Computer-based Personalization as Facilitator
of Mathematics Self-Efficacy and Mental Computation
Performance of Middle School Students
Joseph Patrick Martinez
B.S., University of Colorado, 1979
M.A., Ohio University, 1984
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Administration, Supervision and Curriculum Development
1995 by Joseph Patrick Martinez
All rights reserved.
This thesis for the Doctor of Philosophy
Joseph Patrick Martinez
has been approved
/2- // V / f s'
Martinez, Joseph Patrick (Ph.D., ASCD)
Computer-based Personalization as Facilitator of Mathematics
Self-Efficacy and Mental Computation Performance of Middle
Thesis directed by Associate Professor R. Scott Grabinger
Theoretically, the personalization of instructional context
enables learners to construct deeper meaning by assimilating new and
prior knowledge structures. In addition, personal relevance may
provide meaningful feedback to the learner about his or her own
capabilities with regard to certain academic tasks.
The present experiment tested these theoretical notions by using
microcomputers to personalize an instructional story with the
individual backgrounds and interests of middle school students
(N = 104). Personalization was compared to nonpersonalized and
control conditions. The study found that a single-session,
personalized, short story is an effective method for raising learner
percepts of mathematics self-efficacy. There were no main effects or
interactions on mental computation performance.
The findings lay new groundwork for future studies on
personalization as an instructional design strategy. Implications for
story-based personalization are discussed. Further empirical testing of
Social Cognitive Theory, which asserts that self-efficacy is a major
determinant that mediates the relationship between knowledge and
performance, is proposed.
This abstract accurately represents the content of the candidate's thesis.
I recommend its publication.
R. Scott Grabinger
To my lovely daughter, Alejandra, I wish to express my love and
gratitude for your patience and understanding while I concentrated on
this endeavor. To my wonderful son, Rafael, my love and gratitude for
keeping me young at heart. And, to my mother, Patricia Rebecca
Martinez, for instilling in me the values required to persist in life's
LIST OF TABLES AND FIGURES....................................xi
The General Problem.......................................7
Background of the Problem..............................8
Gender and Self-Efficacy............................8
Self-evaluation and Mathematics Performance........10
Personalization as Vicarious Modeling..............11
Mental Computation Strategies......................16
Sources of Self-Efficacy Information............19
Perceived Self-Efficacy and Performance.........21
The Role of Self-Efficacy in Academic Domains......23
Self-efficacy and Academic Performance..........23
Research from the Social Cognitive Perspective.....27
Purpose of the Study...............................30
2. REVIEW OF RELATED RESEARCH...............................32
Interventions Enhancing Self-Efficacy....................34
Social Comparative Modeling...........................36
Multiple Sources Modeling.............................37
Personalization of Instructional Context...............46
Personalization as Concrete Context....................52
Context and Mental Computation......................55
Chapter Two Summary.......................................58
Levels of Personalization..............................70
Mental Computation Performance.........................73
Data Formatting and Reduction...............................78
Results of Two-Factor ANCOVA................................86
Results of One-Factor ANCOVA................................88
Within-Grade Posttest Self-Efficacy.....................91
Predictive Power of Covariates..............................95
Mental Computation Performance..............................99
Experimental Design Assessment.............................100
Limitations of the Study...................................101
Self-efficacy as a Mediating Mechanism.....................104
Implications for Social Cognitive Theory................105
Implications for Computer-Based Math Instruction........105
Need for Further Research...............................106
Mental Computation Standards.........................Ill
A. Consent/Assent Forms...................................114
B. Self-Efficacy Pretest..................................116
C. Biographical Inventory.................................118
D. StoryTeller Screens....................................119
E. Self-Efficacy Posttest.................................121
F. Mental Computation Posttest............................123
FIGURES AND TABLES
Figure 3.1. Experimental Design....................................64
Table 3.1. Internal Consistency of Pilot Measures..................67
Table 3.2. Frequency Distribution of Participants by Group,
Grade, and Gender......................................69
Table 4.1. Means and Standard Deviations for Group x Gender
on Pretest Self-Efficacy, Posttest Self-Efficacy, and
Posttest Mental Computation Performance............................83
Table 4.2. Means and Standard Deviations for Group x Grade
on Pretest Self-Efficacy, Posttest Self-Efficacy, and
Posttest Mental Computation Performance............................84
Table 4.3. Means and Standard Deviations for Gender on All
Table 4.4. Between-Grade Means and Standard Deviations for
Grade on Both Dependent Variables.......................86
Table 4.5. Two-factor ANCOVA Table of Group x Gender with
Pretest and Grade on Posttest Self-efficacy.............87
Table 4.6. Adjusted Means Tables and Resulting P-Values for
Group x Gender with Pretest and Grade on Posttest
Table 4.7. Two-factor ANCOVA Table of Group x Gender with
Pretest and Grade on Posttest Performance...............88
Table 4.8. One-factor ANCOVA Table of Group with Pretest
and Grade on Posttest Self-Efficacy.....................89
Table 4.9. Adjusted Means, Standard Deviations, and Resulting
P-Values for Group with Pretest and Grade on
Table 4.10. One-factor ANCOVA Table of Group with Pretest
and Grade on Performance...............................90
Table 4.11. Within-Grade ANCOVA Table of Group with Pretest
on Posttest Self-Efficacy of 8th-graders...............91
Table 4.12. Adjusted Means, Standard Deviations, and Resulting
P-Values for Group with Pretest on Posttest Self-
Efficacy of 8th-graders...........................................92
Table 4.13. Within-Grade ANCOVA Table of Group with Pretest
on Posttest Self-Efficacy for 7th-Graders..............92
Table 4.14. Within-Grade ANCOVA Table of Group with Pretest
on Posttest Self-Efficacy for 6th-Graders..............93
Table 4.15. Within-Grade ANCOVA Table of Group with Pretest
on Performance for 8th-Graders.........................93
Table 4.16. Within-Grade ANCOVA Table of Group with Pretest
on Performance for 7th-Graders.........................94
Table 4.17. Within-Grade ANCOVA Table of Group with Pretest
on Performance for 6th-Graders.........................94
Table 4.18. Reliability Coefficients for Pretest and Posttest
First and foremost, I wish to acknowledge my major advisor and
mentor, Dr. R. Scott Grabinger, for his constant encouragement and
commitment to high standards. I also wish to acknowledge a true
scholar, Dr. William Alan Davis, who guided me through the
methodological design and analyses of this study. And to other
esteemed members of my committee, Dr. William Juraschek, for
sharing his wisdom about the dual role of cognition and affect in
mathematics learning, Dr. Duane Troxel, for 18 months of personal
support and motivation, and Dr. Paul Encinias, for his friendship,
encouragement, and enthusiasm.
Others have given support and assistance in many ways along
the way. They include Patricia Sigala, Muriel Woods, Paul and Brenda
Roper, Mike Medina, Steve Johnson, Dr. George Kretke, Dr. Brent
Wilson, Dr. Hanna Kelminson, Dr. Brian Holtz, Suzie Galaudet, Don
Middleton, Mike Morris, Bill Hendricks, and Dr. Martin Tessmer.
There are no doubt many others who have generously given their time
and effort, but I would be remiss not to mention my much-respected
aunt and uncle, Esther and Ernesto Jiron, for Wednesday suppers and
an unfailing commitment to extended family unity.
Self-efficacy, one's self-judgments of personal capabilities to
initiate and successfully perform specified tasks at designated levels,
expend greater effort, and persevere in the face of adversity (Bandura,
1977; 1986), is a relatively new construct in academic research (Multon,
Brown, & Lent, 1991; Schunk, 1991a, 1994). Although self-efficacy is
examined with much greater depth in therapeutic contexts, recent
studies show that self-efficacy holds significant power for predicting
and explaining academic performance in various domains (Lent,
Brown & Larkin, 1986; Marsh, Walker, & Debus, 1991; Schunk, 1989a;
Schunk, 1994; Zimmerman, Bandura, & Martinez-Pons, 1992).
Academic domains in which perceived self-efficacy receives
considerable attention include specific situations of
technological/computer literacy (Delcourt & Kinzie, 1993; Ertmer,
Evenbeck, Cennamo, & Lehman, 1994; Murphy, Coover, & Owen,
1989), writing (Pajares & Johnson, 1994; Pajares & Johnson, 1995),
choice of academic major (Hackett, 1985; Lent, Brown, & Larkin, 1993),
teacher preparation (Ashton & Webb, 1986), and mathematics learning
(Hackett & Betz, 1989; Norwich, 1987; Pajares & Kranzler, 1995; Pajares
& Miller, 1995; Randhawa, Beamer, & Lundberg, 1993). Additionally,
Albert Bandura (1977; 1986), cautions that while self-efficacy is
domain-specific, it is also task- and situation-specific; that is, percepts of
efficacy pertain to criterial tasks and situations in which they are
studied. This perspective enables researchers to gain a deeper
understanding of the interactive relationship between self-efficacy and
The present study examines how developments in the field of
instructional technology, design, and innovation serve to influence
positively self-efficacy and corresponding academic performance. The
conceptual framework for this study follows the perspective of Social
Cognitive Theory, the overarching theoretical framework of the
self-efficacy construct (Bandura, 1986). Within this perspective, one's
behavior is constantly under reciprocal influence from cognitive (and
other personal factors such as motivation) and environmental
influences. Bandura calls this three-way interaction of behavior,
cognitive factors, and environmental situations the "triadic
reciprocality." Applied to an instructional design perspective, students'
academic performances (behavioral factors) are influenced by how
learners themselves are affected (cognitive factors) by instructional
strategies (environmental factors), which in turn builds on itself in
The methods for changing students' percepts of efficacy,
according to Bandura (1977, 1986), are categorically subsumed under
four sources of efficacy information that interact with human nature:
(1) enactive attainment, (2) vicarious experience, (3) persuasory
information, and (4) and physiological state. The present study uses an
instructional design strategythe personalization of instructional
contextto influence the first three of these sources. Enactive
attainment is achieved by modeling experience during the instruction.
Vicarious experiential learning is achieved by personalization of
instructional context, in which characters in an instructional story
reflect the interests and personal relevance of learners. Persuasory
information is achieved through modeling, in which characters
overcome self-doubts and come to realize that effort and the
acquisition of cognitive skills are the primary determinants of
performance. Subsequent analyses gauge the effects of this
intervention on self-efficacy and its relationship to performance.
Personalization of instructional context (personalization) is not a
new instructional strategy. If fact, academics have long been aware that
relating new knowledge to students' existing familiarity with the world
is an effective way for learners to acquire deeper meaning from new
information. Learners' needs, background knowledge, and personal
experiences are thus accommodated in the instruction. The use of the
term personalization, however, has different meanings. The
Personalized System of Instruction (Keller & Sherman, 1974), for
example, applies more to individual pacing and the person-to-person
interaction between students and facilitator. Personalization is also
discussed as a means of incorporating students' goals and choice of
topics into a curriculum, particularly for addressing values (Howe &
Howe, 1975), and as a model of behavior modification for disruptive
students (Mamchak & Mamchak, 1976).
The term is used here in an instructional-design perspective.
From this perspective, the domain context of instruction is adapted to
facilitate increased relevance and familiarity to students with new
content (Ross, 1983). More specifically, the instructional context is
individually tailored to students' interests and backgrounds by merging
information from biographical inventories into the instructional
content. This design model, introduced by Anand and Ross (1987)
increases the personal meaningfulness of the content and is referred to
in the present study as the Anand/Ross model (see also, Ross &
Anand, 1987; Ross, McCormick, Krisak, & Anand, 1985). Miller and
Kulhavy (1991) give a concise definition of personalization that is
compatible with the Anand/Ross model: personalization refers to "the
act of using verbal modifiers and exemplars which have been lifted
directly from an individual's own repertoire of life experience" (p. 287).
Personalizing is used in this study within the context of
mathematics learning. Various forms of personalizing mathematics
learning are shown to be effective for either students of formal, school
contexts (Resnick, 1987; Ross, McCormick & Krisak, 1985) or informal,
non-school contexts (Carraher, Carraher & Schliemann, 1985; Carraher,
Carraher & Schliemann, 1987; Lave, 1985; Lave & Wenger, 1991).
With recent advancements in educational technology, namely
the proliferation of computers in the schools, personalization is made
more practical. Computer software can be programmed to
instantaneously transform data into something meaningful by relating
the data to a form or structure that makes sense and is knowable to the
individual learner (Anand & Ross, 1987; Ross & Anand, 1987). A
major premise of this study is that computer-based personalization
gives the learner greater capability to relate to, and make meaning
from, new information. While this approach has had success in
improving learning, motivation, and attitudes with regard to
mathematics word problems (Cordova, 1993; Davis-Dorsey, Ross &
Morrison, 1991; Davis-Dorsey, 1989; Lopez, 1989; Lopez & Sullivan,
1992), its potential has not been adequately explored in relation to its
effect as a source of self-efficacy information. Particularly, the potential
for vicarious experience is expanded when the computer presents
information with increasing familiarity, such as with familiar models
or characters in an instructional story.
Modeling, in addition to personalization, is also an effective
means of conveying vicarious information in both therapeutic and
academic self-efficacy research. Modeling refers to someone whose
behavior, speech and expressions serve as behavioral cues to the
observer. Early studies by Bandura and colleagues at Stanford
University revealed that observed modeling of therapeutic behaviors
could facilitate changes in percepts of efficacy for clinical patients
(Bandura, 1982; Bandura, Adams & Beyer, 1977). In academic settings,
Dale Schunk of Purdue University and colleagues consistently found
that live and filmed models are effective sources of efficacy
information (Schunk & Gunn, 1985; Schunk & Hanson, 1989)
especially when observed models maintain a high degree of familiarity
to the research participants (Schunk, 1987; Schunk & Hanson, 1985;
Schunk, Hanson & Cox, 1987). Modeling with a high degree of
familiarity is made practical as an instructional strategy by
computer-based personalization of instructional context.
The present study investigates the premise that computer-based
personalized storiesby way of character modelingcan effectively
influence students' mathematics self-efficacy and performance.
Instruction in mental computation strategies is selected as the criterial
subject matter for this investigation for several reasons: (1) knowledge
of strategies in mental computation gives confidence to learners, (2)
lack of strategies in mental computation may reduce learners
confidence and sense of efficacy, (3) school children readily evaluate
their own mathematical capabilities in comparison with peers based on
the ability to compute mentally, (4) unlike estimation, mental
computation requires an exact answer and facilitates a more accurate
view of the hypothesized relationship between self-efficacy and
performance, and (5) the National Council of Teachers of Mathematics
is calling for renewed interest in mental computation as an important
mathematics alternative for the twenty-first century (Reys & Barger,
1994; Reys & Nohda, 1994; Silver, 1994).
Many school children have self-doubts about mathematics. Of
course, the reasons are many, but this study suggests that one of the
reasons is due to low mathematics self-efficacy and a lack of strategies.
Which comes first? A lack of mathematics strategies could certainly
influence one's self-efficacy to perform in the domain. From the social
cognitive perspective (Bandura, 1977, 1986), however, children's' lack
of efficacy to perform can also adversely affect their ability to learn. The
problem must be addressed simultaneously; that is, children must
acquire task-specific knowledge about their capabilities as they
experience learning. This reduces faulty self-doubts and facilitates
more accurate appraisals of one's present capabilities. It also
demonstrates that learning mathematics improves with the acquisition
of strategies and is not solely a matter of innate cognitive ability.
The General Problem
Even average-ability students are sometimes known to do poorly
in specific subject areas while performing up to standard in others.
This phenomena is often reflected in the domain of mathematics. The
reasons for this phenomenon no doubt reflects the multivariate nature
of school learning. We must also take into account the idiosyncratic
nature of diverse learners. When capable learners do not perform up
to their potential despite positive environmental conditions, we must
give more attention to the self-regulatory processes within individuals
that promote or inhibit performance. From the social-cognitive view,
self-efficacy is an important factor that resides within the learner and
mediates between cognition and affect, and results in changes in
academic performance (Zimmerman, Bandura, & Martinez-Pons,
1992). The growth and reduction of self-efficacy is influenced over time
by social comparison with peers and is therefore more pronounced as
one grows older.
Background of the Problem
Gender and Self-Efficacy. By the time children reach middle
school (grades six through eight), the majority of them have made
significant judgments regarding their preferences toward certain
academic domains. These judgments are no doubt influenced by their
perceived capability with regard to the domains, as a result of social
comparison with peers and feedback from teachers. This is particularly
true in the domain of mathematics. At this stage, children are already
making decisions leading to career directions and choice of classes. By
high school, these decisions become more solidified. For educators, the
critical time to reduce or prevent mathematics alienation is in middle
school, or early on in high school.
Elementary school children usually have greater confidence in
their academic capabilities, and this confidence extends equally across
gender to both verbal and mathematical domains of learning. In later
years, however, gender differences regarding mathematics begin to
emerge. Fennema and Sherman (1978) found that there were no
significant differences with gender and mathematics learning, nor with
gender and motivation for learning, for 1,300 middle school children.
There were, however, significant effects on mathematics confidence
and on perceptions of mathematics as a male domain, with boys
reportedly averaging higher on both variables. When these results are
compared to previous research by the same authors, using the same
design but with high school students (Sherman and Fennema, 1977)
the overall results indicate that the gender gap on mathematics
confidence and perceptions begins to widen in middle school and
increasingly widens in high school. Although these studies did not
measure self-efficacy, per se, the significant variables of confidence and
gender stereotyping of a domain are contributing sources of self-efficacy
Expectations about doing well in mathematics (confidence)
relates closely to one's beliefs about personal capabilities for
successfully performing domain-specific tasks (self-efficacy).
Nonetheless, one can maintain high mathematics self-confidence in
general, but low mathematics self-efficacy with regard to specific tasks
such as mental computation of fractions. Likewise, gender stereotyping
of the mathematics domain may raise or reduce one's expectations for
overall success in the domain, but it does not determine precisely one's
beliefs for accurately solving particular mathematics problems. The
interacting perceptual influences of confidence and gender stereotyping
are influential sources of self-efficacy information, but not
determinants of beliefs about capabilities with regard to specific tasks.
Therefore, it is reasonable to examine the effect of gender on
mathematics self-efficacy with regard to task-specific performance
Self-evaluation and Mathematics Performance. Children make
judgments about their mathematical capabilities based on
accumulating knowledge and experience. They tend to see themselves
as either mathematically inclined or disinclined. These perceptions of
mathematics efficacy are shaped by an unlimited array of personal,
environmental, and behavioral factors. In the academic milieu,
learners make judgments about their capabilities based on comparisons
of performance with peers (Brown & Inouye, 1978; Schunk, 1987;
Schunk & Hanson, 1985; Schunk, Hanson, & Cox, 1987), successful and
unsuccessful outcomes on standardized and authentic measures, and
feedback from others such as teachers (Bouffard-Bouchard, 1989;
Schunk & Rice, 1987), parents, and peers. These sources of information
about their capabilities accumulate within individuals to form
perceptions of mathematical competencies. But these judgments are
fluid in that they are altered along the way according to new
experiences and knowledge. Students whose perceptions of their
capabilities are high often go on to challenge themselves, persevere in
the face of difficulties, and expend greater effort resulting in more
successful experiences. Self-doubters on the other hand often resign
early in the face of difficulty, and/or avoid the subject altogether to
preserve self-worth (Bandura, 1986; Brown & Inouye, 1978). A
challenge to educators, therefore, is to adopt instructional
interventions that not only make content more understandable, but
also increase the likelihood that learners will perceive their capabilities
as sufficient to the task.
Personalization as Vicarious Modeling
Cognitive self-arousal can take two forms: personalizing the
experiences of another or take the perspective of another. In the
personalizing form, observers get themselves emotionally
aroused by imagining things happening to themselves that
either are similar to the model's or have been generalized from
previous positive and aversive experiences [...] Research
conducted in this framework has been concerned primarily with
how role-taking strategies develop and affect social behavior.
However, experimental evidence is lacking on how vicarious
arousal can be affected by putting oneself in the model's place.
What little evidence does exist suggests that personalizing
modeled experiences is more vicariously arousing than role-
taking. (Bandura, 1986, p. 313)
If humans gained knowledge only through direct experience
children would be quite limited intellectually. Fortunately, children
can learn from observing others perform and also by observing the
consequences of the given performance. This form of vicarious
modeling is evidenced in the fact that children can learn from
televised depictions of human behavior (Beagles-Roos & Gat, 1983;
Meadowcroft & Reeves, 1989; Thelen, Fry, Fehrenbach & Frautschi,
1979). Children also can make judgments about their own capabilities
by watching models perform and imagining themselves performing
above, equal to, or below the observed level of performance. Children
make these judgments based on knowledge about themselves,
resulting from past experiences, and perceptions of their own
capabilities. The more substantiated evidence individuals gain from
observing others, however, depends on the similarity between
themselves and the model (Brown & Inouye, 1978; Littlefield & Rieser,
1993; Schunk, 1987; Schunk & Hanson, 1985; Schunk, Hanson & Cox,
1987). If children observe persons of obvious greater physical strength
perform a highly physical feat, they do not usually expect that they too
can perform up to that standard. If, however, children observe peers
children developmentally similar to the observerwhom they
perceive to have similar or lesser capabilities perform a requisite act,
then their senses inform them that they too possibly can perform at
that level. Therefore, model similarity, or peer modeling, is an
important source for judging capabilities for performing certain tasks.
We have no peers of greater similarity to us than ourselves.
Despite the number of traits possessed by others who are similar to us,
we gain considerable knowledge of what we can do from what we have
already done. Yet we have neither the time nor the opportunity to do
much in our limited lifetimes. From the social cognitive perspective,
we cannot be expected to gain our entire life's knowledge based on
personal experiences. The resulting dangers alone, experienced by
simple trial and error, would have disastrous consequences on our
well-being and life expectancy. The challenge for designers of
instructional stories is to model learning experiences so the learner
vicariously experiences the feelings and cognitions of the protagonist
or other characters.
The effectiveness of modeling is related to four subprocesses of
the observer: attention, retention, production, and motivation
(Bandura, 1986; see also Bandura, 1971, for more background on these
Attention requires that the observer attend to the actions of the
model. Activities that are modeled should therefore be relevant and
engaging to the learner.
Retention requires that the information be relevant and
meaningful to the observer. Learners must recognize some feature of
new information in order to perceive and classify it as something
meaningful (Sainsbury, 1992). From the social cognitive perspective,
observers can translate symbolic modeling (e.g. from media) into
meaningful behaviors which can be overtly emulated.
Production requires that the observer of a model be
developmental capable of emulating the behaviors of the model.
Children, therefore, adjust perceptions of efficacy depending on their
perceived similarity to the model (Schunk, 1983, 1987).
Motivation processes are often dependent on incentives. Social
cognitivists believe that symbolic incentives, including improved
social functioning and enhanced self-efficacy, inform observers of the
value and effectiveness of emulating modeled behaviors.
One method of modeling that attends to these four subprocesses
and demonstrates some success is personalization of instructional
context (Anand & Ross, 1987; Lopez & Sullivan, 1992; Ross & Anand,
1987; Ross et al., 1985). Computer-based instruction supports
personalization by allowing the learner to determine some of the
personal referents in which the content is situated. Unlike televised
modeling of instructional information, computers are able to
transform the instructional context to reflect individual input. This
capability is currently being explored by interactive strategies of
computer-based learning, in which the learner is addressed by name or
is allowed a certain degree of control in selecting the pacing,
sequencing, and characteristics of the instruction. This kind of
interactive personalization is described in the literature under the label
of learner control (see for a review, Kinzie, 1990).
Personalization, as used in the present study, allows the learner
to control the personal referents of instruction, such as character
names, in an instructional story. The learner transforms textual
information to contain familiar referents. Theoretically, this allows
the learner to envision being in the instructional context being
depicted and observe a model that is highly similar to the learner. This
degree of association enables learners to accommodate new
information with existing knowledge structures (Davis-Dorsey, 1989;
Ross, 1983; Ross & Anand, 1987).
Another potential benefit of personalized context using models
of high similarity is that the learner is able to experience vicariously
the emotions and cognitive representations of the models. Using
personalized characters in an instructional story, learners can gain
significant personal information about their capabilities with regard to
the instructional strategies enacted by the modeling characters.
Hypothetically, if the depiction is of positive gains in self-efficacy and
usage of strategies, then learners are able to picture themselves
similarly, thus gaining efficacy and using the strategies.
Mental Computation Strategies
During my own childhood, I was taught that the one (and
only) way to compute mentally was to imagine a
'chalkboard in my mind' and then to 'see' the numbers
and 'carry' as I would with paper and pencil.
Unfortunately, the numbers always disappeared before I
could finish the calculation. (Richards, 1991, p. 109)
The author of the passage above is now a college mathematics
instructor. To finish the passage, she goes on to write that she was
convinced she was not "good at mathematics," but after learning a
mental arithmetic strategy (left-to-right operation) she gained a
"renewed sense of confidence" in her mathematics capabilities.
Mental computation of mathematics is the deriving of exact
answers to mathematical calculations without the use of recording
devices such as computers, calculators, or writing instruments (Reys,
B., 1985; Reys & Barger, 1994). Up to 80 percent of mathematical
computations performed in non-technical settings, such as the
exchange of money or the determination of times and distances, are
done mentally (Reys & Nohda, 1994). Most often, we do not take the
time or have the opportunity to use recording devices in making
computations. Many times it is simply too embarrassing to use
recording devices at the check stand or restaurant and, therefore,
people avoid carrying calculators everywhere they go (Lucas, 1991).
Mental computation is still important in this age of high
technology. The National Council for Teachers of Mathematics
(NCTM) is calling for renewed emphasis on computational
alternatives, including mental computation and estimation, as
necessary strategies to complement advances in technology (Reys &
Nohda, 1994; Shumway, 1994). It is all too easy to make mathematical
errors using technological devices, and the ability to compute or
estimate numbers mentally assists children in checking calculations.
Mathematics competency is often displayed in the classroom by
those efficacious learners who have acquired mental computation
strategies. Peers performing mathematical computations swiftly and
accurately in their heads is associated with high mathematical
capability. Students, however, who never learned these strategies
compare their inadequate performance with those of skilled
performers and often judge their mathematics capabilities in general as
inferior. Attributing their lack of skill to lack of capability is what
Bandura (1986) calls "faulty self-knowledge." Relevant to this, one of
the goals of the instruction examined in the present research is for the
models to gain in perceptions of self-efficacy as they improve
performance on the criterial task. Mental computation is the criterial
task because it requires an exact answer and therefore reduces guessing
and alleviates the difficulty of judging estimation. Importantly, it also
assists the learner in making accurate self-appraisals of performance.
Mental computation is something the student must perform alone
without the aid of external devices and so a correct answer is clearly the
result of one's own internal processing. In judging the relationship
between levels of self-efficacy and performance, mental computation
provides a fairly exact measure.
There are, therefore, a number of major influences affecting
mathematics self-efficacy of children. These influences include
domain-stereotyping and gender-stereotyping, which often result from
social comparison with classmates. Mathematics capabilities are
usually displayed socially by mental computation performance in the
classroom, making this an important problem area for enhancing
Social Cognitive Theory provides a framework for explaining
how personalization and modeling are used to enhance the capabilities
of human learning. Self-efficacy is a major construct of this theory.
Bandura (1977), sought to address the related question of what
mediates knowledge and action beginning with his seminal work on
self-efficacy. Bandura (1986, p. 391) defines the performance
component of self-efficacy as
people's judgments of their capabilities to organize and execute
courses of action required to attain designated types of
performances. It is not concerned with the strategies one has but
with judgments of what one can do with whatever strategies one
Students feel self-efficacious when they are able to picture
themselves succeeding in challenging situations, which in turn
determines their level of effort toward the task (Paris & Byrnes, 1989;
Salomon, 1983; 1984).
Bandura (Bandura 1977, 1986) asserts that self-percepts of efficacy
highly influence whether students believe they have the coping
strategies to successfully deal with challenging situations. One's self-
efficacy may also determine whether learners choose to engage
themselves in a given activity and may determine the amount of effort
learners invest in a given academic task, provided the source and
requisite task is perceived as challenging (Salomon, 1983, 1984).
Several researchers have since investigated the relationship of
self-efficacy to learning and academic achievement, but research in the
area of academic performance is still developing (Lent, Brown, &
Larkin, 1986; Multon, Brown & Lent, 1991; Schunk, 1994). One
challenge to instructional technologists, therefore, is to investigate new
methods of raising learners' levels of self-efficacy and academic
performance through the use of appropriate technological innovations.
Sources of Self-Efficacy Information. People make judgments
about their capabilitiesaccurate or notbased on enactive experience,
vicarious experience (observation), persuasory information, and
physiological states. In school, children gain knowledge and
experiences through experiential activities. They also gain information
based on seeing how peers they judge to be similar to themselves
perform at various levels and under given circumstances. They also
are told by teachers, peers, family and others about their expected
capabilities. Children give themselves physiological feedback about
their capabilities through symptoms such as soreness or sweating.
These sources of efficacy information are not mutually exclusive, but
interact in the overall process of self-evaluation.
Bandura, Adams, & Beyer (1977) advise that enactive experience
is a highly influential source of efficacy information. Successful
experiences raise self-efficacy with regard to the target performance
while experiences with failure lower it.
Another source of efficacy information is vicarious experience
through observation. Observing peers, or peer models, especially those
with perceived similar capabilities, perform target performances results
in evaluative information about one's personal capabilities.
Verbal persuasion or convincing serves as another source of
efficacy information. Teachers, for example, can raise or inhibit
students' percepts of efficacy by suggesting whether or not they have
the capabilities to succeed in a given task (Bouffard-Bouchard, 1989).
Models can also be used to demonstrate to self-doubters that personal
capabilities are more often a result of effort rather than innate
Students often have physical reactions to anticipated events.
Many a public speaker testifies to sweaty palms and nervous vocal
reactions when performing a speech. These physiological indicators
are sources of self-efficacy information as well.
Social cognitive theory postulates that the aforementioned
sources of self-efficacy information are the most influential
determinants of performance.
Perceived Self-Efficacy and Performance. Early studies by
Bandura and colleagues focused on self-efficacy in therapeutic contexts,
such as investigating training methods to enhance patient self-efficacy
and reciprocal coping behaviors in phobic situations (Bandura &
Adams, 1977; Bandura, Adams & Beyer, 1977). It is only in the 1980s
that self-efficacy pertaining to academic performance began to be
investigated with great depth. To understand how this extension of
self-efficacy and performance unfolded from clinical situations to
academic situations, we can look back at one exemplar case.
Bandura and colleagues (Bandura, Adams, & Beyer, 1977)
administered a multi-level treatment program for snake phobics.
Participants in this case were assigned to a control condition in which
they received assessment but no treatment, or one of two treatment
conditions. In the treatment conditions, the phobics either participated
with or observed a therapist in a fearful situation with snakes.
On one level, participant-modeling, participants first observed a
therapist dealing with snakes then gradually participated in longer
time intervals with a therapist in various activities. In a second
treatment condition, participants only observed a therapist modeling
the requisite coping performances. Results of this study showed that
participant-modeling significantly improved the participants' self-
efficacy for coping with snakes, while controls reported no change. The
second treatment group, who only observed, also reported improved
self-efficacy. But the real test of the treatment effectiveness would be
measured by correspondence between levels of posttest self-efficacy and
the criterial performance tasks, which included various approach
behaviors ranging from entering a room containing a snake through
actually handling a red-tailed boa constrictor.
Correspondence, the positive correlation between judgments of
self-efficacy for being able to perform a given task and then performing
it, was high (86% to 90%) for all conditions. The implications are that
one's self-efficacy for a given situation is changeable through both
enactive experience and vicarious observation, and that one's percepts
of efficacy are strong predictors and explanations for criterion
The Role of Self-Efficacy in Academic Domains
In academic domains, the research on self-efficacy is less
extensive; however, we are now seeing it being applied to such diverse
academic domains as mathematics, computer literacy, writing,
in-service teacher training, choice of academic majors, and so on.
Many of these studies are correlational and describe how self-efficacy
relates to academic outcomes.
Self-efficacy and Academic Performance. Dale Schunk, presently
of Purdue University, is one of the more prolific researchers applying
self-efficacy as an academic construct. He and his colleagues often use a
research paradigm that goes beyond correlational analysis to include
instructional interventions designed to raise learners percepts of
efficacy and corresponding performance on criterial tasks. Schunk's
treatments to influence self-efficacy include variations on modeling,
attributions of success or failure, and goal-setting. Some of his studies
that focused on peer modeling as a source of efficacy information (see
Schunk, 1987 for a review) are related to the framework of the present
study and are therefore detailed in Chapter Two, "Review of Related
Research." Other singular studies that employ similar research designs
are reviewed as well.
Pajares and colleagues often used advanced statistical procedures
to account for the explanatory and predictive variance of self-efficacy in
relation to other personal determinants, such as anxiety, academic
background, self-confidence, and so on (Pajares & Kranzler, 1995;
Pajares & Miller, 1994a; Pajares & Miller, 1994b; Pajares & Miller, 1994c;
Pajares & Miller, 1995). Consistently, Pajares and colleagues find that
self-efficacy maintains high explanatory and predictive power for
In one study of 350 college students, Pajares and Miller (1994c)
examined the hypothesized mediational role and predictive power of
self-efficacy in mathematics problem solving. Using previously
validated measures, the researchers ran several mathematics-related
independent variables in relation to mathematical problem solving.
Results show that self-efficacy held greater predictive power for
problem solving success than did mathematics self-concept,
background in mathematics, perceived usefulness of mathematics, and
gender. The effects of background and gender, however, were
significantly related to self-efficacy, supporting Bandura's assertion of
the mediational role of self-efficacy on performance. Simply put,
background and gender are not independently strong predictors of
mathematics performance, but they are influential sources of
mathematics self-efficacy which is highly predictive and plays a strong
mediational role on performance.
Self-efficacy is a domain-specific construct in academics. Many,
including Bandura, argue that it is also task-specific, and attempts to
measure self-efficacy at the domain level often result in ambiguous or
uninterpretable results (Bandura, 1986; Pajares & Miller, 1994c, 1995).
Many of the studies that show self-efficacy to account for lesser
variance than other personal determinants often stray from Bandura's
prescriptions for a microanalytic strategy. Often these studies assess
self-efficacy globally with just a few scale items; that is, they ask
participants to report on their confidence or efficacy with regard to a
specific academic domain, and not a specific performance task. At this
level of self-reporting, it is expected that self-efficacy can not reliably be
separated from other personal determinants such as self-concept,
anxiety, self-confidence, and background. It thus raises the question of
whether one is actually measuring self-efficacy, or more generally
measuring attitudes and other common mechanisms toward a given
academic domain. Of course, the latter are important in some areas of
educational research, but do not always give us sufficient evaluative
information for performance on specific, criterial tasks. One possible
lens from which to view self-efficacy within the context of
instructional technology is to consider one's judgments of personal
capabilities to authentically accomplish a specific performance
objective. Self-efficacy and performance are inextricably related, and in
the domain of mathematics both are often correlated with gender.
Gender Effects. There is a potential gender effect in mathematics
learning and mathematics self-efficacy. As discussed earlier, Fennema
and Sherman (1977) and Sherman and Fennema (1978) found that
mathematics confidence and gender stereotyping are significant
predictors of mathematics performance for middle and high school
Studies with college students show that gender influences
self-efficacy in mathematics-related actions, such as academic major
and career decisions (Hackett, 1985; Lent, Lopez, & Beischke, 1991;
Matsui, Ikeda & Ohnishi, 1989; Matsui, Matsui & Ohnishi, 1990). Other
studies found that gender is an influential source of efficacy
information in modeling (for example, Schunk, Hanson & Cox, 1987;
Schunk, 1987). In personalization studies, Murphy and Ross (1990)
found gender to be an influential factor in determining mathematics
success for eighth graders. Other researchers (Lopez, 1989; Lopez &
Sullivan, 1992) found that personalization significantly benefited
seventh-grade Hispanic boys in performing mathematics calculations.
Together, these lineages of research suggest that gender maintains a
significant influence on mathematics self-efficacy.
As the foregoing indicates, a gender effect has often been
reported on the dependent variables (mathematics self-efficacy and
performance). In separate studies, a gender effect was reported on the
independent variable (personalization). The present study further
examines the possibility of a gender effect or interaction within a
matrix of personalization, mathematics self-efficacy and mathematics
Research from the Social Cognitive Perspective
This study follows prescriptions for microanalytic research
designs as specified by Social Cognitive Theory (Bandura, 1986, p. 422).
The instructional design dimensions of the intervention also follow
An important assumption of Social Cognitive Theory is that
personal determinants, such as forethought and self-reflection, do not
have to reside unconsciously within individuals. People can
consciously change and develop their cognitive functioning. This is
important to the proposition that self-efficacy too can be changed, or
enhanced. From this perspective, people are capable of influencing
their own motivation and performance according to a model of triadic
reciprocality in which personal determinants (such as self-efficacy),
environmental conditions (such as treatment conditions), and action
(such as practice) are mutually interactive influences. Improving
performance, therefore, depends on changing some of these influences.
Pedagogically, the challenge is to 1) get the learner to believe in
his or her personal capabilities to successfully perform a designated
task, 2) provide environmental conditionssuch as instructional
strategies and appropriate technologythat improve the strategies and
self-efficacy of the learner, and 3) provide opportunities for the learner
to experience successful learning as a result of appropriate action.
Within the model of triadic reciprocality, the ability to influence
various personal determinants is accorded to five basic human
capabilities: 1) symbolizing, 2) forethought, 3) vicarious, 4)
self-regulatory, and 5) self-reflective.
People are generally gifted with the capability of symbolizing. In
an academic context, this allows learners to process abstract experiences
into models that guide their learning and performance. For example,
observing someone on computer or videotape vocalize a
computational algorithm for calculating may serve as an adequate
instructional representation of performing that procedure. One can
learn how to perform the strategy in this manner, and may even gain
in self-efficacy by observing a peer model that this procedure is within
the scope of one's own capabilities.
Forethought, the cognitive representation of future events, is
also a powerful causal influence on one's learning. For example,
watching a self-efficacious model perform a mathematical calculation
using a particular strategy may lead the observer to foresee this within
the scope of his or her own capabilities and consequently expect to
perform the procedure with success.
Vicarious capability occurs by observing others and vicariously
experiencing what they do. According to Bandura (1986), if we had to
directly experience everything we learn, we would not have adequate
time and opportunity to learn very much. Observing a model's
thinking through text-based soliloquy, for example, can direct the
observer on how to conceptualize a mathematics calculation or
overcome self-doubts about successful performance.
Students typically self-regulate by determining what capabilities
they have with regard to a given task and in effect compare those
capabilities against a set of standards they maintain for themselves.
Students who believe that they can achieve a high grade in a
mathematics course may persist in their efforts to achieve the grade.
Conversely, low self-efficacy pertaining to a given task may inhibit
one's effort and persistence (Bouffard-Bouchard, 1989).
People compare their performance with that of their peers in
various contexts, especially the classroom. The accuracy of their
assessments determines whether they overestimate or underestimate
their capabilities. Consequently, accurate self-reflection is critical to the
development of self-efficacy.
The five basic capabilities discussed above are important
guidelines for self-efficacy interventions. In the present study, the
instructional story uses symbolic and vicarious modeling to influence
self-efficacy and expectations (forethought) of success.
One's sense of self-efficacy is determined by an array of personal,
social, and environmental factors. From the social-cognitive
perspective, these factors can be changed not only to influence one's
level of self-efficacy, but also subsequent performance on criterial tasks.
The personalization of instructional context is predicted to be an
effective strategy for raising the learners' percepts of efficacy through
the instructional design strategies of enactive experience, vicarious
modeling, and persuasory information.
Enactive experience is facilitated by enabling learners to
interactively experience what they are learning while they are learning.
Vicarious modeling is facilitated by allowing learners to individually
select personal referents that reflect the learners' interests,
backgrounds, and familiarity with the world. Persuasory information
is facilitated by cognitive modeling of characters in the story, who
cognitively experience how their percepts of efficacy are raised as they
begin to learn and try out mental computation strategies.
Purpose of the Study
The goals of this study are threefold. One goal is to build upon
existing research which show that personalization of instructional
context is one way to increase the mathematics performance of learners
in computer-based instruction (CBI). A second goal is to investigate
whether personalization through character modeling can be used to
raise the self-efficacy of CBI learners. The third goal is to test whether
personalization may be a facilitating strategy for improving the
combined self-efficacy and subsequent performance of CBI learners.
REVIEW OF RELATED RESEARCH
Arguably, the most important role of a teacher is to
communicate effectively with learners, especially in guiding them to
construct meaning from new and unfamiliar subject matter. But
learners only construct meaning if they are able to recognize, classify,
and characterize new information based on their personal
understanding and experiences (Sainsbury, 1992). Ability, though,
remains inert in the absence of the learners' motivations and
perceptions of self-efficacy; that is, how learners judge their capabilities
with regard to performing tasks (Bandura, 1977, 1986, 1993; Pintrich &
De Groot, 1990). There is substantial evidence that learners better
understand subject matter that relates to their existing knowledge and
experiential backgrounds (Brown, Collins, & Duguid, 1989; Bruner,
1990; Reed, 1938). There is also evidence to suggest that learners are
more motivated to activate existing schema if the context of the task is
personally relevant (Cordova, 1993; Davis-Dorsey et al., 1991).
Unfortunately, ability, knowledge, and motivation sometimes
remain inert. If learners are to employ strategies and regulate their
learning, they need to be motivated (Pintrich, Cross, Kozma, &
McKeachie, 1986). Additionally, learners need to perceive themselves
as potentially successful in the learning task or their use of knowledge
and motivation may both remain inert (Bandura, 1986). Equally
unfortunate is the fact that inert use of knowledge and motivation is
often inaccurately perceived in school and society as lack of ability.
Learners' motivational tendencies may preclude performances
appropriate for completion of a task, but their self-percepts of efficacy
mediate both their motivations and performances and may also
preclude both (Bandura, 1986; Brown & Inouye, 1978; Weisz &
Cameron, 1985). Therefore, getting learners to relate to new
information and to believe in themselves are important tasks for
teachers. It usually requires a great deal of interpersonal
communication between the teacher and the learner and is a human
element that is often lost in computer-based instruction.
In contemporary times, many cognitive psychologists are urging
instructional technologists to consider the learner as an active
participant in the construction of meaning, particularly when
designing computer-based instruction. Yet quite often the learner is
viewed only as a passive receptor of communication. Active learners
need to see themselves within the context of new ideas. That in itself
is motivating and promotes understanding. From this perspective, the
processes of constructing meaning are embedded in the combined
social, personal, and emotional context of learning (Lebow, 1991;
Zimmerman, 1990; Brown, Collins, & Duguid, 1989; Collins, Brown, &
Newman, 1989) and it is therefore unwise to disassociate new learning
from an experiential context. It makes sense to situate new learning in
a context that is relevant and familiar to the learner and, when
possible, to demonstrate the learner being successful in the task itself.
Computer-based, personalized storytelling can be viewed as a
means for increasing personal relevance, situated learning,
narrative/story-based learning, distributed cognition, adaptive
learning, vicarious observation, narrative therapy, and symbolic
modeling (as depicted on media). The present thesis narrows this
scope to modeling and personalization research that meets the
following criteria: (1) live or symbolic peer modeling that enhances
academic self-efficacy; and, (2) studies that describe personalization as
an instructional design strategy in order to influence learning, affect,
self-regulatory processes, or performance. These criteria were chosen
in order to maintain a lucid focus on the dependent variables,
mathematics self-efficacy and its corresponding effect on mathematics
Interventions Enhancing Self-Efficacy
Many personal determinants interact to influence the
motivation, cognition, and performance in a mathematics learning
environment. A seemingly endless array would include self-concept,
self-esteem, self-confidence, anxiety, background, socio-economic
status, ability, gender, and self-efficacy. However, according to Bandura
(1986), "any gigantic attempt to study all these reciprocal actions at once
would produce investigatory paralysis. It is the subsystems and their
various interrelations, rather than the entirety, that are analyzed"
Self-efficacy for academic tasks is integral to this research for
several reasons. Self-efficacy is shown to hold greater explanatory and
predictive power for academic outcomes than many other
determinants (Pajares & Miller, 1994a, 1994b, 1994c; Zimmerman,
Bandura, & Martinez-Pons, 1992). Students who foster faulty self-
knowledge about their abilities pertaining to academic tasks can be
helped by personalized models who demonstrate improvement in self-
efficacy during skill acquisition.
Interventions that demonstrate success in raising academic self-
efficacy, include various forms of modeling. The forms of modeling
that pertain most to personalization of instructional context are social
comparative modeling, multiple sources modeling, peer modeling,
same-gender modeling, and vicarious modeling. In many cases, these
interventions overlap where the use of multiple models also serves
the purpose of the others.
Social Comparative Modeling
In an early examination of the power of social comparison,
Brown and Inouye (1978) sought to test whether learned helplessness
one's expectations of inevitable failure due to lack of control over
proposed circumstancescould be induced by vicarious modeling.
The researchers set up live models of differing levels of perceived
similarity of competence to the observer. Observers were either told
that they were of similar competency or superior competency to the
model. A third group did not receive any feedback and a fourth
(control) group did not observe a model. Using performance with
anagrams as the performance task, participants in all groups witnessed
a model demonstrate frustration and failure with the task. Observers
in the superior competency group persisted longer than all other
groups, volunteering to spend more time trying to solve the anagrams.
Observers in the similar-competency group persisted less than all other
groups. The implication of the Brown and Inouye (1978) study is not
only that model similarity can adversely affect one's persistence and
expectations of success, but also that social comparison among peers is
an influential and vicarious source of one's perceived self-efficacy,
which was demonstrated by the greater success of those who expected
to perform better than the model.
The effects of social comparison on self-efficacy and performance
was also tested in mathematics learning. Schunk (1983) provided 40
low-achieving fourth and fifth graders with instruction in performing
division calculations. Four conditions were established in this
experiment: 1) social comparative feedback on the number of
calculations previously solved by peers, 2) a stated goal for solving a
number of calculations, 3) both treatments (multiple sources of efficacy
information), and 4) a control group. Participants receiving multiple
sources of efficacy information demonstrated greater skill as evidenced
by the number of calculations worked and solved correctly, as well as
higher judgments of self-efficacy. Further analysis also showed that
social-comparative-only feedback positively influenced skill use, and
the goal-only condition was significantly related to higher reports of
The combined results of the Schunk (1983) study imply that the
efficacy-performance relationship is influenced by information about
both goals and social comparison. Self-efficacy models should
therefore be comparable in age and development to the observer and
exhibit goals of successful performance. Personalization allows for this
Multiple Sources Modeling
Schunk and Rice (1987) conducted a pair of experiments to test
the effects of strategy value and use-feedback information on self-
efficacy and reading comprehension of low-achieving, elementary
school children. In experiment one, participants received one of four
conditions: 1) specific strategy value information, 2) general strategy
value information, 3) specific and general combined, or 4) no value
information. In the second experiment, participants received one of
three conditions: 1) strategy effectiveness feedback, 2) specific strategy
value information, or 3) or combined effectiveness-specific value
information. Strategy value was conveyed by pointing out how
strategy use improves other children's performances, a source of social
comparison and self-efficacy information. Strategy effectiveness
feedback was operationalized as verbal feedback from the trainer to
participant on how strategy use improves performance. Results from
experiment one showed that both self-efficacy and skill (as indicated by
performance) were significantly and positively altered in both the
specific strategy value and general strategy value conditions; moreover,
the specific-general strategy value group yielded an interaction across
all other conditions. Experiment two tested how effectiveness feedback
might build upon the first set of results. Results of this experiment
showed that combined effectiveness-specific value information was
more effective for improving self-efficacy and skill than either the
strategy value only or strategy effectiveness feedback only conditions.
One interpretation of these results is that multiple sources of
information are more effective than a singular treatment for changing
percepts of efficacy and corresponding performance. Using multiple
sources of strategy effectiveness-value information in an instructional
story increases the likelihood that learners will be able to see the
effectiveness and value of using strategies.
Peer modeling, that is modeling among observers based on
similarity of attributes between the model and observer, is examined
extensively (see Schunk, 1987 for a review). Perceived similarity is
shown to be an effective source of self-efficacy information for
children's learning and performance. Researchers believe that learners
are affected by greater attention, retention, production (enactment of
behaviors) and motivation for learning strategies modeled by peers.
One of the early modeling studies that focused on background
similarity was conducted by Rosekrans (1967). In that study, children
watched films in which they were led to believe that the film model
was either similar or dissimilar to themselves. Children in the
similarity group demonstrated the modeled behaviors more accurately
and were able to recall more of the model behaviors.
Schunk and Hanson (1985) investigated whether they could
positively influence the self-efficacy and mathematics achievement
(with subtractions) of 72 eight-, nine-, or ten-year old children through
peer modeling on videotape. The participants had previously
experienced difficulty learning fractions. The researchers also
investigated whether mastery or coping behaviors were of significant
benefit. There were no significant differences on either the coping or
mastery condition, however, same-gender peer modeling resulted in
significantly higher mathematics self-efficacy and performance than
the other conditions. Observers of the teacher model also reported
significantly higher posttest self-efficacy and performed significantly
higher than controls.
Schunk, Hanson, and Cox (1987) conducted two experiments to
see whether peer model gender attributes affected the mathematics
(fractions) achievement of fourth, fifth, and sixth grade school children
struggling with mathematics learning. In the first experiment, the
researchers investigated whether model gender combined with either
mastery or coping behaviors would affect the achievement behaviors
of the observers. (Mastery behaviors are those where the model
performs faultlessly. Coping behaviors include those where the model
demonstrates overcoming difficulty, fear, or anxiety for the task.) In
the second experiment, they investigated whether mastery or coping
models combined with either a single same-gender model or multiple
same-gender models promoted achievement behaviors. Participants
watched videotaped sessions of female teachers working with the
model(s). Results from the first experiment indicated that observing a
coping model had a significant effect on children's self-efficacy and
posttest performance, regardless of gender. Results of this experiment
significantly favor the coping condition with multiple models.
Implications of these results are that using coping and multiple models
are appropriate instructional strategies for raising mathematics self-
efficacy and performance. It seems that modeling coping behavior is
more effective for struggling students than modeling mastery
behavior. Multiple models, it is postulated, enables greater
opportunity for the observer to identify with at least one of the models.
The study, however, showed no effect on model-observer gender for
elementary school children, which is consistent with research
indicating that gender differences in mathematics do not emerge until
junior high or middle school (Meece, Parsons, Kaczala, Goff, &
Futterman, 1982). Therefore, it seems that self-efficacy and
mathematics performance of young children can be improved through
modeling, in particular by using multiple, coping models. The present
study includes character models who demonstrate varying levels of
coping behavior, thus allowing for greater opportunity for learners to
identify with at least one level of coping.
The effects of model gender on the mathematics self-efficacy of
children is of particular interest because computer-based
personalization enables gender-based character changes.
Unfortunately, this has not been adequately investigated to date.
Schunk and Hanson (1985), for example, found that same-gender peer
modeling is an effective method of raising children's self-efficacy and
improving mathematics performance. That study, however, was not
designed to investigate possible cross-gender effects, such as using
opposite-gender models for observers.
In another study, Schunk, Hanson, and Cox (1987) found that
model gender had no effect on the mathematics (adding and
subtracting of fractions) self-efficacy of mathematics low-achieving,
elementary school children, consistent with other findings for this age
group (Meece, Parsons, Kaczala, Goff, & Futterman, 1982).
Murphy and Ross (1990) investigated whether gender may be a
factor in student preferences and in solving mathematics story
problems. Each of the eight story problems contributed to a larger,
thematic story line. The study allowed participants to select from two
gender-oriented stories: "Angie's Travels" and "Mack's Trip;"
however, participants were then further assigned, without choosing, to
one of three conditions within the selected story: 1) preferred-gender,
2) nonpreferred gender, and 3) mixed gender. Names of characters, as
well as pronouns "he" and "she" enabled a specific gender orientation.
Two "mixed protagonist" treatment versions, "Angie-Mack" and
"Mack-Angie," were also devised. Significant variations on the
problem-solving and attitude posttests generally favored the
preferred-gender treatment over the mixed-protagonist group, but
neither of these groups significantly differed from the nonpreferred
gender group. Posttest results of problem-solving scores also revealed a
gender effect in favor of girls, regardless of protagonist gender.
Implications of that study to the present study are that personalization
which allows for gender-based referents may benefit girls, but this is
certainly not a foregone conclusion.
Schunk and Hanson (1989) conducted three experiments of peer
modeling versus self-modeling on the cognitive skill learning of
children nine to twelve years old. In experiment one, children
classified by the school as low math-achievers were assigned to one of
three conditions: 1) observing multiple peer models of the same gender
solve fraction calculations (peer-modeling), 2) watching themselves
solve calculations on videotape (self-modeling), and 3) a videotape
control group. Results showed that both treatment conditions were
significantly more effective than the control condition. In experiment
two, the children either watched themselves on videotape work easier
or more difficult problems. In this case, both conditions were
significantly more effective than control conditions. In experiment
three, children either watched tapes of the process of learning to
perform calculations of fractions versus their performance after they
had learned to perform the calculations. Significant results of the three
experiments demonstrate that self-modeling is a significant method of
modeling skill acquisition and in raising percepts of mathematics
self-efficacy. In the present study, vicarious modeling is achieved by
adapting the referents of the story protagonist to reflect several
personal attributes of the learner.
When someone with the authority of a teacher, say,
describes the world and you are not in it, there is a
moment of psychic disequilibrium, as if you looked into a
mirror and saw nothing. (Adrienne Rich, quoted in
Rosaldo, 1989, p. ix)
Personalization, as used in the present study, refers to
manipulating the instructional content to contain personal referents of
the learner. Such personal referents may include familiar names,
persons, places, or things. By relating the referents of context to
familiar, everyday conditions of the learner, the content is made
knowable to the learner (Brown, Collins & Duguid, 1989; Resnick,
1987). By modeling a successful learner, the observer believes that the
task is achievable.
The purpose of personalization is to stimulate intrinsic interest
and facilitate personal meaning of new content. This is accomplished
by portraying tasks depicting what real people would do in a realistic
situation. For subject matter that is meant to facilitate the instruction
of learning strategies, it is important that the complexity of the
environments of everyday life not be entirely reduced or abstracted out
of context. At the same time, a narrative must be flexible enough to
disassociate the concepts and principles from the initial learning
context. Encouraging students to construct learning strategies that can
be transferred outside of the classroom requires authentic learning
environments that can be explored from multiple perspectives, with
levels of complexity that approximate the experiential sophistication of
the learners (Spiro & Jehng, 1990; Salomon & Perkins, 1989). Similarly,
employing multiple perspectives increases the likelihood that the
observer can derive multiple sources of efficacy information. To
achieve personalization from multiple perspectives, the story was
constructed to involve discourse between characters who demonstrate
a shared knowledge construction. Here is a contextual example of how
a character in the story (Aisha) offers an explanation to a conceptual
problem that is elaborated by another character (Leroy), followed yet by
a rejoinder from the protagonist (Marcos).
"Okay," Aisha asked, "we're driving 1800 miles. If we
drive 600 miles per day, how long will it take us to get there?"
"That's easy, replies Leroy, "just let me grab a pencil and
"No need," said Aisha. "Some calculations are made for
doing in your head."
"I know," suggested Marcos. "Just keep counting on 600
until you get 1800 and we're there, that's 600, 1200, 1800. Three
"That's right," applauded Aisha. "And there are other
ways you can calculate this. How about breaking down 1800 and
600 to 18 and 6 by canceling an equal number of trailing zeros.
When the zeros are at the end of both numbers, you can do this.
So, really the calculation is simply 18 divided by 6, which equals
"I got another way." added Marcos. "What about counting
back, or subtracting 600 from 1800 until there's nothing left. You
can do that three times, which equals three days."
"Terrific," said Aisha. "You see, there is no right way or
wrong way. Everyone uses mental computation strategies all the
time, but the important thing is to think about which ones will
work best for you in the right situation. You don't need to
remember the names of each strategy, just that numbers are
The example passage provided above exemplifies the gist of how
characters in an instructional story can relate instructional information
in a conversation. Each passage in a story, however, is elaborated with
uniquely personal referents provided by the learner.
Personalization of this form, enables the various modeling
interventions discussed previously, and facilitates other enhancements
to computer-based instruction. These enhancements include
personalization of instructional context and personalization as concrete
Personalization of Instructional Context
Personalization, as used in the present study, follows a lineage of
research on variations of the Anand/Ross Model. This instructional
design model enables learners to transform the characteristics of
learning and instruction by merging familiar referents with abstract
nouns and pronouns as in mathematics word problems and
Anand and Ross (1987) developed three versions of a
computer-assisted lesson for teaching division of fractions to fifth- and
sixth-grade children. Participants were assigned to one of three groups:
1) personalized context, 2) concrete context, and 3) abstract context.
Personalization was facilitated in this experiment by enabling students
to change referents in story problems to personal information, such as
personally favored people, places and activities. In the concrete
version, names and events were hypothetical (realistic, but
unfamiliar). The abstract condition was presented using general
referents such as "objects" in place of specified things (such as candy
bars). The achievement posttest included items from all three
experimental contexts. Attitudes were also assessed on an eight-item
Likert-scale asking about students reactions to their respective
treatment. Achievement results yielded significant effects for both
treatment conditions over the abstract condition, while neither the
personalized nor abstract group differed significantly from the concrete
group. With regard to attitudes, the personalized group also yielded a
significant effect over the concrete group, but did not differ from the
In a subsequent investigation, Ross and Anand (1987) sought to
compare findings from the first study, in which the instruction was
delivered via computer, to printed versions of mathematics story-
problems using essentially the same treatment design. Participants
were again fifth- and sixth-graders. Mathematics achievement was
assessed using a three-section posttest containing "context," "transfer,"
and "recognition" story problems. Attitudes were also assessed. As in
the Anand and Ross (1987) study, overall results on the achievement
subtests showed the personalized treatment to be the most effective,
and was never surpassed by the other conditions. The overall attitude
measure was not significant although item analyses mostly favored
Implications of the two Anand and Ross studies described above
for the present study are that personalization is effective in teaching
mathematics and in learning to solve word problems.
Personalized Learning. Most research that employs the
Anand/Ross Model has been conducted using mathematics word
Lopez and Sullivan (Lopez & Sullivan, 1992; see also Lopez,
1989) demonstrated how personalization of mathematics word
problems could improve the mathematics (one- and two-step
arithmetical operations) achievement and attitudes of rural Hispanic,
seventh graders in Southern Arizona. Participants were grouped by
pretest score and gender, and assigned to one of three groups:
1) individualized personalization, 2) group personalization, and 3)
nonpersonalized. Students then filled in biographical inventories,
detailing familiar nouns and pronouns such as favorite kinds of ice
cream and the names of friends. In the individualized treatment, each
student received mathematics story problems in which generic nouns
and pronouns were merged with personal referents. In the group
treatment, common and familiar referents of the majority were
merged for one set of story problems for the entire group. In the
nonpersonalized version, there was no attempt to familiarize the
problems. Substitutions were made using a computer program,
however the children received print versions of the story problems.
Results show that both the individualized and group personalization
treatments were significantly higher than the non-personalized
version for two-step arithmetic calculations and mathematics attitudes;
although, the treatment versions were not significantly different from
each other. There was also a significant attitudinal effect for only the
individualized treatment. Attitudinal items consisted of interest,
liking, and preference questions. The study suggests that
personalization of mathematics story problems is an effective
instructional design strategy for improving mathematics achievement
In another study (Davis-Dorsey, Ross, & Morrison, 1991; see also
Davis-Dorsey, 1989), researchers investigated whether personalization
of mathematics word problems would benefit elementary school
children. Personalization of context, in this case, was combined with
"problem rewording for explicitness." Treatments were administered
as text. Overall significant results show that second graders benefited
from the combined intervention of personalization and problem
rewording, but personalization itself was not significant. Fifth graders,
on the other hand, benefited from the personalization intervention,
but not problem rewording. Gender also yielded a significant main
effect for fifth graders in favor of females. These results suggest that
older children in elementary school may benefit more from
personalized context of mathematics story problems, having more
developed schemata for processing information in a real-world context.
One interesting way to build upon the findings of Lopez and
Sullivan (1992) and Davis-Dorsey, Ross, and Morrison (1991), however,
is to provide the personalization treatments on computer, and to use
personalization as an instructional method instead of as a testing
method. The present study employs these variations.
Personalized Instruction. There are several studies, as of this
writing, that examined personalization as a instructional strategy for
relating individually to diverse learners.
Herndon (1988) sought to extend on the Anand/Ross Model by
comparing three levels of personalized instruction for understanding
syllogisms. Participants were high school seniors, assigned to one of
three groups: 1) individual personalization, 2) group personalization,
and 3) non-personalized. Students completed an inventory that asked
them to report their most valued possessions, and other personal
referents such as the names of people and things. Individual
inventory items were then merged into personalized syllogisms for
experimental groups one and two. All instructional versions were
delivered to students as text.
Herndon (1988) found that the individual personalization
approach had a positive effect of students' attitudes (i,e. whether the
instruction appealed to students). There were also significant effects for
the two personalization treatments on continuing motivation
(i,e. whether students would like more syllogism instruction), but this
conclusion should be viewed skeptically because it was based on one
"yes" or "no" question. Still, these results suggest that personalized
instruction may contribute to improved learner affect which, like
cognition, has a reciprocal influence on self-efficacy.
Cordova (1993) used a personalization technique designed to
enhance intrinsic motivation and mathematics learning for fourth-
and fifth-grade children. Participants were assigned to one of five
conditions in a 3 (levels of personalization) by 2 (levels of choice)
design. The intervention consisted of a HyperCard-based, computer
program designed to teach children arithmetical rules such as order of
operations and use of parentheses. Personalization was accomplished
by allowing the user to change generic referents in an instructional
fantasy story, such as character names, dates corresponding with the
users' birthdays, teachers' names, and desired birthday gifts. Choice
was accomplished by allowing the user to select the icons representing
the user. Children were posttested on a battery of attitudinal measures
and a 16-item mathematics test. Significant results showed that
students reported liking the personalization and choice features and
scored higher on the mathematics test.
Personalization as Concrete Context
The situated nature of learning, remembering, and
understanding is a central fact. It may appear obvious that
human minds develop in social situations, and that they use the
tools and representational media that culture provides to
support, extend, and reorganize mental functioning.
(Pea, 1991, p. 11.)
Many studies have shown how skills and knowledge are often
better learned, remembered, and understood in the context in which
they are acquired.
Ross (1983) conducted two experiments to test the notion that
adapting the context of instruction benefits students. In one
experiment, 51 college-age, preservice teachers were assigned to one of
three groups: 1) adaptive context, 2) nonadaptive context, and 3)
abstract context. In the adaptive context, participants were given
statistics instruction on probability using explanations and examples
from the domain of education. In the nonadaptive context,
participants received the instruction from a medical-related
perspective. From the abstract perspective, participants learned
statistical rules and formulas without reference to any other content
domain. Posttests included education, medical, and abstract items.
Results of this experiment showed overall significant posttest results in
favor of adaptive context over nonadaptive and abstract contexts on
education items. Adaptive context was also significantly favored over
abstract context on abstract items.
In a follow-up investigation, Ross (1983) sought to replicate the
findings in the above study using 50 nursing students. Therefore, the
medical domain was now the adaptive context. Results of complex
comparisons, using the Scheffe method, showed adaptive context
significantly superior to the others.
Results of the two Ross (1983) experiments showed that
education students performed better in the education adaptive context,
and nursing students performed better in the medical adaptive context.
Implications of these results are that adaptive contexts are more
effective design methods for learning quantitative material. One
contributing reason for these phenomena may be that depth of
learning is greater when new content is assimilated to prior knowledge
However, Ross, McCormick, and Krisak (1985) further examined
the effects of personalization of statistical content on achievement and
preferences of college education and nursing students in other
experiments and found mixed results. The researchers anticipated that
allowing participants to choose their preferred thematic context of
instruction (adaptive) versus being given their least preferred context
(nonadaptive), would result in higher achievement by the adaptive
group. They also sought to examine whether students in the
nonadaptive context would suffer detrimental learning. Attitudes and
recall of critical information were assessed by posttest questionnaire.
In experiment one, nursing students were randomly assigned to
one of four treatment groups: 1) standard-adaptive (automatically
given the medical context), 2) standard-nonadaptive (abstract context),
3) learner-control adaptive (given preferred choice of context), and 4)
learner-control nonadaptive (given least preferred choice of context).
The four instructional contexts were abstract, education, medical, and
sports themes. Effects by group on achievement and recall were
generally not significant. Item analysis of attitudes generally favored
the adaptive context.
In experiment two, 50 education students were used in the same
design as in experiment one. Significant results in this case favored the
adaptive group, however, unlike the experiment with nursing
students, there was no significant attitude effect.
Generally, the mixed results of the two combined studies suggest
that personalization (adaptive context) can be an effective method for
presenting statistical content to college students, but that this is not a
foregone conclusion. The present study, however, changes the
instructional context by replacing abstract and generic referents with
personal ones, thus placing the learner not only in the context of the
story, but in the situated nature of the story problems as well.
Context and Mental Computation. Carraher, Carraher, and
Schliemann (1985) conducted a qualitative analysis of how Brazilian
street-market children invoke effective computational procedures
(algorithms) in real-life contexts in contrast to traditional school
mathematics and abstract computational problems. The Brazilian
researchers predicted that participants would often perform
mathematics computation differently in informal settings than in
school, and this would often be more effective. For example, if
children could efficiently compute the costs of variety of market
produce involving addition, subtraction, and multiplication, how
would they fare in performing abstract, school-based problems?
Participants were poor children ranging in age from nine to fifteen,
with little formal schooling, ranging from one to eight years. The
children first were asked to perform computations mentally by
interviewers posing as customers in their natural (informal) setting,
the street market. A follow-up test asked them to perform word
problems, using market items and the same numbers, but under the
unnatural (formal) condition of being given a pencil-and-paper test in
their homes. Results showed that participants correctly solved 98
percent of the context-embedded problems in the informal setting, but
only 74 percent of imaginary, context-embedded items and 37 percent of
abstract items in the formal setting. It was also learned that children
apply different computational algorithms when presented with
problems orally than with the pencil-and-paper test. Interpretation of
these results argue that mental computational algorithms may be more
effective when applied to a real-life context, but also that the strategies
(oral or written) invoked are context dependent. This analysis supports
the notion of the present study that children are helped when their
employment of mental computational strategies occur in a natural
context. Recall, however, that mental computation strategies are
appropriate for certain kinds of problems under certain conditions,
however, more formal algorithms are appropriate for more complex,
The proposition that the use of written versus oral (mental)
computation strategies is context dependent was further analyzed by
Carraher, Carraher, and Schlieman (1987) in a study involving third-
grade Brazilian school children. The researchers predicted that the type
of mental or written computational strategy children would employ
would depend on whether the context was concrete or abstract.
Specifically, they wanted to test whether formal mathematical
operations would predominate in school-type settings, while informal
operations would predominate in natural contexts. As in a previous
study (Carraher, Carraher, & Schliemann, 1985), children were tested in
different settings. A major difference in this design, however, was that
the street market was simulated, in school, as a store situation. Three
settings were analyzed: 1) the simulated store, 2) problems embedded in
story problems, and 3) computational exercises. Problems were
presented orally, however, children were allowed to perform
operations mentally or with pencil-and-paper. Children more often
chose to perform the operations mentally, however, this was not
significantly different than the number of operations performed with
pencil and paper. Significant results revealed that children in the
simulation setting accurately calculated more often than the other
conditions. Qualitative analysis of posttest interviews with the
participants, also revealed that children used different algorithms to
perform operations mentally than on paper. Specifically, students
often performed mental computation algorithms by decomposing
quantities (e.g. solving portions of the calculation at a time) and
repeated grouping (e.g. using repeated addition instead of
multiplication). That study thus contributes to the idea in the present
study that mental computation strategies are not simply written
algorithms performed mentally, and that these strategies may be more
effective in certain contexts: either real or simulated, as in text-based
Story-based Context. The notion that story-based instruction aids
learning is supported. Anderson, Spiro, and Anderson (1978)
conducted an experiment to test whether text is better interpreted, that
is, learned and recalled, in story structure form. Two story passages
were created in two contexts: One involved dining at a fancy restaurant
and the other shopping in a supermarket. They predicted that the
restaurant context would provide more structure, due to the natural
temporal order of appetizers, main course, and dessert, and would
therefore be more effectively interpreted. Participants were 75 college
students randomly assigned to read either passage followed by a
posttest recalling food items, food order, and character names situated
in the passages. All actors and most food items in the passages were
identical. Posttest results supported the hypothesis that the restaurant
context significantly predicted better recall of food items. The
restaurant context also significantly enabled better recall of characters
attributed to certain food items. Results of the study support the long-
held notion that context schemata significantly aid the interpretation of
textual information; that is, situations in which the presentation of
information occurs in a natural way is a worthwhile aid to learning.
Chapter Two Summary
The effects of personalization on self-efficacy (as the two are
defined in this study) were assessed in only one previous experiment
(Cordova, 1993), using only a few multidimensional items that were
domain-general in nature. In this case, self-efficacy was not analyzed
according to the same guidelines used in the present study, which
purports that self-efficacy is task- and situation-specific. There is,
however, solid evidence that various comparable sources of self-
efficacy information, such as symbolic modeling with a high degree of
personal relevance and using multiple models, can be effectively
incorporated into instructional strategies that promote increased self-
efficacy and improved subsequent performance.
Modeling is an effective means of raising mathematics
self-efficacy, and personalization is an effective means of improving
mathematics performance. The present study converges these lines of
research by specifying an instructional design strategy that personalizes
instructional stories where characters model skill acquisition and
improved personal changes in self-percepts of efficacy.
There are several reasons that peer-modeling and
personalization of instructional context are effective instructional
interventions for raising learner self-efficacy and mathematics
performance. First, peer modeling provides evidence to the observer
that perhaps he or she can too perform successfully at a designated
level of performance. Multiple models, too, enable the learner to relate
to at least some attributes of the models.
Second, personalization extends on modeling in that it also
enables the embedding of instruction in contexts that are familiar and
relevant to the learner. Theoretically, learning in situated contexts
enables the learner to assimilate new knowledge into existing
knowledge structures. Personalization can be viewed as either a form
or extension of modeling, as it allows the learner greater control over
character referents embedded in instructional stories, but also enables
the learner to observe thought patterns of the characters. These
thought patterns, or cognitions, can vicariously portray for the observer
how one's faulty self-knowledge, or low self-percepts or low
self-efficacy, may be corrected through strategy acquisition and
attention to persuasory information.
Third, there is growing evidence that gender differences in
mathematics performance are dissipating; however, questions remain
about how self-efficacy influences mathematics performance. The
present study includes gender as an attribute variable to further gauge
whether personalization is an effective intervention for raising
gender-based percepts of efficacy and mathematics performance.
The treatment (personalization) and attribute (gender) variables
were thus tested on the dependent variables, mathematics self-efficacy
and mental computation performance. This experimental matrix
provides further explanatory and predictive evidence of the effects of
personalized storytelling as an instructional design strategy.
Peer modeling of successful performance is shown to be an
effective instructional strategy for raising self-efficacy. Personalization
strategies are shown to improve learning and mathematics
performance. Given this history of empirical research, it is postulated
that personalization which uses peer models as characters in
instructional stories is likely to improve both self-efficacy and
The hypotheses for significant results were that:
1. The group receiving personalized stories will report
higher mathematics self-efficacy for mental
computation than both nonpersonalized and control
2. The group receiving personalized stories will
demonstrate greater mathematics performance for
mental computation than both nonpersonalized and
3. The group receiving personalized stories will report
higher mathematics self-efficacy and demonstrate
greater mathematics performance than both
nonpersonalized and control groups.
4. Both males and females receiving personalized stories
will report higher self-efficacy and demonstrate greater
performance than males and females in both
nonpersonalized and control groups. Therefore, there
will be no interaction between personalization and
5. The group receiving nonpersonalized stories will
report higher self-efficacy and demonstrate greater
performance than the control group.
Lessons in mathematics mental computation and
computer-literacy were developed and embedded into story form. The
story on mental computation served two conditions: 1) a personalized
story, and 2) a nonpersonalized story. A computer-literacy story served
as a control condition. An instructional computer program,
StoryTeller (Martinez, 1995), was then created to gather biographical
information and randomly select one of the three conditions for the
user. In the personalized version, StoryTeller merges biographical
referents into the story narrative, thus transforming the story to reflect
the backgrounds and interests of the user.
This study was designed as a true experiment to raise the
mathematics self-efficacy and performance of middle school students
using instructional stories as the treatment.
The basic experimental design was a 3 (level of personalization)
by 2 (gender) design with pretest self-efficacy and grade levels as
covariates. Subjects were randomly assigned to one of three group
levels: A) personalized, B) nonpersonalized, and C) control (see Figure
Figure 3.1. Study is a 3 (levels of personalization) x 2 (gender)
Experimental Design with Pretest and Grade as Covariates.
GENDER Personalized Nonpersonalized Control
Male PSE MCP PSE MCP PSE MCP
Female PSE MCP PSE MCP PSE MCP
Dependent variables are shown inside 3oxes.
PSE = Perceived Self-Efficacy, MCP = Mental Computation
StoryTeller was designed to randomly assign participants to one
of three conditions. This enabled the use of existing classes while
providing the opportunity for true random assignment. Group A
(personalized) and Group B (nonpersonalized) were given the same
story about mental computation. Group A used personalized referents
provided by the participant. The story given to Group B contained only
generic and abstract nouns and pronouns. In both treatment versions,
characters in the stories modeled growth in self-efficacy for mental
computation as well as learning various mental computation
strategies. Group C (control) received an instructional story about
computer literacy, and thus received no relevant treatment.
Design of this study for reporting self-efficacy follows a research
model used often by Dale Schunk (1987, 1991) and colleagues in
experiments pertaining to mathematics self-efficacy. In this model,
students are given a short time (usually only a few seconds) to see a
math calculation and make judgments on a Likert scale regarding their
expectations of being able to accurately perform the calculation. Later,
participants are given more time to actually perform the same or
similar calculations. This study asked students to report their
mathematics self-efficacy for a set of mental computations both before
and after the treatment phase. Pretest self-efficacy scores served as a
covariate in this true-experimental design.
Participants were given a mathematics self-efficacy posttest on
the day following the treatment phase, and then a mental computation
performance posttest. Due to the fluid nature of self-efficacy, Bandura
(1986) recommends that performance measures follow closely in time
to posttest self-efficacy measures. Mental computation performance
calculations were identical to those in the posttest self-efficacy measure.
A pilot study was conducted with middle school students
(N = 91) to determine the internal consistency of 71 mental
computation items (symbolic calculations) to be used in this study. All
math calculations were gathered from three primary sources, which
include research forms used in a Japanese study among fourth, sixth,
and eighth graders (Reys & Reys, 1993), a seventh-grade "snapshot" in
the U.S. (Reys, Reys, & Hope, 1993), and for 13-year-olds from parallel
exercises (actual calculations are not released) used in the 1983 National
Assessment of Educational Progress (NAEP, 1993). All items were
selected because they had received some measure of validation for the
approximate age-group in other studies.
Data from pilot testing resulted in three measures that were used
for both self-reports of self-efficacy on a Likert scale and as mental
computation performance test items (correct or incorrect). One item
("50 + 40"), was not included in any resulting measure because it was
correctly answered by every student. The three resulting measures
were analyzed in SPSS for Windows for internal consistency to
compute the Cronbach's Alpha and standard error of measurement
(see Table 3.1).
Content validity for the tasks as a mental activity was supported
by observation. The tester observed that students were not using
calculators or desk space to compute a pencil-and-paper algorithm. The
measures were tested in two sessions for six different classes: 1) for
making self-efficacy judgments, and 2) a performance test.
During the self-efficacy session, it was emphasized that students
not try to compute the calculation at all. Rather, they were instructed
to focus on whether they believed they could accurately compute the
calculation and to circle the answer that best described their belief state.
The performance session immediately followed the self-efficacy
session. Students were instructed to accurately compute each
calculation and write the answer on the answer sheet.
Table 3.1. Internal Consistency of Pilot Measures.
MEASURE NUMBER OF ITEMS CRONBACH'S ALPHA STANDARD ERROR
A. Self-efficacy 20 rxx = -85 Smeas = 3.75
A. Performance 20 rxx = -83 Smeas = 2.59
B. Self-efficacy 30 rXx = -94 Smeas = 4.14
B. Performance 30 rxx =.88 Smeas = 3.15
C. Self-efficacy 20 rxx = .96 Smeas = 3.31
C. Performance 20 rxx = -90 Smeas = 2.58
Note: Three sets of mathematics calculations are used to create
measures A, B, and C. Results were analyzed for internal consistency
as both self-efficacy scale and performance test measures.
Items were presented by a computer-display program that was
transferred to videotape. The tape displayed items at intervals of six
seconds for the self-efficacy measure, and 18 seconds for the
performance measure. These procedures for presenting self-efficacy
and mental computation performance items were also used in the
experiment, except that items were displayed for 13 seconds, instead of
18 seconds, on the performance test. It was determined that 13 seconds,
the amount of time used by the NAEP (1983), would be sufficient for
mentally computing the present list of symbolic calculations.
The measures resulting from the pilot study were used in the
actual experiment, with measure "A" being used as a 20-item self-
efficacy pretest, and measures "B" and "C" being combined for both the
self-efficacy posttest and mental computation performance test.
Participants in this study (N = 104) attend a rural-suburban
middle school, about 10 miles from Denver. Participants were sixth-,
seventh-, and eighth-grade pupils enrolled in six elective computer
classes (see Table 3.2 for frequency distribution by group assignment,
grade, and gender). The sample consisted of 39 females and 65 males
ranging in age from 11 to 14. The difference in total gender
participation may be due to the fact that the classes used were electives,
and compete against other electives such as orchestra, foreign language,
and various arts courses.
Participants were asked to report their age, grade, gender, and
whether they liked math (yes or no). Grade data was needed to produce
a grade covariate. Participants were also asked to report their names so
that they could be tracked from pretest to posttest.
The student body of the school is ethnically diverse, and this was
reflected in the population sample. Approximately 29 (28%)
participants were of Hispanic/Chicano origin, 2 (2%) were
African-American, 3 (3%) Asians, and 70 (67%) were Caucasian.
Table 3.2. Frequency Distribution of Participants by Group, Grade, and
Grade M F M F M F Totals
6 10 7 7 7 5 3 39
7 11 3 7 4 9 9 43
8 5 1 7 2 4 3 22
Column Totals 26 11 21 13 18 15 104
Note: M = male, F = Female.
Because there was no theoretical basis for including ethnicity as a
variable, participants were not asked to designate an ethnic origin, and
the numbers provided are based on qualitative observations by the
experimenter. The sample also consisted of a wide range of student
math abilities. The middle school maintains an "integrated" policy of
inclusion of special needs students in mainstream classes. Students
with special needs include non-English proficiency and
Participants who missed any part of the three-day experiment
(n = 32), were allowed to continue in the study but were excluded from
the data analysis. The large number of exclusions was due to non-
English proficiency and a number of impromptu school activities (e,g. a
general assembly, ice cream social for fund-raisers, and a recall of band
students for re-recording an audiotape).
Participants and their guardians were informed that their
participation was part of an investigation to determine whether
computer-based, personalized instructional stories are an effective way
to improve beliefs about mathematics learning and mathematics
performance. No other agreements or compensation were made.
Consent forms from guardians and participants were required for all
participants (see Appendix A).
The independent variables in this study include personalization
(the treatment variable) and gender (an attribute variable).
Levels of Personalization
For the personalized group, data obtained from computer-
administered biographical inventories were merged with character,
place and thing elements to create personalized stories for every
student in the group. For the nonpersonalized group, the original
generic and abstract referents remained in the story. The control group
received the generic computer literacy story.
The purpose of including gender as an attribute variable is
directly related to previous research on mathematics self-efficacy,
which shows that the relationship between gender and mathematics
performance is often mediated by mathematics self-efficacy (Pajares &
Miller, 1994c). Participants were asked to report their gender on the
Although the present experiment conducted random
assignment, statistical power was increased by adding covariates into
the analyses. This enabled more control for reducing the effects of
some preexisting differences. This procedure allows us to have more
confidence in assessing the actual contribution of the treatment. Using
ANCOVA in a pretest to posttest experimental design is often
recommended (Keppel & Zedeck, 1989).
The 20-item self-efficacy pretest (see Appendix B) was included
in this experimental model to gain statistical power and to control for
large individual differences in math ability and nonequivalent
systematic group assignment that could result by chance.
Many mathematics scholars agree that schools often neglect the
teaching of mental computation skills and that grade-specific
performance outcomes are speculative (Coburn, 1989). In fact, there
has only been one large-scale assessment of mental computation
exercises in the United States, which was conducted with 18
whole-number items by the National Assessment of Education
Progress in 1982-83 (Reys & Barger, 1994). Therefore, the 70
mathematics calculations used in this study were piloted and used for
three middle-grade levels, but grade level served as a covariate in the
ANCOVA design to adjust for preexisting individual differences that
may result as a function of grade.
The two dependent variables in this study were mathematics
self-efficacy and mental computation performance. The self-efficacy
and performance variables were both, individually and in conjunction,
expected to increase as a result of the treatment interventions, with the
greatest increase expected from the personalized treatment.
The operational definition, mathematics self-efficacy, is used
here to denote the self-perceived capabilities of participants to mentally
compute a set of mathematics calculations. Mathematics self-efficacy in
was determined by summed responses for all individuals. Participants
were told to respond immediately about their beliefs to accurately solve
a list of 50 calculations. Participants were then shown a videotape that
displayed mental computation calculations one at a time for six
seconds each. Each item was announced orally on the audio track and
displayed visually on the top third of the screen over a blue
background. The short duration for each item was necessary to
eliminate the likelihood that participants would be able to solve the
calculation mentally and therefore gain concrete feedback about their
capabilities. Concrete feedback biases efficacy judgments by informing
the students of their real, as opposed to perceived, personal capabilities
(A. Bandura, personal communication, April 7, 1995; D. H. Schunk,
personal communication, March 20, 1995).
Mental Computation Performance
Mental computation performance was determined by a summed
score on a 50-item test. Participants were shown the same 50
calculations as on the self-efficacy posttest, but were given 13 seconds,
instead of six seconds, to determine and write their answers. The
13-second interval for each item is usually sufficient time to compute a
mental computation algorithm.
This study utilizes three measurement instruments: A
self-efficacy pretest, a self-efficacy posttest, and a mental computation
performance test. Reliability and validity information for items used
in all measures was obtained in the pilot study.
Self-efficacy pretest. Students were given answer sheets for the
self-efficacy pretest, which served as a covariate in this experimental
study (see Appendix B). The pretest consisted of 20 mental
computation calculations. Responses on the five-point Likert scale
ranged from "Almost Always" to "Almost Never."
Construct validity for the design of the self-efficacy measures was
supported by expert review in personal communication over electronic
mail (A. Bandura, personal communication, April 7, 1995; D. H.
Schunk, personal communication, March 20, 1995). The experts were
given two instruments to review for this study. It was determined that
one instrument consisted of items that were related to self-efficacy but
was not precise enough to measure self-efficacy alone. The other
instrument was approved as a self-efficacy measure and was selected
for use in the present study. Additional validity for mental
computation of the self-efficacy measure was secured by observing
participants in the pilot study to verify that they were not using writing
or calculation instruments to perform non-mental computational
Biographical Inventory. This 13-item, computer-administered
inventory was used to gather data from participants to be merged into
the context of the personalized story. For consistency, however, all
participants in all groups were required to complete the inventory. It
asked for concrete nouns and pronouns that were substituted with
abstract people, places and things in the story (see Appendix C).
The computer program, StoryTeller (Martinez, 1995), was created
in HyperCard 2.2. StoryTeller merges keywords and phrases within
existing stories using a search-and-replace external command. For the
personalized treatment, the program merged personal referents with
abstract and generic referents of the nonpersonalized story (for sample
screens, see Appendix D).
Story. The instructional treatment program, StoryTeller, was
delivered to learners on Macintosh computers. The treatment story
was presented as a lesson in mental computation strategies.
Although stories can follow complicated structures, for this
investigation the story structure was quite simple. The characters in
the story joined together to discuss and perform calculations related to
mental computation strategies and discuss overcoming low self-
percepts of mathematics efficacy, while taking a simulated five-day trip.
Situations were developed in the story so that characters used mental
computation strategies when shopping in a convenience store and
eating in a restaurant.
The 3,636-word story readability level was at or below the
normal reading level of the subjects' age group in order to reduce bias
in favor of reading ability. The Flesch grade level was calculated at 6.3
and contained 1% passive sentences.
Participants in the control condition were given a
nonpersonalized, 2994-word story dealing with computer literacy. The
Flesch grade level for this story was calculated at 8.0 and contained 5%
The posttests were administered as non-graded tests. Because
personalization was used as a new method and efficacy generator, it
was appropriate that the effect of grading as a reward not be used to
influence the effects of the treatment.
Self-Efficacy. Self-efficacy for 50 mental computation items was
measured using the same procedures as the self-efficacy pretest.
Parallel items, as determined from the pilot validation, were used (see
Mental Computation. The mental computation posttest
contained identical calculations as the self-efficacy posttest and was
performed in the same manner, except that participants were given 13
seconds to fill in the answer box for each calculation (see Appendix F).
The experimental setting was the school computer lab,
consisting of 30 Macintosh Plus computers. The computers each had
four megabytes of RAM memory and nine-inch built-on display
monitors. A liquid-crystal display panel with overhead projector was
used for demonstrating how to get started on the computer program.
A video cassette player and television monitor were used to display
mathematics calculations for both the pretest and posttest.
The experiment took place with existing classes over three
regularly scheduled class sessions on separate, consecutive days. Six
classes of varying grade levels participated. Each group received
identical preparation and testing in the experiment.
Session one consisted of conducting the pretest. Participants
were first given a practice exercise for making self-efficacy judgments
on a Likert-scale. A non-math related exercise was selected to avoid
biasing participant expectations about the upcoming pretest. The
experimenter asked participants to take six seconds to decide, using the
semantic differential of "almost always" to "almost never," if they
believed they could name all eight of the U. S. states that begin with the
letter "n." They were subsequently given 13 seconds to think about
whether they could name all eight states. Finally they were asked to
think about the accuracy of their judgments relative to their
performance. The 3-minute pretest was then administered.
Session two consisted of a 40-minute class period. Participants
were shown how to open the computer program, fill in the
biographical inventory, and get started. Participants then read the
stories off the computer screens for the rest of the class duration.
Session three was conducted the day after the treatment phase
and consisted of the 5-minute, 50-item self-efficacy posttest and the
11-minute, 50-item mental computation posttest.
Data Formatting and Reduction
Data analysis procedures followed guidelines set forth by Keppel
and Zedeck (1989). All groups were pretested to be sure that they were
homogeneous enough for the purpose of the study (i,e. that their score
differences were not due to their group characteristics). Tests for
homogeneity of slopes were run for both dependent variables. Result
show that we can be confident that there were no interactions between
the covariates and independent variables (p > .05).
The hypotheses in this experiment are based on sample data, so
we can only assess the probability that the treatment would be
generalizable to a similar population. Alpha levels for all ANCOVA
results were relaxed to .10 to increase power due to the small sample
size and brief treatment. This permitted us to analyze results beyond
the omnibus p-values in the primary ANCOVA model, thus
increasing the chances for Type I error (rejecting the null hypotheses if
they are in fact true) and reducing the chances for Type II error
(accepting the null hypotheses if they are false).
The criticality level of the intervention is low. The intervention
is not an analysis of a potentially harmful drug; it is an investigation of
the usefulness of an instructional design innovation. There are no
known adverse affects associated with this type of instructional
treatment and so the severity of making a type I error is unlikely to
cause adverse consequences. The benefit of making a correct decision
on the hypotheses, of course, is that this study will contribute to a
longer lineage of future research.
One-factor and two-factor ANCOVAs served as the test statistics.
The independent variables for this analysis, "level of personalization"
("group") and gender, were run with two covariates, pretest and grade,
on the dependent variables mathematics self-efficacy and mental
It was anticipated that the results of the two-factor ANCOVAs,
using "group" and "gender," would show a personalization effect
(p < .10) but not a personalization-by-gender interaction (p > .10) on
each of the dependent variables. ANCOVA was used to analyze
whether there was significant variance in the adjusted group means
with regard to the independent and dependent variables.
Secondary analysis consisted of a one-factor ANCOVA on
"group" results, with "pretest" and "grade" as covariates. This was
done to see whether the independent "gender" variable drained
statistical power without accounting for appreciable variance.
The following alternate hypotheses were tested:
1. There is no personalization effect on perceived
2. There is no personalization effect on mental
3. There is no gender effect on perceived mathematics
4. There is no gender effect on mental computation
5. There is no personalization-by-gender interaction on
perceived mathematics self-efficacy.
6. There is no personalization-by-gender interaction on mental
This experiment investigated whether a personalized or
nonpersonalized story would increase the self-efficacy and mental
computation performance of middle school students.
Means and standard deviations were calculated for
group-by-gender for all measures (see Table 4.1), and for group-by-grade
for all measures (see Table 4.2).
To adjust for differences attributable to incoming levels of
mathematics self-efficacy and competence, initial procedures consisted
of a two-factor ANCOVA design, with "gender" and "group" (level of
personalization) serving as independent variables, and "pretest" self-
efficacy and "grade" level serving as covariates. This procedure was
run on both dependent variables, posttest self-efficacy and posttest
mental computation performance.
A second ANCOVA model (one-factor ANCOVA), without the
independent "gender" variable, was then run again on both dependent
Finally, a third ANCOVA model (within-grade model), without
the independent "gender" variable, and without the "grade" covariate
was run for each grade level on both dependent variables.
Table 4.1. Means and Standard Deviations for Group x Gender on
Pretest Self-Efficacy, Posttest Self-Efficacy, and Posttest Mental
Note: SE = Self-efficacy. Pretest and posttest measures consist of
average responses on 5-point Likert scale. Performance consists of
correct responses to 50-item mental computation posttest.
Table 4.2. Means and Standard Deviations for Group x Grade on
Pretest Self-Efficacy, Posttest Self-Efficacy, and Posttest Mental
Note: SE = Self-efficacy. Pretest and posttest measures consist of
average responses on 5-point Likert scale. Performance consists of
correct responses to 50-item mental computation posttest.
Overall results of the two-factor ANCOVA indicate that the
personalized treatment was significantly more effective than the
control condition for raising participants' mathematics self-efficacy.
Across all grades, gender differences for pretest self-efficacy, posttest
self-efficacy, and mental computation did not produce significant main
effects (see Table 4.3 for means and standard deviations for gender on
Table 4.3. Means and Standard Deviations for Gender on All
Pretest Posttest SE Performance
Gender M SD M SD M SD
Males 3.445 .894 3.445 .894 17.662 10.557
Females 3.348 .797 3.348 .797 17.538 11.385
Note: Gender differences across all groups and measures were small.
Results from the one-factor ANCOVA model revealed that the
group effect on posttest self-efficacy was enhanced with gender
eliminated from the second model.
There was no group effect on performance levels in either the
two-factor or one-factor ANCOVA models.
Within-grade analyses produced a group effect, for
eighth-graders only, in favor of Group A over Group C, and Group B
over Group C.
Although an ANCOVA design was not conducted between
grades, analyses of means show that reports of posttest self-efficacy and
mental computation performance ascend by grade level (see Table 4.4
for between-grade means and standard deviations).
Table 4.4. Between-Grade Means and Standard Deviations for Grade
on Both Dependent Variables.
Grade 6 Grade 7 Grade 8
Measure M SD M SD M SD
Self-Efficacy 3.173 .749 3.318 .867 4.003 .767
Performance 14.410 9.312 16.535 9.399 25.409 12.470
Note: Reports of self-efficacy and performance results ascend by grade
Results of Two-Factor ANCOVA
A two-factor ANCOVA, using "group" and "gender" and
independent variables, and "pretest" and "grade" as covariates, was
used to determine whether there was significant variance in the least
squares (adjusted) group means with regard to both dependent
Regarding posttest self-efficacy, the preliminary data analysis
returned a main effect for "group," F (2,103) = 2.631, p. = .0772, but not
for "gender" (see Table 4.5). There was no "group-by-gender"
interaction. Further analyses of pairwise comparisons of adjusted
means revealed that the personalized group, Group A (LSM = 3.524,
SD = .476), reported significantly higher posttest self-efficacy than
Group C, the control group (LSM = 3.386, SD = .443). Group B, the
nonpersonalized treatment group (LSM = 3.386, SD = .443), did not
differ significantly from either Group A or Group C. (see Table 4.6 for
adjusted means, standard deviations, and resulting p-values).
Table 4.5. Two-factor ANCOVA Table of Group x Gender with Pretest
and Grade on Posttest Self-efficacy
Source df Sum of Squares Mean Square F-Value P-Value
Group 2 .971 .486 2.631 .0772*
Gender 1 .113 .113 .612 .4358
Group Gender 2 .165 .083 .447 .6408
Pretest (covariate) 1 47.941 47.941 2.6E2 .0001
Grade (covariate) 1 1.195 1.195 6.473 .0126
Residual 96 17.726 .185
Dependent Variable: Posttest self-efficacy.
* Independent variable, significant at the .10 alpha level.
Table 4.6. Adjusted Means Tables and Resulting P-Values for Group
x Gender with Pretest and Grade on Posttest Self-Efficacy
Group Count LSmean Std. Dev. Std. Error
A 37 3.524 .476 .078
B 34 3.386 .443 .076
C 33 3.272 .435 .076
Group Versus Diff. Std. Error t-Test P-Value
Group A Group B .138 .109 1.267 .2081
Group C .252 .110 2.292 .0241*
Group B Group C .114 .107 1.061 .2915
leant differences are shown in favor of both Group A over
control Group C.
* D< .10