The impact of a mathematics performance-based assessment on elementary preservice teacher development

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The impact of a mathematics performance-based assessment on elementary preservice teacher development
Clark, Julie Ann
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xv, 306 leaves : ; 28 cm


Subjects / Keywords:
Teachers -- Training of ( lcsh )
Student teachers -- Rating of ( lcsh )
Mathematics teachers -- Rating of ( lcsh )
Competency-based education ( lcsh )
Competency-based education ( fast )
Mathematics teachers -- Rating of ( fast )
Student teachers -- Rating of ( fast )
Teachers -- Training of ( fast )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


Includes bibliographical references (leaves 280-306).
General Note:
School of Education and Human Development
Statement of Responsibility:
by Julie Ann Clark.

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|University of Colorado Denver
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Auraria Library
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LD1190.E3 2004d C52 ( lcc )

Full Text
Dip. T., Flinders University, Australia, 1982
B.Ed., Flinders University, Australia, 1983
M.A., Boise State University, 1991
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Educational Leadership and Innovation
Julie Ann Clark

This thesis for the Doctor of Philosophy
Degree by
Julie Ann Clark
has been approved
Michael P. Marlow
William Juraschek
Linda Damon
V- 2 1-QL{

instructional choices on their students. While specific experiences
varied, all of the preservice teachers became strong advocates for a
differentiated pedagogical approach that considers individual student
needs. Data analysis highlighted several main thematic connections
including: (a) philosophy, attitudes and experiences, (b) knowledge,
management and instruction, and (c) assessment and instruction.
This abstract accurately represents the content of the candidates
thesis. I recommend its publication.
Michael W. Marlow

I dedicate this thesis to the memory of my wonderful sister Karen
McCann, who passed away on June 10, 2003. Her zest for life and
determination has become an example for me to model my life on.
I also dedicate this thesis to my parents, Kevin and Dorothy
Heinrich, and children, Kevin and Andrew Clark, for their unfaltering
understanding and support while I was writing this. It is only with the
encouragement and practical assistance of my family that completion of
this thesis has been possible.
Karen McCann

Grateful thanks to my advisor, Mike Marlow, for his
encouragement and support over the past few years. I also wish to
thank my committee members, Carole Basile, Bill Jurashcek, and Linda
Damon for their ongoing support amid many challenges. Appreciation
also goes to the staff of the Graduate School for their assistance and
support during this process.

1. INTRODUCTION..........................................1
Introduction to the Problem.........................1
Mathematics Education Reforms...................4
Preservice Teacher Education Assessment.............7
Impact of Preservice Teachers on Students...........9
Mentoring of Small Groups.......................9
Preservice Teacher Impact as Evidenced by the
Oregon Work Sample.............................10
Conceptual Framework...............................11
Effective Experiences..........................11
Structure of Performance-Based Assessments....11
Dissertation Structure.............................15

Preservice Teacher Historical Context.............17
Current Education Reforms.........................18
Effective Teachers.....................................20
Definition of Effective Teachers..................20
Characteristics of Effective Teachers.............21
Knowledge of Subject Matter.......................22
Knowledge and Use of Pedagogy.....................24
Reflective Practitioner...........................25
Dimensions of Effective Teachers..................26
Student Achievement....................................27
Main Influences...................................27
Class Size........................................30
Effective Mathematics Instruction......................33
National Standards................................33
What is the Current State of Mathematics?.........34
Effective Practice................................35
Preservice Teacher Education...........................38
Quality Programs..................................38
Learning to be a Teacher..........................39
Learning to be a Mathematics Teacher..............41

Quality Concerns.................................43
Learning to be Student Centered..................44
Preservice Teacher Beliefs.......................46
Future Course Directions.........................48
Field Experience.................................50
University and School Partnerships....................51
Effective Teacher Education in Professional
Development Schools (PDSs).......................51
Description of PDS...............................54
Impact of Partnerships...........................55
Assessment of Preservice Teacher Proficiency..........58
Teacher Certification............................58
Traditional Assessment...........................60
Alternative Assessment...........................61
Performance-Based Assessment.....................65
What research is needed in teacher education?.........68
3. METHODOLOGY.............................................70
Study Overview........................................70
Choice of Methods.....................................72
Why Use Case Studies?............................76

Background........................................... 77
Initial Teacher Education Program Information....77
Performance-Based Standards for Colorado Teachers....82
Elementary Mathematics PBA Design.....................84
PBA Requirements.................................85
Study Demographics....................................87
Study Location...................................87
Potential Study Effects...............................91
Teacher Candidate (TC) Roles.....................91
Elementary Students..............................93
Researcher Participation.........................93
Data Collection Methodology...........................95
Data Types.......................................95
Elementary Student Pretests and Posttests.....96
Interview and Focus Group Data Storage...........97
TC Pre Interviews and Post Interviews............98
TC Focus Groups..................................98
Clinical Teacher Interviews......................99
Elementary Student Interviews...................100
TC Journals.....................................100

Reliability and Validity.........................102
Data Analysis Methodology.............................103
Analysis of Interviews, Focus Groups, and Journals 103
Analysis of Student Pretests and Posttests.......108
Standards of Quality and Verification.................108
4. RESULTS AND DISCUSSION.................................110
Organization of Results...............................110
Group Selection.......................................113
The Teacher Candidate Perspective.....................116
TC Integration of PBA Requirements...............116
Bethanys Experiences............................117
Bobs Experiences................................125
Jills Experiences...............................132
Marys Experiences...............................142
Susans Experiences..............................154
Bringing the Teacher Candidates Together..............165
Attitudes and Philosophy.........................165
Instructional Understanding......................169
Use of Assessment................................176
Group Issues.....................................179
The Student Perspective...............................184

Student Characteristics..........................184
Affective Growth of Students.....................185
The Group Experience.............................193
Impact on Students...............................197
Mathematical Understanding.......................203
Student Connections...................................210
Linking Affective Characteristics and Achievement.. 210
Small Group Impact...............................213
5. CONCLUSION AND RECOMMENDATIONS.........................215
Thematic Connections..................................215
Philosophy, attitudes, and experience............215
Knowledge of Students, Management, and Instruction
Assessment and Instruction.......................218
Models Arising from the Study.........................218
Evidence of Teacher Certification Standards......222
PBA Impact.......................................227
Implications Concerning the Value of PBAs.............228
Benefits for Schools.............................228
Benefits for Teacher Education...................229
The Future of PBAs in Teacher Education..........231
Future Directions of Research.........................231

A. Performance-Based Standards for Colorado Teachers 233
B. Colorado Model Content Standards for Mathematics... 243
C. CSAP-Like Fifth Grade Test.....................268
D. CU University of Colorado at Denver............276
E. Teacher Candidate Interview Questions..........277
F. Elementary Student Interview Questions.........278
G. Clinical Teacher Interview.....................279

3.1 Research Process Summary................................71
4.1 TO Management Concerns Over Time.......................180
4.2 IA Test Scores.........................................202
4.3 ELC Test Scores........................................203
4.4 Elementary Student Adoption Rate.......................207
5.1 TC Development.........................................221

2.1 Teacher Licensure Tests in 1998-99............................59
2.2 PBA Models for Effective Teacher Evaluation..................66
3.1 Mountain View Student Demographics............................88
3.2 Mountain View Mathematics CSAP Scores.........................89
3.3 TC Information...............................................91
3.4 Coding Captions.............................................106
4.1 A Comparison of TCs Attitudes and Philosophy.................166
4.2 Content Studied.............................................170
4.3 ELC and I A instructional Choice............................172
4.4 ELC and I A Assessment Comparison...........................176
4.5 Early Learning Center Scores................................198
4.6 Intermediate Area Test Scores...............................199
4.7 Total Test Scores, Variance and Standard Deviation..........200
4.8 Total Test Scores Without Bethanys Group...................201
4.9 Attribute Connections......................................211
5.1 Artifact Evidence of Standard Proficiency...................222

Introduction to the Problem
As society becomes more complex, pressure for educational
change comes from the business community, professional
organizations, and, more recently, from government legislation (Becker
& Jacob, 2000). Teacher education reform is inextricably interwoven
with public school reform. Tyack and Cuban (1995) state that
educational reform is intrinsically political in origin (p. 8): it is generally
viewed as a mechanism for social reform (Cremin, 1988). Education,
like the population, has grown from small local areas to state and
national levels. One tool of reform has been the development of
educational standards. Assessment of these standards has in turn led
to state level tests.
Not only are standards being set for student achievement but
there is a trend for teachers to be held directly accountable for this
achievement as measured by these state tests (Bracey, 2000). Schools
are changing both curriculum and instruction in an effort to improve

these test scores (Brawdy & Egan, 2001). Legislative requirements are
also beginning to affect university teacher education programs
(Wiseman, 1999).
Because teacher preparation programs inherit much of their
culture from the standards and practices shaping life in our public
schools, the politics of high-stakes testing has become increasingly
influential in the decision-making processes associated with curriculum
and instruction in undergraduate teacher preparation programs (Brawdy
& Egan, 2001, p. 438).
As a result of legislation like Senate Bill 99-154 in Colorado, (see
Appendix A), teacher licensure requirements have been changed and
university education programs are to be judged on the basis of the
performance of their teacher candidates (TCs). In particular, preservice
teachers are evaluated on how their own students achieve. All Colorado
universities, including Mountain University (MU, a pseudonym), are
entwined in the new teacher licensure standards and the necessity to
adjust teacher education programs. MU is part of the National Network
for Educational Renewal (NNER), a consortium of liberal arts colleges
and K-12 schools founded by John Goodlad. The main goal of NNER is
the simultaneous and interdependent renewal of schools of education.
Public K-12 schools work with MU as professional development partner

schools (PDSs) to fulfill four functions: new teacher preparation,
professional development, research/inquiry, and the renewal of
curriculum and instruction in the university and the public schools
(Abdal-Haqq, 1995).
In the Initial Teacher Education (ITE) program at MU emphasis
has always been given to effective teaching practices during the
internships of TCs in PDSs. In general, effective teaching practices are
selected according to impact on student learning (McEwan, 2002). One
approach of assessing TC teaching performance is the use of
performance-based assessments (PBAs). The PBA of interest in the
present study includes measures of pedagogical understanding, as well
as measures of the achievement of the students of the TC being
After the introduction of the new PBA teacher education
program, both the partner schools and MU needed to determine the
success of the program. One of the PBAs examines preservice
teachers instruction and assessment in elementary mathematics.
Mathematics is one of the disciplines being tested by the State of
Colorado, and there is significant interest in student performance and
attitude in this discipline. Ironically, mathematics continues to be one of
the most feared disciplines among students and indeed teachers

(Tobias, 1993). There was a genuine need to investigate the impact of
the elementary mathematics PBA on the development of preservice
teachers and their students.
Mathematics Education Reforms
Reforms in mathematics education have been under discussion
for many years. According to Battista traditional instruction ignores
students personal construction of mathematical meaning, the
development of their mathematical thought is not properly nurtured,
resulting in stunted growth (1999, p. 430). Researchers reinforce the
importance of real understanding of mathematics, by emphasizing the
need for students to make connections between ideas, facts, and
procedures based on previous knowledge and newly established
relationships (Hiebert, & Carpenter, 1992).
Acknowledging students previous experiences and creating
meaningful opportunities for new mathematical connections to develop
can lead to more complex understanding. Consequently, students are
more likely to remember what they have learned and to transfer the
knowledge to new problems (Hiebert et al., 1996). Unfortunately many
elementary preservice teachers also fear mathematics and are

concerned about their adequacy as teachers of this discipline (Battista,
Despite increasing pleas for changing practice in mathematics
education, many teacher education programs have failed to conform to
the suggested reforms (Graham, Li, & Curran, 2000). This is further
exacerbated by the mathematics anxiety displayed by numerous
preservice teachers (Battista, 1986; Sliva & Roddick, 2002). Preservice
teachers need to personally experience good mathematics teaching
methods and be given time to process their experiences. Teacher
education that is conducted in a setting that promotes investigation and
inquiry into the problems of mathematics teaching seems to hold
promise for assisting preservice teachers in becoming inquiring,
reflective, mathematics teachers (Mewbom, 1999, p. 39). Mewbom
indicates that preservice teachers need many opportunities to reflect
about the complexities of mathematics teaching.
Preservice teachers have established beliefs about teaching, as
they have been in classrooms most of their lives (Clock, 1999). As
reform often requires a fundamental shift in teaching beliefs and
practice, teachers must be able to reflect on what is happening in their
classrooms. Reflection enables teachers to review, reconstruct, and
critically analyze their students performance (Shulman, 1987; Yager,

1997). Barrow (1991) states that the more opportunities teachers have
to reflect on their practice and to learn from their experiences, the more
comfortable they may feel about the changes they face, and the greater
commitment they may have to quest for educational reform (p. 38).
The latter part of the twentieth century has seen a shift from a
purely scientific approach to teacher education to recognition of the art
of teaching (Bryan, Abell, & Anderson, 1996). Studies indicate that
preservice teachers, when exDosed to reflective mathematics
methodology in a supportive environment, can learn to manage learning
environments effectively, develop sensitivity to students, and engage
students in mathematics inquiry (Clock, 1999; Jaworski, 1992). Clock
suggests that preservice teachers can learn effectively with small
groups of students and need to be given multiple opportunities to reflect
on students mathematical thinking.
Part of the teacher-education reform movement must involve a
renewal of assessment processes. Clearly a changing preservice
teacher education practice will necessitate an examination of
assessment methods. Appropriate assessment of preservice teachers
readiness to begin teaching is an essential element of all programs.

Preservice Teacher Education Assessment
Traditionally, assessment has been designed to compare how
students perform relative to one another (Wolf & Reardon, 1996).
According to Wilson, teacher assessment has generally taken the form
of multiple-choice tests and observational checklists (1995). Licensure
in most states has involved the successful completion of university
courses and field experience, as well as some type of state-mandated
competency tests. Assessments, such as the National Teacher Exam
(NTE), test content knowledge as well as pedagogical knowledge. Rich,
Barikowski, and Boyd (1995) state that while there is some evidence for
content validity of the NTE, there is virtually none in terms of predictive
validity. In addition, traditional assessment tends to assume that
preservice teachers1 classrooms are teacher-centered. Because we
are broadening and changing our conceptions of teaching and learning
we must rethink the questions that frame teacher assessment (Wlson,
1995, p. 194).
Like any good assessment, the assessment of preservice
teachers should inform instruction and have developmental aspects.

Concerns over the effectiveness of traditional tests that emphasize
basic skills continue to be made. Ironically, there is not good research
evidence that the basic skill level of teachers effectively translates into
better instruction, as measured by student outcomes (Brookhart &
Loudman, 1995, p. 20). The aim of preservice teacher assessment is to
determine whether a preservice teacher has the kind of professional
knowledge, ability, and ethics needed to teach effectively (Brookhart &
Loudman). Traditional competency tests are insufficient to examine
content, pedagogy, curricular knowledge, and interpersonal skills, as
well as an ability to work with students.
According to Schalock, Schalock, Cowart, and Myton, (1993) it is
rare for teacher licensure programs to assess on the basis of student
performance, and yet this is clearly an important aspect of teaching. It is
essential for preservice teacher evaluation to emphasize the use of
knowledge (Yarbrough, 1995). In addition, Pasch alludes to the
importance of developmental assessment over time (1995). Preservice
teachers are developing as professionals and need to be assessed
accordingly. Many programs look to the use of PBAs that allow
preservice teachers to show what they can do in authentic situations.

Student learning is the professional touchstone for both
teachers and teacher educators, and the professional status of
either will grow only when teachers are demonstrably able to
nurture the kind and level of learning in students that is deemed
essential for our nation at a particular point in time (Schalock,
Schalock, Cowart, & Myton, 1993, p.108).
Impact of Preservice Teachers on Students
Mentoring of Small Groups
Studies have indicated that interventions by preservice teachers
in small group tutoring can impact significantly on elementary students
reading levels (Hedrick, 1999). Mewbom found that small group tutoring
in mathematics by preservice teachers can actually reveal information
about elementary students learning that was previously unknown by
their classroom teachers (1999). This is probably due to the individual
attention given to students in the group over a sustained number of
weeks. Students in Mewboms study made gains in mathematics
While Betts, Zau, and Rice (2003) found evidence of a
relationship between class size and achievement for elementary
students, other researchers such as Hattie (2003) have not found
significant links. Mulryan and McCaslin claim that grouping does not in
itself improve lessons (1992, p. 168). Research concerning the

relationship between class size and student achievement remains an
ongoing need.
Preservice Teacher Impact as Evidenced
bv the Oregon Work Sample
Information concerning preservice teachers impact on student
achievement aligns closely with a core belief held by many educators
that teachers make a difference to student achievement (Wright, Horn,
& Sanders, 1997). In fact, studies indicate that teachers are probably a
major factor in determining student performance (Schalock, 1998). The
question remains then as to what impact preservice teachers should be
expected to have on student performance.
In Schalocks study, preservice teachers required to design and
teach a unit of work in Oregon have a significant impact on learning
(1998). Pretest and posttest data, as well as ongoing assessment of
various kinds, are used to measure student achievement. This study
indicates that an expectation of student growth as the result of
preservice teacher intervention is reasonable towards the end of their
student teaching experience.

Conceptual Framework
Effective Experiences
A need for reflective practice with a strong research basis is
highlighted by Fendler, (2003). As reflective practice is considered a
key factor in effective teaching, it is crucial, that preservice teachers are
given multiple and sustained opportunities to practice this skill (Rich,
Barcikowski & Boyd, 1995). Many groups and individual researchers
emphasize the importance of reflection on pedagogical theory, beliefs,
and attitudes (NCTM, 1991; National Research Council (NRC), National
Staff Development Council, 1995; 1996; Schmoker, 1996; Wiggins,
1998). It is essential that preservice teachers have many opportunities
to make connections between their academic theory and field
Structure of Performance-Based Assessments
Wilson indicates that preservice teacher assessment should
have certain essential components (1995). In particular, evidence of
active student engagement and resultant achievement must be part of
an effective assessment process. Teacher licensure must demand that

preservice teachers demonstrate application of the knowledge learned
in their university courses (Shalock et al., 1993). Many teacher
education programs are moving to alternative assessment methods
such as PBAs as a means to assess preservice teachers (Turner,
2002). PBAs allow the development of preservice teachers over a
period of time (Baron and Wolf, 1996). The definition for PBAs used in
this study is given in the following quote.
Performance assessments in education frequently attempt to
tap the complex structuring of multiple skills and knowledge,
including basic as well as higher-order skills, embedded in
realistic or otherwise rich problems (Messick, 1997, p. 62).
According to Snyder, Elliott, Bhavnagri, and Boyer, assessment
should provide feedback to assist both the preservice teacher and the
program as a whole (1993-94). The use of PBAs promises to give
important feedback to universities about their teacher education
programs. In addition to assessment information, PBAs can assist in the
development of effective teachers by allowing preservice teachers to
align theory with practical experience.
According to Johnson, using performances in assessment
implies a particular structure (1996). He describes eight main parts to a
1. involves a complex goal and requires good judgment

2. results in a whole that is more than the sum of the parts
3. is personalized by the student
4. allows for refinement during the process, has known criteria, and
gives many opportunities to demonstrate criteria
5. does not involve paf responses and indicates mastery of criteria
6. is judged according to impact, rather than the process used by
the student
7. requires appropriate adjustments when errors occur
8. results in student autonomy, where little assistance is required by
Rich et al. states that while PBAs are promising, there are many
measurement issues that remain unresolved (1995). The reliability and
validity of PBAs can be helped by using multiple sources of evidence.
Some suggestions given by Rich et al. include the use of portfolios,
reflective journals, and case studies. In order to satisfy these concerns
and also maintain the integrity of the performance assessment, a lens
for this study will be based on Johnsons criteria with the addition of
reflective journals that will be kept by the participants.
The PBA in this study required TCs to take control of
mathematical instruction for a small group of students for a period of
three months. TCs began by determining students mathematical needs

and designing appropriate instruction for their groups. Following the
three month implementation, TCs discussed the impact of their
instruction on students achievement and attitudes.
This study was designed to determine the effect of an
elementary mathematics PBA on preservice teacher and elementary
student learning. Four main research questions are addressed in the
How are teacher candidate attitudes towards teaching
mathematics affected by developing and implementing the
lessons from the PBA assignment?
What did teacher candidates learn about pedagogy and/or
assessment from the PBA?
What effect did the lessons have on elementary student
attitudes towards mathematics?
What effect did the lessons have on the elementary student
achievement of the specific content selected for instruction by
the teacher candidates?
As the main purpose of this study was to determine the impact of
the PBA on the development of preservice teachers, a case study of the

participants provided the most information and allowed for confirmation
of results through both quantitative and qualitative data. Given that the
use of PBAs is a relatively new aspect of teacher education
assessment, it was important to evaluate the effectiveness of the
specific PBA in question. Both the university and PDS required
information about TC and elementary student development.
Dissertation Structure
The dissertation follows a traditional structure. Following a
comprehensive literature review, the methodology is carefully detailed.
The results are divided into two main sections within chapter four, (a)
TC development and (b) elementary student achievement and attitudes.
Finally, in chapter five, models arising from the study are presented and
implications and future directions are discussed.

Historically, many different views have been posited concerning
teacher quality. In 1900 virtue and high moral character exemplified an
excellent teacher (National Research Council (NRC), 2001). An
emphasis on ideologies characterized teacher quality in 1950.
According to the NRC, the 1960s and 1970s altered the teacher quality
lens to one of technically visible skills.
The latter decades of the twentieth century heralded an
unprecedented government involvement in defining teacher quality.
Dwyer discusses numerous studies that have found teaching to be a
very complex activity (1994). Acknowledgement is given to the
complexity of teaching; depicting it as both science and art (Tomlinson,
1995; Uhlenbeck, Verloop, & Beijaard, 2002). At the same time
however, considerable value is placed on measurement of teacher
effectiveness by student achievement (Bracey, 2000).

Preservice Teacher Historical Context
Teacher qualifications varied greatly, in the latter part of the
nineteenth century, and were largely unregulated. Most elementary
teachers had no formal education beyond elementary school
themselves (Cremin, 1988). State departments of education and
teacher associations became more important forces during the first part
of the twentieth century (Cremin, 1988). A more scientific approach to
the skills and psychology of teaching characterized this time in
education (Fendler, 2003). In addition, schools, and education in
general, became more standardized across the nation. Subsequently,
teacher associations demanded uniform teacher training, certification,
and improved salaries (Tyack & Cuban, 1995).
During the latter part of the twentieth century, educational policy
recommendations for teacher preparation such as A Nation Prepared:
Teachers for the 21st Century (Carnegie Forum on Education and the
Economy, Task Force on Teaching as a Profession,1986), and What
Matters Most: Teaching for Americas Future (National Commission on
Teaching and Americas Future, 1996) along with the development of
new teacher standards have addressed teacher effectiveness and

accountability. Increasing concern about test scores in the 1990s has
resulted in more state legislative control in schools and teacher
Teacher quality and teacher preparation became the focus of a
1998 amendment to the Higher Education Act of 1965. Student
achievement drove the reform, and preservice teacher education was at
the heart of the Act. As a result, higher education institutions became
more accountable for successful teacher preparation (NRC, 2001).
Current Education Reforms
Accountability at all levels of education has resulted in numerous
reforms. Standards for school children, as well as for teachers,
influence curriculum and assessment (Council of Chief State School
Officers (CCSSO), 2000). In response to demand for proof of program
impact testing and assessment have begun to take precedence in many
educational decisions (National Science Foundation (NSF), 2002). The
impact on teacher preparation of high-stakes testing and increased
accountability is a concern to many teacher educators (Brawdy & Egan,
One of the consequences of such preparation is the production
of teachers who arrive at schools with technically sound instructional

practices but no real sense of who they are as either teachers or
learnerslet alone any real understanding of the material they will be
teaching (Brawdy & Egan, 2001, p. 438).
The ability of teacher education programs to produce quality
teachers is continually being evaluated. Many states have developed
performance standards for teacher certification and some states require
evidence of student achievement as a result of preservice-teacher
intervention (NRC, 2001). Title II, part of the Higher Education
Reauthorization Act of 1998 amendment, legislated reporting of
preservice-teacher performance by universities.
The teacher education standards and revision of teacher
licensure were intended in part to cause a reexamination of effective
teaching practice at all levels (Mason, 1997; Pecheone & Stansbury,
1997). Teacher education programs are being restructured in response
to these demands (Quatroche, Duarte, HufFan-Joley, & Watkins, 2002).
University faculties continue to grapple with the issues of educational
theory and practice and appropriate assessment for preservice
To date, the methods for achieving effective reform through the
standards remains somewhat elusive (Ben-Peretz, 2001). All
stakeholders have the goal of improving student learning but none is

clear about how this is best achieved. To begin this journey an
investigation of effective teacher characteristics will be examined.
Effective Teachers
Definition of Effective Teachers
National organizations such as the Interstate New Teacher
Assessment and Support Consortium (INTASC, 1992), the National
Council for Accreditation of Teacher Education (NCATE, 1997), and the
National Board for Professional Teaching Standards (NBPTS, 1994)
have written standards to describe effective teachers. For purposes of
this discussion, information from these three sets of standards, an NRC
(2001) study of teacher licensure tests, and McEwans (2002)
compilation of effective teacher traits have been combined to provide a
broad national picture.
Traits of effective teachers are commonly measured against
teacher impact on student achievement (McEwan, 2002). Although
different emphases are used and varying language chosen, many
commonalities are found among the standards and characteristics of
the major national groups (NRC, 2001). This discussion will be divided
into four main areas: characteristics, subject-matter knowledge,
pedagogical knowledge, and instruction.

Characteristics of Effective Teachers
McEwan cites research that supports the notion of teacher as a
knowledgeable practitioner as well as caring for students (2002). In fact,
balance of instructional methods and teaching as both science and art
is desirable. Research indicates that student learning occurs effectively
with different methods of instruction in a variety of educational settings
Personal traits of effective teachers as determined by McEwan
include passion, being positive and real, and being a leader (2002). She
also discusses teaching style. The effective teacher exhibits a personal
unique style, bringing drama, enthusiasm, liveliness, humor, charisma,
creativity, and novelty to teaching (McEwan, p.14). Examples of
effective teachers with different and unique styles are given to reinforce
this position.
Hatties meta-analysis of over 500,000 studies found two
affective characteristics to be prevalent in effective or expert teachers
(2003). Firstly, he describes the high respect that expert teachers have
for students. Secondly, he discusses teachers passion for teaching and

Palmer states that the choice of teaching as a profession is
frequently motivated by a desire to make a difference in students lives
(2002). He further iterates the passion of teachers by describing
teachers who buy their own classroom supplies, bring lunch from home
for hungry children, and encourage depressed youth. They do all this
in spite of everything because they have a deep passion to help
children learn and grow (Palmer, 2002, p. xviii).
It is neither desirable nor simple to measure these kinds of
characteristics. However, in a discussion about quality teaching,
recognition of the importance of affective traits is crucial. Without these
characteristics the effective teacher portrait would be incomplete.
Knowledge of Subject Matter
Subject-matter knowledge has been given credibility in
determining teacher effectiveness for over a century (Tyacks & Cuban,
1995). Current concern over standards-based instruction combined with
high-stakes testing, includes fears that the curriculum has become too
broad with little depth. In 2000, the Education Weekly reported that two-
thirds of teachers believe teaching to the test has increased (National
Science Foundation (NSF), 2002). In addition, these teachers attest to

an increase in a knowledge-oriented curriculum as determined by
standards and testing.
An examination of content standards must be included in any
credible discussion of effective teaching. It is essential that teachers
understand and competently implement standards across all subject
areas. An investigation of standards written for the main disciplines
reveals areas of commonality (Pecheone & Stansbury, 1997). The
emphases are as follows:
promoting literacy
developing numeracy
encouraging inquiry
building citizenship skills
life-long learning habits
INTASC, NCATE, and NBPTS all highlight deep subject-matter
knowledge as necessary for effective teachers (NRC, 2001). The
standards indicate that teachers should be able to create multiple paths
of understanding for students according to their own rich
understanding of content (Leinhardt, Putnam, Stein, & Baxter, 1991).
Shulman discussed areas of knowledge that effective teachers should
have (1987). These include knowledge of content, curriculum, and
educational goals.

Strong content knowledge as measured by the NBPTS is linked
to student performance (Blair, 2000; Quatroche et al., 2002). Fennema
and Franke cite research that supports links between teacher
knowledge and student performance (1992). According to Hattie, a
deep knowledge of content and pedagogy is important for effective
teaching (2003). He indicates that knowledge alone is not sufficient;
expert teachers integrate the knowledge across areas according to
students needs.
Knowledge and Use of Pedagogy
The three main groups of standards examined all discuss the
necessity for effective teachers to manage and monitor student
learning (NRC, 2001, p. 28). According to these groups, teachers must
select instructional methods appropriate to what is being taught as well
as in response to student needs. Likewise, suitable assessment
methods must be implemented and used to make good instructional
Quatroche et al. cite research to support a relationship between
student achievement and a teachers ability to develop and teach
engaging lessons (2002). Blair also discusses the positive impact that a
teachers ability to adapt and improvise instruction has on student

achievement (2000). Of the top five statistically significant influences in
student achievement, the teachers role is paramount in four of them
(Hattie, 2003). These four factors include teacher feedback,
instructional quality, direct instruction, and remediation.
According to Fennema and Franke, numerous studies have
determined relationships between pedagogical knowledge and student
learning in mathematics (1992). This research indicates that it is
important for teachers to understand students mathematical thinking
and to have good knowledge of various instructional approaches. In
addition, they describe Petersons conclusions that unless a teacher
can understand his or her own thinking in mathematics, knowledge of
content will not be useful in structuring the classroom so that students
can learn (Fennema & Franke, p. 157).
Reflective Practitioner
INTASC and NBPTS describe the importance of reflection about
pedagogical practice for effective teachers (NRC, 2001). Reflection
should be based on experience, data, and current research. These
national bodies assert that teachers need to model reflective practice in
order to inspire their students (INTASC, 1992; NBPTS, 1994). The tenth
of McEwans traits, based on a broad review of literature including

qualitative and quantitative research, relates to reflection and
metacognition (2002). She describes this trait as elusive because
although its results can be seen, the actual thought processes are not
directly observable.
Lowerys study supported the impact of reflection on teacher
learning (2002). In addition, Lowery cites previous studies by Schon
(1987), Valli (1992), and Kraus and Butler (2000) that also found
evidence of the significance of reflection in teacher development.
Quatroche et al. discuss the strong relationship between teacher
effectiveness and extensive reflection (2002).
Dimensions of Effective Teachers
It is generally recognized that teachers expectations, beliefs,
and knowledge play an important part in determining their classroom
behavior and the nature of learning opportunities they provide (Good,
Mulryan, & McCaslin, 1992, p. 185). Hatties massive meta-analysis
found that expert teachers do make a significant difference in student
learning. In fact, approximately 30% of the variance in student
achievement is attributable to teachers- what teachers know, do, and
care about (Hattie, 2003, p. 2). Through analysis, Hattie has defined
five main dimensions of expert teachers as follows (p. 5):

1. can identify essential representations of their subject
2. can guide learning through classroom interactions
3. can monitor learning and provide feedback
4. can attend to affective attributes
5. can influence student outcomes
Teaching is a continually evolving profession as new content is
constantly emerging, the social environment changing, and recognition
of diverse needs increasing. Teachers need a wide range of
knowledge, skills, abilities, and dispositions to perform these many
complex tasks (NRC, 2001, p. 32).
Student Achievement
Main Influences
As explained, teacher effectiveness accounts for about one-third
of variance in student achievement (Hattie, 2003). A longitudinal study
conducted over eight years by the Center for Research on the
Education of Disadvantaged Students (CDS) found that school and
teacher effects accounted for 24% of student achievement (CDS,
1992). According to Hattie, students account for about 50% of the
variance. Student ability, disposition to learn, and affective and physical
attributes jointly compose the student factor in the meta-analysis. In

addition, home, schools, principals, and peers each account for 5-10%
of the variance of factors affecting student achievement.
A recent study conducted by the Public Policy Institute of
California used a large database of student, school, and district data to
investigate factors that impact student achievement (Betts, Zau, & Rice,
2003). Resource allocation (including class size and teacher training)
and achievement were analyzed according to socio-economic-status
(SES) groupings of schools. Students at lower SES schools received
fewer resources and scored lower on reading tests. Student factors
impacting on achievement were student attendance, average scores of
peers, and other classroom-peer effects. Although some effects were
connected to teacher education, certification, and experience, the
impact for these factors was not totally clear (Betts et al., 2003).
Jacob and Lefgren found inservice-teacher training in the
Chicago public school system to have no statistically significant impact
on elementary student achievement in reading and mathematics
(2003). One reason posited by the researchers for the results was the
mismatch between content of the training and curriculum needs. The
results are consistent with other U.S. studies of the effect of inservice
teacher training on student achievement (Jacob & Lefrgren).

The authors discuss the apparent disparity between their results
and those of another large study. In a Jerusalem study, student
achievement was positively affected as a result of teacher inservice.
The school districts were however quite different. The program in
Chicago involved high poverty, low-achieving schools and was not
closely aligned with school curriculum. The Jerusalem district involved
mainly middle-class schools, closely aligned the inservice with school
curriculum, and provided after school support for students (Jacob &
Lefgren, 2003). This further illustrates the complexity of making
decisions with regards to effective teachers and student achievement.
Darling-Hammond cites National Assessment for Educational
Progress data that demonstrates clear connections between Year 8
performance, in mathematics and science tests, and teacher training
(2003). This research determined that the following teacher-related
factors are influential in student achievement: (1) subject-matter
training, (2) preservice teacher education and professional development
in diversity, (3) training in teaching higher-order skills, and (4) teacher
education or professional development in laboratory skills and hands-on
The above examples highlight the need to contextualize study
results. While it is possible to generalize some research findings, many

educational studies are dependent on local environmental factors. It is
important to screen study results through a filter of methods used,
number of participants, length, and location.
Class Size
Conflicting research findings make conclusions concerning class
size difficult to make. Possibly, analysis procedures in some studies
have failed to separate this factor adequately from other student
achievement influences. Analysis in Hatties study indicated no
significant link between class size and student achievement. Hattie
claims that in many recent studies, where connections between class
size and student achievement have been determined, influencing
factors have been subdivided too much (2003). Good, Mulryan and
McCaslin claim that grouping does not in itself improve lessons (1992,
p. 168). While some research has found connections between student
learning and small groups, most studies have failed to investigate
student thinking and problem solving (Good, Mulryan, & McCaslin).
Betts et al. found evidence of a relationship between class size
and achievement for elementary students but not for middle school or
high school students (2003). A longitudinal study in Tennessee, the
Student Teacher Achievement Ratio Study (STAR), determined four

clear relationships between class size and student achievement
(Student Achievement Guarantee in Education (SAGE), 2002).
According to study data, of students at 79 schools, those in small
graduate on time 72% of students, versus 66% from regular
classes and 65 % from classes with a paraprofessional
complete more advanced mathematics and english courses
complete high school -19 % dropped out, versus 23% from
regular classes and 26% from classes with a paraprofessional
graduate with honors
Research from Wisconsins Student Achievement Guarantee in
Education (SAGE) also seems to confirm the effectiveness of class size
reduction on student achievement (Waymack & Drury, 1999). Study
data indicated that the achievement gap between African-American and
white students narrowed in small classes. In addition, students in small
classes displayed more initiative, greater effort toward their school
work, and less disruptive and inattentive behavior in class, even after
they had returned to normal-size classes in the fourth grade (Waymack
& Drury, p. 3).
According to Good et al. (1992) small groups are not the utopia
of instruction that some educators had predicted. Simply placing

students in a small group does not guarantee improved performance.
Teachers must consider a variety of pedagogical factors including
instructional methods, relevance, social issues, interest, and
Slavin determined that group goals and individual accountability
are both important in small-group success (1990). In addition, giving
students opportunities to tutor each other can be beneficial for older
students (Good et al., 1992). Research is cited however, that found that
tutoring by second and third grade students in small groups was not
associated with achievement gains.
Mulryans research indicated links between student engagement
and small groups (1989). She notes however that this finding is
somewhat dependent on students level of achievement. Other
researchers have stated that low achievers may be reluctant to ask
questions in groups because they do not want to appear stupid (Good,
Mulryan, & McCaslin, 1992). Research concerning the relationship
between class size and student achievement remains an ongoing need.

Effective Mathematics Instruction
National Standards
The NCTM was one of the first national groups, in 1989, to
develop a comprehensive set of standards for both students and
teachers. The revised document in 2000 addressed pedagogical
principles in equity, curriculum, teaching, learning, assessment, and
technology. In addition, main content areas, including number and
operations, algebra, geometry, measurement, data and analysis,
problem solving, reasoning and proof, communication, connections,
and representation, are considered by grade-level groupings.
National standards have been further refined according to each
states interpretation and on the basis of local needs. Forty-nine states
have established mathematics standards (CCSSO, 2000). The
Colorado Department of Education (CDE) Model Content Standards for
Mathematics identify the need for students to be mathematically literate.
The main goal is to enable students to understand and use
mathematics that will be needed for citizenship and employment in the
21s* century (see Appendix B).

What is the Current State of Mathematics?
Many people believe that the 1989 NCTM standards heralded
the biggest mathematics reform of the 20th century (Kaufman, 2001).
Despite all reform efforts, recent studies indicate that mathematics
taught in elementary and middle school mainly consists of basic
arithmetic tasks that use memorization and repetition (Kaufman).
According to Kaufman, U.3. students perform adequately on basic
computational tests but have difficulty in applying mathematical
knowledge to problems.
Data continues to give mixed messages concerning U.S. student
achievement. The National Education Association (NEA) states that
current data indicates promising performance in mathematics by U.S.
students (1998). Assessments have indicated a slow but steady gain in
achievement for 9-year-olds and 13-year-olds (NSF, 1998). The
performance of 17-year-olds is however, close to that of 1973.
Following the release of A Nation at Risk, the majority of states
increased mathematics requirements for graduation. International
comparisons show that U.S. students have much room for
improvement in mathematics (NEA, p. 2).

Proficiency is an ambitious goal, and the United States will never
reach it by continuing to change slowly through education policy, the
Kaufmann report states (2001). In recent years, many states and school
districts have raised academic standards in mathematics, introduced
new assessments, and offered teachers professional development
opportunities. But these efforts have been inconsistent across different
districts and states. (NRC report as described by Kaufman).
According to Good et al., research continues to highlight
significant problems in mathematics education (1992). Many teachers
do not present mathematics in ways that are meaningful for their
students. Furthermore, it is apparent that an inordinate amount of time
is spent on low-level procedures rather than on reinforcing conceptual
Effective Practice
Knowledge of how to teach is as important as knowledge of
what to teach. Effective teachers have the ability to organize
mathematics so that fundamental ideas form an integrated
whole...mathematics teaching is dependent upon appropriate
professional training and certification, as well as ongoing
professional development (Graham and Fennell, 2001, p. 320).
Kaufman cites a recent NRC report that emphasized the
importance of forming a strong foundation for teaching mathematics.
The NRC committee did not favor the use of any one instructional

technique but instead posited that students background and needs
should be the first consideration when selecting a method (2001).
Teaching mathematics is a complex endeavor, and there are no easy
recipes (NCTM, 2000, p. 17). This premise is further supported by
Graham and Fennell, who explain that no one way to approach
mathematics teaching will work with every student in all situations
(2001, p. 319). In addition, Koehler and Grouws state that teaching is
becoming more complex over time (1992).
An expectation for students to act as mathematicians is
encouraged by the NCTM in its new Principles and Standards (2000).
Hiebert and colleagues recommend designing a curriculum around
problem solving in order to meet the reform demands and meet the
needs of all students (1996). Banchoff suggests that authentic projects
allow students to become genuine mathematicians (2000). In addition,
Banchoff states that students need to learn that finding solutions is
frequently not an automatic process.
A project conducted by Kings College in London investigated
characteristics of effective teachers of numeracy (Teacher Training
Agency, 2002). Quantitative and qualitative analysis of data from 90
teachers and 200 students allowed identification of effective teachers
on the basis of student achievement. The major relevant traits that

emerged from the research are combined with those stated by the
NCTM and summarized in the following list:
Highly effective teachers:
believe that all students can learn
believe that all students need equal access to resources
make connections between mathematical ideas
believe that understanding concepts is more important than
low-level procedural understanding
accept and use different approaches to instruction and learning
believe that assessment should guide instruction and be
beneficial for students
continue their own learning
are reflective and willing to refine their practice
Cornell also emphasizes the necessity for teachers to be aware
of individual student needs and problems (1999).
It is crucial for students to feel supported by their mathematics
teachers. Success in math is analogous to a foot race. Once
you fall behind, catching up let alone winning becomes
progressively less likely. Thus, a math teacher needs to
recognize when students become lost and then initiate
corrective action (Cornell, 1999, p. 227).

Preservice Teacher Education
Quality Programs
Traits of effective teachers, factors impacting on student
achievement, standards, and teacher certification requirements all help
to form a picture of a proficient beginning teacher. Darling-Hammond
cites studies that indicate a relationship between teacher education and
student performance (2003). In the Strauss and Sawyer 1986 study, a
1 % increase in teacher quality (as measured by certification exams)
was directly correlated with a 3-5% decrease in student failure rates in
North Carolina. Other studies in California, Texas, and New York have
also found connections between teacher training and subsequent
student performance.
According to Darling-Hammond, the increased expectations for
schools have a direct impact on teacher education programs (2003).
Reform proposals from A Nation at Risk (1983) to What Matters Most
(1996) maintain that teacher education needs to be restructured and
fault traditional forms of teacher education for not thoroughly preparing
teachers for their role in schools (Sandholtz & Wasserman, 2001, p.
54). Darling-Hammond has determined the qualities of effective teacher

education programs through an examination of national organization
standards and research data. Quality teacher education programs
common, clear vision of good teaching
well-defined standards of practice and performance
rigorous core curriculum in learning development, assessment,
and content pedagogy
use of problem-based learning, performance assessment, and
action research
well-supervised, extended clinical experience (30+ weeks)
strong relationships with PDSs
expert mentor/clinical teachers
PDSs that use the same standards as the teacher education
Learning to be a Teacher
According to Ball and Cohen there are four main areas that
teacher education programs need to emphasize (1999):
teachers need to know their subject matter in depth. A deep
understanding allows teachers to make rich and meaningful
connections for their students

teachers need knowledge of how students learn. This includes
knowing that children learn in different ways, being able to
engage students, and being able to challenge all students at
an appropriate level
teachers need a good understanding of diversity and the
impact that it has on students background experiences and
teachers need to differentiate instruction in order to meet the
needs of all students
Teacher education has the difficult task of undoing many
behaviors and restructuring beliefs that have been learned through
many years of schooling (Ingersoll & Kinman, 2002). This must be
addressed, because research has indicated that preservice teachers
background school experiences strongly influence their teaching (Lee &
Krapfl, 2002). What... becomes vital then, for the preservice teacher, is
to undo all that they have learned and observed through their
apprenticeship of observation (Hartman, 2003, para. 6).
Teacher education programs have tended to use a transmission
approach to learning. Research indicates that theoretical knowledge is
not sufficient to ensure a real understanding of pedagogy (Stuart &
Thurlow, 2000). Szabo, Scott, and Yellin describe the current position

on teacher education as encouraging opportunities for preservice
teachers to construct their own knowledge concerning effective
teaching (2001).
While there is general agreement concerning effective teacher
qualities, more research is required to determine characteristics of
optimum teacher preparation programs. What can be concluded is that
learning to teach is a highly complex process that is very personalized
and contextualized. Therefore, while no firm generalizations are
applicable across all education contexts for all preservice teachers,
there is a rich and diverse knowledge base that informs preservice
teacher education programs (Mayer, 1999).
While learning to be an elementary teacher has many general
principles, there are some experiences that relate solely to teaching
mathematics. The following section examines the research about
learning to be a mathematics teacher in elementary schools.
Learning to be a Mathematics Teacher
The last few decades have seen a renewed interest in preservice
teacher education in mathematics. At the same time, however, Graham,
Li, and Curran have found that mathematics programs for preservice
teachers have changed little from the mid-1900s (2000). According to

Brown and Borko, reports indicate that many teacher education
programs do not prepare students to teach mathematics according to
the reform movement (1992). This information has prompted more
research into the nature of teacher education programs with regards to
mathematics. Much of the research has investigated content and
pedagogical knowledge but because of the nature of teachers work it
can be difficult to distinguish between knowledge and cognition (Brown
& Borko). Ball determined that many preservice teachers have
inadequate knowledge of the underlying principles of mathematics
It is important to realize that teacher education is not the
beginning of students exposure to teaching. In fact before they take
their first professional course, future mathematics teachers have
already clocked over 2,000 hours in a specialized apprenticeship of
observation (Lortie, 1975, p. 61) which has instilled not only traditional
images of teaching and learning but also shaped their understanding of
mathematics (Ball, 1989, p. 1). The role of methods courses in
addressing beliefs, ways of thinking, content, and pedagogy is daunting
given the small amount of time allowed for this learning.

Quality Concerns
The difficult task faced in mathematics methods courses is
further highlighted by Hembrees 1990 meta-analysis of 151 studies.
Analysis revealed that preservice elementary teachers have the highest
level of mathematics anxiety of any major at universities. This assertion
is supported by Battistas 1986 study and Rech, Hartzell, and Stephens
1993 study (Quinn, 1997). Research continues to indicate that regular
university mathematics courses have little influence on either
knowledge level or positive attitude towards teaching (Stevens and
Wenner, 1996). This is of particular concern because a strong link has
been established between mathematics anxiety and poor conceptual
understanding (Quinn, 1997). Research has also found links between
school students attitudes and the anxiety of their teachers (Trujillo &
Hadfield, 1999). In addition, evidence of poor mathematical content
knowledge in elementary preservice teachers has been found in
numerous studies (Quinn, 1997; Stevens & Wenner, 1996).
Sloan, Daane, and Gisens study of the relationship between
mathematics anxiety and learning styles in elementary preservice
teachers provided new information (2002). While accounting for only

8% of the variance, a relationship between global learners and
mathematics anxiety was found. Global learners are described as right-
brain dominant people who are holistic, divergent and intuitive. Sloan et
al. suggest that the link with mathematics anxiety may be due in part to
the systematic, sequential approach given to mathematics in many
traditional courses.
According to research, most preservice elementary teachers
have been exposed to traditional mathematics instruction that
emphasized procedures and consisted primarily of direct instruction
(Battista, 1999; OBrien, 1999). Alsup cites research to suggest that
traditional approaches to mathematics learning have contributed to the
poor understanding observed in many preservice elementary teachers
(2003). Consequently Alsup favors an active, student-centered
approach to mathematics methods in university. He cites numerous
studies that indicate an increased likelihood for preservice teachers to
use a constructivist approach to teaching mathematics when they have
personally experienced it.
Learning to be Student Centered
Methods course must prepare preservice teachers in accordance
with the NCTM standards. This includes attention to mathematical

knowledge, commitment to students learning, self-efficacy, professional
attitudes, and pedagogical knowledge that is aligned with the reform
movement (Hughes, 1999). Therefore, part of the role of methods
courses should be to model a mathematics community like that in their
future classrooms (Alsup, 2003; Ball, 1989).
According to Brindley (2000) and Holt-Reynolds (2000), it is
beneficial for preservice teachers when they are provided with an
active, learner-centered environment with authentic activities. Ellsworth
and Buss determined that preservice teachers felt more confidence in
their abilities when they were given some responsibility for their own
learning (2000). These students indicated that class discussion was
extremely valuable in a methods course. Alsup states that exposure to
student-centered teaching increases the likelihood of preservice
teachers using similar methods in their own classrooms (2003).
A study at the University of Nevada investigated the impact of a
mathematics methods course, designed according to NCTM standards,
on preservice teacher content knowledge and attitudes (Quinn, 1997).
Overall there was significant improvement in the number of questions
answered correctly, although there was little improvement seen in four
specific questions that related to fractions, long division, geometry, and
probability. Preservice teachers attitudes towards mathematics

improved significantly during the study. These results provide support
for further research into effective preservice teacher education.
Preservice Teacher Beliefs.
Teacher beliefs have a major impact on pedagogical practice
(Thompson, 1999). For this reason, development of appropriate beliefs
in preservice teachers is vitally important. Emenaker investigated the
impact of a problem-solving based mathematics course on preservice
elementary teachers beliefs (1996). He began his study with a concern
that the beliefs these teachers hold about the nature of mathematics
and what it means to do mathematics actually interfere with their ability
to help students become successful problem solvers (Emenaker, p.
75). The problem-solving approach was associated with positive
changes on the belief scales. Analysis also revealed a relationship
between level of achievement and change in beliefs.
Simply exposing preservice teachers to new ideas is often
insufficient to cause change. Studies have revealed that teachers often
incorporate seemingly conflicting information into their current schema
(Thompson, 1999). However, it is possible to expose teachers to ideas
that alter their thinking and practice. Thompson cites studies by
Carpenter, Fennema, Peterson, Chiang, and Loef (1989) and Cobb,

Wood, and Yackel (1990) that found evidence of a change in beliefs
and action. Cobb and colleagues determined that teachers need to be
able to reflect in and about their own classrooms.
Brown and Borko give evidence, from a longitudinal study at
Michigan State University, that preservice teachers beliefs about
mathematics and teaching can be significantly altered through
participation in methods courses (1992). The impact on actual
classroom teaching, however, was not totally clear in this study. In
addition, Quinn found that preservice teachers developed positive
attitudes towards using manipulatives in their teaching after exposure to
a manipulatives-based methods course (1998).
Preservice teacher beliefs about mathematics learning are
frequently challenged by a constructivist approach (Steele & Widman,
1997). The impact of a mathematics methods course on conceptions
about mathematics teaching and learning was investigated through
ethnography. The preservice teachers in the study broadened their view
of mathematics from being primarily computational to a deeper
conceptual understanding and they became more willing to take risks
during the course. By the end of the study, participants described their
future classrooms using a constructivist philosophy.

Kelly refers to a premise, proposed by the mid-continent regional
educational library in Colorado, that in order to teach in an exemplary
way teachers must understand how students learn (2001). In Kellys
study, 83 elementary preservice teachers undertook an inquiry-based
mathematics and science course using a spiral approach. Data analysis
revealed a significant positive change in the following areas: (1)
confidence to use an inquiry-based science curriculum to support the
learning of mathematics, (2) confidence in teaching mathematics and
science, (3) beliefs concerning the use of hands-on activities.
Conversely negative changes were found in: (1) beliefs concerning the
importance of direct instruction, (2) anxiety towards doing and teaching
Future Course Directions
A recent NRC report details the necessary components of a
preservice elementary mathematics teacher education program.
Universities should create programs or courses that emphasize
thorough knowledge of mathematics and of processes through which
schoolchildren come to understand the subject (Kaufman, 2001, p. 2).
Vacc and Bright report that when mathematics methods courses are
consistent with field experience, preservice teachers are more likely to

alter their beliefs (1999). In their research, preservice teachers adopted
a more constructivist philosophy with regards to teaching mathematics.
Taylor discusses the impact of standards on preservice teacher
education in mathematics (2002). Suggestions are given for students to
create lessons using information from current research data. Taylor
posits that programs should immerse the preservice teachers in both
theory and practice (p. 138). Other educators suggest using an inquiry
approach to learning for preservice teachers (Loucks-Horsley, Hewson,
Love, & Stiles, 1998).
Lampert and Ball discuss the role of methods classes in
preparing preservice teachers as agents of change in mathematics
reform (1998). Many current teacher education programs only prepare
students to teach in current schools with little consideration given to the
future. Lampert further iterates beliefs that preservice teachers must
become part of a professional community (1990). Teacher education
must incorporate mathematics reform and respond to the changing
needs of students (Graham & Fennell, 2001). This implies that there is
a role for methods courses in assisting preservice teachers to become
knowledgeable and reflective practitioners.

Field Experience
While research provides evidence supporting the value of good
field experience, poorly planned internships or experience at schools
with philosophies that differ significantly from the university may have a
negative impact (Gallego, 2001). Promise of positive results is seen in
programs that make connections between field experience and
university courses. Szabo, Scott, and Yellin describe a program that
integrated a classroom management class with a one day a week field
experience (2001). The preservice teachers in this study became more
empowered through this integrated approach.
Field experience has frequently presented preservice teachers
with a traditional teacher-centered pedagogy (Mewbom, 2001). To
address this concern, the University of Georgia developed a field
experience program that directly supported the mathematics methods
class. Preservice teachers were given the opportunity to work closely
with an individual student in mathematics. Mewbom explains that, in
this way, the management concerns were lessened and the preservice
teachers were able to reflect critically on the intricacies of pedagogy.
Follow-up discussions were conducted in the pre-requisite methods

class. In addition, supervisors of the preservice teachers later observed
an increased readiness to teach mathematics.
Field experience has more impact on a preservice teachers self-
efficacy than methods courses do (Hughes, 1999; Lowery, 2002).
Growth and development of preservice teachers during field experience
is enhanced through opportunities for discussion and reflection
(Lowery, 2002). It is becoming increasingly clear that the benefit of field
experience is closely related to the nature of the school/university
Most recently, teacher education institutions have engaged in
discourse concerning the most appropriate mixture of courses and
practicum. While seeking a balance between state enforced teaching
standards and philosophical beliefs, many teacher education programs
are increasing collaboration between K-12 schools and universities
(Goodlad, 1999).
University and School Partnerships
Effective Teacher Education in Professional
Development Schools fPDSs)
The commonly accepted aim of teacher education is to ensure
quality and effectiveness in teachers who receive a license (Schalock,

1998). NBPTS has determined the characteristics that exemplary
teachers should exhibit (2000). These include knowledge of students
needs, knowledge of content and curriculum, meaningful applications of
knowledge, learning environment, and assessment.
Achieving a balance between theory and practice has long been
the aim of teacher education programs, but research has indicated that
graduates from many traditional programs have difficulty in transferring
what they have learned in university classes into their own teaching
(Osterman, 1991). Abdal-Haqq states that many traditional field
experience programs fail to provide substantive experiences that
promote appropriate development of effective teaching principles
(1992). Effective programs allow preservice teachers to practice the
methods they are being expected to adopt (Blume, 1998; Blumenfeld,
Krajcik, Marx, & Soloway, 1994; Ferry, 1995; Radford, 1998; Tetley,
According to Darling-Hammond, Griffith, and Wise, discrepancies
between what preservice teachers learn in theory and what they then
actually practice can be partly addressed through closer relationships
between universities and schools (1992). Korthagen and Russell
suggest that integrative teacher education programs with extensive field
experience will assist in transfer of knowledge into practice (1999). It is

considered essential for preservice teachers to experience first hand
the theory linked with practice within classrooms (Darling-Hammond,
An increasingly popular response to the need for closer
school/university relationships has been the implementation of
professional development schools (PDSs). PDSs provide a unique
collaboration between universities and schools, giving preservice
teachers continuous opportunities to learn: by teaching, by doing, and
by collaborating (Darling-Hammond, 1994; Hillkirk, 2000).
School/university partnerships that follow the PDS model provide
opportunities in research, professional development, and teacher
training beyond anything that an individual institution can offer.
Traditional teacher education mainly involves transmission of
learning from a professor to a preservice teacher. As research
indicates, teachers tend to teach using methods they are most familiar
with. It is therefore important for teacher education to model a variety of
methods in the preservice period. For example, according to Darling-
Hammond, teachers will be more likely to use cooperative groups if they
themselves have had similar learning experiences (1994).
School/university partnerships can provide a variety of learning
experiences in authentic settings.

Description of PDS
According to Improving Americas Schools (1996) there are
about 200 PDSs in the U.S.. There is diversity among the schools but
there are some commonalities (Abdal-Haqq, 1995). In general, PDSs
are public schools, involved in reform efforts, working in collaboration
with districts, universities, and professional associations, and using
research to guide their decisions (Abdal-Haqq).
Sandholz and Wasserman summarize the four main goals of
PDSs, as defined by the Holmes group and Levine: (1) maximize
student learning, (2) support professional teaching practice, (3)
enhance the professional education novice and veteran teachers, and
(4) encourage research and inquiry related to educational practice
(2001). Darling-Hammond describes the role of PDSs as joining theory
and practice in a way that changes the character of beginning teachers
Sandholtz and Wasserman studied preservice teachers and their
cooperating teachers in a traditional program and a PDS program
(2001). Analysis of the programs revealed that PDS preservice teachers
displayed more confidence in teaching and less anxiety about working

with students. Cooperating teachers from both programs had extra
responsibilities and disruption of their classes, but also gained from an
examination of their own pedagogy (Sandholtz & Wasserman, 2001).
PDS teachers, through the partnership structures, received support to
solve problems and discuss concerns. All preservice teachers dealt with
problems associated with learning to teach. PDS preservice teachers,
however, seemed to have more support in the form of school seminars,
peer observation, and through general school involvement.
Impact of Partnerships
The impact of PDSs is a relatively new research area. Available
data concerns preservice teachers, schools and students. Clearly,
PDSs are beneficial for both the schools and universities (Sandholtz &
Wasserman, 2001). A five year study found that teachers who had been
in a one year internship during their teacher education rate had a higher
retention rate (Darling-Hammond, 2003). Conversely, studies continue
to show that teachers employed under emergency licensure or in
programs like Teach For America have low retention rates.
Cobb (2000) described the need for research concerning the
impact of PDSs on students, teachers, preservice teachers, and
universities. Cobb indicated that the few studies completed in this field

all attested to the positive impact PDSs have on participants. According
to Cobb, Morrows 1995 study and Tusins 1992 study indicated that
PDS preservice teachers outperformed their non-PDS peers in
pedagogical practices such as inclusionary practices.
Wimsatt found that cooperating teachers in PDSs stated that
their own teaching practices had changed as a result of program
participation (1996). In addition, Cobb stated that most study findings
have connected PDS participation with positive preservice teacher and
cooperating teacher attitudes (2000). A few studies, however, such as
the Long and Morrow study (1995), have found no significant
connection between attitudes and beliefs and PDS participation.
An Illinois university incorporated INTASC principles into their
PDS teacher education program (Nierstheimer, Lloyd, Taylor,
Moore,and Morrow, 2000). The role of the preservice teachers during
their first year varied according to each cooperating teachers needs but
did involve various kinds of instruction in both small groups and with
entire classes. Cooperating teachers reported more individual time and
feedback available for all students.
The preservice teachers stated that the program provided
invaluable orientation to teaching in schools. Given that the preservice
teachers will be at the same school during the four year program,

invaluable relationships will be established with staff and students. It is
believed that the elementary students will benefit increasingly with each
passing year of the program. In addition, the preservice teachers will
have an excellent knowledge of the districts standards-based
curriculum and be well prepared for their first year of teaching
(Nierstheimer et al., 2000).
Cobbs survey at an elementary school investigated cooperating
teachers perceptions of impact of the PDS at the end of the third and
fourth years participation (2000). While teachers believed that the PDS
had an overall positive impact on students, there was uncertainty
concerning any connections with standardized test scores. Cooperating
teachers stated that the PDS program provided excellent teacher
preparation and many indicated that the interns were better prepared
and more capable as beginning teachers than they had been. While
some cooperating teachers indicated that PDS participation had
impacted their teaching philosophies and practice, many believed that
the impact had been negligible.
Walling and Lewis described part of a PDS study that
investigated professional identity development (2000). Data concerning
self-efficacy and general education beliefs were collected from
preservice teachers in traditional and PDS programs through

questionnaires and open-ended written responses. PDS preservice
teachers indicated more concern for system-wide education issues, as
well as a belief that they would remain classroom teachers during the
foreseeable future.
Walling and Lewis summarize their findings that data presented
here provide some evidence that the types of experiences
which preservice teachers have in professional development
school programs may indeed foster beliefs and attitudes that
represent a more mature professionalism than that of traditional
preservice teachers (2000, p. 4).
Assessment of Preservice Teacher Proficiency
Teacher Certification
The main purpose of teacher certification is to control teacher
quality (NRC, 2001). Licensure tests are largely intended to engender
confidence concerning teacher competence (American Education
Research Association, 1999). The tests, however, are only one part of
the process; preservice teachers are assessed in university courses
and in field experience (NRC).
A two year project by the NRC examined teacher certification at
every possible level. Data from a meta-analysis and case studies
informed the committee. The type of licensure testing demanded by
various states is summarized in Table 2.1. The tests originate with the

Educational Testing Service, the National Educational Service, and
individual states.
Concerns continue to be raised about the ability of tests to
distinguish teachers who are proficient from those who are not (NRC,
2001). Given the complexity of teaching, doubts have been raised as to
whether tests can appropriately evaluate teacher knowledge. This
concern is particularity pertinent for multiple-choice tests.
Although tests have been validated for content, there have been
few efforts made to validate the tests against other variables such as
student achievement (Pullin, 1999; Sired & Green, 2000). The NRC
states that new forms of testing provide opportunities for validation
against multiple criteria (2001). In addition, a recommendation is made
for states to use multiple forms of evidence for teacher licensure.
Table 2.1
Teacher Licensure Tests in 1998-99
Test Type No. of States
Basic Skills 38
General Knowledge 14
Subject Matter Knowledge 21
Pedagogical Knowledge 28
Subject-Specific Pedagogical Knowledge 7

There is certainly resistance to enforced change. Berlak and
Emery describe the negative impact that the Teacher Performance
Expectations have had in California (2003). Preservice teachers no
longer have the luxury of reflection; instead they are to be prepared for
an environment of high-stakes testing and accountability. In fact, Berlak
and Emery believe that, in an effort to comply with legislation, teacher
educators have made curricular decisions that are contrary to their
better judgment (p. 32).
Traditional Assessment
According to Hughes, while most teacher educators attempt to
provide experiences that will help prepare preservice teachers for the
many complexities of teaching, until recently they have assessed
competencies in notably traditional ways (1999). Kennedy and Parks
also discuss the disparity between effective teacher characteristics and
assessment in teacher education programs (2000). Affective skills in
particular are given little attention in preservice teacher assessment
(Goodlad, 1994).
While testing and assessment have been viewed
interchangeably by many educators in the past, there are some
important distinctions. Using Galsers work, Lin describes the purpose

of testing as being selective and predictive (2002). Assessment, in
contrast to testing, measures learning against particular criteria. While
standardized tests have good reliability they can lack validity.
Alternative assessments such as performance measures potentially
have more validity than tests (Shepard, 2000; Wiggins, 1993). This
occurs because performances are actual examples of the learning in
context (Shepard).
Alternative Assessment
Assessment reform began in K-12 schools and has bubbled over
into teacher education programs. The NCTM has advocated the use of
multiple assessment tools in schools and, consequently, their use
should be modeled for preservice teachers (Thompson, 1999).
Alternative assessment methods have actually been discussed since
the 1940s but several factors have proved to be enormous barriers.
These include acceptability of non-test data, the fact that alternative
assessment methods are time consuming, and the need for greater
organization (Lambdin, 1993).
Cohen discusses the need for assessments that address the
complexity of teaching in all of its many facets (1995). Alignment of
assessment with instruction is crucial in teaching mathematics

effectively (Brosnan & Hartog, 1993). According to Dwyer, research has
indicated that teaching is an intricate process and it is therefore
necessary to have assessment that matches this sophistication (1994).
According to The National Commission on Teaching and Americas
Future, assessment of teachers must emphasize classroom
performance (1996).
Many teacher educators have endeavored to model teaching
practices such as constructivism in their classes, but this has often not
included assessment (Knapp, 2000). Knapp posits that assessment
should facilitate course learning goals and be consistent with the
teaching philosophy and beliefs espoused in the class. Thoughtful
assessments that include cases studies, journals, and open-ended
tasks tend to promote deep thinking and understanding (Knapp).
Assessment that models K-12 expectations and provides rich
learning opportunities for a preservice teacher is ideal (Hughes, 1999).
In an attempt to provide more authentic assessment, many teacher
education programs have adopted portfolios (Krause, 1996; Schalock,
1998). Portfolios have the potential to provide valuable ongoing
information concerning a preservice teachers development of effective
teaching practices (Lyons, 1998; Stone, 1998; Willis & Davies, 2002;
Wolf, Whiney, & Hagerty, 1995). Houston and Warner state that a

major responsibility of teacher education is to facilitate the self-
monitoring necessary to function as critical and progressive thinkers
(2000, p. 74).
Following the implementation of a standards-based portfolio at
an Indiana teacher education program, faculty and students were asked
to provide feedback (Quatroche et al., 2002). The faculty described the
portfolio as resulting in more meaningful learning for students. They
noted that the process of putting the portfolio together encourages
thought, reflection, and analysis...(and) the culmination becomes a
professional growth experience rather than a final test (Quatroche et
al., p. 271).
Students involved in the study had mixed reactions to the
portfolios (Quatroche et al., 2002). A number of students found that
analysis and reflection was difficult and time consuming. Students with
good writing abilities appeared to be more positive about the portfolios
than those with poor writing abilities. Despite the difficulties, many
students expressed pride in their finished products and more
confidence about being interviewed for teaching positions.
Willis and Davies quantitative analysis of the impact of portfolios
on preservice teachers found strong evidence in favor of their use in
teacher education programs (2002). A majority of students rated the

portfolios as very worthwhile and believed that the process encouraged
deep reflection. Most students believed that the portfolio process had
better prepared them for seeking jobs. The researchers concluded that
program-wide portfolios provide opportunities for ongoing reflection and
are more effective than those used for individual courses (Willis &
Doebler, Roberson, and Ponder cite Shulman and Colberts
1989 work that discusses the need to connect theory and practice in
classrooms, particularly when solving problems (1998). Using case
studies, another alternative assessment form, has assisted students in
making their own connections between course work and classroom
experience (Doebler & Roberson). In particular, preservice teachers
engaged in case studies were found to be more likely to (a) use
appropriate technical terminology, (b) identify solutions to practical
educational problems, and (c) apply theory to practice.
PBAs incorporate multiple forms of authentic assessments and
have become popular in teacher education programs. In response to
educational reforms and the concerns expressed about licensure tests
many states are considering PBAs (NRC, 2001). In 2001, only one
state actually used PBAs for licensure decisions. Instead, many
universities have incorporated PBAs into their reorganized programs.

Performance-Based Assessment
PBA Description. Tenopyr defines PBAs as primarily alternative
measurement tools such as portfolios, work samples, and interviews
(1997). Wiggins emphasizes the task or process that results from such
an assessment (1993). Messick states that performance assessments
in education frequently tap the complex structuring of multiple skills and
knowledge, including basic as well as higher order skills, embedded in
realistic or otherwise rich problems (1997, p. 62).
Linn and Baker (1996) describe PBA characteristics as follows:
open ended
involve the use of higher order complex skills
often require long time periods
allow for some student choice
require human judgment for scoring
In addition, Linn and Baker depict PBAs and instructional goals
as being completely interconnected (1996).
Johnson and Jones analyzed two PBA models used for teacher
evaluation by Teach for America and the California Department of
Education (1998). Both models take a holisitic view of effective teaching
and examine actual classroom performance. In addition, the models

allow beginning teachers to become aware of strengths and
weaknesses in their teaching. The models are summarized in table 2.2.
Table 2.2
PBA Models for Effective Teacher Evaluation
Teach for America
Management for student learning and
Organizing curriculum for student learning
and growth
Teaching/lnstruction for student learning
and growth
Assessing for student learning and growth
Utilizing external resources for student
learning and growth
Teacher professionalism for student
learning and growth
California Department of Education
Creating and maintaining an effective
environment for student learning
Understanding and organizing subject-
matter knowledge for student learning
Engaging and supporting all students in
powerful learning
Assessing student learning
Planning and designing learning
experiences for all students
Developing as a professional educator
Clearly the models incorporate the main characteristics
advocated by effective teaching research. This includes best practices
in pedagogy (such as differentiating instruction), appropriate
assessment, and maintaining professionalism (using current research
data) (NRC, 2001).
Disadvantages of PBAs. Reliability and validity issues are of
concern when using PBAs (Messick, 1997; Moss, 1994). When PBAs
only include a few tasks, there is low reliability. Therefore, it is ideal to
implement multiple long and short term tasks (Messick). The question
also arises concerning transferability and generalizabiiity (Linn & Baker,

1996) . This can be helped by assessing standards over several tasks.
Clearly, this is both time-consuming and expensive on a large scale.
Uhlenbeck, Verloop, and Beijaard discuss the problem of grader inter-
reliability (2002). Consistency among graders of PBAs is a concern and
can be expensive to ensure.
In addition, the nature of PBAs can make construct validity
difficult to define (Messick, 1997). It may not be possible to define the
knowledge and skills that will comprise the performance. In general,
while PBAs can provide a comprehensive picture of students, they can
also be costly, open to judgment errors, and highly influenced by
situational factors (Mumford, Baughman, Supinski, & Anderson, 1997).
Advantages of PBAs. PBAs provide authentic contexts for
students to demonstrate their ability and to apply knowledge and skills
(Baron & Wolf, 1996; Darling-Hammond & Ancess, 1996; Sackett,
1997) . Preservice teachers are able to demonstrate complex
pedagogical skills in appropriate contexts and at the same time
experience methods of assessment that are advocated for their use in
the classroom (Thompson, 1999). Quatroche et al. attested to the
meaningful learning associated with authentic assessment (2002).
PBAs honor the individuality and professionalism of teaching.
The holistic outlook promoted by engaging in PBAs encourages

preservice teachers to understand the big picture of teaching as a
profession. Acknowledging multiple perspectives on good teaching
offers the opportunity for teachers to define their practice according to a
model of teaching consistent with their beliefs (Mason, 1997, p. 102).
PBAs give feedback to the assessor and assessed. Whereas
traditional assessment has commonly focused on pockets of unrelated
knowledge in a vacuum, PBAs incorporate theory and practice in
authentic settings (Uhlenbeck, Verloop,& Beijaard, 2002). Furthermore,
PBAs provide a more accurate picture than traditional assessment of a
preservice teachers potential as an effective teacher (Uhlenbeck et al.).
The effective use of PBAs addresses the concerns made by teacher
educators over the past few decades.
What Research is Needed in Teacher Education?
Despite the belief that teachers have a significant impact on
student achievement, the U.S. continues to invest in new curriculum
and testing rather than in teacher quality (Darling-Hammond, 2003). To
satisfy the requirements for quality teacher education programs,
increased funding is needed. Policy makers require convincing
research data before more funds are committed (Darling-Hammond,
2003). In reference to PDSs Cobb writes it is essential that the

research literature begin to reflect not only our strong philosophical
commitment to these collaborative ventures, but also an awareness of a
responsibility for accountability (2000, p. 64).
A need to determine the optimum components of teacher
education programs that produce effective beginning teachers is clear.
Accountability issues have increased pressure on all schools, and
districts require evidence of the impact of preservice teachers on
students. Previous studies have used quantitative and qualitative data
but have left a gap for more needed research. Cobb also writes that
from the limited PDS research available, which specifically measured
effectiveness, it is apparent that this field is still in its infancy stages
(2000, p. 65).
Concern about achievement in mathematics continues to prompt
calls for reform. Improvement in mathematics results dominates
professional development and public expectations. When this concern
is combined with teacher education reforms, there is a clear need for
research in teacher education that examines:
1. the impact of field experience on preservice teacher
development as elementary mathematics teachers
2. the impact of preservice teachers on elementary students
achievement and development in mathematics

Study Overview
This chapter gives details of the methods used for data collection
and analysis. In addition, a description of the PBA requirements, study
location, and participants are provided. The main research questions
addressed in the investigation are:
1. How are teacher candidates (TCs) attitudes towards
teaching mathematics affected by developing and
implementing the lessons from the PBA assignment?
2. What did TCs learn about pedagogy and/or assessment
from the PBA?
3. What effect did the lessons have on elementary student
attitudes towards mathematics?
4. What effect did the lessons have on the elementary student
achievement of the specific content selected for instruction
by the TCs?

The research process, summarized in Figure 3.1, was adapted
from Jarvinen, 2001 Figure 4.
Figure 3.1
Research Process Summary

Choice of Methods
Numerous research projects about preservice teachers have
collected data suitable for case studies. In studies examining preservice
teacher beliefs about teaching, data about attitudes and reflections
have been collected through interviews, written reflections, journals,
and focus groups (Boger & Boger, 2000; Gibson, Brewer, Magnier,
McDonald, & Van Stat, 1999; Gould, 2000; Mewbom, 1999).
Other studies have focused on preservice teacher effectiveness
and development, preservice mathematics teaching, and quality field
experience, and collected data in the forms of reflective journals, group
and individual interviews, professor observations, written responses to
open-ended questions, lesson video tapes, tests, lesson plans,
portfolios, e-mail messages, and artifacts (Brett, Woodruff, & Nason,
1997; Grisham, Laguardia, & Brink, 2000; Hedrick, 1999; Nicol, 1999;
Oldham, Van Der Valk, Broekman, & Berenson, 1999; Walling & Lewis,
2000; Willard-Holt & Bottomley, 2000). Formal case studies have been
chosen to investigate preservice teachers evolving beliefs about
mathematics teaching and to determine the nature of excellent
mathematics teachers (Mapolelo, 1999; Vacc & Bright, 1999).

A case study is an in-depth study, described from the
participants viewpoint, of a particular instance of a phenomenon in a
natural setting (Gall, Borg, & Gall, 1996). Whether viewed as an object
or method, Creswell defines a case study as an exploration of a
bounded system or a case overtime through detailed, in-depth data
collection involving multiple sources of information rich in context
(1998, p. 61). Soy explains that case studies provide understanding of
a complex issue (1997). In addition, Cousin and Jenkins describe case
study research as neutral between qualitative and quantitative
methods, and can be appropriately conducted within any paradigm
capable of studying an exemplary instance (2003, para. 2).
Some general weaknesses of qualitative research include: (a)
fewer participants, (b) less easily generalized, (c) difficult to aggregate
data and make systematic comparisons, and (d) difficult to confirm the
researchers ability to analyze data (Ratcliff, 2003). In addition, Brock,
Kukulski, and Tanis describe weaknesses pertinent to specific
qualitative data forms (2002). Interviews, for example, may (a) be
biased by the interviewer, (b) be influenced by the participants beliefs
of the interviewers expectations, and (c) involve incomplete recollection

(2002). Direct observation can be (a) time consuming, (b) rely on
selectivity of the recorder, and (c) data may be influenced by observer
presence. These potential weaknesses are partly addressed through
the advantages covered in the following section, but some of the
concerns warrant further discussion.
Determining generalizations based on single instances is difficult
(Cousin & Jenkins, 2003). While this can be partly assisted by the
representativeness of the case selected, as well as the use of multiple
data sources, it remains an ongoing problem (Miles & Huberman,
1994). Issues of validity and reliability are discussed in the latter part of
this chapter.
Another concern is the written report that summarizes results. It
may not be clear to readers how examples were chosen for inclusion in
the findings. Bachor suggests that the use of visuals can assist in giving
readers broad pictures of the information (2000). In addition, Bachor
describes analysis techniques where quantification allows the reader to
quickly determine the typicality of the evidence being reported (para.

Ratcliff describes some of the strengths of qualitative research
as follows: (a) provides depth and detail, (b) openness, (c) allows
readers to "view the experiences rather than have categories imposed
on them, and (d) attempts to avoid prejudgments (2003). Brock et at.
describe some additional strengths of case studies (2002). Interviews
allow information to be focused and insightful. Direct observation brings
reality and context to the data.
Guba and Lincoln posit that the complexities of human behavior
necessitate situational analysis of research data (1994). Case studies
provide avenues to study authentic classroom practice in context
(Stenhouse, 1988; Patton, 1990). The intricacies associated with
teaching can be realistically explored through the varied data formats
and multiple perspectives made possible through case studies.
In comparison to large samples with few variables, case studies
provide depth that results in a product (that) is a sharpened
understanding of why the instance happened as it did, and what might
be important to look at extensively in future research (Davey, 1991,
para. 1). Davey posits that case studies assist readers to become

familiar with an unfamiliar topic; in effect, learning a new language in
The great strength of case-study method is that it allows the
researchers to concentrate on a specific instance or situation
and to identify, or attempt to identify, the various interactive
processes at work. These processes may remain hidden in a
large scale survey but may be crucial to the success or failure
of systems or organizations. (Bell, 1993, p. 8).
Whv Use Case Studies?
Creswell (1998) recommends qualitative research for studies
involving how or what questions. Case studies are becoming an
increasingly popular choice for educational researchers seeking to
address complex questions (Kelly & Sloane, 2003). Kelly and Sloane
refer to statistics from 1992 to 1997 where case studies accounted for
41% of educational research. In order to answer this studys questions
satisfactorily, it was important to gather extensive data concerning
participants actions and attitudes at every stage, including interviews,
observations, tests, and artifact collections.
This particular study is based on a system that is bounded by
both time and activity parameters. Given the research questions,
investigating beliefs, attitudes, achievements and actions, a case study
approach was selected as the basis for study design. The choice of a
mixed-method case study follows a trend, described by Ratcliff, for

studies to blend aspects of quantitative and qualitative methods
Multiple qualitative methods including interviews, focus groups,
journals, and observations were employed to provide data from varying
perspectives. Quantitative methods, pretests and posttests, provided
definitive data concerning student achievement gains. The combination
of qualitative and quantitative methodologies chosen for this study,
provided rich data and strengthened the validity and reliability.
Initial Teacher Education Program Information
Program Structure. The study developed as a result of the
revision of the Initial Teacher Education Program (ITE) at MU. At the
time of this study, the 18-month teacher licensure program was for
students with a bachelors degree. Students, or TCs, as they are
generally referred to, took education courses at the university to fulfill
licensure requirements throughout the first year of the program. During
the second semester, TCs interned two days per week at a partner
school in addition to taking classes. Most TCs divided their experience
equally between primary and intermediate grades. The final semester of

the program required a full time commitment as interns at the same
partner school for a total of 56 days in a single classroom.
Demographically, 23% of elementary TCs were male and 77%
female. The average age was 31 years and the age range was 22-56
years. Ethnicity was described as 93% European American, 4% Asian
American and 3% Hispanic. All TCs had worked with children for a
minimum of 30 hours prior to acceptance into the program and 85%
had worked or volunteered more hours.
Program Philosophy. The knowledge base established by the
planning committee represents a new model of teaching, one that holds
that teaching includes attention not only to students learning and well
being but also to the renewal of K-12 schools (ITE Program Handbook,
1998, p.6). The teacher education program was based around five
teaching responsibilities as follows:
1. Teacher as Scholar
Teachers have a responsibility to understand the central
concepts, tools of inquiry, and structures of the disciplines they
2. Teacher as Instructor
In collaboration with others, teachers have a responsibility to
provide for effective instruction.

3. Teacher as Student Advocate
In collaboration with others, teachers have a responsibility to
create a healthy educational and social climate that nourishes
students intellectual, social, and emotional growth.
4. Teacher as Professional
Teachers have a responsibility for professional and ethical
behavior, continual professional development, and to engage in
professional activities related to teaching.
5. Teacher as Leader
Teachers have a responsibility to facilitate school improvement
through democratic participation, collaboration, and leadership.
In addition, TCs were intended to move through their program as
a cohort group, thus increasing collegiality and accountability within the
group. The program is based on the premise, that teacher education is
a joint effort of university and school faculties and both faculties share
ownership and responsibility for the success of teacher candidates
(ITE Program Handbook, 1998, p.7). An interdisciplinary approach was
emphasized in ITE courses and a number of courses were taught jointly
across disciplines.

Background Experiences. All TCs took a four-credit course
entitled Mathematics for Elementary Teachers. This course was taken
either before commencement of the program or during the first
semester. While this course focused primarily on content, methodology
was interwoven throughout each topic. A problem-solving approach
combined with hands-on learning characterized much of the class
During the first and second semesters, all TCs took a 12-credit
interdisciplinary course entitled Integrated Learning for the Elementary
School. Topics included curriculum, instruction, and assessment in
each of the four major areas of instruction (i.e. mathematics, science,
literacy, and social studies), as well as classroom management, use of
technology, inclusionary practices and differentiated instruction,
multicultural education, and trends and issues in education (Basile,
Nathenson-Mejia, & Olson, 2003, p. 295).
The program employs a gradual release of responsibility model
which utilizes co-teaching between teacher candidates and clinical
teachers to ensure that teacher candidates gradually assume teaching
responsibilities (ITE Program Handbook, 1998 p. 11). This gradual
release may follow the suggested sequence:

the TC observes a teacher who is teaching a type of lesson
that the TC will later co-teach
the TC and clinical teacher co-teach lessons like those
previously observed using plans prepared by the clinical
teacher or provided/prepared under a professors direction in
university classes
the TC and clinical teacher co-teach a variety of lessons, using
the clinical teachers plans
the TC and clinical teacher co-teach lessons, using plans the
teacher candidate and clinical teacher have coilaboratively
the TC and clinical teacher co-teach lessons, using plans the
teacher candidate has developed
the TC teaches alone with coaching and other support as
needed from the clinical teacher
The program fosters critically reflective inquiry about teaching
and learning, the development of collaborative skills necessary to work
effectively with other adults on schooling issues, the ability to meet the
needs of an increasingly diverse population of students, the ability to
function as change agents in schools, and the ability to apply

democratic principles in educational settings (ITE Program Handbook,
1998, p. 7).
Performance-Based Standards for Colorado Teachers
PBAs were designed to assess TC proficiency in 45 standards
that the state now requires for teacher licensure in Colorado (see
Appendix A). At MU, the 45 standards are incorporated into seven
PBAs, including the one on elementary mathematics. Successful
completion of the seven PBAs indicates proficiency in both knowledge
and action of the teacher education standards. In particular, the
elementary mathematics PBA addresses TC proficiency in the following
2.1 students develop an understanding and use of, number
systems, geometry, measurement, statistics and probability, and
functions and variable use
2.2 use Colorado Model Content Standards in Mathematics for
the improvement of instruction
3.1 design short and long-range standards-based instruction
3.3 develop and utilize a variety of informal and formal
assessments, including rubrics

3.5 use assessment data as a basis for standards-based
3.6 provide effective verbal and written feedback that shapes
improvement in student performance on content standards
5.3 apply appropriate intervention strategies and practices to
ensure a successful learning environment
5.5 understand the cognitive processes associated with various
kinds of learning and ensure attention to these learning processes so
that students can master content standards
5.7 accurately document and report ongoing student
6.2 design and/or modify standards-based instruction in
response to diagnosed student needs, including the needs of
exceptional learners and English language learners
6.3 utilize his/her understanding of educational disabilities and
giftedness and their effects on student learning in order to individualize
instruction for these students
6.5 develop and apply individualized education plans
6.6 collect data on individual student achievement and be
accountable for each childs learning

Elementary Mathematics PBA Design
The structure for the PBAs was partly selected to allow TCs to
implement Gardners notion that students ought to deal with the
answers to complex questions over time and those who guide their
learning must document increasing sophistication in the learners
thinking (1991). The complex question being addressed specifically for
mathematics instruction is Darling-Hammonds notion of: How do
teachers effectively balance the need to establish learner-centered and
learning-centered classrooms simultaneously? (1996). That is, how do
teachers accommodate the needs, interests, and abilities of individual
students while also ensuring that each develops new mathematical
Given the emphasis on mathematics in elementary schools, TCs
need opportunities to strengthen both content and pedagogical
knowledge. The main purpose of this study was to research how TCs
utilized the elementary mathematics PBA to assess instructional needs
of students at various grade levels. The researcher used methodology
of formal and informal observations, tests, interviews, and analysis of
student work. During the study, TCs had the opportunity to administer

and assess students with Colorado School Assessment Proficiency
(CSAP)-like questions and to determine subsequent instructional plans.
Prior to the PBA pilot, TCs had the opportunity to complete the following
tasks, as part of their teacher education program:
examine and interpret standardized test results, including
CSAP in their first semester
learn classroom observation skills
observe and analyze elementary mathematics classes
use informal and structured interviews regarding students
mathematical dispositions and knowledge
analyze samples of student work across all main mathematics
content areas, in both isolated and integrated formats
understand the essence and purpose of each mathematics
draw conclusions regarding important mathematics learning
and teaching issues from current research (e.g. tracking
students, skill acquisition versus conceptual development)
PBA Requirements
TCs needed to complete a number of tasks in order to satisfy the
PBA requirements. The following lists details the PBA tasks:

administer CSAP-like tests to elementary students
grade the tests with clinical teachers
select students for the group with clinical teachers and the site
interview students
examine examples of students work
write case studies of each students mathematical strengths
and weaknesses
determine individual and group goafs
design instruction for the group
administer instruction
conduct ongoing assessment of students achievement
administer CSAP-like tests after three months of instruction
grade tests
reinterview each student
write summaries of each student's progress during the three
write a summary of the PBA experience