadequate preparation of less than sufficient reading and math skills catch up with them. Another problem related to mastery experiences is that ninth graders generally do not see these experiences as yielding immediate gratification and rewards. End of course grades, gaining academic credit, and earning a diploma are dist al goals, especially for relatively immature ninth graders. Dropouts report scarce experience with proximal academic tasks, let alone with increasingly challenging tasks. There is little opportunity

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59 to gain a taste of success and the resulting self effic acy that is generated with appropriate and rewarding mastery experiences in high school ( Schunk, 1984 ) The second source of efficacy is vicarious experience. Mastery sources are strong sources of self efficacy, and so are vicarious sources, which sometimes act independently of mastery sources. Vicarious sources become especially prominent when mastery experiences are limited or absent. Vicarious experience i s knowledge that is gained indirectly from oneself and from the behaviors of others. With respect to oneself, v icarious experiences allow for reflection on a new situation. It gives the learner the chance to mediate what they can do based on what they ha ve observed themselves do in the past. Modeling by others can take on a substantive role in forming self efficacy beliefs and in the subsequent affect on behaviors ( Bandura, 1986 1995 ; Pajeres, 199 6a 2002 ; Schunk, 1984 ) Ba ndura (2012) has shown the importance of vicarious sources in shaping self efficacy through the findings of two meta analysis studies that demonstrated vicarious sources of self efficacy had lowered or raised self efficacy independent of performance outcom es ( Bandura, 2012 ) One of the strongest vicarious experiences originates from the modeling of behaviors by others who are similar, with the strength of this source of self efficacy increasing accordingly to the degree of close ness of the model to oneself. The successes of models similar to oneself raise self efficacy, while their failures lower it ( Bandura, 1986 1995 ) In a study of sources of self efficacy beliefs, shaped by social modeling and its relationship to mathematical outcomes, Schunk and Hanson (1989) found that efficacy beliefs were stronger when influenced by observing

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60 peers than when influenced through observing teachers, even though the same mathematical operations were modeled by both groups. The power of peer models is already strong in elementary students, as discussed within this study. This influence grows stronger throug h the school years until a student enters high school (Schunk & Hanson, 1989). The ninth grade is a point at which students elevate the significance of social interactions and relationships and the influence of peer groups grow accordingly ( Neild, 2009 ) Adolescents tend to deliberately choose peers who share similar interests and values. Peer groups start taking on increasingly important roles and become progressively more influential on the high school student ( Bandura, 1994 ; Nichols & Whit e, 2001 ) These peer models may or may not facilitate academic achievement and subsequent graduation. Academic achievement may suppress identification with certain peer groups and success in school; 42% of the over 400 dropouts interviewed in Baltimor e and Philadelphia stated they spent time with people who were not interested in school ( Bridgeland et al., 2006 ) On the other hand, the lack of belonging to a peer group has been also positively correlated to high school graduation for some students. Struggling middle school students who enter high school with a different peer group actually do better once in high sch ool. Those with lower ability, who continue to identify with similar students, may be the victim of interactions that oppose academic achievement ( Neild, 2009 ; Nichols & White, 2001 ) It is possible to gain self efficacy via vicarious experience without similar role models, but this is o nly when it is gained from competent models who adequately convey

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61 knowledge, skills, and strategies of competency ( Oettingen, 1995 ) Unfortunately, many dropouts do no t benefit from such role models in their high school educators. Students ( Bridgeland et al., 2006 ) Social persuasion is the third source of self efficacy. Social persuasion is the response one receives from others, such as teacher feedback and encouragement; and it is usually displayed in the form of spoken judgments from others. Verbal advice often p rovides students needed reinforcement of positive remarks on their efforts and work, but social persuasion can be a two edged sword. Withholding feedback is harmful, and it is as equally destructive to self efficacy to provide negative and unrealistic fee dback ( Bandura, 1986 1995 ; Pajeres, 1996a ) Constantly berating a ninth grade algebra student for not being able to solve a problem on the board is just as negative as pra ising a student Social persuasion as a self efficacy booster requires precision in use and origin. Social persuasion, which is tied to individual accomplishments and not just tas k completion, is most effective in bolstering self efficacy ( Schunk, 1984 ) It is of use only when the source is believed to be trustworthy and sincere ( O ettingen, 1995 ) ; and as reported by many dropouts, there seems to be a lack of such sources in the high school environment. Poor attitudes, little caring, modest help, and shallow expectations are examples of the lack of social persuasion self efficacy sources in the high school experience of many dropouts. Dropouts have expressed that teachers themselves appear disengaged with the class and clearly emit signals showing their dislike of teaching. Students further

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62 perceive that their teachers, for the m ost part, care only about presenting the material and just standing in front of the board. The teachers talk past the students instead of communicating with them, rarely involving the students in their education ( Bridgeland et al., 2006 ; Gewertz, 2006 ; Gleason & Dynarski, 2002 ; Orfield, 2 006 ; Schornick, 2010 ) Interviewees from Baltimore and Philadelphia focus groups examining the high school ( Bridgeland et al., 2006 ) The fourth source of efficacy is the affective states, which are the emotions, attitudes, and physical and emotional responses from items in an environment These sources impact judgments of own self efficacy. Negative moods and attitudes, stress, and frustrations generally yield lower self efficacy. When someone is in a positive mood a nd holds positive attitudes, they tend to be more self efficacious. What is important is how one perceives their physical and emotional responses to their environment ( Bandura, 1986 1993 1995 ) There are many origins of negative physical and emotional responses for the typical high school dropout, and it starts immediately upon entering their freshman year. Ninth grade has long been associated with feelings of anxiety and confusion; and many students quickly believ e they do not belong in school, due in large measure to lack of individual caring on the part of school adults. In addition, nint h graders are faced with frustrating school policies that trigger mandatory grade failure, such as too many absences. This may discourage students from persisting in school for the remainder of

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63 the quarter, semester, or year, eroding their commitment to s chool success ( Neild, 2009 ; Wheelock & Miao, 2005 ) As they continue on in school growing boredom and irrelevancy causes many feelings of negativity and lack of motivation in some high school students. As a collective group, high school dropouts are unmotivated, not challenged enough, and bored by classes. As reported in a survey of 467 dropouts, these negative responses to school were cited as more central to a decision to leave school than actual grades. Only 35% of students who took the survey cited academic failure as the reason for dropping out of high school ( Bridgeland et al., 2006 ; Gewertz, 2006 ) Course material was pointed out as irrelevant to the real world, and 70% to 80% of students stated that they wanted more authentic world learning opportunities ( Bridgeland et al., 2006 ; Gewertz, 2006 ) Self efficacy is a fluid construct holding individualistic, domain, and time altering efficacy beliefs are a product of what they personally believe to be true in a giv en environment and at any one time. This involves the weighing and particular epoch of time ( Oettingen, 1995 ) Individuals perceive their beliefs of self efficacy differently throughout their lifespan and across diverse domains. The older a student gets, the more pessimistic they grow in their self perception of individual abiliti es and expectancies of performance. Older students tend to expect much less from their behaviors with far less positive outcomes; and sometimes, to them, it accordingly does not make sense expend much of an effort in school ( Archambault, Janosz, Morizot, et al., 2009 ; Kloostermann & Cougan, 1994 ) Younger students, especially in the earlier elementary school years, report a much

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64 greater sense of belief in their academic abilities than do older students ( Pajeres, 1996b ) Additionally, research has shown that by the fifth grade students develop separate verbal and math self efficacy measures, due to their increasing ability to distinguish their competency in different academic areas ( Zimmerman & Martinez Pons, 1990 ) Specific student self efficacy beliefs in high school are likely to not be the same as they were in earlier school years, with a noticeable change from the eighth to ninth grade. Finally, since the typical high school student is old enough to distinguish competencies in different academic areas, self efficac y beliefs that are formed through experience in algebra are likely to be different from those formed in other subjects, such as English or other math courses (e.g., geometry). Self efficacy and the high school algebra experience. Algebra, as mentioned ear lier, is an academic discipline encompassing distinctive characteristics that affords students a school experience that is unlike that provided by other classes. Because of this, self efficacy beliefs, and the sources of these self efficacy beliefs, devel oped from algebra involvement are distinctive from the self efficacy beliefs that originate in other areas of high school. With the increasing emphasis on algebra as a core mathematics course that must be mastered in order to successfully earn a high scho ol diploma, the concept of algebra and mathematics self efficacy, the sources of algebra and mathematics self efficacy, and their resultant influence on successful algebra outcomes with the subsequent high school performance are appealing areas for explora tion. Mathematics self efficacy is strongly related to and is predictive of mathematics performance outcomes ( Hackett & Betz, 1989 ) Unfortunately, many students lack the necessary beliefs in their ability to succeed in mathematics, particularly in algebra. For instance, s tudents were asked to rate their perceptions of competency in nine academic

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65 areas on a self efficacy questionn aire that was given to over 100 ninth and tenth graders in a large Eastern city. Algebra was rated at almost the lowest self efficacy. Only the ability to learn foreign language was judged to be lower ( Zimmerman et al., 1992 ) Mastery experiences and social persuasion sources have been found to be especially critical to t he formation of algebra self efficacy. Lopez et al. (1997) used path analysis to demonstrate that mastery experiences and social persuasion were each significant predictors of algebra self efficacy. Algebra self efficacy was then predictive of algebra ou tcomes, in this case passing grades. Although vicarious experiences and affective states were not significant predictors in themselves, the path from self efficacy, which was a summation of paths from al l four sources, continued to demonstrate that the ov erall self efficacy concept was significant in predicting algebraic results. This group of experiences and actual test scores in predicting algebra self efficacy. They showe d that while perceived mastery experiences significantly predicted self efficacy, actual standardized math scores did not. Clearly, self efficacy beliefs from their mastery experiences were more important on future algebra outcomes than actual perfo rmance. experiences in mathematics. However, due to the uniqueness of algebra, earlier mathematics mastery experiences may be inadequate toward advancing the new mastery exp eriences required to promote algebra self efficacy. Sufficiency in past mathematics mastery experiences and getting good grades in earlier math courses may not be enough to maintain mathematics self efficacy. Students who previously coasted by and were n ot challenged enough are prone to not persevere when it gets harder to maintain good grades

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66 in a high school mathematics course such as algebra ( Kloostermann & Cougan, 1994 ; Stipek, 1981 ) Additionally, gaining mastery experiences in algebra can be very different from gaining other mathematics mastery experiences, again, abstractness, structure, and language ( Helfand, 2006 ; Kollars, 2008 ; Pajeres, 1996a ; Rand Corporation, 2008 ) Students who have experienced mathematics only at a concrete or procedural level, characteristic of many U.S. classrooms, must negotiate the complex gap from concrete to abstract reasoning without the benefit of prior mastery experiences in this new way of thinking ( Rakes et al., 2010 ) Teaching for content and not mathematical reasoning o r meaning is a practice that starts in elementary school education and continues into the high school years of mathematics instruction. In the U.S., as compared to Russia, China, and Japan, our students rarely write arithmetic sentences to represent situa tions. Consequently, when our students move to algebra, they have developed the predisposition to understand algebraic formulas as something they need to calculate rather than as expressions that represent something ( Thompson, 2008 ) Too much instructional emphasis is placed on this mechanical and rote learning of the steps of algebra. Teaching methods that focus on the rigid, procedural aspects, in reality, interfere with the student gaining the requisite conceptual understanding skills ( Lobato, 2008 ) and falls short of ensuring that students understand the meanings behind the algebraic concept ( Rakes et al., 2010 ; Thompson, 2008 ) Once they waver from a scripted path as defined by a mechanical method of instruction students are not given the tools necessary to find their way and gain successful mastery experiences ( Sfard, 1991 )

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67 As one example, when studying how to use functions and graphs in algebra, students must begin to understand and interpret one set of algebraic objects in terms of another (e.g., a function equation with its graph, data set by its equation, or data set by its graph). Leinhardt, Zaslavsky, & Stein (1990) found that students generally have no problem with gaining procedural fluency at plotting points and equations but do have problems with the ability to extract meaning from graphical representations. The difficulty lies in the connection of a graph to the construct being represented. Specifically, students are readily capable of demonstrating procedural fluency; but this procedural understanding is unable to guide students through problems involving interpretation ( Leinhardt, Zaslavsky, & Stein, 1990 ) Further complicating the issue, teaching methods used to convey content often exacerbate the trouble with g aining mastery algebra experiences. Both students and teachers expect immediate rewards for teaching and learning efforts. However, the understanding of abstract mathematical ideas often requires a lengthy, iterative process, and innovative instructional methods, something that is rarely addressed in our country. In reading through the 2008 Final Report from the National Mathematics Advisory Panel about instructional changes needed in our algebra curriculum it was dismaying to see a lack of information about what is required with respect to instructional pedagogy to ensure that our high schools do a better job of conveying abstract reasoning and meaning skills ( Rand Corporation, 2008 ) Teachers of high school algebra, espec ially, must be success.

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68 It stands to reason, given the a general lack of belief in algebraic self efficacy, that students need sources of strong social persuasion in ad dition to credible mastery experiences. Teachers must effectively model, illustrate, and explain with multiple instructional methods. They then need to provide for effective feedback mechanisms, making certain students understand the content of the algeb raic material ( Borko & Whitcomb, 2008 ) And to be most efficient for increasing algebra ic self efficacy, this feedback needs to be task a nd situation specific ( Kloostermann & Cougan, 1994 ) Disappointingly, and similarly to the shortage of mastery experiences, there is a deficiency of Mathematics teachers are using boring and uninspiring teaching methods. They simply lecture, provide notes, and work problems on the board, rarely engaging their students. Former high school mathematics students in one study pointed out the lack of social persuasion in their high school algebra courses. They could not ask teachers for help, and teachers failed to make the classes relevant or applicable to real life. Students also alleged that their teachers had low expectations of them, which negatively affected ( Schornick, 2010 ) Vicarious experiences are a third form of self efficacy at play in algebra classes. These vicarious algebra sources can be teachers or other youth and can add to or decrease from existing levels of algebra ic self efficacy. Children who have observed verb al modeling of division operations by proficient teachers were noted to have gained more confidence and increased skill in performing division operations than those who did not observe this modeling ( Zimmerman et al., 1992 ) Students who were part of like student,

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69 clique groupings experienced less variance (good and bad) i n Algebra I achievement results in a study of 230 high school students in two high schools in the mid south ( Nichols & White, 2001 ) In any case, an encouraging and participatory set of vicarious sources in mathematics classes can lead to increased student engagement and, thus, more positive outcomes ( Weiss, Carolan, & Baker Smith, 2010 ) On the other hand, poor vicarious sources, such as math teachers who simply lack their own mastery knowledge of or a demonstrative interest in the algebra content area, can be damaging to mathematics self efficacy, an other issue highlighted by the National Mathematics Advisory Panel ( Rand Corporation, 2008 ) Many algebra teachers do not model or show a real interest in gaining algebra knowledge for the sake of learning and preparation f or the future. What is largely demonstrated is that teaching algebra is equated to merely earning a credit or passing the standardized state tests at the end of the year ( Roschelle, Singleton, Sabelli, Pea, & Bransford, 2008 ) Algebr a is replete with the fourth source of self efficacy, the affective state sources Many of these algebraic self efficacy. Two such affective state sources are mathematics anxiety and mathematics frustration. Taking part in al gebra instruction holds a good potential for intensifying previously rooted feelings in these two areas ( Hackett & Betz, 1989 ) The feelings of anxiety do not diminish upon high school entry as students grow increasingly frustrated and less confident about their ability to grapple with the unique learning challenges of algebra, especially in lieu of the emphasis on procedural le arning. One critic of the 2008 Final Report from the National minded focus on proficiency is likely

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70 ( Thompson, 2008 ) Student perceptions of relationships with mathematics teachers also do not facilitate a reduction in anxiety and frustration from algebra participation Schornick (2010) discussed several ne gative perceptions, as revealed from phenomenological, first person accounts of the high school math experience of several students. They described their high school math classes as something they just had to take in order to get by. Math learning and cl ass efforts were not valued by their teachers. High school math teachers were perceived as uncaring, and they did not demonstrate the relevancy of algebra to other academic courses and real world applications. The lack of teacher demonstrated relevancy t o the real world especially stood out in the memories of these interviewees. One student remembered that the only class where he saw relevancy was in trigonometry, when the teacher showed students how it was used in construction ( Schorn ick, 2010 ) These negative student perceptions allow for students to dislike mathematics, lose interest in learning and performing in math classes (e.g., algebra), and become jaded in class. Algebra then grows beyond the desire of some students to work hard to master the contents ( Allensworth et al., 2009 ; Gamoran & Ha nnigan, 2000 ; Kaput, 2000 ) Students who believe algebra is useful in their lives and who enjoy mathematics are more confident of their mathematical prowess. Students who are confident in their abilities will do better in their algebra courses ( Kloostermann & Cougan, 1994 ) Step t hree: The relationship between s ocial l earning t heory and self efficacy and the b ehaviors l eading to h igh s chool d ropout certain

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71 behaviors in their differing environments, such as the environments of a high school algebra class and or greater high school atmosphere. Social learning theory proposes that to understand human behavior one needs to exam the interplay or interac tion, that takes place between an individual and the environment The environment i s comprised of both the physical setting and other individuals who are part of th e setting to behave in a particular fashion can be attributed to percei ved reinforcements and expectancies from their individual interactions with the environment ( Mearns, 2004 ; Rotter, 1982 ) Rotter (1982) represents th e relationship of interaction within a particular environment with this equation: Behavior Potential = f (Reinforcement Value + Expectancy) This formula illustrates that the likelihood of a person engaging in a particular behavior, the Behavior Potential (BP), is a function of the relationship between the Reinforcement Values (RV) and Expectancies (E). Reinforcement value refers to the personal desirability of different sources of activities, and expectancy is the subjective probability that a given behavior will lead to a partic ular outcome. The formula does not literally equate to a one way relationship but is reciprocating. Engaging in certain behaviors will influence how one perceives the reinforcements and s about the reinforcements and expectancies in their respective environments will shape the potential to employ specific behaviors ( Pajeres, 1996a ) People frequently generalize their reinforcement values and expectancies and their potential to engage in associated behaviors from one situation to another and from past experiences to new experiences. People take notice of which reinforcements lead to

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72 certain outcomes and then form expectancies about other, similar situations ( Schunk, 1984 ) Individuals are especially prone to generalize when a new experience occurs in a similar environment as before, when there is no comparable basis from which to form new reinforcement values and expectancies, and when there is nothing in the similar environment that facilitates the construction of a new set o f reinforcement values and expectancies ( Mearns, 2004 ; Rotter, 1982 ; Rotter, Geller, & Hamsher, 1968 ) Consid school. If this earlier educational environment has yielded the expectancy that employing only minimal effort and nominal investment in time is necessary to receive the reinforceme nt that enough content was mastered to pass a core class, such as mathematics, and automatically advance in school, what will stop them from anticipating the same reinforcements and expectancies in a new but similar situation? The students will continue t o employ the same minimal effort and nominal investment in time behavior, thinking they are in a comparable school environment once they reach the ninth grade. Sorry to say, algebra students, especially in the ninth grade, are quickly faced instead with a dissimilar environment with opposing reinforcement values, among them an inability to easily master content and unpleasant expectancies, such as the requirement to work harder without automatic progression. They can either change their behavior of puttin g forth minimal effort and nominal investment in time or they can choose to engage in a new class of behaviors, such as avoiding the effort and time needed to pass the course. This i n turn can lead to solidification of the new reinforcement value of an in ability to easily master content and the expectation of needing to work harder to progress, values and expectancies that might generalize and carry forth to other high

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73 school subject classes, repeating, until one gets hopelessly behind and drops out of hig h school altogether Conversely, if one has previously come to expect feelings of anxiety and irrelevancy from math classes and has been rewarded by negative feedback and because of this has been engaged in behaviors of schoolwork avoidance then these patt erns may also carry forth to the new environment of ninth grade and into an algebra class. This will continue, unless the environment offers a new set of reinforcements and expectancies and the resultant development of desired behaviors. I ndividual perceptions are central to the relationship between behavior potential, reinforcement values, and expectancies and an absolute classification of these concepts is impossible. However this does not preclude the identification of several themes of perceived reinforcement values and expectancies associated with the potential to engage in behaviors characteristic of a high school dropout. Before examining the role of social learning theory and themes of reinforcement values and expectancies leading to dropout in high school, it is desirable to highlight the c oncept s of s elf e fficacy Learning Theory Since these two theories are so tightly coupled, I assert it is possible to equate and substitute the four sources of self efficacy mastery experiences, vicarious experiences, social persuasion, and affective states for the reinforcement values within predictive formula of behavior and to combine the theories into one fo rmulaic representation. The re lationship between self efficacy and social learning theory. Self efficacy beliefs are often associated with both behavior s and expectancies. When faced with task demands, most people behave in a way to make desired things

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74 happen rather than let things merely happen to them. They act in ways to either please others or themselves. People with high self efficacy expect to succeed and see reinforcement value in their surroundings; they maintain or increase their efforts. Peo ple with low self efficacy expect failure and see negative reinforcement values in their environment; they withdraw or give up ( Bandura, 2001 ) Self efficacy shows a demonstrated relationship between the potential to engage in certain academic behavio rs and academic success. People avoid activities and behaviors they believe exceed their abilities or coping skills, and low self efficacy leads to evasion of behaviors that are perceived as threatening and emotionally draining. Student beliefs about what they have accomplished and what they expect they can accomplish affect their academic performance, as individuals have a tendency to participate in actions that they deem desirable. They often avoid behaviors and groups of actions where they receive nega tive reinforcement and, thus, expect little results in areas such as algebra tasks, no matter what the benefit even if they persevered ( Bandura, 2012 ; Kloostermann & Cougan, 1994 ) T asks, such as algebra school work often supply unpleasant feelings and negative teacher feedback and actions and are, thus, to be ignored and avoided. There is little motivation to change a behavior, if one believes they have no personal power over the o utcome or have scant expectations that they can manipulate outcomes with their action s ( Bandura, 1995 ) Students may know they will benefit from passing classes, such as algebra; but they will avoid the actions needed to reach this academic milestone, if they really do not believe in the value of this benefit. Strong self efficacy beliefs can change this avoidance behavi or and bring about desired outcomes as it can catalyze a student to be more engrossed in the subject matter,

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75 which, in turn, prompts the exertion needed to succeed ( Bandura et al., 2001 ; Pajeres, 1996a ) Archambault et al. (2009) found a relationship between student engagement and low grades in high school mathematics courses, although the research did not specify if this was in algebra course work A d ecreas e in student engagement significantly correlated to low grades in math. We will see shortly that the particular behavior of disengagement from high school academic work is a one that is typical of, and a strong predictor of, the eventual high school dropo ut ( Archambault, Janosz, Fallu, et al., 2009 ) Bandura (1985, 2005, 2012) maintains that self efficacy b eliefs are not only connected with behaviors but are also tightly linked with expectancy outcomes. People anticipate likely outcomes of their actions and act upon what they believe they can do well at in addition to the outcomes they expect from their a ctions. Low self efficacy beliefs leads to low expectancies and low expectancies reciprocate by resulting in low self efficacy beliefs; these two concepts also jointly affect behavior. One must have faith in their ability to and expect they can manipulat e outcomes with their behaviors Again, those with high self efficacy envision doing well in many situations; those with low self efficacy predict failure ( Bandura, 1986 1995 2012 ) If students come to expect low rewards from their environment, then they may not engage in behaviors appropriate for the environment. Stude nts disengage from and avoid high school work not only because they have been reinforced with signs that they cannot do well but because they come to expect the lack of immediate gratification and rewards. They may come to expect the need to exert too muc h effort and experience too much of a struggle to get a passing grade in core courses in order to earn credits toward graduation.

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76 Lopez et al. (1997) demonstrated how self efficacy produced negative expectancies, which, in turn, affected interests and lowe red outcomes in algebra performance at a Midwestern high school. Self efficacy beliefs directly affected interests and outcomes; but these interests and outcomes were also mediated by, and strongly correlated to, student expectancies ( Lopez, Lent, Brown, & Gore, 1997 ) As a general rule, therefore, many orchestrate the potential to take on certain behaviors includ ing individually perceived self potential to engage in specific academic behaviors. Individuals make judgments about the effects of their actions. These judgments are regulated heavily by their self efficacy beliefs, as framed within a given particular environment. They also need to anticipate the probable effects of differ ent courses of actions and events (expectancies). In the end, this set of judgments and anticipation allows one to adjust behavior accordingly to the situation at hand ( Bandura, 1977 ; Bandura et al., 2001 ; Hackett & Betz, 1989 ; Kloostermann & Cougan, 1994 ) Since the literature shows that self efficacy is related to the potential to engage in certain as follows, to a new formula which adds the factor of self efficacy to the previously mentioned relationship : BP = RV + E + SE where SE = Self efficacy However, self efficacy beliefs are developed from the four sources of self efficacy also previously discussed These four sources can be perceived differently by ( Bandura, 1977 ) Since these four sources can be perceived differently, they are basically the same

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77 the personal desirability of different m BP = RV + E +SE to BP = ME + VE + SP + AS + E where RV = ME + VE + SP + AS, and ME = Mastery Experiences VE = Vicarious Experiences SP = Social Persuasion AS = Affective States This resultant formula depicts my pro posal to use the sources of self efficacy to to represent the relationship of both sources of self efficacy and expectancies to the potential to engage in behavior within a given environment I also find it unnecessary to use the concept of self efficacy twice in this formula, since it is judged to be adequately represented by its four sources. This formula will form the foundation for the conceptual framework that will guide the qualitative ph ase of the study. Before presenting this conceptual framework, it is preferred to detour for a bit to discuss some of the major findings pertaining to representative behaviors of high school dropouts and general categories of sources of self efficacy and expectancies associated with these groupings of behaviors. Behaviors, expectancies, and self efficacy sources of the high school dropout. The literature makes the case for three major groupings of behavior which are exhibited by high school dropouts. These three categories are academic disengagement, rule non compliance, and self regulatory deficiency. All three categories have a

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78 significant role in predicting high school dropout, and all three are associated themselves with themes of sourc es of self efficacy and student expectancies ( American Psychological Association, 2010 ; Archambault, Janosz, Fallu, et al., 2009 ; Bridgeland et al., 2006 ; Hand, 2010 ) I will discuss the three categories of behavior and briefly touch upon some of the sources of self efficacy and expectancies that can be coupled with these categories. A more comprehensive and resultant showing of these findings will b e synthesized and subsequently presented in the final step of the literature review along with its accompanying table, Table II.2. Academic disengagement can be thought of as a lack of involvement and withdrawal from school activities and demands. Studen ts who are disengaged report feelings of disconnectedness with their classes, other individuals, and the school eventual educational outcomes. Students who are engaged in school perform better academically and avoid problem behaviors, such as dropping out ( Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Weiss et al., 2010 ) In a survey of over 13,000 Canadian high school dropouts, Archambault, Janosz, Fallu, and Pagani (2009) found that low student engagement significantl y predicted high school dropout. Academic disengagement is a multi factored concept possessing three dimensions: behavioral, affective, and cognitive. The behavioral aspect includes aspects such as participating in and preparedness for class willingness to engage in effortful learning and to what extent a student demonstrates these behaviors in classroom work, discussions, and activities. The affective component refers to notions to include students

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79 feelings, interests, perceptions, and attitudes towar d school. This embraces liking school and the subject matter, feeling involved in school, perceiving one can do the work, and a general enthusiasm for learning. The cognitive aspect encompasses the students thinking they want to learn more about a topic and their use of self regulation strategies ( Archambault, Jan osz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Weiss et al., 2010 ) There are a number of student perceived sources of self efficacy and expectancies that are associated with the behavior of disengagement. Affective states, for example, that students perceive as unrewarding, un challenging, and uninteresting become associated with expectancies that active participation in school produces little benefit. Several researchers have discussed how dropouts report restricted opportunity for rewards from school participation realize li mited association between high school attendance and life goals and have, generally, lowered expectancies about the value of a high school diploma. Young adults in general, have a hard time with long term planning and delayed satisfaction. More so, d ro pouts, as a group, expect to realize immediate rewards for their efforts, lack the ability to consider any distant goals, and are able to see only what is right ahead of them ( Bridgeland et al., 2006 ) Because of these negative sources of self efficacy and expectancies, dropouts report they tended avoid participating in class activities or homework ( Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Eckstein & Wolpin, 1999 ; Gewertz, 2006 ; Hickman et al., 2008 ; Tyler & Lofstrom, 2009 ) In interviews with 467 students from a diverse racial/ethnic group, concerning the reasons for dropping out of school, 47% stated that the classes were not interesting and 69%

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80 stated that they were not motivated or inspired ( Bridgeland et al., 2006 ) Others reported they withdrew from class participation in large part due to another perceived self efficacy sou rce social persuasion. Feedback from teachers frequently demonstrated a lack of ( Roderick & Camburn, 1999 ; Weiss et al., 2010 ) In the same study, 68% of the students interviewed stated that they would have worked harder if they had been subject to higher expectations for meeting academic standards ( Bridgeland et al., 2006 ) On the other hand, while they would have worked harder, they also needed, and expected, more support to fulfill these higher standards ( Bridgeland et al., 2006 ) Similar to their general high school experience, studen ts confront sources of self efficacy and expectancies that result in academic disengagement from high school algebra classes. In addition to negative self efficacy sources such as difficult content, a mechanical learning process, and uncomfortable feeling s (i.e., frustration, confusion, inadequacy, and anxiety) students have described that algebra classes were simply boring. They were taught directly from the textbook, and there was no opportunity to reflect on how and why algebra was done in a n exacting fashion. If technology was used, a method more commonly used to capture the interest of youth; it was not the kind of technology that enticed students to actively participate in class. Technology usage typically consisted of merely using overhead projectors and computers to do textbook work, and was no student interaction with the computers themselves ( Kollars, 2008 ; Roderick & Camburn, 1999 ; Schornick, 2010 ) Students have come to expect an algebra experience that is bland and irrelevant and that does not require them to pay attention, especially since there is scarce opportunity to

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81 actively participate. In addition, just throwing work at them without extra help from out of touch teachers has made it difficult for students to expect to succeed. This all facilitates the tendency for students to disengage from algebra. Another general category of behavior displayed by dropouts is a lack of rule compliance. This is typified by actions, such as skipping school and classes, not accomplishing homework and class work, disrupting class room activities and acting rude toward teachers ( Archambault, Janosz, Fallu, et al., 200 9 ; Tyler & Lofstrom, 2009 ) All of these behaviors impact school performance and ultimately graduation success Research shows that a ttendance and completing homework is eight times more predictive of failure to graduate than test scores ( Allensworth & Easton, 2007 ) Attendance, in partic ular, is an early indicator of dropping out of high school, since dropouts display a greater incident of absenteeism beginning with their first semester of ninth grade ( Hickman et al., 2008 ) magnitude of these behaviors 26% of the dropouts interviewed did no homework, and 80% did one hour or less of homework each day ( Bridgeland et al., 2006 ) Several students in an interview protocol concerning the high school math experiences of college students reported similar behavior with respect to their high school math classes and declared that they often did no math homework ( Schornick, 2010 ) Sources of self efficacy, such as a lack of belief in ability to complete h omework or a lack of quick mastery of the subject material especially when combined with a do not comprehend the material, fuel a potential to engage in rule non compliance. This is associated with the

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82 expectanc y of an inability to ever catch up with class work and its corollary of severe penalties for failure to do so. Students quickly expect that if they fall behind they will not be able to catch up. Therefore, they report it is easier to just gi ve up and not try. As an example, i n one survey, 43% of dropouts stated that they missed too many days of school and could not hope to catch up ( Azzam, 2007 ; Bridgeland, et al., 2 006 ) The third category of behavior that is representative of the high school dropout regulating in a given school environment means that a student assumes a degree of individu al responsibility for their own learning. This includes motivating oneself to get school work done, resisting distractions in the surroundings, planning and organizing school work, and self initiating the taking and reviewing of class notes, all of which assist in the learning process. Students who self regulate set near term goals for themselves and are personally committed to initiating actions consistent with learning. They also start to rely less on some adults (e.g., parents) but are aware of persona l learning roadblocks and know when they need to seek help from other adults (e.g., teachers). Students who employ self regulation strategies have a higher sense of self efficacy and personal control over their high school surroundings ( Zimmerman et al., 1992 ; Zimmerman & Martinez Pons, 1990 ) In one study of high school students, use of self regulated learning strategies was positively correlated with mathematical self efficacy, while their reliance on adu lts to facilitate learning was negatively correlated with mathematical self efficacy ( Zimmerman & Martinez Pons, 1990 )

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83 Unfortunately, use of self regulatory strategies is somewhat counter to the nature of an adolescent, especially in the immature and younger student of early high school years. Teenagers have a hard time taking on pers onal responsibility for their environment. This is particularly true in the case of dropouts ( Bridgeland et al., 2006 ) Zimmerman and Martinez Pons (1990) did research that showed students in grades 8 through 11 declined in their efforts to seek adult assistance (e.g., parents) but increased in is trend. They decline in their efforts to seek assistance from both adults and teachers ( Zimmerman & Martinez Pons, 1990 ) Besides the immaturity of younger high school students, the sudden appearance of freedom counters the interest in self regulatory strategies. Dropouts have stated that they had too much freedom and not enough rul es in their lives (38% of the dropouts in one study). Additionally, 51% of this same dropout group reflected back and stated that they should have accepted some personal responsibility for their educational successes. Unfortunately, they recognized this well after they had left school, when they were older and likely more mature ( Bridgeland et al., 2006 ) Fueling the n life social activities, being free on the streets, and hanging out with friends as reported by high school students taking part in the Philadelphia and Baltimore focus group of dropouts ( Bridgeland et al., 2006 ) This lack of self regulation capacity, combined with too much freedom and not enough rules, leads to dropout expectancies that the re is no need for them to control their own destiny ( Gleason & Dynarski, 2002 )

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84 Step 4: Self e fficacy, reinforcements, and expectancies in high school algebra exacerbating the potential to engage in high school dropout behaviors. Review of the literature has provided a flavor for the types of themes of student perceived sources of self efficacy and expectancies that originate from both high school algebra classes and the greater high school experience. These themes of sources of self efficacy and expectancies, which ar e common or unique to both domains, can then be associated to the three themes of behavior which the literature shows are characteristic of the high school dropout population. Because of this, student perceived sources of self efficacy, expectancies, and behaviors are the pivotal factors in the conceptual framework I propose as a lens through which to qualitatively study the hypothesized relationship between algebr a and graduation. The proposed conceptual framework takes advantage of efficacy, and the relationship between the two, and is depicted as follows: BP dropout = (RV algebra + E algebra ) (RV high school + E high school ), where RV = ME + VE + SP + AS, ultimately finalizing the conceptual framework a s BP dropout = ((ME + VE + SP + AS) algebra + E algebra ) ((ME + VE + SP + AS) high school + E high school ) This equation represents the cumulating supposition reached from the literature review synthesis. It is a supposition that demands further analysis from the gathering of first person perception s of high school dropouts: the sources of self efficacy and expectancies resulting from high school algebra experiences increase the severity of those gained from the greater high school experience. In other words, the multiplicative power of high school a lgebra sources of self efficacy and expectancies with the greater high

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85 school environment sources of self efficacy and expectancies aggravate the potential to engage in behaviors characteristic of a high school dropout. To re emphasize, this formula should not be interpreted literally as a mathematical formula but as a representation of a proposed concept. Whereas the plus symbol, (+), does symbolize the additive power of four perceived sources of self efficacy eater high school experiences, the multiplication symbol (*) does not connote a straight multiplicative process. It signifies the capacity of the sources of self efficacy beliefs and expectancies deriving from high school algebra experiences to intensify t hose from the greater high school experience which eventually results in the potential to engage in behaviors leading to high school dropout. Do not make the same mistake that many of our beginning algebra students make: read ing the above formula with a pr edisposition to understand it as something you need to calculate rather than as an expression one needs to understand in represent ing a specified concept. Bandura (1995, 2012) claims that global viewpoints of self efficacy can be useful when considering i ts relationship to individual behaviors in broad areas, such as the expansive high school arena. However, he also cautions that one needs to use judgment of self efficacy in the correct contextual settings to avoid generalized personality trait conclusion s and to provide predictive accuracy about a certain situation. Domain specific assessments of self efficacy are more predictive than general predictions. It is even suggested that within a domain one needs to constrict further and look at task specific contexts. Algebra is one such task specific setting within the content of both mathematics and high school ( Bandura, 1995, 2012) Pajeres (1992b) reinforced

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86 power, self efficacy judgments should be consistent and tailored to the domain of On the other hand, Bandura and Pajeres discuss how self efficacy can generalize or cut across different tasks and domains. A general sense of self efficacy allows one to transfer beliefs of competency and behaviors from one domain to the next. This occurs when the environment offers expectancies for similar judgments of competency or skill sets across domains ( Bandura, 1986 ; Pajeres, 1996b ) Therefore, if a student is encouraged to develop persistence and self regulatory skills in algebra, i t should generalize to behaviors in other academic areas in a high school career. However, the reverse is true as well. Sources of self efficacy and expectancies developed in algebra may very well transfer to general high school behaviors ( Oettingen, 1995 ) The proposed conceptual framework strives to use a domain specific assessment of student perceived sources of self efficacy (along with expectations) to drive a lev el of specificity when studying the relationship between high school algebra and high school graduation. It also provides for student perceived sources of self efficacy and ly (or positively) to the greater high school experience. Finally the framework additionally allows me to explore the use of self efficacy beliefs in the qualitative portion of the study to explain findings from the quantitative component. Moving Forward: Why the Research i s Needed and Potential Themes o f Self Efficacy, Expectancies, a nd Behaviors t o Explore i n t he Qualitative Aspect o f t he Research As discussed and presented, the literature offers arguments for factors that singularly support, explain and predict poor high school algebra performance and poor

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87 high school graduation rates. There is scant research that spotlights the issues jointly or looks at the hypothesized relationship between these two issues, much less provide a persuasive argument about the reason for its existence. The few sources that directly referred to the influence of high school algebra performance on high school graduatio n were either non research based or were in one compared with other predictors. Two sources were non research based documents one from a large city newspaper article and t he other from an online school administrat or journal website. As previously cited, former Colorado Governor Roy Romer expressed his opinion that algebra was the largest contributor to the high school dropout problem. He had observed this correlation whil e serving as superintendent of the Los Angeles successful completion of algebra serve as a requirement for high school graduation ( Helfand, 2006 ) presents a non research based supposition that the initial high school algebra class was likely to have a large impact on drop out, because of the causal effects of frustration and disengagement ( Vogel, 2008 ) The one piece of quantitative factor located in the research is a 2006 doctoral dissertation that used logistic regression to determine that Algebra I was an important p redictor of high school dropout and that students who failed Algebra I were 4.1 times more likely to drop out than those who passed the course. Unfortunately, this study was extremely narrow in focus and did not include any qualitative aspect. The sole research question and focus of this research was to only examine the academic and demographic

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88 predictors of high school dropout, which included Algebra I as one of the predictors ( Orihuela, 2006 ) There is a good amount of research that justifies the importance of algebra to a diploma wa s examined. Algebra is a core content course, taken by an increasing number of s tudents in their first year of high school the ninth grade. Unfortunately, failing a core course quickly puts a student behind in the number of core content course and overall number of credits required to earn a high school diploma. This is especially re levant in the school district under study, since most students take algebra in the ninth grade transition year. If students fail the course, even if they end up not taking algebra in subsequent semesters, they must still make up for the shortage of mathem atics core course credit and total number of credit required for high school graduation. There is certainly scant research about the relationship from the perspective of the students themselves. Several researchers have shared their research on insights f rom the ( Bridgeland et al., 2006 ; Dynarski & Gleason, 1998 ; Gewertz, 2006 ) Some research has also been on student perspectives of their high school mathematics and algebra experiences ( Hackett & Betz, 1989; Schornick, 2010) However, this research has not extended to the examin ation of the relationship between student experiences in high school algebra and subsequent

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89 Asse ssments of sources of self efficacy and expectancies offer the opportunity provide unique insight from the first person accounts of the students themselves In the student interviews, I hope to identify clusters of themes of sources of self efficacy and e xpectancies, either to confirm what has been derived from the literature or to identify fresh themes from accounts of their experiences in high school algebra, which, when multiplied by sources of self efficacy and expectancies from the grea ter high school experience, lead to the three major reported categories of behavior leading to high school dropout. provide the most efficient manner of synthesizing the themes gather ed from the literature review. This review represented how themes of student perceived self efficacy sources and expectancies lead to the potential to engage in the three categories of behavior typical of high school dropouts. This is a way of organizing data in a conceptual manner that ( Miles & Huberman, 1994 ) and were the themes initially sought by using phenomenology as the method of conducting the qualitative portion of the study Table II.2 is a visual matrix of the findings from the lit erature, as mapped onto the conceptual framework, and the relationship of the three main themes of dropout behaviors, which resulted from sources of self efficacy and expectancies in high school algebra class and the greater high school environment The s ources may be common or may be unique to the domains of the greater high school experience and high school algebra classes. The commonality or uniqueness of these sources will be explored during the qualitative analysis of the interview results. Disengag ement in the table is

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90 represented in three aspects: by its behavioral, affective, and cognitive dimensions. This representation of clusters of a priori themes identified in the literature review synthesis does not necessarily equate to literal mappings acr oss each row but offers possible examples of relationships Table II.2 A summary of the sources of self efficacy and expectancies developed from the review of the literature common to experiences in both high school algebra and high school experiences to examine in the qualitative analysis that can lead to the potential to engage in the three behaviors characteristic of the high school dropout. Behavior Mastery Experiences Vicarious Experiences Social Persuasion Affective States Expectancy BP dropout = ((ME + VE + SP + AS) algebra + E algebra ) ((ME + VE + SP + AS) high school + E high school ) DISENGAGEMENT Lack of interest in being part of the school/class environment (BEHAVORIAL) School activities show no relevance to real world Lack of positive attachments with school adults and/or other peers Misdirected influence of like peer groups Lack of clear feedback about individual accomplishments or help from teachers Boredom with class activities Different expectancies for rewards and value of educati on and having to fit into a academic environment Low expectations for performance Lack of liking school or feeling like they belong in school (AFFECTIVE) Procedural vice meaning activities Disconnectedness from adult role models Lack of relevancy Tasks are boring and uninteresting; Lack of perseverance and willingness to engage in effortful learning (COGNITIVE) Repeated failures, despite providing effort Lack of proximal goals Observing peer groups making little effort Lack of adult effort in convincing students to master needed skills Lack of caring: No one in position of responsibility takes an observable interest Frustration with academic abilities Disappointment with low expectations from adults Too much effort for too few rewards

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91 Table II.2 (cont.) Behavior Mastery Experiences Vicarious Experiences Social Persuasion Affective States Expectancy RULE NON COMPLIANCE Rudeness, tardiness, absences, and incomplete class and homework Severe consequences for non compliance, such as not doing homework Peer group influence Too much freedom Inability to challenge irreversible policies (i.e., homework, showing work with only one way of doing it) Inability to make up work Penalties SELF REGULATORY DEFICIENCY Little knowledge of unit culture (unit=high school or algebra class) Ill placed peer influence Lack of adult supervision Anxiety, confusion Teacher or school initiates help versus student requesting help Lack of support combined with low expectations

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92 CHAPTER III. METHODS The mixed methods research design for the study is a two staged, sequential analysis that is partially mixed and explanatory, with the quantitative part preceding the qualitative segment. Even though there are the same number of quantitative and qualitati ve research questions (two apiece), the principal and personal desire for the research was to understand some aspect of the nature of the hypothesized relationship between high school algebra performance and high school graduation, whether proven to be quantitatively significant or not after statistical investigation. This led to my deliberate choice to elevate the qualitative portion to dominant status within the research ( Creswell & Plano Clark, 2007 ; Creswell & Plano Clark, 2011 ; Leech & Onwueg buzie, 2007 ) The design is partially mixed since most of the quantitative and qualitative work was executed separately prior to mixing the results of each in the data analysis stage ( Leech & Onwuegbuzie, 2007 ) The combined quantitative and qualitative results we re then examined in tandem du ring the data analysis stage for the purpose of showing that results from only one technique of research yield ed an incomplete and misleading answer. This yield ed practical significance that enhance d the educational value of the results ( Onwuegbuzie & Leech, 2004 ) It is of little value to merely prove there wa s a statistically significant relationship between high school algebra performance and high school graduation. One need ed to elucidate and offer ins ight into the nature of this relationship to understand why it exists so that one may eventually use the results to

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93 counter with effective and pointed interventions. This wa s especially relevant given the onment. Accordingly, the rationale for using mixed methods research wa s significance enhancement, specifically to use qualitative data to expand on and enhance the results of the quantitative stage of the study. Although the results of inferential statist ics usage may be statistically significant, measurement techniques alone are insufficient to allow for insight into the human aspect of the interaction between high school algebra performance and high school completion ( Moustaskas, 1994 ) It is this unquantifiable, human aspect that begs for the use of qualitative techniques to truly highlight the consequenc es associated with the quantitative results. The mixed method research purposes of this study we re to: 1. elaborate and enhance the significant findings of the quantitative results, 2. clarify why a student who performs poorly in high school algebra may have a likelihood of failing to graduate high school on time, 3. shed new light on the nature of the interaction between student high school algebra performance and high school completion by understanding some of the factors causing this relationship, and 4. delve into first person student experiences that provide a compelling way in which to communicate and illustrate further significance to the relationship between student high school algebra performance and failure to graduate high school on time ( Collins et al., 2006 ; Onwuegbuzie & Leech, 2004 )

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94 Demographics and Context of the School District U nder Study The school district under study is a large urban schoo l district in the Rocky Mountain region of the United States. It is one of the largest in its state, with an enrollment of almost 30,000 students. The student population is about 53.2% White, 29.9%, Hispanic, 7.1% Black, 1.9% Asian, 0.9% American Indian, and 7.6% other. For the past five years, the Hispanic population has grown steadily, while most of the other free and reduced lunch status. There is a significant population of English language learners, although it is unknown what the exact percentage of the total student population this is The school district encompasses several distinct neighborhoods that range from middle class pockets to economically disadvan taged areas. The school district also includes a large military population and their school aged children, and several retirees with no household children. There are several traditional and alternative methods of obtaining a high school diploma in the school district. Besides the five traditional high schools, there are six alternative programs that offer a mixture of night, digital, and specialized instructional programs, all leading to the granting of a regular high school diploma. Despite this plet hora of pathways to a high school diploma, during recent years, the district continues to maintain disappointing and steady graduation statistics. The h igh s chool d ropout s ample c lassification There is inconsistency across the nation in the definition of how the high school dropout is calculated and reported. Many states use dissimilar formulas for defining their high school graduation statistics. The National Center for Education Statistics (NCES)

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95 uses four definitions, which in turn relies on two diffe rent data sources. Each of the four definitions yields its own respective graduation rates and totals ( Stillwell, 2010 ) The definition that guided the sampling choices for both the quantitative and qualitative designs within this research is the Average Freshmen Graduate Rate (AFGR). The AFGR is the percentage of high school students who graduate with a regular diploma on time, specified as four years after starting the ninth grade. This calculation uses a data source known as the Core of Common database and includes information for the confluence of all public elementary and secondary schools, school districts, and other educational administrative and operating units across the United States ( Tyler & Lofstrom, 2009 ) The primary reason why the AFGR was chosen for sampling decisions in this study versus another rate such as Status Completion Rate (the percentage of people who eventually earn a diploma at some point in their life), which is based on representative American household survey data is simple: The lo nger a student stays in high school and the older a student gets, the less probability that the student will eventually earn a high school diploma ( Chapman et al., 2010 ; Tyler & Lofstrom, 2009 ) A secondary reason why AFGR wa s the guideline for sampling within this study standard by which all of its schools and districts are reviewed with respect to accr editation judgments and other state educational policies. AFGR is of critical importance to the school district under study and, thus, was to my research as well. The United States has realized an average AFGR of 70% over the past couple of years ( Tyler & Lofstrom, 2009 ) However, the district under study has no t performed as

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96 well, averaging approximately 65% AFGR over the past couple of years. Table III.1 shows this AFGR and the demographical breakdown for the school district under study at the end of the 2010 academic school year. Table III.1 The demographica l AFGR breakdown for the school district under study at the end of the 2010 academic school year. Demographical Characteristic AFGR White 69.8% American Indian 56.5% Asian 64.7% Black 51.3% Hispanic 58.5% Female 68.3% Male 61.8% English Language Learner 52.7% Free and Reduced Lunch 56.0% Total District 64.7% Research Study Design The four research questions predisposed the quantitative and qualitative methods selected for use in the study ( Onwuegbuzie & Leech, 2006 ) a dictatorial approach to designing a research study ( Plano Clark & Badiee, 2010 ) The first stage objectives were to provide answers to the quantitative resea rch questions and to prove or disprove the three hypotheses associated with these research questions. Accordingly, this part featured quantitative sampling design, data collection, and data analysis techniques. The second stage objectives were to contrib ute enlightenment of and practical significance for the results provided in the quantitative stage, whether or not it wa s established to be statistically significant, while seeking to resolve the qualitative research questions. It was also to confirm the one qualitative hypothesis associated with the study. The second stage

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Page 97 of the study, thus, incorporate d qualitative sampling design, data collection, and data analysis techniques. Overall Research Sample Design and Data Collection Procedures The overall mi xed methods sampling design was a sequential design using parallel student sample sets ( Onwuegbuzie & Collins, 2007 ) The sample data sets were selected and analyzed at different times in the study, with the samples for the quantitative part collected and examined before the qualitative part. Consequently, the samples for each stage of the study were not the same. Students for the quantitative portion of the study started high school about an average of two to four years e arlier than those from the qualitative sample. Most of the students from the quantitative set of samples were unavailable, and it was deemed impractical to expend the effort to locate them. However, sampling error was minimized, since the population sour ce for both samples share d similar demographic and academic characteristics, to include attendance and performance in algebra courses in the same high schools. All university and school district research protocols were stringently followed during both stag es of collecting and analyzing data. Entry to the school district for the conduct of the research study and collection of data was facilitated through the district leadership to include the Deputy Superintendent and the Director of Research and Evaluation I was also assisted by the Director of Alternative Schools, who made contact with the principals at each school where student interviewees were identified. Further, guidance counselors at each school supported the research effort by personally introduc ing me and showing visible favor for the research to the candidate student interviewees.

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Page 98 Quantitative Methods Descriptive statistical techniques were used to provide exploratory information about the quantitative student sample. It was desired to underst and the nature of the sample data in terms of numbers of each in gender, ethnicity, socioeconomic status, disability, and native language or English language learner status. Descriptive statistical methods were also applied to identify how many students f irst took an algebra course in high school, particularly in the ninth grade transition year. Other demographics examined are attendance rates, English grades, and reading and writing state standardized tests scores, all for the same year in which a studen t first takes high school algebra. Finally, descriptive techniques are used to calculate the number of students who graduated and who failed to graduate. Recommendations suggested by Kinnear (2009) and Leech, Barrett, & Morgan (2008) were used to select t he appropriate statistical tests by which to explore the quantitative research questions and hypotheses ( Kinnear & Gray, 2009 ; Leech, Barrett, & Morgan, 2008 ) The particular statistical tests used, as mapped to each respective quantitative research question, are described below. school algebra course serve as a significant predictor of fa ilure to complete high school? The first research question is a basic, two variable associational research question with nominal, dichotomous in dependent (first high school algebra course performance) and dependent (high school completion) variables. The Chi Square and Phi statistics were used to test for the association and strength of association, respectively, between these two variables, since these statistics are especially suited for testing the presence and strength of an association between nominal variables when there is a large sample size

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Page 99 and the levels are mutually exclusive, e.g., pass or not pass algebra, graduate or not graduate high school ( Gay, Mills, & Airasian, 2006 ; Kinnear & Gray, 2009 ; Leech et al., 2008 ; Morgan, Leech, Gloeckner, & Barrett, 2007 ) Poor performance is represented by a failing algebra course grade since a student does not earn credit for a failing grade. One additional sub research question evolved from the review of the literature The literature review established that algebra should be examined in its own right as a predictor of hig h school graduation due to its relative importance within a successful high school education. Recall that algebra is increasingly a core course required for a diploma; and more students across the United States, and in the school district at the center of this study, are required to take this core course during the ninth grade transition year. Ninth grade is a transition year full of inherent difficulties; and if students get behind because of failing coursework, it endangers their ability to catch up and graduate on time ( Allensworth & Easton, 2005 ; Allensworth & Easton, 2007 ; Neild, 2009 ; Neild et al., 2008 ) Especially relevant to the school district under study are budgetary constraints that may force elimination of programs designed to counter the negative effects of the ninth grade transition year on future high school success. I was, thus, curious to establish whether or not the grade level in which algebra is taken was a significant factor in student high school graduation. The additional sub research question is: Does the high school grade in which algebra is first taken serve as a predictor of high school graduation success?

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Page 100 This question was also examined using the Chi Square statistic and its strength of the Chi Square test has more than two lev els of measurement. The grade in which a student first takes algebra has five levels of measurement ( Kinnear & Gray, 2009 ; Leech, et al., 2008 ) Research Question Two: Wil l poor performance in high school algebra be more likely than the other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English grade, English language learners, ethnicity, sex, and special educa tion status to lead to subsequent failure to complete high school? This research question involves several independent variables and one dependent variable and, thus, calls for a complex associational statistic test. Logistic regression is well suited for conducting data analysis when the hypothesized model composed of predictor causes can lead to only two states of a dependent variable and when these predictors are a combination of nominal and scale variables ( Wright, 2008 ) Logistic regression basically asks the questions: What is the affect of X in terms of the probabilities on Y, and what is the change in odds in the occurrence of Y for a unit change in X? Logistic regression calculates these answers in a model of one or more different Xs, which may or may not be significant in predicting Y (Pampel, 2000). While discriminant analysis could also be used to predict a relationship between s everal predictor variables and a dichotomous outcome variable, the use of logistic regression has increasingly become the statistical tool of choice in the field of social sciences. Logistic regression has fewer assumptions than discriminant analysis and is, thus, a preference when one wants to estimate the probability of an event occurring, especially in the health and social services (Peng, So, Stage, & St. John, 2002; Wright, 2008).

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Page 101 In particular, there have been several instances of using logistic reg ression in studies concerning higher and secondary education. It is especially useful for matters involving retention, since many have found the school retention issue to be a complex one comprised of many potential concerns (Battin Pearson et al., 2000; Peng et al., 2002). Zhang, Anderson, Ohland, Carter, and Thorndyke (2008) used logistic regression in one representative study to determine predictors of student retention and graduation in a college engineering program (Zhang, Anderson, Ohland, Carter, & Thorndyke, 2008). Predictor variables, in this case, were pre existing demographic and other academic characteristics; and the end state was graduating or not from a college engineering study. I have similarly taken an equivalent path in the approach to constructing a model of predictors of high school graduation success in the quantitative aspect of this study. I used many pre existing demographical and other concurrent academic characteristics to identify factors that may have an impact on success. In this case, was with respect to high school graduation, with a special interest in the role of algebra. Logistic regression is also well suited for disclosing the relative significance of predictor variables to the dependent variable in a model and w ould s how if individual predictors are significant individually or collectively in a model predicting high school graduation. A correlation matrix was executed in conjunction with the logistic regression test, between the predictor variables used in the logist ic regression high school graduation model in order to determine which were significantly correlated to each other. Even though it wa s not the intent within th e study to examine why other predictor variables may have be en significant, I wanted to determin e if bivariate correlations existed between

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Page 102 the independent variables in the predictor model of graduation that may have raised concerns for the results of the logistic regression tests. I also desired to leave the door open for future study of other corr elations as per relationships to graduation. The use of both logistic regression and a correlation matrix shows which predictor variables may be significant, either by themselves or in combination with others. This would provide insight for what follow o n work and analysis is needed in order to study the high school graduation success rate within the school district under study. Effect size and power. The statistical tests in the quantitative study considered two essential aspects of quantitative analys is, effect size and power. Effect size allow s for measurement of the strength of a relationship, and power indicate s the probability that the relationship truly exists. While effect size is an indicator of practical significance in statistical findings, large effect sizes do not necessarily mean that the statistical findings are an estimation of practical significance. This is a judgment call on the part of the researcher ( Leech et al., 2008 ) For this study, even a small effect size for the Chi Square and logistic regression t ests was deemed worthy of attention and examination, since to ignore any suggestion of a relationship between high school algebra and high school graduation yields the possibility of missing an opportunity to identify and help more students who are in dang er of not completing high school. I argue that the second aspect, statistical power, held even more weight for why this research was needed. The sample size used was much larger than recommended for adequate power in order to ensure presence of sufficient statistical power for both the Chi Square and logistic regression tests. In addition the logistic regression test is a

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Page 103 parametric test a characteristic that also increases power (Leech, Barrett, & Morgan, 2008). An increase in statistical power minimize s the chances of a Type II error in the test results. A Type II error means that we accept the null hypothesis when the alternative hypothesis is true. There is also some concern for a Type I error, which equates to rejecting the null hypothesis when it is true. The level of significance set in the statistical tests minimize d the possibility for a Type I error. I was more concerned about preventing a Type II error, which is minimized with sufficient statistical po wer, since it is my belief that increased student harm would be realized with a Type II error. In this study, if a Type I error wa s made, it is interpreted that the independent variables (high school algebra performance, high school algebra grade level, a nd the total high school graduation predictor mode) have an effect on high school graduation success (the alternate hypothesis), when in actuality it does not (the null hypothesis). Thus, we will have rejected the null hypothesis, when we should have acce pted it. If one acts upon the results of research in the incidence of a Type I error, one might put in place interventions that are not needed and, thus, expend scarce resources in an inefficient manner. On the other hand, it is stoutly argued that realiz ing a Type II is a more severe consequence or accepting the null hypothesis that algebra performance either alone or in a model of predictors does not have an effect when it actually does. This is a weightier error in terms of human cost, since it can le ad us to discount an important pointer of eventual failure to graduate. If this occurs, w e may not only miss the opportunity to help more students achieve high school graduation but also focus scarce

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Page 104 resources and attention on predictors that may not be a ny more effective in preventing high school failure than algebra performance outcomes. Choosing to pay attention to a that wa s personally motivated, from my viewp oint as the mother of a high school dropout who might have been helped by this research effort into a previously ignored but possible predictor of failure to graduate high school. Quantitative sample and data collection. The quantitative data collection was comprised of a cohort of students who started high school in 2004 and who did graduate, or would have graduated, on time four years later in 2008. I identified the students who either dropped out or who did not graduat e on time in one of two ways. The first was through the district provided withdrawal codes indicating who voluntarily dropped out of school. The second was through a lack of a hdrawal and as a graduate within this study Only one cohort data set was used for two reasons. The f irst and most important reason is that the time and effort expended by school district personnel in gathering this cohort sample data was considerable. It was desired not to place an excessive burden on the school district with the request for data. Seco ndly, since the predominance importance was placed on the qualitative aspect of the research, it was felt that one cohort would provide enough data by which to answer the quantitative research questions and hypotheses.

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Page 105 The design for quantitative sampling was purposeful, featuring a combination of criterion and convenience sampling schemas. The sampling was criterion because criteria were established in using a cohort of students who started in the school district in the year 2004 and continued through 200 8 Students who transferred into district high schools during the cohort boundary years were not included in the analysis in order to minimize the influence of possible extraneous variables, such as the influence of other district educational practices. Th e sampling was convenient because of the characteristics of the data that was provided. The district tracks an immense quantity of student performance and demographical data, partially due to the need to fulfill requirements of federal and state complianc e regulations and chiefly due to the desire to use such categorical data when analyzing school and district improvement efforts. This allowed for convenience in requesting the type of data desired for the quantitative portion of the study. This data was conveniently available, although the school district did expend considerable effort in packaging the data for my research. The data was provided in an Excel format to provide the ability to sort the data as needed and to permit the data to be formatted fo r entry into the Statistical Package for the Social Sciences (SPSS) software. Individual student privacy was maintained through the use of dummy student identifiers. Quantitative variables The choice of predictor variables for the quantitative portion of the study was driven by three factors. The first factor was my personal interest and desire for answers to the two original and one additional quantitative research question as it particularly relates to the school district under study.

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Page 106 The second crit erion, as pertaining to the logistic regression model, was the desire to stay synergistic with the school district at the focus of this study. Many of the independent or predictor variables are attribute variables deliberately chosen for inclusion because they we re the same variables used when disaggregating data for accreditation and assessment purposes and for state and federal reporting requirements. These variables are cognizant of a preference to have statistical results that are congruous with langu age similar to academic performance data. The third criterion was the findings from the literature review. The literature on factors leading to failure to graduate high school justifies using a model with several pr edictor variables of high school failure. As reported previously, there are multiple avenues to study when analyzing which pre existing factors are significantly related to high school dropout, to include socioeconomic status, lack of attendance, course c ompletion progression, freshman grades, ethnicity, and gender ( Allensworth & Easton, 2005 ; Alspaugh, 1998 ; Orfield, 2006 ) algebra, it wa s practical to use them in a statistical model of gra duation success. Additionally, as reported, I did locate one study in which many of these predictor variables were used. Orihuela (2006) used in her logistic regression based study a model of many of the same variables that I choose to use. These variab les look into the predictors of whether students will drop out or not. I wanted to determine if the same trends are repeated with data and similar variables in this school district. The predictor variables she used were (a) Algebra I grade, (b) Florida C omprehensive Assessment Test

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Page 107 level, (c) language proficiency, (d) gender, (e) race/ethnicity, (f) Exceptional Student Education program membership, and (g) socioeconomic status. Given these factors, using Chi Square and Phi to examine the relationship betw quantitative research question are the independent dichotomous variable algebra grade (passing or failing) and the dependent variable high school graduation. The additional sub re search question, using Chi independent nominal v ariable grade in which the first algebra course is taken and the dependent variable high school graduation The demographical, independent variables for the second quantita tive research question are gender, ethnicity, English language learners (non English speaking students), free and reduced lunch students (indicates socioeconomic status), special education status, attendance rate in the same year in which the first high sc hool algebra course is taken, and the grade level in which the first high school algebra course is taken. The other independent, academic performance variables are the grade received (pass or fail) in the first high school algebra course and the following factors during the same year in which the first high school algebra course is taken: (a) English grade, (b) proficiency in the state math assessment test, (c) proficiency in the state reading assessment test, and (d) proficiency in the state writing asses sment. As stated previously, proficiency in the state standardized test is a score that is judged to be advanced or proficient. Not proficient is scored as partially proficient or unsatisfactory. Levels of measurement of the independent variables were st ructured to be scale or dichotomous in order to be able to make use of the logistic regression statistic. Since

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Page 108 ethnicity could be one of five values, four separate dummy variables were created to delineate the ethnicity of the students one less than the number of categories which is recommended in logistic regression ( DeMaris, 1995 ; Pampel, 2000 ; Wright, 2008 ) The ethnicity variable delineating Caucasian the largest in number, was t hus omitted from the calculations. The dependent variable is high school graduation success, which is defined as receipt of a regular high school diploma. Table III.2 lists the independent and dependent variables used in the statistical tests along with the levels of measurement of each. Table III.2 Independent and dependent variables and their corresponding levels of measurement. Name Level of Measurement Values INDEPENDENT VARIABLES Gender Dichotomous Female, Male Indian Dichotomous Indian, Not Indian Asian Dichotomous Asian, Not Asian Black Dichotomous Black, Not Black Hispanic Dichotomous Hispanic, Not Hispanic Free and Reduced Lunch Dichotomous Yes (Free or Reduced), No Attendance Scale 0 100% Special Education Dichotomous Yes, No CSAP Algebra Dichotomous Proficient, Not Proficient Grade in which first algebra course is taken Nominal 8, 9, 10, 11, 12 Grade in first algebra course Dichotomous Pass ( D or above), Fail English grade Dichotomous Pass ( D or above), Fail CSAP Reading Dichotomous Proficient, Not Proficient CSAP Writing Dichotomous Proficient, Not Proficient DEPENDENT VARIABLE High School Graduation Dichotomous Yes, No Missing data. The total cohort size for students in the years 2004 to 2008 was 1,757. Missing data was excluded from analysis in the Chi Square and logistic regression statistical tests. Pertaining to the logistic regression test, d ata was missing for 492 students, or about 28%

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Page 109 of the original data set, leaving a sample size of 1,305 students with complete data within the logistic regression model of predictors. It was felt that the remaining data, or 72%, was of sufficient size by which to execute the test and retain adequate statistical power Looking at the missing data for the Chi Square test for association of the grade in which algebra was first taken with respect to graduation, there was data missing for 162, or approximately 9%, of the students who did not have information about the grade in which they first took algebra. This could repres ent either missing data or the fact that some of these students did not take algebra at all. When looking at the overall affect on graduation of either passing or failing algebra, a total of 366, or approximately 21%, of the 1,757 total students had missi ng algebra grade data. As before, this could represent either missing data or the fact that some of these students did not take algebra at all in high school. Again, due to the large sample size, it was judged that, and the results did confirm this, the missing data would not affect the results and power of either Chi Square test. Qualitative Methods tradition of phenomenology ( Moustaskas, 1994 ) and it was based on the desire to find meaning in the phenomenon of student experiences in high school algebra and its potential conse quence on and association with high school graduation.

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Page 110 Research Questions Three and Four: How do student held expectancies and perceived reinforcements from their experience in first high school algebra class lead to the potential of the student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school? How efficacy beliefs class, influence the potential of a student to enga ge in behaviors that are symptomatic of a student who will probably fail to finish high school? Qualitative data design and collection techniques, in an effort to identify student perceived sources of self efficacy and expectancies, as per the conceptual f ramework, followed the recommendations of Creswell about conducting a phenomenological study Creswell recommends that phenomenological data be gathered by holding in depth interviews of ten subjects. He states that ten interviews are sufficient for find ing meanings and themes from those subjects that have experienced the phenomenon under study ( Creswell, 1998 ; Creswell & Maietta, 2002 ) Although I followed the guidelines of Creswell with respect to recommendations for phenomenological data collection, the actual analysis of the qualitative data was consistent with the teachings of Moustakas (1994) concerning how to best identify meaning from the phenomenon under investigation I chose the tradition of phenomenology, since it is a metho d that is ideal for person experiences in a particular environment. I felt that it would allow me to spotlighting the presence of student perceived sources of self chool algebra class and greater high school environment and permit me to determine congruen cy of the proposed conceptual framework that dr o v e the qualitative portion of the study. ( Moustaskas, 1994 )

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Page 111 Qualitative sample and data collection. Similarly to the quantitative sampling procedures, the qualitative sampling was purposeful with criterion and convenienc e characteristics. It was criterion because I requested interviewees that meet certain standards (failure in first high school algebra course and high school graduation dropout or still in high school after four years). Additionally, the use of purposefu l qualitative sampling was necessary to ensure the accessibility to those interviewees with personal knowledge of the qualitative research questions ( Rubin & Rubin, 2005 ) The sampling was convenient, since it was drawn from a population of mature students who were readily available in the Another aspect of the convenience sampling technique was the request to identify and interview students who were all over the age of eighteen. This was done to avoid the need to obtain parental permission f or student participation, although I was granted permission by both the Institute Research Board (IRB) and the school district to use minors with proper parental permissions. Additionally, based on my literature review of publications that discussed the f irst person experiences of high school dropouts, I believed that students, age 18 and over, would provide more mature and honest answers to my interview questions ( Azzam, 2007 ; Bri dgeland et al., 2006 ) The qualitative sample consisted of 11 students who were enrolled in four different alternative high school programs and who were all working toward earning a regular high school diploma. Five of the students had previously dropp ed out of school but have since returned to school. Five of the students have not dropped out but have been in school for five or more years. There was one student, in the fourth year of high school, who was attending both a regular school and an alterna tive school in an attempt to

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Page 112 make up credits. This student will undoubtedly need a fifth year of school in order to earn a high school diploma. The alternative schools where the students are attending were a mixture of a digital high school, a night scho ol, and two non traditional high The demographical and academic characteristics of the inter viewed students were diverse. Table III.3 summarizes these demographical and academic characteristics. I was unable to obtain socioeconomic or special education status for the students. All of the students were required to take four core courses (mathem atics, English, science, social studies) plus an elective or physical education in their freshmen year; and all have the same number of core course and total course credit requirements for graduation. Selection of the student interviewees was facilitated w Principal and guidance counselors. An I RB approved script and consent form that explained the purpose of the study and associated risks and which requested permission to conduct the interviews was reviewed with each interviewee. Interviewees were given the opportunity to withdraw their participation at any time. I asked for, and was granted, permission from each student to digitally record the interviews. This allowed me to pay full attention, ac tively listen, and watch the body language and facial expressions of the students. I also used the digital recorder to record my impression of each interview immediately following the conclusion of each interview. Interviewees were assured that they woul d remain anonymous and were assigned letter identifiers (A through K) for use in later discussion about the findings. I also received permission from each participant to review their academic transcripts.

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Page 113 I began each interview with an effort to establish rapport and empathy with the student. I started with my background as a parent of a student who dropped out, failed his first algebra course, and finished high school in an alternative program. I discussed that I sincerely wanted to help their fellow st udents have an easier path to high school graduation by uncovering their personal thoughts about high school algebra and high school experiences. I told them I would be using this information to work with the school district to make improvements to high s chool education. This opening approach to the interview appeared to put the students at ease in my presence; and as a result, they all seemed to be open and honest with me. Table III.3 Demographical and academic characteristics of the student interviewee s. Demographical Characteristics Academic Characteristics White 6 Dropped out completely and returned to school 5 Black 4 Still in school, in year 4, but will need a fifth year 1 Hispanic 1 Still in school, in year 5, with 2 of the 3 to most likely need year 6 3 Male 10 Still in school, in year 6 2 Female 1 Failed English and first algebra course 5 Age: 18 5 Failed one course, algebra, first semester, grade 9 4 Age: 19 4 Failed two courses, first semester, grade 9 2 Age: 20 2 Failed three core courses, first semester, grade 9 4 Took algebra for the first time in ninth grade 11 Failed four core courses, first semester, grade 9 1

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Page 114 Qualitative measures. The measures I constructed to use during the analysis of the interview data were designed to c apture student perceptions of their experiences in high school algebra classes and in their greater high school experience in terms of sources of self efficacy and expectancies. The three measures are the student perceived sources of self efficacy and exp ectancies within high school algebra and the greater high school experience and behaviors characteristic of the high school dropout. These were identified by asking the students to discuss their experiences in, feelings toward, and opinions about both high school algebra and the greater high school environment. In addition, they were asked to compare their experiences in and feelings from algebra classes with other courses. They were also asked to contrast their experiences in and feelings from attend ance in a traditional high school program with that of their current participation in an alternative program. Finally, the students were bluntly asked to express their honest opinions about a relationship between their algebra experiences and their decisi on to either dropout of school or their inability to finish high school after four years. It was anticipated that these measures, plus the communication of honest opinions, would allow me to use the interview transcripts to derive student perceptions about sources of self efficacy beliefs and expectancies in high school algebra and the high school environment. This would then demonstrate that there was an association with one, two, or all three of the themes of behavior characteristic of dropouts. In addi tion, by recording the interviews, I was freed up to look for voice tones facial expressions, and body language that would also convey meaning and essence. I have had several years of training as an interviewer for various U.S. Navy responsibilities, whi ch

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Page 115 assisted me in my attempt to look for verbal and non verbal cues of important student responses The interview questions which were used are attached as the appendix. Qualitative analysis. The qualitative analysis of interview data was conducted using the recommendations of Moustakas (1994) on how to identify significance from the interview process. The qualitative analysis involved reviewing the transcripts at least twice to derive the sources of self efficacy and expectancies. I first analyzed the i nterview data to search for themes of sources of self efficacy in both high school algebra and the greater high school environments, using a qualitative analysis technique known ( Onwuegbuzie & Leech, 2007 ) Constant comparative analysis is typically characteristic of a grounded theory study and is a method referred to as coding of the data. It is a systematic way by which to identify distinct meaningful statements from i nterview data. Although traditionally used in grounded theory, it was well suited for the purpose of providing meaning by allowing me to look for chunks of data within each interview and then coding these chunks into descriptive terms or themes for the di fferent sources of self efficacy. in expectancies from experiences in high school algebra and the greater high school environment. This technique is one that is typically used t o focus in on words that are used before and after a keyword ( Onwuegbuzie & Leech, 2007 ) In this case, the keywords wer e the sources of self efficacy that are developed from the student interviews by usi ng constant comparative analysis.

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Page 116 Validity of qualitative analysis. The validity of my analysis of the qualitative interviews was verified through member checking. One of my colleagues from the Northern Rocky Mountain Education Research Association assist ed me by reading through the student interview transcripts and confirming that we both reached similar conclusions concerning the themes of student perceived self efficacy sources and expectancies. This was an especially important step in this study, to h find a predictor of high school dropout that was not tied to race, ethnicity, economic status, or my failure as a parent. Mixed Methods Data Analysis The mixed methods analysis that follow ed the collecti on and analysis of the quantitative and qualitative stages relied on two parts of a seven stage mixed methods data analysis model advocated by Onwuegbuzie and Teddlie ( Onwuegbuzie & Teddlie, 2003 ) These we re data comparison and data integration. Data comparison allows for a comparison of the data from the quantitative analysis against the data from the qualitativ e analysis. Data integration seeks to view the data in its totality, searching for the significance of the combined sets of data, thereby permitting a holistic view of the depth and breadth of the relationship between high school algebra and high school g raduation. Limitations There are natural limitations with using self reported interview data. Although the conclusions reached through analysis of the qualitative data (as reinforced with member checking) can provide us themes, we cannot measure traits s uch as student perceptions of self efficacy and expectancies with absolute certainty ( Eckstein & Wolpin, 1999 ) I also chose to not use an established and reliable se lf efficacy questionnaire that

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Page 117 might have provided a better measurement of sources of self efficacy and its relationship to high school algebra or high school graduation outcomes. Instead, I desired to obtain a first person account of student perceived so urces of self efficacy that would yield meanings and essence not obtainable from answers on a questionnaire. The qualitative sample used for the study may not be reflective of the entire body of dropouts. All of the students interviewed for this study wer e engaged in an ongoing effort to finish high school. As a result, they may have a somewhat different perspective about their experiences than others who are completely divorced from an educational environment.

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Page 118 CHAPTER IV. RESULTS The quantitative resul ts revealed that passing the first high school algebra course is significantly correlated which a student initially takes algebra is also significantly correlated to high school graduation. Finall y, algebra performance is significant within a model of predictors of graduation; and failing algebra is more likely to lead to failing to graduate high school than any other predictor variable in the model, except for the English grade a student receives in the same year of their first algebra course. The qualitative results from the student interviews told a story of several sources of self efficacy beliefs and expectancies from their experiences in high school algebra classes and the greater high school environment. The students did not state that their experiences in high school algebra were a direct cause for their choices to drop out of school or to not graduate in time. This makes it difficult to attribute student experiences in algebra as the overw helming reason for failure to graduate high school However, the results from the interviews show that student perceived sources of self efficacy and expectancies can lead to the potential to engage in behaviors that are characteristic of high school drop outs Therefore with respect to this student sample experience in algebra wa s a factor to be considered when examining the causes of high school failure. Lastly the value of mixed methods research wa s confirmed, since the quantitative results we re not necessarily consistent with the qualitative integrated results. To rely on one source over the other can lead to faulty findings and misleading conclusions. The

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Page 119 data samples analyzed in this study provi ded integrated results that not only disclosed the performance and high school graduation but expanded upon and enhanced this knowledge with some reasons as why this relat ionship may exist. Quantitative Results Descriptive statistics on the quantitative data set of students in the cohort for years 2004 to 2008 revealed several items of interest. Of the 1,797 students who started in the school district in 2004, 405 student s, or just over 22%, had not yet graduated after the standard four years, which would have been by the end of the 2008 school year. Although this is much different than the reported district AFGR of 65% over the past couple of years, I attributed this dis crepancy to three characteristics of my sampling data: (a) I did not use any students who transferred into the district during these four years, (b) I eliminated missing data from my statistical models, and (c) I counted those students the district did not have as a formal withdrawal as students who just stopped going to school and were, thus, non graduates. The district has a large ambulatory population of students who transfer into and out of the district during any one school year and had no way of trac Descriptive statistics confirmed that more males failed to graduate than females. Of the 405 students who failed to graduate, 232 (57%) were males; and 173 (43%) were females. More students took algebra for the first tim e in the ninth grade than in any other grade, a total of 53%, or 866, of the 1,635 students for which there was data provided. Table IV.1 represents this breakdown by the grade in which algebra was taken for the

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Page 120 first time. The missing data also includes a small percentage of students who never took algebra at all, since this is not a school district required course. Table IV.1 Grade in which a student first takes algebra for the 2004 to 2008 student cohort. Grade of First Algebra Course Number Percent 8 424 25.9 9 866 53.0 10 240 14.7 11 93 5.7 12 12 .7 Total 1,635 91.0 Missing Data 162 9.0 Total 1,797 100.0 Seven percent of the students for which course grades were available failed their first high school algebra courses 101 of 1,330 students. Twice as many students also received a failing grade in English during the same year in which they took their first algebra cours e 220 of 1,521 students. There were many more missing grades in a first algebra course (366) than there were missing English grades (56) in the same year that algebra was first taken. Therefore, the fact that more students failed English in the same year in which they failed their first algebra course should be considered with caution. Table IV.2 shows the numbers of students in who received failing grades in their first algebra course and their English course in this same year.

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Page 121 Table IV.2 Numbers of a lgebra pass/fail and English pass/fail students in the same year as their first algebra course for the 2004 to 2008 student cohort. Pass/Fail First Algebra Course Number Percent Pass/Fail English in same year as First Algebra Course Number Percent D and above, Pass 1,330 92.9 D and above, Pass 1,521 87.4 Below D, Fail 101 7.1 Below D, Fail 220 12.6 Total 1,431 100.0 Total 1,741 100.0 Missing Data 366 Missing Data 56 Missing Data Total 1,797 Total 1,797 Total Other demographical data for the 2004 to 2008 student cohort is summarized in Table IV.3. Table IV.3 Descriptive statistics for the student cohort of school years 2004 to 2008. Variable No Variable No Variable No Gender: Male Female 881 916 Special Education: No Yes 1,689 108 Graduate: No Yes Missing 405 1,392 56 Ethnicity: Indian Asian Black Hispanic White 22 37 151 305 1,282 Free & Reduced Lunch: No Yes Missing 1,233 223 341 Grade in 1 st Algebra Course: Fail Pass Missing 101 1,330 366 Grade in English Course: Fail Pass Missing 220 1,521 56 Graduates by gender: Male Females 649 743 Non Graduates by gender: Male Females 232 173

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Page 122 Does poor performance in the first high school algebra course serve as a significant predictor of subsequent failure to complete high school? algebra course serves as a significant predictor of subsequent failure to complete high school, a Chi Square statistic was used. Assumptions were checked and met. Table IV.4 shows the Pearson Chi high school algebra course does significantly predict if a studen t will graduate high 2 = 223.04, df = 1, N = 1431, p < .001). Students who fail algebra are less likely to successfully graduate high school than students who pass algebra. Phi, which indicates the strength of the association between the t wo variables, is .395; and thus, the effect size is considered to be medium to large, according to Cohen ( Leech et al., 2008 ) Table IV.4 Chi Square analysis of high school algebra performance association to high school graduation. Graduate Variable n Graduate Not Graduate Performance in First Algebra Course 223.04 <.001 D and above, Pass 1,191 1,161 30 Below D, Fail 240 169 71 Total 1,431 1, 330 101 Does the high school grade in which algebra is first taken serve as a predictor of high school graduation success? To investigate whether or not the high school grade in which algebra is first taken serves as a predictor of high school graduation success, a Chi Square statistic was used. Assumptions were met and checked. Table IV.5 shows the Pears on Chi Square results and indicates that the grade in which algebra is first taken is significantly associated with 2 = 73.32, df = 4, N = 1635, p < .001). Students who

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Page 123 took algebra for the first time in the ninth grade wer e more likely to fail to graduate high school than students who took algebra in grades 8, 10, 11, or 12. Students who took algebra for the first time in the tenth grade were more likely to fail than students who took algebra in grades 8, 11, and 12. Stud ents who took algebra for the first time in the twelfth grade were least likely to fail to graduate high school, with the next least which indicates the strength of th e association in 5 x 2 cross tabs, is .212, which is considered to be a small to medium effect size ( Leech et al., 2008 ) Table IV.5 Chi Square analysis of the high school grade in which algebra is first taken and its association to high school graduation. Graduate Variable n Graduate Not Graduate Grade in Which Algebra is First Taken 73.32 <.001 8 th grade 424 391 33 9 th grade 866 666 200 10 th grade 240 174 66 11 th grade 93 59 34 12 th grade 12 6 6 Total 1,635 1,296 339 Is poor performance in high school algebra more likely than the other predictor variables of reading comprehension ability, writing performance ability, low socioeconomic status, English language learners, ethnicity, sex, and special education status to lead to subsequent failu re to complete high school? Logistic regression was conducted in order to assess if the predictor variables of grades in first high school algebra course, gender, ethnicity, socioeconomic status, and English grade during the year of first high school algeb ra course significantly predicted high school graduation or not. The assumptions of observations being independent and categories that are mutually exclusive and collectively exhaustive, were checked and met;

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Page 124 and the model contains all relevant and no irr elevant predictors ( Wright, 2008 ) When all of the predictor variables were considered together, they significantly predict whether or not a 2 = 32.59, df = 11, N = 1305, p < .001. Table IV.6 presents the results of significance and the odds ratios, which suggest the odds of graduating high school are increasingly greater as one passes their first high school algebra course and also passes their English course in the same year as taking their first algebra course. Additionally, being female and having a better attendance rate during the same year as taking a first high school algebra course also increas es the odds of graduating high school. Finally, the negative association and less than one ratio between free and reduced lunch shows that being on free and reduced lunch decreases the odds of graduating high school. The Cox and Snell effect size, an eff ect size typically used in education (personal communication, 2009), was approximately .31, which is a medium or typical effect size (Leech et al., 2008). Multicollinearity was checked in two ways. A linear regression model was used, which showed that thr ee of the intended predictor variables violated multicollinearity. These three predictor variables were the three ninth grade state standardized tests (writing, English, and math). Because they violated the multicollinearity assumption, these variables were not included in the logistic regression model. This also minimized a concern about the reliability and validity linked to the large errors in measurement and sampling of the state standardized tests (personal communication, 2009) Additionally, the state is in the process of changing the format and content of the state standardized assessment tests. Therefore, these predictor variables may reflect student outcome in a

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Page 125 different manner in the future, thereby minimizing the use of a current model of g raduation with the variables included. during the same year of a failing first high school algebra course grade was almost twice as likely to lead to high school graduation failure, I ran a correlation matrix between all of the predictor variables as another method of determining multicollinearity. As expected, the and the grade in English during the s ame year. The correlation was about .35. Although .35 is generally associated with two variables accounting for more than 10% of shared variance between these variables, Leech, Barrett, and Morgan (2008) do not recommend combining or eliminating such var iables, unless the correlation is at .50 to .60 or above (Leech et al., 2008). Important for statistical power considerations, all of the predictor, independent variables had a sample size greater than 20. Logistic regression requires a large sample size. According to Leech, Barrett, & Morgan (2008), use of the logistic regression statistic test requires a large sample size of a minimum 20 cases per predictor, for a minimum total of 60 cases. According to Wright, a minimum of 50 cases per predictor is ad vised ( Wright, 2008 ) As evident in Table IV.1 through Table IV.3, there was a more than adequate sample size, including accounting for any missing data, for all of the predictor variables and for the total number of cases.

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Page 126 Table IV.6 Logistic regression predicting graduation success from the combination of independent variables. Variable B SE Odds Ratio p Gender .46 .21 1.59 .031 Indian .73 .84 1.07 .930 Asian .34 .52 1.40 .516 Black .07 .12 .93 .556 Hispanic .05 .08 1.05 .540 Free and Reduced Lunch .70 .28 .50 .011 Attendance .05 .02 1.04 .012 Special Education .42 .40 .66 .284 Grade of First Algebra Course .01 .12 1.01 .921 Grade in First Algebra Course 1.50 .38 4.50 .000 English Grade 2.14 .27 8.49 .000 All of the statistical tests showed good power, due to the large sample sizes, hyp othesis that algebra has an effect on a student high school graduation, either by itself or in a model of predictors. The avoidance of a Type II error also showed that the grade in which high school algebra is taken is important. Two of the three quantita tive hypotheses were proven to be true. Performance in compared to other f actors in a model that includes several predictor variables of high school graduation. Hypothesis two was proven partially true: Ethnicity is not a class to high scho ol graduation. The hypothesis that socioeconomic status and gender

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Page 127 algebra class to high school graduation is false. Qualitative Results Analysis of the student int erview data did not allow me to pinpoint the experience out of or for attending high school for an extended period of time in order to earn a diploma. Only one of the 11 students interviewed actually attributed experience in algebra as the motive for dropping out of school. However, analysis of the student interviews did uncover in this study, several student perceived sources of self efficacy and expectancies in the first high school algebra course that, when multiplied with sources of self efficacy and expectancies in the greater high school experience, led to the potential to engage in all three groups of behaviors typical of the high school dropout. In particul ar yielded negative perceptions of all four sources of self efficacy: mastery experiences, vicarious experiences, social persuasion, and affective states. The impact of social persuasion sources and affective states sources, however, are more severe and prevalent within this qualitative sample analysis, which will be explained further and with some collective experience in high school reflects an abundance of self efficacy sources of social persuasion and affective states that negatively impact the greater high school experience. The results, thus, have prompted me to modify the originally proposed conceptual framework, from

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Page 128 BP dropout = ((ME + VE + SP + AS) algebra + E algebra ) ((ME + VE + SP + AS) high school + E high school ) t o BP dropout = (( ME + VE + SP + AS) algebra + E algebra ) ((SP + AS) high school + E high school ) The ME and VE are shown in sub script to depict that, although it is present in student perceptions of their algebra experiences, it is not as heavily weighted as the sources of social persuasion and affective states. The most prevalent representative behavior was that of disengagement from both the first high school algebra class and the greater high school experience, which led to remaining in school beyond the normal four years it should take to earn a diploma or to dropping out of school altogether. Once more I caution the reader th at the above formula is a representation of a proposed framework and is not to be considered to be a literal mathematical equation. One interesting note concerning the qualitative results: I was provided an interview sample that consisted almost entirely of males. I expected to have a sample of more males, since dropout rates in the United States are higher for males than females ( Archambault, Janosz, Fallu, et al., 2009 ; Buchmann, 2009 ) In addition, with respect to mathematical performance, unlike earlier times in the United States, females are now performing as well as males, as measured by results on standardized math tests to include tes ts that measure skills such as complex problem solving, which is part of the high school algebra curriculum ( Buchmann, 2009 ; Hyde & Mertz, 2009 ) The logistic regression results also showed that females were more likely to graduate than males in my predictor model of graduation. Even given the research and logistic regression results, I did not anticipate such a high concentration of males in the

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Page 129 i nterview sample. In discussing this with one of the high school counselors, she had mentioned that in her experience, most of the females who had dropped out or were continuing on past four years did so primarily due to pregnancy and were still under the age of 18 Most of the female students in the alternative high school programs were also under 18, a condition that I deliberately choose to avoid in the convenient sampling choice. They were also more inclined to pass algebra, stay motivated, and contin ue to aggressively pursue completion than their male counterparts. How do student held expectancies and perceived reinforcements from their experience in first high school algebra class lead to the potential of the student to engage in behaviors that are s ymptomatic of a student who will probably fail to finish high school? How efficacy beliefs experience in his/her first high school algebra class, influence the potential of a student to engage in behaviors that are symptomatic of a student who will probably fail to finish high school? The findings in answer to the two qualitative research questions are presented first and then for t he greater high school experience. Major themes of identified sources of self efficacy and expectancies in algebra and high school are discussed and then followed up with identified themes of the behaviors. A matrix depicting a more expansive summary of findings is provided at the conclusion of this chapter. This matrix also depicts relationships between student perceived sources of self efficacy and expectancies in a first high school algebra course and the greater high school experience and the potenti al to engage in the three themes of behaviors representative of high school dropout. One note for the reader: references to students are always referred to in the masculine tense to ensure that the one female student in the interview populace remains anony mous.

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Page 130 Perceived sources of self school algebra class. Analysis of the student interview transcripts revealed negative consequences in the form of behaviors representative of the high school dropout as a result of all four sources of self efficacy from participation in their first high school algebra class. I have themed these sources of self efficacy as: (a) misdirected social persuasion, (b) intense affective states, (c) insufficient mastery experiences and (d) ill suited vicarious experiences. The most pervasive and impacting sources of self efficacy within the re the misdirected social persuasion and intense affective states. Two major themes of student expectancies first high school algebra class were uncovered in the interviews: (a) Algebra classes previously held positive expectancy into a negative one, an d (b) Algebra class is a course where if one does not catch on quickly or gets behind, there is no expectation of catching up. Misdirected social persuasion in a first high school algebra class. Social persuasion is the response one receives from others, e .g., teacher feedback and the actions of peers. It is often displayed in the form of spoken judgments from others ( Bandura, 1986 1995 ; Pajeres, 1996a ) Self efficacy will increase when sources of social persuasion, i.e., feedback and others actions, are positive and appropriate in relationship to the task. Feedback is also most useful if it comes from a source that is credible in the eyes of the receiver. Unfortunately, the interviews disclosed that a preponderance of student algebra feedback was perceived by the students as negative and incongruous with the task at hand

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Page 131 algebra course was supplied by school adults (teachers, coaches, counselors) who were unsuccessful in establishing themselves as credible originators. As such, I have themed this as misdirected social persuasion. Negative feedback was in the format of criticism of student work efforts and abilities and in a manner that some students alleged as being teacher viewpoints that were wrong stereotypes of academic ability Incongruous feedback was provided when teachers merely checked work for completion and not comprehension or accuracy. One especially poignant case in point of misdirected social persuasion in the shape of negative teacher feedback was communicated by Student H. He vividly explained that his algebra work and efforts were continually criticized by his high school math teachers, one of wh however, he believed that he had really tried and put forth the effort in order to comprehend the subject material. Checking homework assignments for the sole purpose of ensuring the y were complete was the most common variety of incongruous feedback. Another was administering tests to students only after providing large chunks of instruction and failing to check for student comprehension during the teaching of the material. Student D had his homework checked for completion only and was never provided feedback on the accuracy or his grasp of the assignment. Assessment of his performance was usually provided through failing grades on tests. Student H amplified with an explanation of the homework grading practice in his first high school algebra class:

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Page 132 was like three or Stereotypical attitudes held by school officials toward individual capacity to do well in algebra were perceived by three students. Students B and J told me they felt th ey were stereotyped as athletes first and students second. This perceived stereotype really affected Student B, who had been an honors math student all through middle school and had been eager to continue in honors classes in high school. Student H felt h students. H expense of the rest of the class. Intense affective states in a first high school algebra class. Affective states are the feelings and the physical and emotional responses that lead to judgments of self efficacy. Features of a particular environment that produce unpleas ant moods, attitudes, and frustrations result in a lowered sense of self efficacy ( Bandura, 1986 1993 1995 ; Pajeres, 1996a ) feelings and emotional responses high school algebra cla ss. In fact, sharp, judging by their body language and tone when asked about experiences in their first high school algebra class. Many of the students spoke about the boredom and tediousness associated wi th participation in high school algebra, a common theme from both students who liked and disliked math. Boredom resulted in a lack of attention and caring about the subject

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Page 133 material; and therefore, caused class work and homework to be left undone. Studen t I stated: -sometimes the ---Student I was also quite frustrated at the monoto ny of his algebra classes, but he and one way only. Student I, the one student in the group of 11 students who stated that he not only loved mathematics and but actu ally recognized he needed to learn algebra for his desired profession, was never encouraged to discuss how he solved algebra problems differently. Instead, he was continually criticized and told he was wrong. He made this point over and over again throug hout the interview: When asked if his teacher encouraged him or discussed his methods w hen he found other ways of doing an algebra problem, Student I further stated: hurt for his teacher to ask Student I to demonstrate how he solved the given algebra problems differently? Teacher reliance on the mechanical and procedural aspects of algebra instruction caused much of the boredom and lack of attention, especially relevant in the case of the students who lik ed math. Two students talked about their algebra class as merely lecture

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Page 134 provision and note taking, with no discussion about what they were learning. Two students also mentioned that they are hands on learners and, thus, really found algebra boring, sinc e there were no hands on learning opportunities. Student frustration was usually linked to the inability to keep up with the pace of instruction, chiefly when failing to understand a concept or missing a class. If a student did not comprehend a specific concept or missed a class, there was little chance of being able to stop and review, since the teacher just kept moving on to the next concept. Once this occurred, students would disengage from algebra learning and express indifference to even attempting to keep up. Some students told me they, therefore, believed they had no math ability because of having trouble keeping up Student D failed only one core class, algebra, during the first semester of his freshman year. Algebra was his biggest source of frustration in school, and it was due to his inability to keep up in class. He stated: -plus to pass in th at Student H also communicated frustration with his inability to keep up with the material and his subsequent belief that he just could not do it. rt new. I

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Page 135 Frustration and disappointment at repeating failed algebra courses or being placed in multiple types of math courses, to include the next class in the algebra sequence whether ready or not, were a highlight of several student interviews. When asked about bu exceptionally frustrated at being placed in multiple types of math and algebra courses, with no explanation as to why h is placement in a specific course occurred. A dislik e of algebra and math was another common attitude displayed by the interviewed students. About half of the students stated that they simply dislike math. do it. S Finally, at least three of the students reported anxiety over taking algebra tests, including students who reported liking math. One of the students who stated he was good at math, Stude nt E, expressed that he would forget things a couple of months later. Student J, who also reported test anxiety, stated he did much better at an alternative school where assessments occurred frequently instead of after large chunks of instruction. Insuffi cient mastery experiences in a first high school algebra class. The interviews provided evidence of insufficient mastery experience s that algebra class. This was not attributed to poor mastering of algebra and math content prior to entering high school. When asked about middle school math preparation, only two

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Page 136 students stated they thought their middle school math preparation had an impact on their abilities to master high school algebra. Not surprisingly, the structure of algebra itself prevented mastery of the content thought the material of algebra was harder than any other class. Student C confirmed this and you have to write the equations the exact right way. Student K reported needing the teachers to decompose the concepts a little furth er so it was understandable, and it was The unique structure of algebra, the non stop pacing of instruction, and the course. Chec king work solely for completion and not accuracy also prevented mastery. In the end, students tuned out and frequently believed they had little or no hope of ever gaining the ability to pass their algebra courses. Ill suited vicarious experiences in a fir st high school algebra class. high school algebra course. Vicarious experience is knowledge that is gained indirectly and permits individuals to learn oneself or from the behav iors of others. Modeling by others can particularly assume a substantive role in forming self efficacy beliefs ( Bandura, 1986 1995 ; Pajeres, 1996a 2002 ; Schunk, 1984 ) The vicarious experiences perceived by the students in their first algebra course were ill suited, labeled so since poor attitudes and a dislike for the job was implicitly modeled within some algebra environments. The students spoke of observing lackadais suited

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Page 137 attitudes to model in front of students, who then assume some of the same attitudes themselves. One student, Student G, reported liking algebra and feeling good about his his teachers toward high school algebra. Another teacher modeled with her body language effort, further eroding his belief in his ability to accomplish algebra. Student H stated: step by ste Other ill suited vicarious experiences in algebra came from classmates. For the most part, these were other student s who possessed poor attitudes; but in the case of Student C, he was discouraged by observing how other students could do the algebra work, which he could not. Expectancies in a first high school algebra class. Two themes of expectancies emerged from the a nalysis of the student interviews. Both themes can be associated with the behavior of disengagement. The first is the expectancies of math learning or reverses a prev iously held positive expectancy into a negative one. All of the students who reported they disliked math or that they could not still existing frustrations, an inability to do math, or continuance in disliking math. Their first algebr a classes and all of the remaining algebra and math classes in high school generally maintained this expectancy of not liking and not enjoying

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Page 138 math instruction. This was primarily caused by the misdirected social persuasion and intense affective states so urces of self efficacy in the algebra class environment. On the other hand, these same sources of self efficacy worked to quickly reverse feelings and expectancies of math learning opportunities from a positive to a negative perception for those students w in class, low expectations from the teachers, and poor modeling during instruction in algebra classes were mostly to blame. Students G and I reported they were really good at math and that math was one of their favorite subjects. However, their first high school algebra classes did not nurture this interest. Student G quickly became bored in class, perceived that his teacher really did not try, and decided to similarly not try as well. Student I w anted to go faster than the other students in class, but was not permitted to do so, since he felt he easily grasped the material and because he got bored with the tedium of material that he mastered before the other students. Low expectations from teacher s and school officials in the first high school algebra class affected two of the students exceptionally hard. Student B liked math and believed he had a high mathematical skill level. He was extremely disappointed in not being put in an algebra honors c lass in the ninth grade. He had been in honors math further algebra and math learning due to this disappointment. He said that he felt like he was merely repeating much o f what he had learned in middle school. In fact, this experience with algebra was what he called the turning point in his high school education; it turned a positive expectancy for his high school experience into a negative expectancy at an early time in his high school tenure (the ninth grade).

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Page 139 school because -my mom and dad always had a high expectation for my education. And, you know, when they put me back I was like, wh oa. But not The other student, Student H, did not like math and did not expect much out of the course. He stated that he went to class, did his class and home work, but still struggled. His perception was that the teachers d id not believe he could pass, nor would his teacher expected little results from continual effort, so he basically gave up. e the kids that are always making it, always getting As, say you got a D or you keep an F all the way to the end of the ed to help the more This same student finally expected to never pass algebra, and he succinctly expressed: course is: Algebra class is a course where, if one does not catch on quickly or gets behind, there is no expectation of catching up. It was clear from th e interview conversations that many students anticipated not being able to catch up once they fell behind. It did not matter if it was due to their not comprehending the material quick

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Page 140 enough or to circumstances they felt were out of their control (i.e., missing class due to illness or family commitments). More than half of the students complained they never had the opportunity to find a way to stop and re examine a concept they did not understand the first time. They would often just get completely and hopelessly behind. In addition, attempting to understand the material took too much effort and an exceptional amount of exertion for most of these students. At least seven of the 11 students I interviewed mentioned the inordinate amount of effort it took them for very little reward and perceived benefits. They realized it was just too hard; and as one Perceived sources of self efficacy and expectancies in a st school experience. Analysis of the interview data revealed two primary sources of self efficacy that misdirected social persuasion and intense affectiv e states sources of self efficacy. There that emerged from the analysis of the qualitative sample interview data : (a) High school requires too much effort for too little rewards, and (b) high school is for socialization and not for academic pursuits. Misdirected social persuasion in the greater high school environment. Appropriate social persuasion was lacking in the greater high school environment, especially from teacher s and other adults, such as athletic coaches or guidance counselors, who could have but did not establish themselves as a legitimate authority for the students. Only one of the students mentioned a trusted adult to whom they could talk;

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Page 141 and almost all tal ked about the lack of caring from teachers, guidance counselors, or administrators. In contrast, most of the students pointed out the accessibility and helpfulness of an adult at their alternative high school programs. The need for positive feedback or ju st an adult cheerleader was greatest and, thus, most obviously perceived as missing for the students who were struggling with less than desirable home life situations. Student G, especially, was getting bounced around from home to home. When I asked if a nyone at the high school provided support for him or even knew of his home life during his Another student helped his mother deliver papers in the morning an d, as a result, was habitually late for his first period class. Instead of support to allow him to catch up or an alternative class schedule he was penalized for tardiness and was not provided the opportunity to catch up on missed class activities. As a n aside, he was late for my interview time, because he was still helping his family by transporting a sibling to school. A second misdirected source of social persuasion in the greater high school environment was the influence of peer groups. Some of the students and their peer groups saw high school as a place to hang out to together. Rather than attend high school to participate in learning, high school was a place to go and socialize. In a couple of cases, the students reported they enjoyed high schoo l simply because of the friends with whom they hung out. They further amplified that they needed to be with friends more than they needed to attend class.

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Page 142 Others actually admitted to the less than stellar influences of their peers. When reflecting back o n high school they confessed that many of their peers were negative influences. Student C stated, -I guess the best way to put it is like a lot of them were -ut that. I Intense affective states in the greater high school environment. For the few students who reported they had unpleasant experiences in high school, these experiences were pretty int ense and emotional. The intensity and emotionality of these negatively perceived experiences were conveyed not only by words but by their tone of voice and by their body language (e.g., facial expressions). One student reported feeling lost in high school not knowing any of the kids or teachers. A third student was upset at the constantly changing school administration, which resu lted in a constantly changing set of overall school expectations. Sadly, a couple of these students were also dealing with some rather unpleasant life issues, outside school influences, and/or negative personal characteristics. One of the adolescents was being from a single parent home, was in legal trouble and had a physical disability. Unfortunately, coming to school not only did not provide relief from these outside experiences but ins tead compounded an overall bad situation with negative, misdirected social persuasion and affective states sources of self efficacy

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Page 143 Expectancies in the greater high school environment. Several of the students indicated that high school required too much effort for too little resultant rewards, and many of them actually alluded to the hard work required in order to pass courses. This expectancy frequently evolved into a why bother at al l an expectation of too little rewards was an expectation of severe penal ties for some behaviors, such as not turning in work or being absent from school. In the viewpoint of three students, poor grades led to exceptionally severe and uncalled for punishment along with the withdrawal of a positive affective states source of self efficacy This was the inability to academically qualify for varsity sports participation because of their low grades. These three students truly loved the sports and looked to high school as a place where they could participate in sports, but they were not allowed to play due to low grades Once taken away, it further soured them on expecting anything of value from their respective high school experiences. While I recognize there may be exis ting rules that handcuff the schools with regard to eligibility to participate in high school athletics, it is still noteworthy that for these three students this was perceived as a drastic and demoralizing punishment. The second theme of student expectanc ies in the greater high school experience was that high school participation was for socialization purposes and not for academic pursuits. A couple of the students emphatically pointed out they enjoyed school because attend classes, since it was an occasion to be with friends. One student went so far as to say that he saw high school not only as an opportunity to socialize but to also join every

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Page 144 club he possibly could. Not one of the students mentioned that high school provided them the prospect of learning new things or gaining intellectual knowledge. Behaviors leading to high school dropout or not graduating on time The most prevalent behavior caused by the interplay between student perceptions of negative sources of self efficacy and expectancies in their first high school algebra class and their greater high school experience is disengagement. This is followed by self regulatory deficiency and rule non compliance in about equal proportions. Disengagement was characterized by lack of interest in algebra (8 students), lack of interest in the general high school academic environment (7 students), and the lack of perseverance and the unwillingness to engage in effortful learning in algebra (11 students) and other high school classes A majority of students (8) reported they liked or of the school environment. Only three of the stude nts reported feeling like they did not belong in school. On the other hand, many of the students reported a genuine dislike for algebra (8 students). Self regulatory deficiency itself was characterized by student laziness, lack of preparation for tests, a nd lack of initiative in getting help, even when they recognized they needed help, which was exceptionally prevalent with their algebra classes. Seven of the 11 students mentioned they did not seek help in high school, when they knew they should have. Th e students were either too embarrassed or too independent to seek help. A couple of students went to parents and siblings who, not knowing the material themselves could not rea dily assist in mastery particularly in the subject of algebra. After not recei ving help or being unable to master algebra themselves, it was easier for the students to just give up. Student F, who asked for help from his family, said:

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Page 145 So I practically Student K was embarrassed at how much he really needed help and stated: In addition, larg e high school classes further dissuaded students from seeking help. Student F stated that there were more kids in high school than in middle school, so he really could not get help. Student E was not going to ask for help in front of all his peers. Unfo rtunately it was too late when the students were mature enough to realize they needed to take the self initiative for their respective learning. By the time they had come to this realization, they were hopelessly behind in credits and had either left scho ol or were in an alternative high school program. Rule non compliance manifested itself primarily by unexcused absences and failure to complete homework and class work. All of the students reported skipping algebra and other classes at one time or another and all reported a failure to complete homework and class work. However, although I have no direct proof of this, I suspect that rule non compliance was caused partially by disengagement and self regulatory deficiency. The sudden presence of freedom an d the increased importance of social activities facilitated the behavior of rule non compliance. Summary of Qualitative Findings A comprehensive summary of the findings of themes of student perceived sources of self efficacy and expectancies in algebra and the greater high school experience, as they relate to the three categories of behavior typical of the high school dropout and to

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Page 146 the proposed conceptual framework, is visually presented as recommended, in a conceptually ordered matrix fashion in Table IV. 7 ( Miles & Huberman, 1994 ) The rows do not necessarily represent a strictly linear relationship across each set of sources of self efficacy, expectancies, and behaviors. Sources of self efficacy and expectancies can lead to multiple behaviors, for example. However, the themes are bin ned in rows according to the most likely relationships between the three components of the conceptual framework, as uncovered from the qualitative analysis Bold type indicates answers as related to high school experience only; i talics indicate answers from algebra only; all capitals indicate answers from both high school algebra classes and the greater high school experience. Table IV.7 Summary of themes of st udent perceived sources of self efficacy, expectancies, and behavio rs, as derived from analysis of the interview data and as related to the conceptual framework. Bold type indicates answers related to high school experience only, italics indicate answers from algebra only, and all capitals indicate answers from both high sc hool algebra classes and the greater high school experience. BP dropout = (( ME + VE + SP + AS) algebra + E algebra ) ((SP + AS) high school + E high school ) Behavior Insufficient Mastery Experiences Ill Suited Vicarious Experiences Misguided Social Persuasion Intense Affective States Negative Expectancies DISENGAGEMENT Lack of showing interest in being part of the academic high school environment (BEHAVIORAL) Algebra activities show no relevance to real world Algebra teachers modeled poor attitudes toward algebra learning Stereotyped as athlete before math scholar Stereotyped as slow learner Lack of clear feedback about individual accomplishments or help from teachers Frustration, dislike, not my thing Irrelevant BOREDOM WITH CLASSES t care about school LACK OF POSITIVE ATTACHMENTS WITH SCHOOL ADULTS AND/OR OTHER PEERS Perpetuation of previous negative math experience or reversal of previous positive math experience Low expectations for performance

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Page 147 Table IV.7 (cont.) Behavior Insufficient Mastery Experiences Ill Suited Vicarious Experiences Misguided Social Persuasion Intense Affective States Negative Expectancies Lack of liking algebra or high school (AFFECTIVE) Procedural vice meaning activities Algebra teachers modeled poor attitudes toward algebra learning Lack of relevancy Procedural vice meaning; only one way to do the algebra content FEELINGS OF BEING LOST IN SCHOOL AND/OR ALGEBRA CLASS LOW EXPECTATIONS FOR LIKING WORK Lack of perseverance and lack of willingness to engage in effortful learning (COGNITIVE) Repeated failures, despite providing effort Lack of proximal goals The structure of algebra itself Algebra teachers modeled poor attitudes toward algebra learning Lack of feedback from credible sou rces, such as coaches Lack of effective and/or frequent feedback about performance Frustration at repeating algebra classes or not being able to keep up Anxiety when taking tests Lack of caring: No one in position of responsibility takes an observable interest FRUSTRATION WITH ACADEMIC ABILITIES DISAPPOINTMENT WITH LOW EXPECTATIONS FROM ADULTS Cannot catch up if they fail to comprehend quickly or get behind TOO MUCH EFFORT FOR TOO LITTLE REWARDS RULE NON COMPLIANCE Tardiness, absences, incomplete class work, and homework Severe Consequences for non compliance, such as not doing homework Peer group influence High school was fun and a place to hang out with friends TOO MUCH FREEDOM Inability to challenge irreversible policies (i.e., homework, show ing work with only one way of doing it) NOT BEING ABLE TO MAKE UP LATE AND/OR MISSING SCHOOLWORK WITHOUT PENALTY SELF REGULATORY DEFICIENCY Lack of self initiated learning Little knowledge of unit culture (unit=high school or algebra class) Ill placed peer influence Poor modeling by algebra teachers LACK OF ADULT SUPERVISION LAZINESS, EMBARRASSED TO SEEK HELP Confusion; too proud to ask for help Cannot catch up if they fail to comprehend quickly or get the one to request extra help Teacher or school initiates help versus student initiated High school is for socialization and not for academic pursuits.

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Page 148 CHAPTER V. DISCUSSION This mixed methods research study was instigated by the existence of nearly identical student high school algebra and graduation failure rates in a large urban school district in the Rocky Mountain region of the United States Additionally, based on personal observations, I strongly believed that the school district had put too much emphasis and resources int o understanding these dual problems from the perspectives of teachers, administrators and parents but had failed to do the same from the eyes of the quantitative analysis of data in order to unders tand these two problems, while commendable for the effort, is also an observed s hortcoming. Because of the unsatisfactory algebra and graduation rates and these personal observations, I embarked on a journey to not only prove that student performance in al gebra is a statistically significant predictor of high school failure but also to understand some aspect of first person perspective. Failing to viewpoints or a too heavy reliance on statistics may inhibit the identification of a number of potential predictors of high school dropout. T here is numerous and widespread research on several risk factors associated with the high school dropout issue, but no one promine nt risk factor causing dropout has emerged and some offered predictors have been proven to be ineffective in their prognostic abilities ( American Psychological Association, 2010 ; Gleason & Dynarski, 2002 ; Tyler & Lofstrom, 2009 )

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Page 149 The outlook is not all that hopeless. There are steps we can take to reduce the incidence of failure to complete high school, and I deliberately say we, because solving the dropout issue is one which students, educators, parents, community members and our national consequences. We can focus on identifying more effective predictors of dropout, such as student perceptions, and we can take action in response to those we can be change with identification and intervention. While the results of this study showed that it is nearly impossible to pinpoint student performance in high school algebra as a singular predictor of failure to graduate hi gh school, it did illustrate in the data samples used in this study, that algebra is a major contributor to the problem The holistic consideration of both quantitative and qualitative results painted a picture that student performance and experiences in their high school algebra classes contribute to, and can be used as a predictive risk factor for d ecision to drop out of school. Reiterating, the high school dropout definition used in this study was the Average Freshmen Graduation Rate (AFGR), a calculation of the numbers of students who graduate high school within the traditional four year time frame (Stillwell, 2010; Tyler & Lofstrom, 2009). Failure to graduate in this study is synonymous in meaning with failure to complete high school in four years.

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Page 150 Bottom Line up Front: Although M ixed M ethods R esearch does not P rove C onclusively that A lgebra P erformance is the S ingular C ause of F ailure to C omplete H igh S chool, the F indings from A nalysis of this S Q uantitative and Q ualitative D ata S a mples P oints toward A lgebra P erformance a s a N oteworthy C ontributor to the S chool D istrict D ropout P roblem algebra class is significantly correlated with failure to gra duate high on time and after four years. The quantitative results also showed that the high school grade in which algebra is taken is significantly related to failure to graduate high school. o Algebra performance is more likely to predict failure to gradua te within four years of high school than many other predictor variables used by the school the same year as the first algebra course. Within the qualitative study sample, all four sources of self efficacy were present in ; it was largely the s elf efficacy sources of misdirected social persuasion and intense affective state and negative expectancies that were experienced by students in high school algebra classes Within the qualitative study sample, the self efficacy sources of misdirected social persuasion and affective states combined with negative expectancies in the greater high school environment intensified those which were experienc ed in a first high school algebra course. The result of the interplay between sources of self efficacy and expectancies in high school algebra and the greater high school experience: A potential to engage in behaviors characteristic of the high school drop out.

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Page 151 o The r esultant c onceptual f ramework developed from the analysis of this : BPdropout = (( ME + VE + SP + AS) algebra + E algebra ) ((SP+AS) high school + E high school ) Taking algebra in the ninth grade transition year can be a ri sky endeavor for students The integrated quantitative and qualitative results confirmed that the relationship between the failure to pass algebra and high school dropout crossed ethnic boun daries. algebra class is significantly associated with failure to graduate high on time and after four years. The quantitative results also showed that the high school grade in which algebra is taken is significantly related to failure to graduate high school. The quantitative results showed a significant relationship between algebra performance and high school graduation. Students who fail their first high school algebra class are four times more likely to fail to graduate high school. The results showed that algebra performance alone and in combination with other specified variables predicted high school graduation. All of the quantitative tests had good statistical power, minimizing the occurrence of a Type II error or accepting the null hypothesis when th e alternative hypothesis is true. In other words, the fact that the tests had good statistical power ensured that the results of algebra outcomes having a significant relationship to high school graduation alone and with other predictors was not due to ch ance ( Foley, 2008 ; Murphy, Myors, & Wolach, 2009 ) This is a result which if ignored, can lead us to miss an opportunity to use algebra performance to add to the toolbox of predictor s of the potential to drop out of high school.

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Page 152 The grade in whic h a student first took and failed high school algebra was also shown to be a significant predictor of high school graduation. Nationwide, more of our students are taking algebra in the 9 th grade but more are also failing algebra after this first attempt t hrough the course ( Allensworth et al., 2009 ; Stein et al., 2011 ) This pattern was evident in this school district as the majority of students, more than half of the quantitati ve sample, took algebra for the first time in ninth grade and correspondingly almost a quarter of them failed th is course. The logistic regression results revealed that the performance outcomes from the two core courses, algebra and English, included wit hin the model of predictors of high school graduation, were not only significant but were the two most likely variables to lead to high school dropout. The odds of graduating high school are increasingly greater as one passes both their first high school algebra and English course in the same year as taking the first high school algebra course. Failing English in the same year that one fails their first high school algebra course almost doubl ed the chance of high school dropout than failing the algebra co urse alone. The significance of algebra and English performance in the logistic regression results make sense in lieu of the current research on the impact of failing core courses in high school. When a student fails a core course, he or she gets behind in both the subject course credits and total credits needed to earn a high school diploma. Failing core courses and getting behind in credits is a significant predictor of failure to graduate, a term which the Chicago Public School System ( Allensworth & Easton, 2005 ; Allensworth & Easton, 2007 ; Allensworth e t al., 2009 ; Neild, 2009 ) At one point in time, the Chicago Public School System reported that 40%

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Page 153 of their freshmen failed a major subject and 20% failed two or more core courses during the first semester of their freshmen year. Students who fell off track in the Chicago Public School System had only a 22% on time graduation rate and only 28% graduation rate after five years of high school ( Allensworth & Easton, 2007 ; Roderick & Camburn, 1999 ) In Philadelphia, about one third of their freshmen failed to accumulate enough credits to gai n stature as tenth graders and only 20 percent of these graduated within six years of high school ( Neild & Bal fanz, 2006 ; Roderick & Camburn, 1999 ) All 11 of the students in the qualitative sample failed their first algebra course in the ninth grade of school Four of the eleven students I interviewed fail ed only their algebra course, but seven students failed other courses as well. None of these students will graduate on time, even the one student who was attending an alternative school while concurrently enrolled in a traditional high school, in order to make up credits. Consider too, that the longer one attends school or the older one is when attending high school, the less likely they are to ever earn a high school diploma ( Chapman et al., 2010 ; Tyler & Lofstrom, 2009 ) The two oldest of the interviewed students were 20 years old and there were two students in their sixth year of high school. The two students who were 20 years old were getting close to finishing their core class credits and total credit requirements but both will still need at least one to two more semesters of school in order to earn their diplomas. Four other students were 19 and in the fifth year of school and were at different stages of progression toward a diploma, but almost all will require a t least one more year of school.

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Page 154 Within the qualitative study sample, all four sources of self efficacy were elf efficacy sources of misdirected social persuasion and intense affecti ve state and negative expectancies that were experienced by students in high school algebra classes motivation to succeed and willingness to stay involved in school activities and w ork. As students get older, these beliefs grow increasingly negative S ome researchers claim that this negativity reaches its peak in eighth or ninth grade ( American Psychological Association, 2010 ; Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Bandura, 2012 ; Zimmerman, 1995 ) Weak beliefs in aptitude and academic abilities originated from the combination of all four sources of self efficacy in the students first algebra classes However, misplaced social persuasion and intense affective state sources when pooled with negative algebra course expectancies were particularly strong in flavoring student experiences in the first high school algebra classes Sources of insufficient mastery experiences and ill suited vicarious experiences we re no t we re overtaken by the potency of the other two sources of self efficacy. This qualitative finding is counter to the assertion put forth by Lopez et al. (1997), whic h argued that mastery experiences and social persuasion sources of self efficacy were stronger and especially decisive in the formation of algebra self efficacy and algebra outcomes in high school ( Lopez et al., 1997 ) I make this counter argument for the predominance of social persuasion and affective state sources of self efficacy not only due to the frequency at which students discussed items such as negative or missing

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Page 155 feedback (social persuasion) but by the intensity with which they discussed their attitudes about and feelings (affective states) toward their algebra experiences Social persuasion in the form of positive feedback holds the power of providing needed reinforcement, and to accordingly bolster student self efficacy in their school efforts and academic abilities. Effectiveness of social persuasion increases when it comes from sources that establish themselves as credible in the perceptions of the receivers of feedback. Reversely social persuasion can also be provided in such as way sense of self efficacy. It is destructive to provide negative and unrealistic feedback and as equally harmful to withhold feedback or when it com es from perceived non credible persons ( Bandura, 1986 1995 2012 ; Pajeres, 1996a ) At least two students discussed cases of negative feedback in their high school algebra classes. The first was a student who already had a low sense of m athematical ability and the reality that one of his teachers pointedly told him that he would never pass algebra only served to worsen his mathematics self efficacy. The other student had a strong belief in his mathematical ability but was dismayed at the constant criticism of his methods in solving problems. While I sensed this did not impact his algebra self efficacy, it did serve to preclude a desir e to pay attention and participate in class activities. Social persuasion was frequently provided in an ineffective manner, primarily by the instructional practice of checking homework for completion but not for accuracy. Student D when describing the g rading and review of homework stated his algebra teacher did not check it for correctness :

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Page 156 get graded. It was more of a self Originator of sources of social persuasion in algebra classes failed to establish themselves as credible providers of feedback. More discouragingly, at least three of the students felt that their math educators held stereotypical opinions of their mathematical abilities. They strongly perceived that th ese stereotypes quenched their desire to participate in and pass algebra classes. Research in the field of cognitive psychology has shown that covert stereotypes and prejudices play a large role in the education of students. In the case of gender, mathematical stereotypes held by individuals predicted greater student negativity towar ds math, less participation in class and weaker self ascribed individual ability ( Nosek & Smyth, 2011 ) If this has been shown to be the case with respect to gender, can it also apply to stereotypical attitudes held by instructors with regard to other student characteristics such as ra ce, athleticism, and natural math ability? type, you know. It was more of a -Student B, a very articulate and athletic Black man, wanted to take honors cla sses, but he was discouraged from doing so. His perception was that no one encouraged him in this goal because he was perceived as Black athlete with a decreased ability as a student. In fact, the athletic director in his school discouraged him from hono rs since it was thought to be too hard for him to complete such a class and compete in varsity sports

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Page 157 at the same time. Student B was in math honors classes throughout middle school and one of his personal goals was to continue in such honors classes in h igh school. Although he described himself as an avid athlete, his main goal while attending high school was to do well enough to earn an academic scholarship to college, following in the foot steps of his mother, father and two older siblings. Now maybe I only have the truly not adequate to allow for successful participation in high school honors math classes that this perception became the turning point of his high school experience: -at that point, it was just like I stopped caring about school, -athlete. So t could hardly see -Another student was visibly agitated about a different stereotype when telling me about his algebra class. He stated that hi s algebra teacher labeled him as a slow math Pointedly, he is passing algebra and other math classes in his alternative high school program. The feedback in this high school environment is directed in a positive fashion. Teachers and guidance counselors alike ar e constantly encouraging him with

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Page 158 remarks about how he can and will master and pass his math and other high school courses. Student algebra experiences as exemplified within this qualitative analysis, resulted in perceptions of numerous negative intense affective state sources of self efficacy. Algebra has long been touted by the research as frustrating and discouraging, causing acute emotions of anxiety and exasperation in high school students ( Helfand, 2006 ; Vogel, 2008 ) Some of the interviewed student s confirmed this stating they were so frustrated and discouraged by algebra and math in general that believed they could not do the work which lead to them not car ing about doing the work, fail ing to pay proper attention to class activities, and frequently giving up all effort to keep up or stay involved with class. Student C told me that math was never his thing ; he did not care and so he just never paid attention in his algebra classes. He was frustrated at never being able to ng and claimed that algebra was never something he looked forward to during the day. Student J stated his feelings toward algebra more emphatically: Negative attitudes and emotions toward algebra only increased when, in addition to being frustrated at their lack of algebraic abilities, students became more agitated and discouraged at having to repeat the s ame failed algebra course or being placed in multiple brands of math courses, to include the next class in the progressively harder algebra sequence whether ready or not. Reenrolling students who failed high school algebra the first time decreases their c hances of graduating and reinforces their beliefs that they

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Page 159 cannot do the work ( Helfand, 2006 ) One of the students in this study failed the same beginning high school algebra class four times! Another was bounced around his particular high school algebra and math course offerings as the school attempted to figure out the correct mathematical placement for him. This student first repeated a failed algebra class, then went to a different type of class which included algebraic content, and then was pushed up to the next course in the algebra sequence which was he promptly failed. A third student passed his second time though the first algebra course but failed the next algebra course in the seque nce. And several barely passed a repeated algebra course doing so High school math is typically structured in a sequential pattern but perhaps this is a poor educational practice for the majority of high school st udents, especially those who either dislike math are struggling in classes, or have no intention of entering a science, engineering or math collegiate program or profession ( Stein et al., 2011 ) None of the students I interviewed reported they intended to go onto one of these disciplines Students were also frustrated by an inability to recognize relevancy of algebra to their lives or career ambitions and t eachers of algebra were perceived as not making any effort to show relevancy of the subject matter to real life. ( Schornick, 2010 ) discussed this lack of irrelevancy and i ts detrimental effect on student outcomes in algebra in her study of student perceptions about high school algebra classes ( Schornick, 2010 ) This irrelevancy yielded indifference about class performance on the part of the interviewe d students who stated they liked a lgebra as well as the students who offered that they d like algebra.

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Page 160 I would question whether any of the algebra teachers noticed and or endeavored to connect algebra knowledge and usage to the career aspirations of Student J who desires to be a musician or sound engineer, student K who wants to enter the culinary arts, Student I who wants to be a Marine Corp sniper, and Student H who wants to be a diesel mechanic. Algebra usage can be incorporated into all of these professions and in fact Student I recognized this: The reliance on teaching for procedures and on the mechanical aspects, symptomatic of the American high school mathematics classroom ( Borko & Whitcomb, 2008 ; Lobato, 2008 ; Rakes et al., 2010 ; Thompson, 2008 ) was alive and well in the district algebra classrooms of the students I interviewed. Boredom and still more frustration prevailed as student attitudes and emotions formed as a result of this stagnant instructional practice. The interviewed s tudents were quick to showcase dissatisfaction with such teaching processes and a resultant inability to pay attention in class as their instructors usually relied on lackluster note taking and lecture methods There were few opportunities to engage in mathematical sense for procedural competency and a corresponding lack of teaching for meaning did not help many of the students understand the algebra content material This was especially impacting when a student missed a class and tried to catch up. Student D reinforced this: you know, that they tried to convey. I guess -

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Page 161 -they taught it, because, you know, you have to sit there and they try to burn it into time paying attention to. So like I said, she gave you where you had to pay attention, whole, through and through, got every concept took all your notes down. And if you missed one day of notes, it was like -you came to school the There was certainly no opportunity to express how one learns differently or to be able to model how a s tudent might have solved an algebra problem differently than the teacher, another condition of affairs that reinforced attitudes and emotions of boredom and frustration. Student I repeatedly referred to being pigeonholed into one way of doing things even when he came up with the correct answer in his algebra classes. His algebra teachers never encouraged him to elucidate how he came to his conclusions using his way of thinking. He further related that not only did he get frustrated but became extremely fed up and would thus stop paying attention in class. We have got to find a way to capture the interest of students for who both like and dislike math and we must do this in more than one way Student experiences in algebra classes either perpetuated a expectancies of math learning or turned a previously held positive expectancy into a negative one. Often times dislikes, frustrations and lack of beliefs in abilities in the subject of mathematics start long before a student enters high school and decreases proportionally as an adolescents progresses in school ( Kloostermann & Cougan, 1994 ; Vogel, 2008 ) Students also invest in academic tasks to the extent in which they find the

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Page 162 task valuable an d interesting ( Archambault, Janosz, Fallu, et al., 2009 ) Students entering high school holding a previ ous dislike for and anxiety over math start off in algebra with not expecting to get much out of participating in this high school class. Shortly thereafter the struggles and frustrations incurred by the toughness and uniqueness of a beginning high school algebra class serves to perpetuate this negative expectancy. Algebra is a course that is very different from other courses, including other math courses. It has a unique language with symbols and has a distinctive structure of equations and variables. And the symbols and expressions used in a previous math course are not always used similarly in algebra. For example, within the discipline of algebra the equal sign (=) may represent either equality of two expressions or it may represent an equation to be solved ( Rand Corporation, 2008 ; Stein et al., 2011 ; Vogel, 2008 ) Understanding the abstractness associated with algebraic expression is required for the construction of meaning, generalized relationships and management of multiple representations of algebraic objects (Rakes, Valentine, McGatha, & Ronau, 2010; Vogel, 2008). If one does not catch on quickly to the uniqueness of algebra, then not only is one frustrated but nurtures an expectation of not being able to ever grasp ing the material nor catch up if falling behind. Several of the students were dismayed and po inted out their inability to catch on the first time or to catch up when they did not understand a concept or missed a class. Those students I interviewed with low expectations for positive performance outcomes in math had th o se expectations maintained or reduced to an even lower level during their first algebra course. More than a handful of the interviewees explained that they did not like math, had never liked math, and that algebra was a struggle for them. It

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Page 163 provided the impression that their math ex pectations followed them into high school and they therefore did not expect to achieve positive results in their high school algebra courses. This was added to the fact that accomplishing algebra work was perceived as needing an inordinate amount of perso nal effort in return for failing grades. They and d id On the other hand, students, who entered high school liking math, shor tly became disinterested, bored and unfulfilled by algebra class participation. They soon expect ed to receive very little enjoyment and benefit from participating in and attending class. Their experiences in the first high school algebra class instead ca used them to expect rote learning patterns, little ingenuity and interest in class proceedings and lead to a general expectancy of boredom. The positive expectancy quickly became a negative expectancy. It is possible for a school and a teacher to turn a n egative expectancy about algebra class into a positive one and thus prompt a student to believe that they can comprehend and conquer passed algebra on the third try. I asked what the differe nce was between this class and the previous two classes. He said it was the teacher taking the initiative to ensure that each and every student understood the material. The teacher actually noticed when the students in the algebra class were having troub le. She broke it down into comprehensible bits and pieces and would check with the students: -

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Page 164 Within the qualita tive study sample, t he self efficacy sources of misdirected social persuasion and affective states combined with negative expectancies in the greater high school environment intensified those which were experienced in a first high school algebra course. In general, in the majority of students, social persuasion in the greater high school environment was either provided in the form of punishment or was missing from a school adult. Negative sources of social persuasion also came from high school peers. Rese arch on the dropout problem has reported that many high school students believed that they did not have an adult who they could turn to when they faced difficulties in school and home (Azzam, 2007; Bridgeland et al., 2006). This was a disturbing pattern w ith some of the students I interviewed. Student lack of positive attachments with school adults in high school could either be classified as a missing source of positive social persuasion or an intense affective state. High schools are traditionally a pla ce where a student repeatedly travels from one class to another and to one teacher to another. Classes are taught by teachers who have allegiance to the subject matter and not to a group of students ( Neild, 2009 ) This type of atmosphere is not conducive to forming attachments between students and school adults. Lack of relati onships are a major cause of lack of attachment to a school and of dropping out ( Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Bloom, 2010 ; Bridgeland et al., 2006 ) A couple of students talked in a very passionate manner about the lack of caring in their high school environments. Other students had personal struggles in their lives during high school which they claimed an adult in the school knew about yet provided no extra support or assistance. In these cases, no one seemed to pick up on the distressing

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Page 165 outside influences which clearly distracted from the high school experience I n some instances student s were not only missing support but were penalized instead. One student helped his single mom deliver papers in the morning, so he came late to his first period class during freshman year. He was not allowed to make up the work. Another, Student G, was struggling in freshmen year. He was not getting along with his single mother gotten kicked out of his home periodically and was often faced with the burden of caring for a younger brother before and after school. Even if he did want to get extra help f or class work he could not do it before and after school. This is the same student who reported that no one in the school but Mr. Bxxxx cared about my situation. Yet another student got into legal trouble and also had a hearing problem. His perception s were that his teachers did not care about these two problems even after he told his teachers about his physical limitation. The school district is not alone in this perceived lack of caring. In the Silent Epidemic report on the United States dropout pr oblem, only 56% of the students surveyed said they could talk to a high school faculty or staff member about a school problem and even less, 41% could do the same with respect to a personal problem ( Azzam, 2007 ; Bridgeland et al., 2006 ) The influence of peer groups can be of benefit or detriment as a source of social persuasion, especially in the ninth grade. Peer groups start taking on increasingly important roles and become p rogressively more influential on the high school student. These peer models may not be conducive to academic success in high school and peer pressure can create norms that facilitate or inhibit academic achievement ( American Psychological Association, 2010 ; Neild, 2009 ; Nichols & White, 2001 )

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Page 166 Forty two percent of the over 400 dropouts interv iewed in Baltimore and Philadelphia stated they spent time with people who were not interested in school (Bridgeland et al., 2006). Student D reinforced this notion when he stated his belief that a large part of his dismal high school performance was due to the influence of the kids he hung out with. High school experiences conveyed two negative expectancies in the student qualitative sample The first is that in order to succeed in school, a student needs to put forth an unreasonable amount of effort f or a disproportionate return of rewards. A second negative expectancy is the belief that high school participation was for the benefit of socialization and not for academic learning. Bandura (2012) advocated that proximal goals are necessary to assume pos itive expectancies about the rewards to be gained from any given environment ( Bandura, 2012 ) Proximal goals are usually scarce and knowing that you will receive a diploma four years later is too far out of a goal for most st udents. Without more immediate goals and rewards, it is unlikely that the majority of ninth graders understand that they must earn both core course and total credits in order to stay on track toward graduation. For most, it is often too late by the time they realize this fact ( Neild et al., 2008 ) Even promotion to the next grade is a long time away for entering ninth graders. Some students do not need to earn credits to chronologically advance to the next grade. The school district in this study places students in the next grade even if they have not earned enough credits to technically reach this status. This makes the goal of earning enough credits for graduation yet further away for student s

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Page 167 Starting in the ninth grade, studen ts are faced with increasing academic demands and the need for better organizational skills both of which require increased personal effort They also encounter a state of decreased structure where they have autonomy in personal choices; the burden is on them to choose to do the work. Additionally, in many cases, the student population is so large that teachers often fail to notice when stu dents do not complete and hand in work ( Roderick & Camburn, 1999 ) Several of the students mentioned that they had not put forth the attempt s n eeded to succeed in class, the teachers had not noticed when they did not do their work, and that it required too much effort to succeed in their early high school years. The expectancy that high school was a place for socialization was a second expectancy prominent in the interviewed student sample. The influence of peer groups combined with the sudden appearance of freedom, while turning high school into a personally pleasant experience, did just that, it turned high school into an expectancy of a social gathering versus a work environment. In one survey of high school dropouts, 38% reported that they had too much freedom in high school. Student E reaffirmed this when he stated: -not really going to class. Yeah, like I would go to school, but I would just never The result of the interplay between sources of self efficacy and expectancies in high school algebra and the greater high school experience: A potential t o engage in behaviors characteristic of the high school dropout. The occurrence of misdirected social persuasion and intense affective state sources of self efficacy, combined with the negative expectancies in both high school algebra and the greater high school environment led to a propensity for students in this

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Page 168 qualitative sample to enter into themes of behavior s characteristic of the high school dropout. The most prevalent of these themes of behaviors was that of disengagement in high school academic work. This is not unlike previous research which has shown that disengagement is positively correlated to high school dropout both in the United States and in Canada ( Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ; Bridgeland et al., 2006 ) Disengagement was present i n all of its three dimensions: behavioral, affective and cognitive. Behavioral disengagement was characterized by lack of interest in being part of the academic high school environment. Students reported skipping class and not doing class work and home w ork. Affective disengagement was exemplified by lack of liking algebra/school or feeling like they belong ed in school and lastly, cognitive disengagement was embodied by a lack of perseverance and willingness to engage in effortful learning. While most of the interviewed students actually liked high school, the majority did not like participating in their algebra classes There was also a notable incidence of self regulator y deficiency. Compared to when they become seniors, adolescents are relatively immature during their freshm e n year. They are additionally coming from a previous educational environment (elementary and middle school) where they had much less opportunity a nd encouragement to assume personal re sponsibility for their own learning. By the time high school students ask for help they may have buried themselves in a hole of failed courses and lost credits which is nearly impossible to climb out from ( Allensworth et al., 2009 ; Neild et al., 2008 ; Wheelock & Miao, 2005 ) Similarly to the findings within the Silent Epidemic study of h igh school dropouts ( Bridgeland et al., 2006 ) many of the

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Page 169 interviewed students reported having regrets years later about not taking self initiative for their education. At least half of the students stated they had wished they sought help since rarely did their teachers voluntarily attempt to provide or notice when they were struggling. They almos t all reported that they should taken individual responsibility for their education, but were either embarrassed, too immature or would not take the initiative to ask. By the time they had taken the effort to ask for help it was too late. They were hopel essly behind on work and credits. Student K: everything done with a have to make the Even when they recognize d that there wa s a physical limitation which require d action on their part such as sitting in the front of a class so that they can hear better (Student E), they we re often reluctant to take the initiative to do so

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Page 170 The r esultant c onceptual f ramework developed from qualitative sample BPdropout = ((ME + VE + SP + AS)algebra + Ealgebra) ((SP+AS)high school + Ehigh school) Taking algebra in the ninth grade transition year can be a risky endeavor for ative and qualitative results. A smooth transition to the high school environment during the ninth grade has been proven to be a pivotal point in the academic trajectory and ultimate success of many transition year requiring considerab le adjustment for many students and adolescents may not be ready socially or academically for the move to high school. They leave behind earlier formed attachments with peers and teachers and enter a new world of academic rigor and challenge ( American Psychological Association, 2010 ) .. The degree to which students have difficulty in adjusting to this transition is positively correlated to high schoo l completion. Students are relatively immature in the ninth grade and many do not realize until it is too late that completing work, passing tests and getting acceptable grades are requirements to remain on track toward an on time graduation ( Alspaugh, 1998 ; Bridgeland et al., 2006 ; Neild, 2009 ; Neild et al., 2008 ; Roderick & Camburn, 1999 ) Consistent with the research, many of the interviewees reported that entering high school was tru ly a wake up call. They alleged they were not ready for the transition to

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Page 171 Others stated they got into irreversible academic trouble during their freshman year. passe One student mentioned that he immediately felt lost in the crowd when he arrived in high school. Another stated he was not lost in the crowd, but lost in the work. While not totally negative on his high school experience, he believed he lost his way once he got to high school, and that algebra and all of the work was too hard, which motivate d him to eventually quit school. One additional student thought the influence of his new peers caused difficulty in his transition into high school: The integrated quantitative and qualitative results showed that the relationship between the failure to pass algebra and high school dropout crossed ethnic boundaries. The l ogistic r egression results showed that ethnicity was not a significant factor in the model of predictors of high school graduation when first high school algebra class was included (along with the other predictor variables in

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Page 172 The interviewed students who met my selection criteria of failing their first algebra class and either dropping out or failing to complete high school were likewise a blend of Caucasians, Blacks and Hispanics, with over half of the students self reporting they were Caucasians. The relationship between the failure to pass algebra and high school drop out is an issue that wa s not related to ethnicity in the Interventions put into place which address this relationship therefore, holds the possibility of having utility to meet the needs of students from all e thnic backgrounds. The Value of Mixed Methods Analysis in Investigating the Relationship between High School Algebra Performance and High School Graduation Mixed methods research provide d insight into the depth and breadth of the problem, and the results of the qualitative analysis significantly enhanced the results of the quantitative analysis by showing practical and economic significance. Results from the quantitative method of study were not totally congruent with the results from the qualitative method of study Mixed methods research provide d insight into the depth and breadth of the problem and the results of the qualitative analysis significantly enhanced the results of the quantitative analysis by showing practical and eco nomic significance. The quantitative research successfully demonstrated the significant relationship of algebra alone and in combination with other variables to the failure to graduate high school. But this quantitative examination revealed the existence only of the significant relationship between these two variables. It did not provide any insight to why the relationship exists or why performance in high school algebra may be a significant predictor of failure to graduate or complete high school in four years. The quantitative analysis thus provide d statistical significance but fail ed to provide practical significance

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Page 173 about the relationship between high school algebra performance and high school graduation in the district under study ( Onwuegbuzie & Leech, 2004 ) It is relatively easy to reduce matters to an analysis of numbers. It is also as related to the high school dropout problem It is impossible to calculate a hypothesis about the cognitive and affective motives behind the prompt ing of a student to leave high school before earning a diploma ( Archambault, Janosz, Fallu, et al., 2009 ; Archambault, Janosz, Morizot, et al., 2009 ) The addition al use of qualitative method ology with quantitative techniques allowed me to expose some of the cognitive and affective thoughts of students when experiencing thei r first algebra course from the eyes of the students themselves. Using the tradition of phenomenology to explore the first person experiences of high school algebra and the greater high school experience provide d a unique perspective not realizable by loo king at statistical data only. Several of the students were eager to tell me their stories. Without using phenomenology, would I have been able to tell you that a majority of the students had negative perceptions of the ir algebra teachers in the ninth grade and of the uncaring atmosphere of their high school s ? Would we have realized that we have a number of students who actually enjoyed math until they reached their boring and frustrating ninth grade algebra class? Would I have been able to tell y ou that some students perceived they were erroneously stereotyped in their algebra courses and that this soured them on their high school experience and may have had an effect on their resultant failure to earn a

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Page 174 diploma after four years of high school? An d that these various sources of self efficacy and expectancies may have had an impact on their decision to continue on in school. There is additionally a potential financial consequence from depending on the results of only one research method one which can thought of as reduc ing the economic significance of a research study ( Onwuegbuzie & Leech, 2004 ) The school district under study, like many across the nation, is facing severe budget cuts and has been facing interventions that are ineffective nor can we cut interventions that may help to increase student achievement. Depending on the results of only one research meth odology may prevent us from obtaining an opportunity to help more students achieve high school graduation and also may encourage us to focus scarce resources and attention on predictors which may not be that effective or inclusive of enough students in try ing to prevent high school failure. Results from the quantitative method of study were not totally congruent with the results from the qualitative method of study The results from l ogistic r egression are not totally congruent with those from the qualitative analysis. Only about half of the interviewed students had failing English grades during the same semester in which they failed their first algebra course. The l ogistic r egression results showed that the odds of failing English were twice as i mpacting on graduation than did a lgebra in the model of several predictors. Had we looked only at the quantitative results we would have been lead to believe in this half truth. Yet, a review of the student transcripts showed that less than half (5) of t he eleven students I interviewed failed both algebra and English at the same time. Again this

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Page 175 argues for the use of both quantitative and qualitative methods to provide us a more comprehensive and holistic answer to research problems. Concluding Thoughts We need to pay attention to the teachers of ninth grade algebra. Alternative schools are onto something. Economically and logically we cannot create methods at least in the ninth grade and especially in core classes such as algebra. We need to pay attention to the teachers of ninth grade algebra. High schools greet their newest and most inexperienced teachers by having them teach ninth year classes ( Neild et al., 2008 Our newest teachers may have the algebraic content fresh in their mind and may be eager to spread their knowledge to students, but are they often unprepared to teach a unique course such a s algebra, especially when the class is filled with students of different abilities. They lack the intuition that one pick s up from experience and time spent in a classroom instructing children of varied levels Putting a new teacher in a ninth grade alg ebra class may be acceptable, but only if the school administration ensures they are indeed teaching effectively and reaching the ir many students. The contrast between the effectiveness of a new and experienced student was apparent in three of the student interviews. One student, Student D, who only failed his algebra class in his first semester of his freshmen year, talked negatively about the fact that the teacher in this class was right out of college and did not provide him effective feedback: s mean. She -she was just right out of college too. The first Algebra teacher I had, in that first Algebra

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Page 176 getting it. Just every test I wou ld take I would fail it. And I thought I was doing Contrarily, one student, Student B, had failed algebra until he reached an algebra class with the Math Department Chair of h is high school when h e started comprehending and passing algebra. And now at an alternative campus he was positively glowing when he relayed that his teacher wa s challenging him and holding high expectations for his learning. The school put him in the hi ghest Algebra class they had, which although it was positively perceived as a n honors class and one which was challeng ing and motivating Student K when he finally passed algebra after three tries, in addition to h aving a teacher who continually asked if the students needed help, put the class into pods or smaller instructional units, in a cooperative learning fashion and additionally actively monitored the learning of each group. Alternative schools are onto some thing. Economically and logically we the delivery methods at least in the ninth grade and especially in core classes such as algebra. School organization has been shown to have a significant impact on dropout. Large urban schools traditionally ha ve had a lower graduation rate than smaller, suburban or rural schools. Schools which reflect hierarchical organization with highly defined roles for staff and students have al so yield ed lower rates of dropouts ( Neild et al., 2008 ) Schools that minimize the impact of largeness and hierarchy are succeeding ( American Psychological Association, 2010 ; Tyler & Lofstrom, 2009 )

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Page 177 The school district at the center of this study is a large, urba n and hierarchal school district. However, the district offers several alternative high school education programs and a number of ninth grade intervention programs The students I had interviewed were at various stages in their progress toward completing the requirements for a diploma. They were also still not passing every course. Yet they were starting to taste success and all seemed to now be working positivel y and steadily toward their objective of earning a diploma. A large part of finally being successful and on th e path toward graduation may be ascribed to the fact that these kids are now more mature and can realize the consequences of not earning a diplom a. But s omething else is going on and is working at these alternative campuses and the manner in which these alternative programs create success which should be examined for utility and possible replication in our regular high schools. Budgets are tight but tinkering with instructional and interventions such as alternative schools and Freshmen Academy and other ninth grade interventions that replicate the practices of small schools and success alternative programs may be a foolish decision. Future Rese arch and Recommended Prescription for Success I used a student interview process and analysis to identify student perceived sources of self efficacy and expectancies whi ch lead to the behaviors representative of the high school dropout. Although my findings were verified through the use of member checking, the next recommended step would be to use a proven and reliable self efficacy questionnaire to provide further quant ification of the role and comparative influence of

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Page 178 each of the sources of self efficacy in a first high school algebra course and its subsequent relatio nship to performance outcomes. My sample set was confined to those students who had failed algebra and f ailed to graduate high school on time, after the traditional four years. Future study should be executed to compare the qualitative f indings from this study with those held by students who have successfully completed high school in four years. Is there a d ifference in the perceived sources of self efficacy and expectancies that come from high school algebra classes between students who fail to graduate high school on time and those who are successful in doing so? Further study would also be extended to a l a rger sample size to determine if the qualitative results would generalize to a greater proportion of the high school dropout sample. My qualitative results are representative of the perceptions of the student size and unique composition of the criterion s ample used in this study and may not necessarily be replicated in a larger and different sample. It would be informative to determine if the results are similar in an investigation of students with different characteristics, for example younger students or those who are still attending a traditional high school. A deeper investigation into the role of implicitly held stereotypes on the part of both students and teachers and its resultant affect on the relationship between student algebra performance and hig h school graduation would be another recommended area for future study, especially since three of the students in the qualitative sample perceived they were the recipient of such stereotypes. Hypothesis two of the study was not fully proven: The hypotheses that socioeconomic status, and gender were not significant factors in the relationship of

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Page 179 proven. Research is needed to determine the significance of gender and socioe conomic status one the relationship of student performance in a first algebra class to high school graduation. Finally, does the actual grade one receives in their first high school algebra course matter and is this significantly correlated to high school graduation? There were over a 100 student with grades of D, to include 7 students with a barely passing grade of D Future research would be insightful into the role of a minimal passing grade in the chance of high school graduation?

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Page 189 APPENDIX Interview Protocol Project: Predicting High School Completion using Student Performance in High School Algebra Location/Date/Time of interview: Interviewer: Interviewee: What current program the interviewee is enrolled in: Which high school did you originally attend (if applicable)? Grade of first h igh school algebra course (from school district): Other courses/grades in year of first high school algebra course (from school district): Briefly describe the project: Commend the interviewee for decision to return to and making the effort to earn a high school diploma Research involves attempt to identify factors which may contribute to a decision to drop out of high school Ultimate goal is to help prevent others from dropping out of high school Questions: 1. What years did you attend High School? Wha t year did you drop out? 2. When did you enroll in the (Name) program? 3. What was the experience of high school/middle school before you decided to drop out? 4. What has been the experience of attending this (name) program compared to your previous experi ence at high school? 5. What was your experience in your first high school algebra course? 6. Are you or have you taken a follow up algebra course? What is/was the experience in this course? 7. Tell me how you felt when you did not do well in algebra the first time you took it? 8. Did it affect your decision to finish high school?

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Page 190 9. Did any other class affect your subsequent feelings toward high school attendance and participation? 10. Going back to algebra what was it specifically about algebra that may have caused your troubles if any? (i.e., language, terminology, lack of preparation)? 11. Were any of the specifics mentioned about the algebra course also issues in other classes. Explain. 12. Can I contact you again if I need to clarify some of the de tails in your answers? Thank you, your insights will be helpful in understanding the things that contribute to a student dropping out. Perhaps together we can begin making High School a better place for all students. (Assure him or her of confidentiali ty of responses and potential future interviews. Would this interviewee like to be debriefed on the results and analysis of this research?)