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Effects of an integrated content and methods course on preservice teachers' beliefs and efficacy toward mathematics

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EFFECTS OF AN INTEGRATED CONTENT AND METHODS
COURSE ON PRESERVICE TEACHERS ’ BELIEFS
AND EFFICACY TOWARD MATHEMATICS
Copyright 2005
by
Darlene R. York
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of
the Requirements for the Degree of
DOCTOR OF EDUCATION
May 2005
Darlene R. York
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UMI Number: 3180327
Copyright 2005 by
York, Darlene R.
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Signature Page
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ACKNOWLEDGEMENTS
I wish to acknowledge and express my gratitude to the faculty and staff of the
Rossier School of Education at the University of Southern California for their
dedication and professionalism. In particular, I wish to than Dr. Carl Cohn, Dr.
William Rich, and Dr. Dennis Hocevar, my dissertation chairperson, for his
consistent support, expertise, and encouragement.
Endeavors such as this are not accomplished alone. I would like to thank my
friend and colleague, Dr. Paula Selvester, for inspiring and guiding me. Also, thanks
to my colleagues in the Mathematics and Statistics Department at Chico State
University for their willingness to be a part of this research study.
Finally, I dedicate this work to my husband, Bill, who supported and
encouraged me throughout this educational journey.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS..................................................................................... ii
LIST OF TABLES.................................................................................................... v
LIST OF FIGURES.................................................................................................. vi
ABSTRACT............................................................................................................... vii
Chapter Page
1. PRESERVICE TEACHERS AND MATHEMATICS
EDUCATION..................................................................................... 1
Introduction.......................................................................................... 1
Background of the Problem............................................................... 4
Purpose of Study................................................................................. 7
Research Questions............................................................................. 8
Hypotheses........................................................................................... 9
Significance of the Problem............................................................... 9
Definition of Terms............................................................................. 15
Beliefs............................................................................................ 15
Constructivism.............................................................................. 15
Self-efficacy.................................................................................. 15
T eaching-efficacy.......................................................................... 15
2. REVIEW OF THE LITERATURE.......................................................... 16
Introduction......................................................................................... 16
Mathematics Education and Reform.................................. 18
Teacher Knowledge and Practice....................................................... 19
General Pedagogical Knowledge and Beliefs............................ 23
Subject Matter Knowledge and Beliefs ........... 23
Pedagogical Content Knowledge and Beliefs............................ 24
Social Cognitive Learning Theory..................................................... 24
Implications ............................. 28
3. RESEARCH METHODOLOGY............................................................. 31
Introduction.................. 31
Research Questions.................................. 32
Research Design.................................................................................. 32
Population and Sample ..... 33
Description of Intervention .................... 35
Instrumentation................................................................................... 38
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iv
Chapter Page
Procedures............................................................................................ 43
Data Collection.............................................................................. 43
Data Analysis.......................................... 44
Research Question 1............................................................................ 45
Research Question 2.............................................. 46
Research Question 3............................................................................ 46
Research Question 4............................................................................ 45
Delimitations................................................................................. 47
4. FINDINGS................................................................................................. 48
Introduction.......................................................................................... 48
Research Question 1............................................................................ 49
Research Question 2............................................................................ 56
Research Question 3............................................................................ 58
5. DISCUSSION............................................................................................ 62
Summary of the Purpose..................................................................... 62
Summary of the Design....................................................................... 62
Discussion of the Results................................................................... 63
Limitations........................................................................................... 66
Implications.......................................................................................... 66
Conclusion........................................................................................... 69
Recommendations............................................................................... 70
REFERENCES.......................................................................................................... 73
APPENDIX................................................................................................................ 82
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V
LIST OF TABLES
Table Page
1. NAEP 2003 Percentage below Basic Proficiency in Mathematics 5
2. Research Experimental Design.................................................................. 33
3. SAT 1 Math Score for Entering Freshman Students................................ 34
4. Means and Standard Deviations for Total Mathematical Beliefs
Instrument (MBI) Pre- and Postscores for All Subjects................... 49
5. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI) Part A Pre-and Posttest Measures
(All Subjects)...................................................................................... 52
6. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI)Part B Pre- and Posttest Measures
(All Subjects)...................................................................................... 54
7. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI) Part C Pre- and Posttest Measures
(All Subjects)...................................................................................... 56
8. Posttest Mathematic Beliefs Instrument (MBI) ANCOVA Effects
for Experimental and Control Groups (Unadjusted Means) 59
9. Posttest Mathematical Beliefs Instrument (MBI) ANCOVA Effects
for Experimental and Control Groups (Adjusted Means)................ 60
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vi
LIST OF FIGURES
Figure Page
1. Pretest frequencies for the Mathematics Beliefs Instrument (MBI)...... 50
2. Posttest frequencies for the Mathematics Beliefs Instrument (MBI) 50
3. Pretest frequencies for Part A of the Mathematics Beliefs
Instrument (MBI)................................................................................ 52
4. Posttest Frequencies for Part A of the Mathematics Beliefs
Instrument (MBI)................................................................................ 53
5. Pretest Frequencies for Part B of the Mathematics Beliefs
Instrument (MBI)................................................................................ 54
6. Posttest frequencies for Part B of the Mathematics Beliefs
Instrument (MBI)................................................................................ 55
7. Pretest frequencies for Part C of the Mathematics Beliefs
Instrument (MBI)................................................................................ 57
8. Posttest frequencies for Part C of the Mathematics Beliefs
Instrument (MBI)................................................................................ 57
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v ii
ABSTRACT
There is much concern about the state of mathematics education in the U. S.
Student achievement, beliefs, and efficacy are influenced by the elementary teacher,
stressing the importance of quality mathematics teacher preparation. This
quantitative study examined the effects of an integrated content and methods
mathematics course taught using constructivist principles on preservice teachers’
beliefs and efficacy toward teaching and learning mathematics. The sample included
124 preservice teachers enrolled in mathematics content or integrated mathematics
content and methods at a mid-sized California university. This study generating
descriptive and comparative data had a nonequivalent control group design with two
groups, the experimental group who received the intervention and a second group
used as the control receiving no treatment.
Data were collected using the Mathematics Beliefs Instrument, a survey
consisting of 33 Likert-style items. Descriptive analysis and analysis of covariance
were conducted to answer three research questions. Results indicated that there
were statistically significant differences in beliefs about teaching and learning
mathematics shifting in a direction more consistent with reform-like beliefs. There
was also a significant enhancement of the preservice teachers’ efficacy toward
learning and teaching mathematics. The analysis of covariance between the
experimental and the control groups revealed no statistically significant difference in
the variables measured.
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This study has important implications regarding the mathematics preparation
of preservice elementary teachers at similar institutions. Recommendations for
further research include studies with random assignment of treatment groups and the
removal of the researcher from the study. Continued research is needed to determine
which course design has the greatest effect on preservice teachers’ mathematical
beliefs and efficacy. Finally, it is recommended that longitudinal studies track
preservice teachers beyond the preparation phase into the first years of teaching to
examine how the change impacts teachers’ practice.
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1
CHAPTER 1
PRESERVICE TEACHERS AND MATHEMATICS EDUCATION
Introduction
Teacher quality has come under increased scrutiny in the past few years. The
No Child Left Behind (U.S. Department of Education, 2002) legislation requires that
teachers be highly qualified, defined as demonstrating competency in the subject
matter areas. The education community now recognizes teacher quality to be the
most important school-related factor affecting student achievement (Darling-
Hammond, 1996a).
These circumstances have led institutions involved in teacher preparation to
examine their programs ensuring that they are preparing preservice teaching candi
dates appropriately. Reform efforts have set ambitious goals for the teaching and
learning process. Teachers and schools have been given the charge to develop deep
comprehension of content, teach students to construct and solve problems, evaluate,
synthesize and critique content (NCTM, 2000; NEGP, 2001). In an effort to meet
these goals, teachers must know and understand deeply the subjects they teach and
be able to draw on that knowledge with flexibility. Effective teaching requires
teachers who understand their students as learners and be skillful in choosing from a
variety of pedagogical and assessment strategies (National Commission on Teaching
and America’s Future, 1996). There is no question among legislators or education
professionals, that teacher quality matters more than ever.
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2
One of the national education goals is that every student receives an
equitable, high quality education. The largest concern is closing the minority
achievement gap. According to the results of the National Assessment of
Educational Progress (NAEP), we are not achieving that goal in mathematics
education. Although all racial/ethnic groups of students have shown improvement
since 1990, the 2003 scores show that white and Asian/Pacific Islander students
continue to outperform black, Hispanic, and American Indian/Alaskan native
students at every grade level (NCES, 2003). Other test results show U. S. students in
all groups are underachieving in mathematics, at the State level the California
Standards Test (CST), and at the international level the Third International
Mathematics and Science Study (TIMSS). This could be attributed to many factors:
In some cases, students lack a commitment to learning, some students have not had
the opportunity to learn important mathematics, and in other instances, the
curriculum and instruction offered to students is not engaging or rigorous.
Students learn mathematics through the experiences that teachers provide.
Their understanding, abilities, confidence in, and disposition toward mathematics are
all shaped by the teaching they encounter in school. The improvement of mathe
matics education for all students, requires effective teaching of mathematics in all
classrooms. In response to these increased demands and pressures, schools have to
choose a direction in which to go. There is a possibility that excellent teaching could
be replaced by “teaching to the test” ignoring resources or tools in order to reach
their goal of improved test scores. Another possibility is that schools will seek out
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educational research and tools that will improve instruction, enacting new and more
effective practices such as “teaching for understanding for all.”
Gamoran, Anderson, Puiroz, Puiroz, Secada, Williams, and Ashman (2003)
suggest this encompasses two goals for students in our schools: (a) understanding of
science and mathematics content; and (b) equity among student of different races,
cultures, social classes, and levels of ability. Evidence from both TIMSS and NAEP
suggests that neither of these goals is being achieved presently. American students
score lower on international tests than do students of other industrialized nations. A
restructuring of instruction and curriculum designed to engage students in collective
pattern finding and sense making is suggested by these test results analyses. Accord
ing to the TIMMS research analysis, U. S. teachers are expected to cover far more
content than teachers in other countries. The study indicates that U. S. mathematics
textbooks address 175% as many topics as do German textbooks and 350% as many
topics as do Japanese textbooks. Yet when it comes to achievement, German and
Japanese students significantly outperform U. S. students in mathematics (Schmidt,
McKnight, & Raizen, 1996). Another major finding from the TIMMS research was
that teachers in the United States exhibit an over reliance on textbooks for decisions
about content and pacing (Stevenson & Stigler, 1992; Stigler & Hiebert, 1999).
Currently, the math curricula in use in most American schools is fact and procedure
based, not designed to develop a deep understanding of concepts in real world
contexts.
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Background o f the Problem
The classroom teacher is primarily responsible for helping students meet the
expected outcomes instituted by NCLB and state legislation. The problem facing
more and more districts every year is the lack of qualification of the tens of thousand
of teachers hired each year. The research is clear— good teachers are critical. “No
other intervention can make the difference that a knowledgeable, skilled teacher can
make on student achievement” (Center for the Future of Teaching and Learning,
1999, p. 5).
Understanding the relationship between teacher quality and student achieve
ment is a critical element of the nation’s agenda because the data on student achieve
ment is unsettling. The performance of our public school students as it has been
reported by the National Assessment of Educational Progress (NAEP) mathematics
proficiency results in 2000 and 2003 indicate that students in California, for
example, scores are well below the national average (Table 1). Mathematics profi
ciency levels have been improving since 1990, which seems encouraging; however,
when disaggregated by race/ethnicity, the scores have not improved and in some
cases have dropped. More importantly for teacher educators, these statistics
reinforced speculation that teacher quality might be one of the causes for these
perceived failures. National concern over the underlying causes of low achievement
has led researchers to examine the quality of teacher preparation programs and to
determine how preservice teachers learn (Bransford, Brown, & Cocking, 2000;
Darling-Hammond, 1996/2000/2001; Zimmerman, 1994).
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Table 1 . NAEP 2003 Percentage Below Basic
Proficiency in Mathematics
Public Grade 4 Grade 8
Nation 24% 33%
California 33% 44%
Teacher education programs are not standardized, and therefore, vary greatly.
Each has its own recipe, a combination of theory and practice, some with not enough
of one or the other, some with uninspired pedagogy, such as faculty who give
lectures about group work with out ever practicing the strategy. One of the critical
components of a high quality teacher education program, is that they allow teachers
to learn about practice in practice (Ball& Cohen, 1999).
Teachers learn just as students do: (a) by studying, (b) doing, (c) reflecting,
and (d) collaborating with others. Extraordinary programs according to Darling-
Hammond (2000), involve many elements working together including a strong
grounding in content areas and in how to teach them to children of particular ages.
Another element of an exemplary preparation program is a commitment to a broad
repertoire of strategies to meet differing needs of learners. Darling-Hammond adds
that teachers who lack knowledge of content and/or teaching strategies cannot offer
their students adequate learning opportunities. In today’s high stakes education
climate, students may not be able to compete because the system failed to provide
them with adequate teachers. Although the preparation phase of teaching is not the
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only place along the professional continuum that needs to be strengthened, it is the
focus of this study.
There is evidence that the teacher’s own beliefs about teaching math are
significantly influenced by their experiences as a student. Additionally, research has
found success in solving math problems is not based solely on one’s knowledge of
math, but also on metacognitive processes related to math strategy usage and
personal beliefs of one’s math abilities (Schoenfeld, 1985).
A teacher’s own efficacy and beliefs are critical factors in determining
student achievement (Armor et al., 1976; Ashton & Webb, 1986; Gibson & Dembo,
1984). Beliefs are part of the foundation upon which behaviors are based. Beliefs
have been strongly associated with behavior in Bandura’s (1981) theory of social
learning. Behavior is enacted when people not only expect specific behavior to
result in desirable outcomes (outcome expectancy), but they also believe in their own
ability to perform the behaviors (self-efficacy) (Bandura, 1986).
When this theory of self-efficacy is applied to the study of teachers, it has
been found that various measures of the construct are related to student achievement
(Anderson, Greene, & Loewen, 1988; Armor et al., 1976; Ashton, 1985; Berman,
McLaughlin, Bass, Pauley, & Zellmon, 1977; Gibson & Dembo, 1984; Ross, 1992),
motivation (Midgley et al., 1989) and students’ own sense of efficacy (Anderson et
al., 1988). Several behaviors that affect student learning such as a teachers’ willing
ness to try new instructional techniques, their affect toward students, and their
persistence in trying to solve learning problems appear to be related to a teachers’
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sense of efficacy (Allinder, 1994; Ashton, Olejnid, Crocker, & McAuliffe, 1982;
Gibson & Dembo, 1984; Guskey, 1984; Ross, 1992). Teachers’ sense of efficacy
has also been related to their practices such as their use of more effective hands-on
science and mathematics techniques (Enochs, Scharmann, & Riggs, 1995; Enochs,
Smith, & Huinker, 2000). Research shows that views of self-efficacy appear to form
fairly early in the career, and are relatively difficult to change thereafter (Tschannen-
Moran, Wolfolk Hoy, & Hoy, 1998). They make the case for the importance of
developing teachers’ knowledge, skills, and sense of their ability to influence
teaching outcomes early on.
There is research supporting the theory that teachers’ efficacy and beliefs
affect their teaching and learning and that we have a tendency to teach the way we
were taught (Bandura, 1997; Darling-Hammond, 1996a; Darling-Hammond, Chung,
& Frelow, 2002; Enochs & Riggs, 1990; Enochs et al., 2000; Schoenfeld, 1989).
Teacher educators need to be cognizant of this and model practices and methods that
they are attempting to foster in their students. It is expected that a majority of these
students will become K-12 classroom teachers and their own learning experiences
will be influential in the types of mathematics programs they offer to their students.
Purpose o f Study
The purpose of this study is to examine the relationship of an integrated
content and methods mathematics course and preservice teachers’ beliefs and
efficacy toward the teaching and learning of mathematics. It is an examination of the
direct effect of participation in a course designed using constructivist principles and
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employing multiple instructional strategies, on the affective variables (construct)
being measured. It is expected that results from this study will inform the practice of
higher education instructors in determining whether the content and methods
included in their courses are effective, or if changes are needed in order to improve
mathematics education.
Research Questions
The questions this study is designed to answer are as follows:
1. What is the effect o f participation in an integrated mathematics
content and methods course on preservice teachers ’ beliefs toward mathematics?
2. What is the effect o f participation in an integrated mathematics
content and methods course on preservice teachers ’ efficacy toward mathematics?
3. Does the integrated content and methods (experimental) mathematics
course have more influence on the affective variables being measured than the
content alone or traditional format (control) mathematics course?
Hypotheses
1. There is a relationship between preservice teachers’ beliefs toward
mathematics and their participation in an integrated mathematics content and
methods course.
2. There is a relationship between preservice teachers’ efficacy toward
mathematics and their participation in an integrated mathematics content and
methods course.
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3. There is a difference between the integrated and the traditional
mathematics courses and their effect on the affective variables being measured.
Significance o f the Problem
This study emphasizes the importance of the integration of content and
methods in courses designed for preservice teacher learning. The liberal studies
program examined in this study is typical of most teacher education/preparation
programs; the requirement for mathematics is only 6 units or credits. With this
limited number of courses, it could be suggested that the problem with teaching and
learning mathematics in elementary classrooms is caused by the limited content
background of the teachers. However, lack of content knowledge is only part of this
complex issue. Graham et al. (2000) found that at most higher learning institutions,
content and methods courses have two different homes. Content courses are
typically taught in the math department and methods courses by faculty in the
education department. “Such a division would seem to work against any effort to
establish a view of mathematics teacher preparation as a coherent process. In
addition, the division may foster a prospective that methods are unrelated to content,
or that content is more important than methods” (p. 20).
Fennema and Franke (1992) found that there is growing consensus in the
mathematics education community that prospective teachers need to acquire several
types of knowledge; content, pedagogical, and pedagogical-content. Affective
variables related to the learning of mathematics also play an important role in the
development of preservice teachers. Throughout the process of learning mathe
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matics, preservice teachers collect a wide variety of experiences-both positive and
negative. These experiences have led to the development of their beliefs about
mathematics. Beliefs can be defined as personal assumptions from which indivi
duals make decisions about the actions they will undertake (Bandura, 1986).
Teacher education programs must sometimes take on the task of helping students/
participants to deconstruct and reconstruct their beliefs about teaching and learning
mathematics. There is also research (Anderson et al., 1988) on the relationship
between teacher and student beliefs about mathematics. Teacher beliefs influence
what occurs in the classroom and although the scope and magnitude of the effect has
not been clearly defined, and is not the only factor influencing teaching practices and
related student outcomes or achievement, it most certainly plays an important role.
Graham and Fennell (2001) point out that because elementary school mathe
matics looks different than the mathematics preservice teachers learns in a college
classroom, the most significant challenge in the preparation of the preservice teacher
is to forge a relationship between their own deep understanding of mathematics, the
mathematics instructional needs of children, and appropriate strategies for teaching.
Since student achievement in mathematics appears to be at least linked to, if not
somewhat dependent upon the teacher’s own beliefs, efficacy, and conceptual under
standing and knowledge of mathematics; it is important for preservice teacher
preparation programs to evaluate their own instruction and course offerings (Armor,
1976; Ashton & Webb, 1986; Darling-Hammond et al., 2002; Gibson & Dembo,
1984).
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The possibility of future teachers using a greater variety of teaching methods
in their own mathematics teaching will be greatly increased if the processes,
methods, and strategies are modeled in their college courses. This provides the
opportunity for the preservice teachers to experience construction of learning,
building their confidence in their mathematical abilities.
A teacher’s role is not limited to the delivery of information, but to scaffold
and respond to students’ learning efforts (Brophy, 1989; Borko & Putnam, 1996). In
light of this view, powerful lessons do not require rote memory and recitation but
critique and problem-solving. Powerful learning communities are characterized by
reflection, discourse, and assistance (Holmes Group, 1990; Rogoff, 1990). Brophy
(1989) has asserted that teaching for understanding and learning takes time. Topics
need to be meaningful to students and studied in depth which would mean that
teachers have to sacrifice breadth for depth in the curriculum, yet, recent research
shows that with the current mandated testing of mandated curriculum, teacher
education and student curricula may suffer.
NCTM (2000) promotes the view of mathematics teaching as a “complex
endeavor” with no “easy recipes” for helping students learn or for helping all
teachers become effective. They suggest that effective teaching requires knowing
and understanding mathematics, students as learners, and pedagogical strategies.
Fennel and Graham (2001) believe that this has implications for preservice teacher
preparation— representing a shift from a major emphasis on mathematical content
knowledge particularly at the secondary level, with a minor emphasis on the methods
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of teaching to the recognition that mathematics content preparation alone is not
sufficient.
Constructivism, an approach to learning developed from cognitive psycholo
gy has focused educator’s attention on the central role learners play in constructing
new knowledge. Influenced by the work of Jean Piaget (1966) and Lev Vygotsky
(1978), it is an eclectic view of learning that emphasizes four key components
(Kauchak& Eggen, 2003):
1. Learners construct their own understanding rather than having it
delivered or transmitted to them
2. New learning depends on prior understanding and knowledge
3. Learning is enhanced by social interaction
4. Authentic learning tasks promote meaningful learning
New knowledge can almost never be simply imparted from an adult to a
child, but rather, new knowledge must be actively constructed by the child (Piaget,
1966). The challenge then is to design a mathematics program that introduces
children to mathematics in a way that will make contact with their own knowledge
construction or schema-teaching that builds conceptual bridges (Griffin & Case,
1997).
Constructivist models are often compared to transmission models that assume
that students acquire knowledge by having it transmitted to them by a teacher or a
text (Cognition and Technology Group at Vanderbilt [CTGV], 1996). There is much
confusion over constructivist theories and their implications for instruction. Some
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assume that constructivist approaches focus on student exploration and discovery
without teacher guidance. Most constructivists reject this view and would instead
argue for guided discovery, and scaffolded inquiry as instructional models (Brown &
Campione, 1994; CTGV, 1996).
The guided discovery or inquiry approach is appropriate especially for future
elementary teachers because it is learning by doing or a hands-on method. In the
elementary classroom, the inquiry approach has long served to engage students in
learning actively, not passively asking them to discover scientific knowledge rather
than being told answers by the teacher or textbook. Using the inquiry method,
mathematics content is covered in greater depth compared to a traditional textbook
approach. Elmore, Peterson, and McCarthey (1996) suggest that this view of
curriculum and instruction fundamentally places greater emphasis on students’ deep
understanding of content, rather than on rote memorization of facts. Classrooms that
represent this view of learning are characterized by students interacting around diffi
cult and interesting problems that engage their interest while the teacher facilitates or
guides them, rather than by passive assimilation of knowledge purveyed by a teacher
standing in the front of the room.
Most beliefs are formed through experience over time, pedagogical practices
that support constructivist theory can be nurtured by engaging pre-service teachers in
learning and in teaching mathematics. It has been shown that learning with concept-
tual understanding makes subsequent learning easier. Mathematics makes more
sense and is easier to remember and apply when students construct new knowledge,
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14
connecting to existing schema or existing knowledge in meaningful ways. The
experience alone does not insure change, but facilitates it.
Much of the research on teacher quality centers around experiential factors
that have an effect on student achievement. Years of experience and levels of
certification are cited as accounting for a high percentage of the variation in student
achievement at the school level. In the report by the National Commission on
Teaching and America’s Future (1996c) the metaphor of a three legged stool of
teacher quality assurance was used. The legs represented teacher education program
accreditation, initial teacher licensing, and advanced professional certification.
Teacher subject matter knowledge has not been found to have a direct relationship
with student achievement, but pedagogical knowledge has. Darling-Hammond
(2000) points out that the positive effects of subject matter knowledge are augmented
or offset by knowledge of how to teach the subject to various kinds of students.
Marzano et al. (2003) notes that it is important for teachers to have a certain amount
of subject matter knowledge, but perhaps more important for them to have peda
gogical knowledge of how best to teach that subject matter content.
Definition o f Terms
Beliefs
Part of the foundation upon which behaviors are based. Behavior is enacted
when people not only expect specific behavior to result in desirable outcomes
(outcome expectancy), but they also believe in their own ability to perform the
behaviors (self-efficacy) (Bandura, 1986).
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Constructivism
Theory suggesting that students learn by constructing their own knowledge,
incorporating what they already know, especially through hands-on exploration
(models include guided discovery and inquiry approach).
Self-efficacy
A person’s expectation that he or she has the ability to deal effectively with a
particular task (Bandura, 1986).
Teaching-efficacy
A situation specific expectation that teachers can help students learn
(Bandura, 1986).
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CHAPTER 2
REVIEW OF THE LITERATURE
Introduction
During the last 2 decades, the tenets of self-efficacy as a component of social
cognitive theory have been widely researched in educational settings (Ashton &
Webb, 1986; Bandura, 1997; Paqares, 1996; Pintrich & Schunk, 1996). One focus
of the inquiry into self-efficacy beliefs is concerning the efficacy beliefs of teachers
and the relationship to their instructional practices and to various student outcomes
(Armor et al., 1976; Gibson & Dembo, 1984).
According to Bandura’s (1987) social cognitive theory, the self-beliefs that
individuals use to exercise a measure of control over their environments include self-
efficacy beliefs, “beliefs in one’s capability to organize and exercise the courses of
action required to manage prospective situation” (Bandura, 1997, p. 2). Strong
efficacy beliefs enhance human accomplishment and person well-being in many
ways.
Preservice teachers learn to teach through multiple means, both individually
and socially. They learn over time using different knowledge, skills, and abilities in
different contexts (Anderson et al., 2000). Preservice teachers learn by observation
during years of being in classrooms building upon beliefs about teaching and
learning developed over years of being students themselves (Bandura, 1977; Lortie,
1975). They learn by developing subject matter knowledge, pedagogical knowledge,
and the meta-language to talk and think about what they and other teachers do in the
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classroom (Shulman, 1986). Finally, they leam by practicing under supervision and
guidance, situated in real classroom contexts (Anderson et al., 2000).
To better understand how preservice teachers leam from and are influenced
by these multiple contexts, this study focuses on the preservice teachers’ self-
efficacy and epistemological beliefs about the teaching and learning of mathematics
and their participation in a mathematics course integrating content and methods
which was designed using social cognitive and constructivist principles.
Mathematics education has undergone several waves of reform in the past 40
years, each with the goal of improving mathematics learning. Yet change has been
difficult and many student experience math classes where math is no more than a set
of arbitrary mles and procedures to be memorized. Math education has remained
much the same as it was in 1950 or even 1900. No single cause accounts for the
failure of the past reform efforts, but it has been the focus of many studies (Cohen &
Ball, in press; McLaughlin, 1990). Some of the most frequent explanations for what
has impeded the progress of math reform efforts are the misrepresentations of
mathematics; culturally embedded views of knowledge, learning, and teaching;
social organization of schools and teaching; curriculum materials and assessments;
and teacher education and professional development (Cohen & Ball, 2001).
The purpose of this literature review is to examine teacher education,
especially as it relates to mathematics education and to define self-efficacy and
beliefs and their relationship to teaching and learning mathematics.
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Mathematics Education and Reform
Teacher education programs are challenged with the task of preparing
students to become educators in a time when standards and expectations are more
rigorous than ever before. The NCLB legislation requires every classroom teacher to
be “highly qualified” a term that has been the focus of much reform in California and
throughout the United States. The national goal is that every student receives an
equitable, high quality education. A focus of closing the minority achievement gap
is cited in much of the documentation. According to the results of the NAEP, we are
not achieving that goal in mathematics education. Although all racial/ethnic groups
of students have shown improvement since 1990, the 2003 NAEP scores show that
White and Asian/Pacific Islander students continue to outperform Black, Hispanic,
and American Indian/Alaskan Native students at every grade level (NAEP, 2003).
Stigler and Hiebert (1999) assert that by analyzing NAEP results we know
that almost all students leam to add, subtract, multiply, and divide whole numbers,
and the majority leam to do very simple arithmetic with fractions, decimals, and
percents. We also know, however, that students’ knowledge and skills are very
fragile and apparently learned without much depth or conceptual understanding.
In an effort to overcome racial inequities in mathematics instruction, the
NCTM (2000) developed its Principles and Standards fo r School Mathematics
suggesting these six principles of practice: (a) high expectations for all students;
(b) a coherent curriculum of important mathematics, articulated across grade levels;
(c) teachers who understand what students need to leam and then challenge and
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support them; (d) instruction that builds new knowledge from experience and prior
knowledge; (e) assessment that supports learning and provides useful information to
both teachers and students; and (f) technology that influences the mathematics taught
and enhances students’ learning.
There is research suggesting that the application of these principles and
standards to practice can improve equity. Schoenfeld (2002) examined a large urban
school district to find how a mathematics reform curriculum based on the application
of the NCTM standards and principles supported minority students. He found that
with the new curriculum in place the proportion of minority students performing well
in terms of skills doubled in the years 1997 to 2000. Fewer than one-third of the
minority students met or exceeded the skill standard of a reference examination
under the former traditional curriculum, described as one that does not emphasize the
kinds of mathematics that would enable students to make sense of the world around
them. With the new curriculum, more than 50% of the minority students met or
exceeded the standard. The conclusion of this study was that to ensure sustained
improvement in mathematics instruction, schools must provide a high-quality
curriculum; a stable, knowledgeable, and professional teaching community; and high
quality assessment aligned with the curriculum.
Teacher Knowledge and Practice
Defining what teachers need to know and be able to do in order to be part of a
knowledgeable and professional teaching community that can effectively instruct
mathematics is difficult and complicated. Although much debated amongst
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researchers, practitioners, policymakers and professional developers for more than
four decades, the answer continues to be elusive and continually changing. Each
stakeholder defines it differently based on their perspective. Ma (1999) conducted
research comparing Chinese and U. S. elementary teachers’ mathematical knowl
edge. A portrait of dramatic differences between the two groups was produced. This
study has attracted and renewed interest in the issue. Ma used her data to develop a
theory of “profound understanding of fundamental mathematics” a kind of
connected, curricularly structured, and longitudinally coherent knowledge of core
mathematical ideas.
Claims that teachers’ knowledge matters are commonplace, but the empirical
evidence to support this supposedly obvious fact has been elusive. Fennema and
Franke (1992) assert that although no one questions the idea that what a teacher
knows is one of the important influences on what takes place in the classroom, and
ultimately what students leam, there is no consensus existing on the mathematical
knowledge that is required to teach. This lack of consensus is problematic for both
policymakers and teacher educators. In order to identify what teachers need to know
or leam, it is important to know how much knowledge of mathematics it takes to
teach effectively. Teacher certification requirements, teacher education policy and
practice, recruitment of teachers, and curriculum development all rely on this idea.
One approach to solving the problem of defining what and how much
mathematics a teacher needs to know has been a policy response, the creation of
multiple documents and lists of what teachers should know. Another approach is
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centered on the research regarding the characteristics of teachers. It is associated
with the assumption that knowledge of and skill with mathematics content is
essential to teaching. Factors such as courses taken, degrees earned, or certification
received is representative of the required knowledge, and that it can be measured
using straightforward indices. This assumption has led researchers to attempt to
empirically validate the theory that the more mathematical knowledge teachers have,
the more mathematical knowledge their students will have (Begle, 1979; Darling-
Hammond, 1999; Monk, 1994).
Knowledge of the subject matter appears to be essential to good teaching to a
certain point, a basic level of competence beyond which it is a less important
influence. When the relationship between subject matter knowledge and teacher
effectiveness is studied, it does not appear to be strong. Begle (1972) found in a
review of mathematics teaching that the absolute number of mathematics course
credits was not linearly related to teacher performance. Monk (1994) in a longitu
dinal study of mathematics and science student achievement found that teachers’
content preparation, as measured by the coursework inn the subject field, is
positively related to student achievement in mathematics and science but the
relationship is curvilinear, with diminishing returns to student achievement above a
threshold level found to be five courses in mathematics.
Knowledge of teaching and learning appears to have a stronger and more
consistent relationship with teacher effectiveness. Begle (1979) found that the
number of credits a teacher had in mathematics methods courses was a stronger
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correlate of student performance than that of the number of credits in mathematics
courses or other indicators of preparation. This finding was validated by Monk
(1994) when he found that teacher education coursework had a positive effect on
student achievement in mathematics and science and was sometimes more influential
than additional subject matter preparation.
Effective teaching requires teachers to demonstrate high levels of subject
matter knowledge, pedagogical knowledge, and pedagogical content knowledge
(Borko & Putnam, 1996; Shulman, 1986). Like students, teachers interpret experi
ences through the filters of their existing knowledge and beliefs (Cohen & Ball,
1990). This is particularly relevant when discussing the sort of teaching that is being
called for in mathematics reform movements like that of NCTM (2000) Principles
and Standards for School Mathematics. The approach emphasizes the importance of
students’ thinking and the development of powerful reasoning and understanding
within subject domains. The reform asks that teachers replace their “direct instruct-
tion” models of teaching with a model that provides opportunities for students to
explore ideas in rich contexts rather than relying primarily on teacher presentation
and student rehearsal (Cohen & Ball, 1990; Schoenfeld, 2002).
In order for teachers to be successful in this shifting of instructional methods,
their knowledge and beliefs must become the targets of change. Shulman et al.
(1989) developed a framework, which categorizes teacher beliefs and knowledge
into general pedagogical knowledge and beliefs, subject matter knowledge and
beliefs, and pedagogical content knowledge and beliefs. General pedagogical
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knowledge includes a teacher’s knowledge and beliefs about teaching, learning, and
learners not specific to particular subject matter domains. Current math reform
movements are grounded in social cognitive learning theory, where learners are
viewed as active problem solvers who construct their own knowledge. The teacher
is responsible for setting up situations or contexts where students’ cognitive activities
are stimulated for learning (Anderson, 1989; Brophy, 1989).
General Pedagogical Knowledge and Beliefs
The instructional and assessment practices necessary for teachers to support
and foster students’ thinking and facilitating their cognitive activities are quite
different from traditional practices, which constitute much of their educational
experience. Preservice teachers’ views of learning can shift toward more construc
tivist perspectives, but how much these views are translated into practice is influ
enced greatly by beliefs that were formed over time.
Subject Matter Knowledge and Beliefs
A number of studies have suggested that teachers with a rich understanding
of subject matter tend to emphasize conceptual, problem-solving, and inquiry aspects
of their subjects, whereas less knowledgeable teachers tend to emphasize facts and
procedures (Ball, 1990; Fennema & Franke, 1992). Other research shows that the
number of courses taken in a particular subject is not a strong determinant of
teaching effectiveness.
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Pedagogical Content Knowledge and Beliefs
Shulman (1986) interprets pedagogical content knowledge and beliefs as
“understanding of what makes the learning of specific topics easy or difficult: the
conceptions and preconceptions that students of different ages and backgrounds
bring with them to the learning of those most frequently taught topics and lessons”
(p. 9). Pedagogical content knowledge and beliefs are distinguished for subject
matter knowledge in that it focuses on the knowledge and skills specific to teaching
particular subject matter.
Social Cognitive Learning Theory
Preservice teachers as learners arrive in their education courses with
established beliefs that influence their learning and what they belief about teaching
(Calderhead, 1996; Lortie, 1975; Munby, Russell, & Martin, 2001). These
preconceptions and misconceptions may also influence their learning during the
leaming-to-teach process, challenging teacher educators’ to impact these beliefs,
which are widely understood to be highly resistant to change (Pajares, 1992a/1992b;
Richardson, 1996).
Yet, new learning requires a change in previously held understandings and
beliefs. Learners need to be engaged and engage themselves in the learning process
cognitively, metacognitively, and motivationally (Pintrich, 2003).
Social constructivist theory views learning as an active process in which
people interpret experience through their existing conceptual structures, or schema,
to modify and expand their knowledge (Piaget, 1985). Like students in their future
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classrooms, preservice teachers should be encouraged to impose meaning and organ
ization on incoming information by relating them to their existing knowledge and
beliefs. These beliefs about learning and learners often remain implicit and serve as
filters through which preservice teachers interpret new instructional approaches
(Ball, 1989; Brown & Borko, 1992).
Historical developments in constructivist learning theory can be concept
ualized in two waves. The cognitive constructivism popular in the 1960s to the
1980s focused on the individual. This era viewed the individual as an active learner
who constructed meaning through multiple opportunities to interact with the
environment of people, properties, and processes (Piaget, 1963/1970). This
perspective described learning as an active construction and restructuring of prior
knowledge. Information processing models were developed emphasizing the human
mind as a symbol processing system (Woolfolk, 2001).
Building on information processing theory, social views of learning began to
expand the psychological process to include the impact of social forces. In the
1990s, the emphasis on the individual’s cognition alone has broadened to include a
greater emphasis on social context. Social cognitive views describe how people
leam through interactions with others and how observation, dialogue, and culture
affect learning. Social cognitive theory (Bandura, 1977/1986/1997) emphasizes the
importance of such cognitive factors as beliefs, self-perceptions, and expectations.
Motivational factors such as self-efficacy beliefs, goal orientation, and task value
also play a fundamental role. The social interaction and the forces both inside and
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outside of the learner become critical in understanding how learners can become
agents of their own learning. The constructivism of the 1990s emphasizes the impact
and influence of sociocultural contexts on thinking and learning (Cole, 1996; Rogoff,
1990).
Social cognitive theory suggests that beliefs about learning are a product of
the activity, culture, and context in which they are cultivated (Jehng, Johnson, &
Anderson, 1993; Schoenfeld, 1983/1985). These beliefs are epistemological, or
socially shared intuitions about the nature of knowledge and learning. Schoenfeld
(1985) states that epistemological beliefs establish a context within which intellect-
tual resources are accessed and utilized, they are related to but distinct from meta
cognition. In his study, Schoenfeld compared the problem solving protocols of
mathematicians and students of mathematics. His method for teaching mathematical
problem solving involved formulating a set of heuristic strategies for how to
approach problem solving that was similar to strategies used by expert mathema
ticians. He observed that students assumed that they would be able to solve a
problem quickly if they were going to solve it at all, that when the procedure for
solving a problem did not come readily to mind the students would resort to blindly
trying answers and checking them.
The students, he believed, had classroom experiences that supported this type
of behavior, most problems they encountered were intended to be solved within an
average of two minutes. Their experience led to their beliefs that the way to solve a
problem was to remember the right procedure, which did not require extensive
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thought or work. He modeled the strategies, provided scaffolding and coaching, and
gradually faded his facilitation in the problem solving process. As students assumed
responsibility for the process, they began to act in less predictable ways, relying
more on their own ideas and reasoning instead of the rules or procedures they have
previously relied upon. This study and other related research gives the perspective
of learning being just as much a matter of enculturation into particular ways of
thinking and dispositions as it is a result of explicit instruction in particular concepts,
skills, and procedures (Resnick, 1989; Schoenfeld, 1985).
This historical perspective gives us a view of learning taking place as a
means of assisted performance in which the learner internalizes the types of
problem-solving behaviors they have been assisted in using by more capable others.
Vygotsky (1978) suggests the learner transforms those behaviors and internalizes
them towards independent control while participating in learning events that take
place within the zone of proximal development.
Taken together these views can be seen as complementary perspectives of
learning positioning preservice teachers as active learners situated in academic,
social and cultural contexts who can reflect upon their learning and teaching and
make informed decisions (Pintrich, 2003). They are, as teachers, also placed in an
important position to assist their students by modeling effective learning strategies
and behaviors.
Until recently, most teacher education program courses on subject matter
were disconnected from their courses on methods, which in turn, were disconnected
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from their courses on learning and development. Darling-Hammond (1996a) argues
that preservice teachers need to leam active, hands-on and minds-on teaching and to
do this they must have experienced it for themselves. Traditional lecture still
dominates in much of higher education where faculty does not practice what they
profess. The absence of powerful, effective teacher education is problematic when at
a time when the nature of teaching is changing and those entering the profession may
never themselves have experienced the kind of challenging instmction they are
expected to offer. Darling-Hammond (1996a) concludes that this cycle must be
broken “It is difficult to improve practice if new teachers teach as they were taught
and if the way they were taught is not what we want” (p. 32).
Implications
Much of the research in mathematics education has grown out of theoretical
work in cognitive psychology (Pajares, 1992; Resnick, 1989; Schoenfeld, 1983/
1985). Social cognitive and constructivism emerged as the epistemological founda
tions for mathematics education and a new vision developed of what it means to
know and do mathematics. In response, NCTM has spearheaded a reform move
ment (NCTM, 1989/2000) that has influenced mathematics education research.
NCTM argues that mathematics students and classrooms should be active and
involved with learners communicating their thinking, defending and justifying their
arguments, and working with others to construct mathematical knowledge.
In response to this new vision, a related view of how to teach within this
theoretical framework was formed. The research community began to explore
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aspects of teacher education and change from this perspective (Borko & Brown,
1992; Borko & Putnam, 1996). This research has provided a substantial body of
evidence that teachers’ efficacy, self-beliefs, and epistemological beliefs about
teaching and learning drive their teaching of mathematics (Enochs et al., 1995/2000;
Parjares, 1992, Schoenfeld, 1983/1985). In order to change teachers’ practices, their
beliefs and efficacy need to be considered. Beliefs are difficult to change and it
takes time. Preservice teaching students bring established beliefs to their preparation
program, they have ideas about what it takes to be an effective teacher that have been
formed through experience over time (Brown & Borko, 1992; Pajares, 1992).
It appears that pedagogical practices or methods that support social cognitive
and constructive theory can be nurtured through contextual experiences in the
learning of mathematics. These experiences alone do not ensure change in beliefs,
but change is limited when preservice teachers leam mathematics methods separately
or differently than content. Given the limited amount of time preservice programs
have to influence teacher development, teaching the methods along with the content
of mathematics increases the chance for change.
Graham and Fennell (2001) suggest that the structure of most university
programs needs to change to accommodate the shifting nature of teacher leaning and
more connections that are explicit between undergraduate and school mathematics
should be provided. This study is designed to measure the effects of such an inte
grated mathematics methods and content course on beliefs and efficacy variables
which play an important role in the development of preservice teachers. Although
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there have been numerous studies attempting to understand the relationship of beliefs
and efficacy and teaching preparation, the subjects of the studies are generally in the
fifth year of study in the preparation program.
This study examines the preservice teachers during their undergraduate
degree experience with the intent of providing students with authentic learning
experiences, of the nature discussed in this chapter, earlier in their preparation. This
study will provide quantitative data that will illustrate that participation in a liberal
studies course designed to provide social cognitive and constructivist experiences
can effect change in preservice teachers’ beliefs and efficacy toward mathematics.
There has also been a lack of control in the methodological design in some of the
previous studies. This study has a quasi-experimental control group design so the
results may have increased validity as differences can be attributed more closely to
the intervention.
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CHAPTER 3
RESEARCH METHODOLOGY
Introduction
The purpose of this study was to examine preservice teachers’ beliefs and
efficacy about teaching and learning mathematics and how these variables relate to
participation in a mathematics course integrating content and methods. With calls
for reform in both teacher preparation and public schooling, the quality of teaching,
curriculum requirements, and achievement standards of teachers and students are
facing increased scrutiny. There is research supporting the theory that teachers teach
the way they were taught and that their efficacy and beliefs affect their teaching and
learning (Bandura, 1997; Darling Hammond et al., 2002; Enochs & Riggs, 1990;
Enochs et al., 2000; Schoenfeld, 1989).
This study was designed to evaluate the effectiveness of an integrated course
combining methods and content in providing positive mathematics learning experi
ences that impact efficacy and beliefs of preservice teaching students. The theory is
that teaching behavior is related to the teachers’ mathematical beliefs and that these
beliefs which are formed over time influence their instructional practice (Brown &
Borko, 1992; Thompson, 1984). Examination of the methods and content included
in teacher preparation courses and the effects they can have on changing beliefs and
efficacy informs instruction for these preservice courses.
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This chapter includes the research questions and description of the research
methodology. The latter includes the sampling procedure and population, instrumen
tation, and procedures for data collection and analysis.
Research Questions
The questions this study is designed to answer are as follows:
1. What is the effect o f participation in an integrated mathematics
content and methods course on preservice teachers ’ beliefs toward mathematics?
2. What is the effect o f participation in an integrated mathematics
content and methods course on preservice teachers ’ efficacy toward mathematics?
3. Does the integrated content and methods (experimental) mathematics
course have more influence on the affective variables being measured than the
traditional instruction (control) mathematics course?
Research Design
A quantitative study using a quasi-experimental design was conducted during
the fall semester of 2004. The total duration of the study was approximately 15
weeks. This study took place in a medium-sized rural state university in northern
California; 164 preservice teaching students will participate in the study. The
research plan was completed by collecting data from preservice teachers who plan to
teach at either the elementary, middle, or secondary level and are enrolled in Math
50A, a required course for students seeking teaching certification. This study has a
nonequivalent (pretest and posttest) control group design (Table 2). Group A is the
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experimental group who received the intervention (methods modeled while teaching
content) and group B is the control group (traditional content instruction).
Table 2. Research Experimental Design
Group A (experimental) Pre-test X Post-test
Group B (control) Pre-test Post-test
A pilot study was conducted during the spring semester of 2004. The survey
instrument was completed by 24 students in one section of the experimental
mathematics course being examined prior to and upon completion of the course. The
analysis of the responses is discussed in the instrumentation section of this chapter.
Population and Sample
This study was conducted on the campus of a medium-sized rural state
university in northern California. The preservice teaching students participating in
the study are enrolled in the Liberal Studies program at the university. They are
working toward a Bachelor of Arts degree and predominately choose teaching as
their future profession. The liberal studies program at this university is the largest on
campus. Approximately two-thirds of the liberal studies graduates go on to enter a
5th year preservice teacher program in the Department of Education to prepare for
careers as classroom teachers.
Students enter the program with a variety of backgrounds. There are several
pathways through the program to choose from, all of which require the Math 50A
and B courses. The sample will consist of students enrolled in Math 50A Concepts
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of Mathematics course. There were two groups, one receiving traditional instruction
(n = 22) and one receiving the integrated content and methods course (n = 102).
There was no random sampling or assignment.
The freshman students entering the university have an average high school
GPA of 3.27, an average age of 18, and 99% of them are from California.
Their scores on the SAT I math are as follows (Table 3):
Table 3. SAT 1 Math Score for Entering Freshman Students
Range of scores Percent of students
700-800 1
600-699 16
500-599 48
400-499 30
300-399 5
200-299 1
The Entry Level Mathematics Placement Examination (ELM) is required for all CSU
students unless they can demonstrate proficiency by scoring 550 or above on SAT I.
The emphasis of the test is on assessing understanding of mathematics concepts and
problem solving skills rather than recall of facts and equations.
Description o f Intervention
The required mathematics courses are taught by many different instructors/
professors with differing styles. Although efforts are made to have a somewhat
standardized content, courses differ widely. In order to provide courses that
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influence preservice teachers’ mathematical beliefs and efficacy, it is important to
model with appropriate methods and content.
The focus of this study is the teaching and learning approach and its effect on
beliefs and efficacy. The primary purpose of this research was to investigate the
beliefs and efficacy that preservice teachers had, while learning mathematics and
whether participation in an integrated content and methods course will have any
effect on these affective variables.
The specific course within the liberal studies program investigated in this
study is Math 50 A. It is one of two mathematics courses required for all liberal
studies students. The integrated course focuses on the Number Sense strand, specif
ically, the structure of the number system, operations, and theory. The elements of
the course include: (a) a constructivist approach, (b) inquiry model, (c) multiple
strategies, and (d) cooperative/group learning. The Number strand has historically
been the cornerstone of the entire mathematics curriculum. It is especially empha
sized in pre-K through Grade 2 and receives less instructional attention in Grades
9-12.
The experimental course was planned with content and methods integration.
Two textbooks are required for the semester course, mathematics content for
elementary teachers {Mathematical Reasoning for Elementary Teachers, Long &
DeTemple, 1997) and mathematics methods for elementary teachers {Math at Hand,
1999). In addition, a course reader with several select articles that are mainly
focused on strategies and methods is required. Topics included in the course are:
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1. California mathematics framework
2. National Council of Teachers of Mathematics Principles and
Standards
3. Alternative algorithms
4. Multicultural perspectives
5. Informal strategies
6. Invented procedures
7. Mental computation
8. Children’s natural processes
Other course requirements include two classroom observations of 1 l A hours
each at local elementary schools followed up with written reflections of the
observations. Also, there are two lab sessions scheduled with local elementary
students coming to campus to practice hands-on, cooperative mathematics activities
that the integrated course students are responsible for planning and facilitating.
Throughout the course there are assessments (quizzes and exams) designed to
evaluate both mathematics content and methods. They were conducted both indivi
dually and in small group settings. Assignments consisted of doing or applying
mathematics, and creating and planning mathematics activities and lessons for
students. Other assessment measures were designed to examine the teaching and
learning process including oral reflections, discussions, and written responses.
The Mathematics Framework for California Public Schools (1999) organizes
the content standards for Kindergarten through Grade 7 by grade level and are pre
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sented in five strands: (a) Number Sense; (b) Algebra and Functions; (c) Measure
ment and Geometry; (d) Statistics, (e) Data and Probability; and (f) Mathematical
Reasoning. According the Principles and Standards (2000) set forth by the National
Council of Teachers of Mathematics (NCTM), “The number and operations standard
describes deep and fundamental understanding of, and proficiency with, counting,
numbers and arithmetic, as well as an understanding of number systems and their
structures” (p. 32).
The instruction of this course was constructivist in manner, using a hands-on,
manipulative based approach. The course instructor modeled good pedagogy while
teaching mathematics content. The students had the opportunity to experience
learning mathematics within the constructivist environment, and then were asked to
plan and practice teaching lessons that support constructivist learning and reflect on
the experiences encouraging deeper assimilation of knowledge and cognition. The
integrated course syllabus description is as follows:
The course content will emphasize Number Sense and deep understanding of
operations with whole numbers and fractions. Most activities will be done in
cooperative group settings where the processing is just as important as ‘the
answer’; thus attendance and thoughtful, active participation are required and
valued. (Math 50A course, personal communication, 2003)
Marzano et al. (2001) in their meta-analysis research combined a number of
studies, to determine the effects of instructional strategies or techniques on K-12
student achievement. Using effect size, which easily translates into percentile gains,
the study aims to identify instructional strategies that “have a high probability of
enhancing student achievement for all students in all subject areas at all grade levels”
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(p. 7). The following are nine categories of strategies that have a strong effect on
student achievement: (a) identifying similarities and differences, (b) summarizing
and note taking, (c) reinforcing effort and providing recognition, (d) homework and
practice, (e) nonlinguistic representations, (f) cooperative learning, (g) setting
objectives and providing feedback, (h) generating and testing hypotheses, and (i)
questions, (j) cues and advance organizers.
Many of these strategies are emphasized in the integrated course including
(a) cooperative learning, (b) nonlinguistic representations, (c) higher level question
ing, and (d) generating and testing hypotheses. Marzano et al. (2001) points out that
although these strategies are good tools, they should not be expected to work well in
all situations and that other aspects of classroom pedagogy also affect student
achievement such as management and curriculum.
Instrumentation
Efficacy and beliefs in relation to teaching can be measured as affective
variables. Questionnaires are a common quantitative protocol for measuring these
constructs (Bandura, 1997; Enochs et al., 2000; Enochs & Riggs, 1990; Schoenfeld,
1989; Zollman & Mason, 1992).
The Mathematics Beliefs Instrument (appendix) is a self-report questionnaire
designed to measure beliefs and efficacy about teaching and learning mathematics.
Respondents were directed to answer the questions as accurately as possible. Sub
scales were divided between two broad categories: beliefs and efficacy.
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Belief subscales have two sections: (a) NCTM standards beliefs and
(b) general beliefs about teaching and learning mathematics. Efficacy subscales
were also divided into two subscales: (a) self-efficacy and (b) teaching efficacy.
The instrument contains 33 items worded in statements such as, “Children should be
encouraged to justify their solutions, thinking, and conjectures in a single way,” and
“I understand mathematics concepts well enough to be effective in teaching mathe
matics.” The instrument was scored on a 4-point Likert scale (1 = true, 2 = more
true than false, 3 = more false than true, 4 = false).
The MBI was formed using a combination of items which were developed
and tested as part of other instruments and some of which the researcher developed
or modified based on the pilot study results, population of study, and the context of
the research. The first 16 items from the MBI were from the Standards Belief
Instrument developed by Zollman and Mason (1992) to measure teachers’ beliefs
about the NCTM Standards (1989). The NCTM standards were written to initiate a
national reform in mathematics education. The document presents NCTM’s vision
of how K-12 mathematics should be taught, learned, and evaluated. The Standards
call for greater emphasis on conceptual development, reasoning, and problem
solving than was provided in previous mathematics curricula. The Standards
portrays mathematics as a connected, cohesive body of knowledge and calls for
teachers to encourage active engagement of students in making conjectures and
discussing ideas. Research shows that teaching behavior and practice is influenced
by what teachers believe mathematics should be (Thompson, 1984) suggesting that
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4 0
important relationships exist between a teacher’s beliefs and their behavior. An
acceptance of the NCTM standards would depend then on a teacher’s beliefs.
The 16 items were either nearly direct quotes or the inverse of direct quotes
from the Standards document. There were 8 that agreed with, and 8 that disagreed
with the Standards. The items were intended to be representative of the standards,
not inclusive. The purpose was to assess the beliefs underlying the Standards not
knowledge of them. The items were randomly chosen on the basis of theme rather
than grade level. Items met three criteria. First, the implication of each item was
intended not to be intuitively obvious to avoid a socially desirable (or correct
according to the Standards) pattern of responding. Second, the item could be clearly
stated in a positive or negative manner. Third, the central idea of the item could be
incorporated into a single sentence (Zollman & Mason, 1992).
The instrument was tested for construct validity by analyzing the convergent
and discriminant nature of correlations between the SBI and other variables. The
data analyzed came from two groups of teachers, one of experienced teachers and the
other of preservice students (n = 133) who were administered the instrument. The
correlations supported the integrity of the construct underlying the SBI. The
variables that should have highly correlated with the instrument did, and lower
correlations were found for those variables that should not highly correlate with the
instrument.
Reliability of the SBI was investigated with internal consistency procedures.
Spearman Brown reliability and a coefficient alpha were used to compare the two
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41
groups. The analysis showed a much higher reliability (SB .803 with coefficient
alpha approaching .79) for the experienced group, while the preservice group showed
(SB .493 and coefficient alpha approaching .65). The reliability evidence rating for
the SBI shows more consistency from subjects more knowledgeable about the
Standards and thus suggests evidence for validity of the instrument as well.
Only one item from the SBI was modified by the researcher for the purposes
of this study. The item from the SBI was worded; “Appropriate calculators should
be available to all students at all times.” It was changed to reflect the updated
Standards (2000) and now reads, “Calculators should be available at appropriate
times as computational tools.”
The next 12 items from the MBI were selected from The Problem Solving
Project Questionnaire designed to measure epistemological beliefs about mathe
matics teaching and learning (Schoenfeld, 1989). The means and standard deviations
for all 70 original Likert scale items were analyzed. The sample of 230 participants
provided data that were analyzed using correlation coefficients, which found strong
correlations between items measuring the same construct. The item factor analysis
supports the reliability and validity of the instrument.
The last five items from the MBI were designed to measure efficacy, both
teaching (outcome) and self efficacy. The items were selected from the Mathematics
Teaching Efficacy Beliefs Instrument (Enochs et al., 2000; MTEBI). Bandura’s
(1981) theory of social learning suggested that people develop a generalized
expectancy concerning action-outcome contingencies based upon life experiences.
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They also develop specific beliefs concerning their abilities to cope with change.
Bandura (1986) called this self-efficacy. Efficacy when applied to teaching is
broadly defined as a situation-specific expectation that teachers can help students
learn (Ashton & Webb, 1986; Bandura, 1997). Research supports the existence of
two relatively independent factors of teaching efficacy that relate to the two
dimensions of Bandura’s theory of self-efficacy which are personal efficacy and
outcome efficacy (Ashton & Webb, 1986; Enochs & Riggs, 1990; Gibson & Dembo,
1984). The MTEBI for preservice teachers consists of 21 items, 13 of which
measure the personal mathematics teaching efficacy (PMTE) subscale and eight
which measure the mathematics teaching outcome expectancy (MTOE) subscale.
Reliability analysis produced an alpha coefficient of 0.88 for the PMTE scale and
0.75 for the MTOE scale (n - 324). Confirmatory factor analysis indicated that the
two scales were independent adding to the construct validity. The developers
suggest that validation of instruments is an ongoing process, but that consistent
research utilizing the MTEBI shows it to be a valid and reliable instrument.
To establish reliability for the MBI, data from the pilot study (n = 24)
conducted during the fall semester was analyzed for internal consistency. Each item
was compared using paired item sample t-test for pre- and posttest values, comparing
means and standard deviations for the group and subscale score totals.
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Procedures
Data Collection
Four intact groups of liberal studies students (n = 102) enrolled in the Math
50A integrated content and methods course designed using constructivist principles
participated as the experimental group in this study. One other intact group of liberal
studies students (n - 22) enrolled in the Math 50A course taught using a traditional
approach participated as the control group in this study. Quantitative data were
collected from both pre- and postmeasures of the Mathematics Beliefs Instrument
(MBI). Reliability and validity information and an explanation of the types of
questions used in the MBI have been addressed in the instrumentation section of this
chapter.
When the semester began, participants were informed of the nature of the
study and assured that their participation in the study was completely voluntary and
anonymous. The intervention for the control group occurred within a routine
educational experience for the liberal studies students. The control group experi
enced routine instruction with no intervention. All participants in both groups were
assured that their participation or nonparticipation would in no way affect their
academic success.
The courses were taught by Mathematics and Statistics department faculty,
whose fields of study vary, but all have some degree of mathematics specialization
and only a few of them have a mathematics education specialty. They were
contacted by the researcher, informed of the study and invited to participate. During
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the first and last weeks of the semester course, the MBI was administered to the
participating enrolled students. The survey took approximately 15 minutes of
instructional time.
This study was approved by the California State University, Chico’s and the
University of Southern California’s Institutional Review Board for the Protection of
Human Subjects. Participation was informed, voluntary, and anonymous. The
preservice teacher participants in the study were identified by their date of birth for
coding purposes, comparing pre- to postresponses, no student was identified by
name.
Data Analysis
This study is quantitative, generating descriptive and comparative data for
measuring efficacy and beliefs about teaching and learning mathematics. As
discussed earlier, quantitative measures of MBI have been used in past studies and
have made advances in understanding this phenomenon (Enochs et al., 2000;
Schoenfeld, 1989; Zollman & Mason, 1992). Most often, survey methods are used
for their efficiency and economy in terms of time, labor, and generalizability to
populations (Denzen, 1978).
This research study attempts to add to the body of knowledge by examining
the effects of participation in an integrated content and methods mathematics course
on preservice teachers’ beliefs and efficacy toward mathematics. Cited throughout
this chapter are several studies investigating the constructs of teaching beliefs and
efficacy and the effects of participation in a course or training on these variables.
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45
The design of this study was quasi-experimental utilizing pre and post measures with
an intervention in between for the experimental group. Some of the other studies had
a similar methodology, but the addition of a control group in this study will increase
the validity of the findings.
Research Question 1
What is the effect o f participation in an integrated mathematics content and
methods course on preservice teachers ’ beliefs toward mathematics?
The Mathematics Belief Instrument (MBI) was administered as a general
measure of beliefs about teaching and learning mathematics. As discussed earlier,
some of the items are specifically designed to measure the participants beliefs and
how closely aligned they are with NCTM Principles and Standards (2000). The
results of each perservice teachers’ MBI were calculated. The total scores for each
group’s pre and post-test measures were correlated. The subscale scores were also
determined. The MBI data was entered into the Statistics Program for the Social
Studies (SPSS) to find total average score and an analysis of variance was performed
to determine if there is any difference between the two groups: Control and experi
mental. Specific subscale scores for efficacy, general beliefs about mathematics, and
beliefs aligned with NCTM were also analyzed. A dependent t-test analysis was
used to determine the degree of shared variance between efficacy and beliefs and
participation in an integrated content and methods mathematics course.
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Research Question 2
What is the effect o f participation in an integrated content and methods
mathematics course on preservice teachers ’ efficacy toward mathematics?
Using subscale scores for self-efficacy, a dependent t-test analysis
determined the degree of shared variance between preservice teachers’ mathematics
self-efficacy prior to and upon completion of the course.
Research Question 3
Does the integrated content and methods (experimental) mathematics course
have more influence on the affective variables being measured than the content
alone or traditional format (control) mathematics course?
A total score for frequency of responses for each group was calculated. An
analysis of covariance (ANCOVA) was calculated to determine if there is a
statistically significant difference in the affective variables, efficacy and beliefs,
between those participating in the integrated course (experimental) and those in the
traditional course (control).
Delimitations and Limitations
Initially, this study confined itself to testing the preservice teaching students
enrolled in Math 50A courses at a rural California State University.
The purposive sampling procedure decreases the generalizability of the
findings. This study is not generalizable to all areas of preservice teaching students.
In this quantitative study, the findings would be subject to other interpretations.
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CHAPTER 4
FINDINGS
Introduction
The purpose of this study was to examine preservice teachers’ beliefs and
efficacy toward teaching and learning mathematics and how these variables relate to
participation in a mathematics course integrating content and methods. This study
was a quantitative study generating descriptive and comparative data for measuring
efficacy and beliefs of preservice teachers and the relationship of these constructs to
participation in a mathematics course designed using constructivist principles.
Four intact groups of liberal studies students (n = 102) enrolled in the Math
50A integrated content and methods course designed using constructivist principles
participated as the experimental group in this study. One other intact group of liberal
studies students (n = 22) enrolled in the Math 50A course taught using a traditional
approach participated as the control group in this study. Quantitative data was
collected from both pre and post measures of the Mathematics Beliefs Instrument
(MBI).
The following three sections of this chapter report the results for each
question this study was designed to address. The overall scores for the MBI and
subscale scores for efficacy and beliefs are reported.
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Research Question 1
What is the effect o f participation in an integrated mathematics content and
methods course on preservice teachers ’ beliefs toward mathematics?
One hundred two preservice teaching students (experimental group, n = 102)
were administered the MBI prior to and upon completion of the integrated mathe
matics course as a general measure of math efficacy and beliefs. Twenty-two
additional preservice teaching students (control group, n - 22) were administered the
MBI prior to and upon completion of the traditional mathematics course as a general
measure of math efficacy and beliefs. The pre- and postmeasures were matched
using subjects’ date of birth. The MBI data was entered into to Statistics Program
for the Social Sciences (SPSS).
The means and standard deviations for total scores on the Mathematical
Beliefs Instrument (MBI) for the overall group are presented in table 4. Frequency
distributions for the total MBI are presented in figures 1 and 2.
Table 4. Means and Standard Deviations for Total Mathematical
Beliefs Instrument (MBI) Pre- and Postscores for All Subjects
Pre Post
N M SD M SD
124 2.21 .20 2.04 .28
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TOTALPRE
Std. Dev = .20
Mean = 2.21
N = 124.00
1.75 1.88 2.00 2.13 2.25 2.38 2.50 2.63 2.75
T O T A L P R E
Figure 1. Pretest frequencies for the Mathematics Beliefs Instrument (MBI).
TOTALPO
30 ■
Std. Dev = .29
Mean = 2.05
1.38 1.63 1.88 2.13 2.38 2.63
1.50 1.75 2.00 2.25 2.50 2.75
T O T A L P O
Figure 2. Posttest frequencies for the Mathematics Beliefs Instrument (MBI).
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5 0
Reliability for the total MBI was investigated with internal consistency
procedures. An item total correlation for the 33 items was analyzed and a coefficient
alpha was used to compare the two groups. The analysis produced a coefficient
alpha approaching .49 suggesting evidence of a lack of reliability for the total
instrument.
A paired sample t test statistical analysis was completed on the MBI total to
determine the difference between pre and posttest scores for the overall group. The
observed probability of the observed difference between the pre and post test scores
(p < .05) provides evidence of a statistically significant difference between the means
for the scores on the pre- and posttests.
Subscale scores for general beliefs about teaching and learning mathematics,
beliefs consistent with National Council of Teachers of Mathematics Standards
(NCTM, 2000), and mathematics efficacy were also calculated and analyzed. Part A
of the MBI consisting of 16 Likert-style items was a form of the Standards Beliefs
Instrument (Zollman & Mason, 1992), which determines the level of consistency of
an individual’s beliefs with the philosophy of the National Council of Teachers of
Mathematics Standards (2000). Eight of these items were consistent with the
standards and eight of the items were inconsistent with the standards. The means
and standard deviations for the standards beliefs subscale of the Mathematical
Beliefs Instrument (MBI) for the overall group are presented in table 5. Frequency
distributions for the standards beliefs subscale are presented in figures 3 and 4.
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Table 5. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI) Part A Pre- and Posttest Measures (All Subjects)
Pre Post
N M SD M SD
124 2.26 .22 2.14 .27
STPRE
5-
C
< D
= 3
cr
a >
l L
Figure 3. Pretest frequencies for Part A of the Mathematics Beliefs Instrument
(MBI).
Reliability for the standards beliefs subscale of MBI was investigated with
internal consistency procedures. An item total correlation for the 16 items measuring
this construct was analyzed and a coefficient alpha was used to compare the two
groups. The analysis produced a coefficient alpha approaching .21 suggesting
evidence of a lack of reliability for Part A of the instrument.
Std. Dev = .22
Mean = 2.26
N = 124.00
1.63 1.75 1.88 2.00 2.13 2.25 2.38 2.50 2.63 2.75
S T P R E
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52
STPOST
1.50 1.75 2.00 2.25 2.50 2.75
1.63 1.88 2.13 2.38 2.63
STPOST
Figure 4. Posttest frequencies for Part A of the Mathematics Beliefs Instrument
(MBI).
A paired sample t test statistical analysis was completed on the standards
beliefs subscale to determine the difference between pre- and posttest scores for the
overall group (n = 124). The observed probability of the observed difference
between the pre- and posttest scores (p < .05) provides evidence of a statistically
significant difference between the means for the scores on the pre- and posttest
scores.
Part B of the MBI consists of 12 Likert-style items which were adapted from
the Problem-Solving Project (Schoenfeld, 1989), and was used to assess change in
preservice teachers’ beliefs about teaching and learning mathematics within and
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53
outside the school setting. The means and standard deviations for the general beliefs
subscale of the Mathematical Beliefs Instrument (MBI) for the overall group are
presented in table 6. Frequency distributions for the general beliefs subscale are
presented in figures 5 and 6.
Table 6. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI) Part B Pre- and Posttest Measures (All Subjects)
Pre Post
N M SD M SD
124 2.89 .37 3.07 .45
BEPRE
401------------ ------
Std. Dev = .38
M ean = 2.89
N = 124.00
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
BEPRE
Figure 5. Pretest frequencies for Part B of the Mathematics Beliefs Instrument
(MBI).
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54
BEPO ST
Std. Dev = .45
Mean = 3.08
N = 124.00
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
B E P O S T
Figure 6. Posttest frequencies for Part B of the Mathematics Beliefs Instrument
(MBI).
Reliability for the general beliefs subscale of MBI was investigated with
internal consistency procedures. An item total correlation for the 12 items measuring
this construct was analyzed and a coefficient alpha was used to compare the two
groups. A coefficient alpha approaching .61 was determined. In order to increase
consistency and reliability, item 22 was trimmed. The remaining 11 items were
analyzed producing a coefficient alpha approaching .65 suggesting evidence for
reliability of Part B of the instrument.
A paired sample t-test statistical analysis was completed on the general
beliefs subscale to determine the difference between pre and post test scores for the
overall group (n - 124). The observed probability of the observed difference
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55
between the pre- and posttest scores (p < .05) provides evidence of a statistically
significant difference between the means for the general beliefs subscale on the pre
and post test scores.
Research Question 2
What is the effect o f participation in an integrated content and methods
course on preservice teachers ’ efficacy toward mathematics?
Part C of the MBI consists of 5 Likert-style items, which were selected from
the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) (Enochs et al.,
2000) designed to measure preservice teachers’ mathematics efficacy. Subjects were
asked to indicate their beliefs about their effectiveness in learning and teaching
mathematics. The means and standard deviations for the efficacy subscale of the
Mathematical Beliefs Instrument (MBI) for the overall group are presented in table
7. Frequency distributions for the efficacy subscale are presented in figures 7 and 8.
Table 7. Means and Standard Deviations for the Mathematics Beliefs
Instrument (MBI) Part C Pre- and Posttest Measures (All Subjects)
Pre Post
N M SD M SD
124 2.16 .69 2.02 .63
Reliability for the efficacy subscale of MBI was investigated with internal
consistency procedures. An item total correlation for the five items measuring this
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56
SE PR E
Std. Dev = .70
Mean = 2.16
N = 124.00
S E P R E
Figure 7 . Pretest frequencies for Part C of the Mathematics Beliefs Instrument
(MBI).
S E P O S T
50
40
30
20
S' 10
< D
1 3
S’
it o
S E P O S T
Figure 8. Posttest frequencies for Part C o f the Mathematics Beliefs Instrument
(MBI)
Std. Dev = .63
Mean = 2.02
N = 124.00
1.00 1.50 2.00 2.50 3.00 3.50
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57
construct was analyzed and a coefficient alpha was used to compare the two groups.
The analysis produced a coefficient alpha approaching .79 suggesting evidence of
reliability for Part C of the instrument.
A paired sample t-test statistical analysis was completed on the general
beliefs subscale to determine the difference between pre and posttest scores for the
overall group (n = 124). The observed probability of the observed difference
between the pre- and posttest scores (Sig. 2-tailed = .000,/? < .05) provides evidence
of a statistically significant difference between the means for the efficacy subscale
on the pre- and posttest scores.
Research Question 3
Does the integrated content and methods (experimental) mathematics course
have more influence on the affective variables being measured than the traditional
instruction (control) mathematics course?
Analysis of covariance (ANCOVA) was calculated to determine if there was
a statistically significant difference in the dependent variables between subjects in
the control and experimental groups. The analyses of covariance will be presented
for each of the dependent variables: NCTM Standards beliefs, general mathematical
beliefs, and mathematical efficacy.
Table 8 shows the unadjusted means and standard deviations of the post test
scores for both the experimental and control groups on the MBI standards beliefs
subscale. The standards beliefs subscale effect size was calculated by finding the
difference between the means of the experimental group and the control group and
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Table 8. Posttest Mathematical Beliefs Instrument (MBI) ANCOVA Effects for
Experimental and Control Groups (Unadjusted Means)
Group Variable n
General math
Standard beliefs beliefs Efficacy
M SD M SD M SD
Experimental 2.15 .28 3.06 .47 1.99 .62 102
Control 2.11 .22 3.11 .33 2.15 .66 22
dividing that difference by the standard deviation of the control group. The effect
size was found to be . 18 indicating the treatment may not have had a practical impact
on the preservice teaching students’ standards beliefs (p > .50), but the result has to
be qualified due to the differential drop out rates and the resulting pretest differences
between the experimental and control groups. ANCOVA analysis was used to
statistically control for pretest differences.
The adjusted posttest scores for the standards beliefs subscale were used as a
dependent variable to determine, whether or not participation in the integrated
content and methods mathematics course had a statistically significant impact on the
preservice teaching students’ beliefs. The adjusted posttest means and standard
deviations for the standards beliefs subscale are shown in table 9. The ANCOVA
results for the test of effect between groups on the post standards beliefs subscale of
the MBI revealed no statistically significant difference [F = .21 indicating observed
significance at the .64 level, p > .05].
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5 9
Table 9. Posttest Mathematical Beliefs Instrument (MBI) ANCOVA Effects for
Experimental and Control Groups (Adjust Means)
Group Variable n
General math
Standard beliefs beliefs Efficacy
M SD M SD M SD
Experimental 2.15 .02 3.07 .03 2.00 .03 102
Control 2.12 .05 3.08 .08 2.08 .08 22
Table 8 shows the unadjusted means and standard deviations of the posttest
scores for both the experimental and control groups on the MBI general mathe
matical beliefs subscale. The general mathematical beliefs subscale effect size was
calculated by finding the difference between the means of the experimental group
and the control group and dividing that difference by the standard deviation of the
control group. The effect size was found to be .15 indicating the treatment may not
have had a practical impact on the preservice teaching students’ standards beliefs (p
> .50), but the result has to be qualified due to the differential drop out rates and the
resulting pretest differences between the experimental and control groups.
ANCOVA analysis was used to statistically control for pretest differences.
The adjusted posttest scores for the general mathematical beliefs subscale
were used as a dependent variable to determine whether or not participation in the
integrated content and methods mathematics course had a statistically significant
impact on the preservice teaching students’ beliefs. The adjusted posttest means and
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60
standard deviations for the general beliefs subscale are shown in table 8. The
ANCOVA results for the test of effect between groups on the post general
mathematical beliefs subscale of the MBI revealed no statistically significant
difference [F= .013 indicating observed significance at the .91 level, p > .05].
Table 8 shows the unadjusted means and standard deviations of the post test
scores for both the experimental and control groups on the MBI mathematical
efficacy subscale. The mathematical efficacy subscale effect size was calculated by
finding the difference between the means of the experimental group and the control
group and dividing that difference by the standard deviation of the control group.
The effect size was found to be .24 indicating the treatment may not have had a
practical impact on the preservice teaching students’ efficacy ip > .50), but the result
has to be qualified due to the differential drop out rates and the resulting pretest
differences between the experimental and control groups. ANCOVA analysis was
used to statistically control for pretest differences.
The adjusted post test scores for the mathematical efficacy subscale were
used as a dependent variable to determine whether or not participation in the
integrated content and methods mathematics course had a statistically significant
impact on the preservice teaching students’ efficacy. The adjusted posttest means
and standard deviations for the efficacy subscale are shown in table 9. The
ANCOVA results for the test of effect between groups on the post mathematical
efficacy subscale of the MBI revealed no statistically significant difference [F= .79
indicating observed significance at the .37 level, p > .05].
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61
CHAPTER 5
DISCUSSION
The basic premise of this study was that participation in an integrated content
and methods course would have an effect on preservice teachers’ efficacy and beliefs
toward teaching and learning mathematics. The discussion in this chapter includes a
summary of the purpose and design of the study. Following the design of the study,
are the results followed by the limitations of the study and the implications of the
results. In the final section of the chapter, conclusions are drawn and suggestions for
future research are presented.
Summary o f the Purpose
The purpose of this study, is to examine the relationship of an integrated
content and methods mathematics course and preservice teachers’ beliefs and
efficacy toward the teaching and learning of mathematics. It is an examination of the
direct effect of participation in a course designed using constructivist principles and
employing multiple instructional strategies, on the affective variables (construct)
being measured. It is expected that results from this study will inform the practice of
higher education instructors in determining whether the content and methods
included in their courses are effective, or if changes are needed in order to improve
mathematics education.
Summary o f the Des ign
A quantitative study using a quasi-experimental design was conducted in the
fall semester of 2004. This study was framed by a social cognitive theoretical
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model, which offers a view of academic learning that may begin to help teacher
educators understand more about how preservice teachers learn, how their efficacy
and beliefs are affected, and how this may affect their classroom practice. The total
duration of the study was approximately 16 weeks. This study took place with 124
preservice teaching students in the liberal studies program in a medium-sized rural
state university in California. Four intact groups (n - 102) enrolled in the Math 50A
integrated content and methods course designed using constructivist principles
participated as the experimental group in this study. One other intact group in = 22)
enrolled in the Math 50A course taught using a traditional approach participated as
the control group in this study. Quantitative data was collected from both pre and
post measures of the Mathematics Beliefs Instrument (MBI). The overall and
subscale scores from the MBI were analyzed to determine differences in the
variables form pretest to posttest. Subscales included beliefs about NCTM
Standards (2000), general beliefs about teaching and learning mathematics, and
mathematical efficacy.
Discussion o f the Results
The preserrvie teaching students who participated in this study, are college
students having completed multiple mathematics courses required for enrollment in
the program/course. It might be assumed that with this level of experience and
interest in education, that most or all of these students would have strong beliefs and
efficacy toward their own learning and teaching of elementary mathematics. Indeed,
researchers have illustrated how critical these constructs are to successful
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mathematics teaching. Teachers’ efficacy, self beliefs, and epistemological beliefs
about teaching and learning drive their teaching of mathematics (Enochs et al.,
1995/2000; Parjares, 1992, Schoenfeld, 1983/1985). In order to change teachers’
practices, their beliefs and efficacy need to be considered. Beliefs are difficult to
change and it takes time. Preservice teaching students bring established beliefs to
their preparation program, they have ideas about what it takes to be an effective
teacher that have been formed through experience over time (Brown & Borko, 1992;
Pajares, 1992).
However, instead of being a homogenous group in this regard, the results
showed that these preservice teachers possessed a range of mathematics beliefs and
efficacy as demonstrated by the MBI. When differences were calculated between
pretest and posttest measures for the overall score and subscale scores for standards
beliefs, general beliefs, and efficacy they all indicated a statistically significant
difference. These differences were significant for both the control and experimental
groups.
Differences between the two measures were in direction more consistent with
the philosophy of current reform efforts in mathematics education and of enhanced
efficacy toward teaching and learning mathematics. These differences may be
attributed to increased content knowledge attained over the duration of the course.
The more knowledgeable preservice teachers become about mathematics content and
standards, the more confident they may become concerning their ability to teach
mathematics. Since the preservice teachers have limited experience actually
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6 4
teaching, they believe if they know the content they can teach it successfully. There
is little awareness of the complexity of the teaching and learning cycle and possible
pedagogical weaknesses that need further development. Marzano et al. (2003) assert
that it is important for teachers to have a certain amount of subject matter knowl
edge, but perhaps more important for them to have pedagogical knowledge of how
best to teach that subject matter content.
It appears that pedagogical practices or methods that support social cognitive
and constructive theory can be nurtured through contextual experiences in the
learning of mathematics. These experiences alone do not ensure change in beliefs,
but change is limited when preservice teachers learn mathematics methods separately
or differently than content. The students’ preconceptions influence their learning
and their new learning requires a change in these previously held understandings and
beliefs. In order to accomplish this, learners need to be actively engaged in the
learning process cognitively, metacognitively, and motivationally (Pintrich, 2003).
Given the limited amount of time preservice programs have to impact teacher
development, teaching the methods along with the content of mathematics increases
the chance for change. Graham and Fennell (2001) suggest that the structure of most
university programs needs to change to accommodate the shifting nature of teacher
leaning and more explicit connections between undergraduate and school
mathematics should be provided.
When the posttest measures were calculated to determine if there were
differences in the dependent variables between subjects in the control and
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65
experimental groups, the analysis revealed no statistically significant differences for
any of the dependent variables: (a) standards beliefs, (b) general mathematical
beliefs, and (c) mathematical efficacy. The results of the test of differences between
groups on the posttest variables indicate that participation in the integrated or
experimental course did not have more impact on the preservice teaching students’
beliefs and efficacy than did the traditional or control course.
Limitations
Due to the nature of the subject selection, implications of this study are
particular to the preservice teaching students who participated in this study. Studies
with larger sample sizes and across several types of institutions of higher education
that offer credential programs, i.e., private, state, small, and large universities would
strengthen the results. In addition, using the Mathematical Beliefs Instrument over a
longer period of time and with large, diverse samples of subjects will strengthen the
reliability of the instrument.
Implications
In order to learn mathematics content during their teacher preparation,
preservice teaching students need to be engaged in the learning process and just as
importantly, they need to be able to engage their own students in their learning of
mathematics. There is research supporting the theory that teachers’ efficacy and
beliefs affect their teaching and learning and that we have a tendency to teach the
way we were taught (Bandura, 1997; Darling-Hammond, 1996a; Darling-Hammond
et al., 2002; Enochs & Riggs, 1990; Enochs et al., 2000; Schoenfeld, 1989). Teacher
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6 6
educators need to be cognizant of this and model practices and methods that they are
attempting to foster in their students.
Preservice teachers come to their preparation programs with their own
theories about learning and teaching mathematics built by years of observing and
participating in classrooms. There is evidence that teacher’s own beliefs about the
teaching of math are significantly influenced by their experiences as a student. Some
of these experiences may not have been exemplary examples or models of good
pedagogy, focusing on content knowledge not understanding of concepts. These
experiences may lead them to believe they already know a lot about the teaching and
learning of mathematics. Consequently, they may hold simplistic notions about
teaching and learning and disregard, for example, the impact of learner-centered,
constructivist pedagogy. One of the critical components of a high quality teacher
education program is that they allow teachers to learn about practice in practice (Ball
& Cohen, 1999). Preservice teachers learn just as students do: (a) by studying,
(b) doing, (c) reflecting, and (d) collaborating with others. Research has found
success in solving math problems is not based solely on one’s knowledge of math,
but also on metacognitive processes related to math strategy usage and personal
beliefs of one’s math abilities (Schoenfeld, 1985).
Extraordinary teacher preparation programs according to Darling-Hammond
(2000), involve many elements working together including a strong grounding in
content areas and in how to teach them to children of particular ages. In addition,
Darling-Hammond (1996a) maintain that what teachers know and can do in the
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67
classroom with students, has the most impact on what students leam. Teacher
educators must assist preservice teachers by designing relevant, authentic learning
opportunities that will serve as rich, challenging contexts to develop competence in
becoming metacognitively, motivationally, and behaviorally active participants in
their own learning process while at the same time, using their new learning to
understand their role in teaching their students.
A teacher’s own efficacy and beliefs are critical factors in determining
student achievement (Armor et al., 1976; Ashton & Webb, 1986; Gibson & Dembo,
1984). It has been found that various measures of the construct are related to student
achievement (Anderson et al., 1988; Armor et al., 1976; Ashton, 1985; Berman et al.,
1977; Gibson & Dembo, 1984; Ross, 1992), motivation (Midgley et al., 1989), and
students’ own sense of efficacy (Anderson et al., 1988). Several behaviors that
affect student learning such as a teachers’ willingness to try new instructional
techniques, their affect toward students, and their persistence in trying to solve
learning problems appear to be related to a teachers’ sense of efficacy (Allinder,
1994; Ashton et al., 1982; Gibson & Dembo, 1984; Guskey, 1984; Ross, 1992).
Teachers’ sense of efficacy has also been related to their practices such as their use
of more effective hands-on science and mathematics techniques (Enochs et al., 1995;
Enochs et al., 2000). Research shows that efficacy appears to form fairly early in the
career and is relatively difficult to change thereafter (Tschannen-Moran et al., 1998).
They make the case for the importance of developing teachers’ knowledge, skills,
and sense of their ability to influence teaching outcomes early on. In order to make
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6 8
an impact and meet the demands of the teaching profession, teacher preparation
programs and colleges of education will need to create and sustain learning cultures
and curriculum that supports, nurtures, and provides new teachers with opportunities
to develop deep understanding of mathematical concepts and strategies or methods.
Since novice teachers do not become experts in their preparation program of
in the first year of teaching (Darling-Hammond, 1999), they must continue to grow
and develop over the course of their careers in order to keep informed of current
research, legislation, and curriculum changes.
Conclusion
The results of this study indicate that preservice teachers vary in their beliefs
and efficacy toward mathematics and that participating in an integrated content and
methods mathematics course can influence these variables. This study revealed that
preservice teachers showed differences from pretest to posttest measures of beliefs
about mathematical standards, general beliefs about teaching and learning
mathematics, and mathematics efficacy in a direction more consistent with the
philosophy of the program and current reform efforts in mathematics education.
When control and experimental groups were compared, there were no statistically
significant differences between them. This may have been due to the fact that the
instruction provided in the course sections varied even within the control group. The
content was consistent for both groups, the methods were meant to be different.
Each instructor has their own way of teaching the course, and the researcher could
not control for the actual pedagogy employed, only the philosophy the instructor
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6 9
claimed as an influence. Additionally, the students come to the program with such
varying backgrounds, knowledge, experience, and motivation, it is not possible to
control for the level of effect each will exhibit.
From a program evaluation perspective, the students were more prepared to
teach mathematics in a manner consistent with the philosophy of the program after
participating in the course. They have shown evidence of possessing pedagogical
and content knowledge necessary to begin teaching. However, these observations
should be considered with caution. The resilience of these newly formed beliefs and
efficacy is vulnerable to change and has not been tested in the classroom. How
resilient the change will be is not easily predicted or measured. The culture of the
school and legislative pressures of the actual teaching practice may have an influence
on these constructs. Only through continuing support, learning, mentoring, and
professional development, will the success of the course and program be determined.
Recommendations
This study sought to determine that participation in a liberal studies course
designed to provide social cognitive and constructivist experiences effects change in
preservice teachers’ beliefs and efficacy toward mathematics. Recommendations
can be made on the basis of these findings.
In these first few years of the 21st century, the nation has turned its attention
toward teacher quality creating legislation that requires teacher be highly qualified.
The education community recognizes teacher quality to be the most important
school-related factor affecting student achievement. These circumstances have led
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7 0
institutions involved in teacher preparation to examine their programs ensuring that
they are preparing preservice teaching candidates appropriately. Reform efforts have
set ambitious goals for the teaching and learning process. Teachers and schools have
been given the charge to develop deep comprehension of content, teach students to
construct and solve problems, evaluate, synthesize and critique content (NCTM,
2000; NEGP, 2001). Teacher education programs should consider creating and
sustaining courses that address these needs. Preservice candidates should be exposed
to experiences that foster deep understanding of concepts and actively engage them
in their learning. Since teachers have a tendency to revert to an automated schema,
they will likely teach the way they were taught (Bandura, 1997; Darling-Hammond,
1996a; Darling-Hammond et al., 2002; Enochs & Riggs, 1990; Enochs et al., 2000;
Schoenfeld, 1989). Therefore, teacher education programs should be designed with
this in mind. There should also be efforts made to provide preservice teachers
opportunities to practice the theories, methods, and content they are learning with
classroom students. These experiences need not be relegated to their final year of
preparation, but rather should be part of their program in the undergraduate degree
process.
Teacher education research on learning to teach mathematics will be
supported by knowing more about preservice teachers’ beliefs and efficacy toward
mathematics. A teacher’s beliefs and efficacy affect their teaching and learning and
views of self-efficacy appear to form fairly early in the career and are relatively
difficult to change thereafter (Tschannen-Moran et al., 1998). This helps to make the
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71
case for the importance of developing teachers’ knowledge, skills, and sense of their
ability to influence teaching outcomes early on. In addition, future studies need to be
conducted to determine which instructional strategies or methods have the greatest
affect on preservice teachers’ beliefs and efficacy toward teaching and learning
mathematics. Longitudinal research with preservice teachers who become classroom
practitioners could provide more evidence of the effects of their coursework on their
classroom practice.
Finally, more studies need to be conducted to determine which mathematics
instructional methods are most effective with preservice teaching learners. More
research should focus on the variety and appropriate use of these strategies within the
context of teaching and learning mathematics. Since they cannot become experts
during their preparation, ongoing professional development and support should be
provided and encouraged throughout their teaching careers.
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REFERENCES
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APPENDIX
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8 2
D.O.B.
This survey has been designed to measure preservice teachers’ beliefs
and efficacy regarding the learning and teaching of Mathematics. The
data will be collected prior to and upon completion of this section of
Math 50A. Your voluntary participation is greatly appreciated.
For each of the following statements, circle the response that is most closely
aligned w ith your beliefs.
1. Problem solving should be a separate, distinct part of the mathematics
curriculum.
true more true than false more false than true false
2. Students should share their problem-solving thinking and approaches
w ith other students.
true more true than false more false than true false
3. Mathematics can be thought of as a language that must be meaningful if
students are to communicate and apply mathematics.
true more true than false more false than true false
4. A major goal of mathematics instruction is to help children develop the
belief that they have the power to control their own success in
mathematics.
true more true than false more false than true false
5. Children should be encouraged to justify their solutions, thinking, and
conjectures in a single way.
true more true than false more false than true false
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83
6. The study of mathematics should include opportunities for using
mathematics in other curriculum areas.
true more true than false more false than true false
7. The mathematics curriculum consists of several discrete strands such a
computation, geometry, and measurement which can best be taught in
isolation.
true more true than false more false than true false
8. In K-5 mathematics, increased emphasis should be given to reading and
writing numbers symbolically.
true more true than false more false than true false
9. In K-5 mathematics, increased emphasis should be given to use of clue
words (key words) to determine which operation to use in problem
solving.
true more true than false more false than true false
10. In K-5 mathematics, skill in computation should precede word problems,
true more true than false more false than true false
11. Learning mathematics is a process in which students absorb information,
storing it in easily retrievable fragments a s a result of repeated practice
and reinforcement.
true more true than false more false than true false
12. Mathematics should be taught as a collection of concepts, skills, and
algorithms.
true more true than false more false than true false
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84
13. A demonstration of good reasoning should be regarded even more than
students’ ability to find correct answers.
true more true than false more false than true false
14. Calculators should be available at appropriate times as computational
tools.
true more true than false more false than true false
15. Learning mathematics must be an active process.
true more true than false more false than true false
16. Children enter kindergarten with considerable mathematical experience, a
partial understanding of many mathematical concepts, and some
important mathematical skills.
true more true than false more false than true false
17. Some people are good at mathematics and some aren’t.
true more true than false more false than true false
18. In mathematics something is either right or it is wrong.
true more true than false more false than true false
19. Good mathematics teachers show students lots of different ways to look at
the same question.
true more true than false more false than true false
20. Good mathematics teachers show students the exact way to answer the
question they will be tested on.
true more true than false more false than true false
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85
21. Everything important about mathematics is already known by
mathematicians.
true more true than false more false than true false
22. In mathematics you can be creative and discover things by yourself,
true more true than false more false than true false
23. Mathematics problems can be done correctly in only one way.
true more true than false more false than true false
24. To solve most mathematics problems you have to be taught the correct
procedure.
true more true than false more false than true false
25. The best way to do well in mathematics is to memorize all the formulas,
true more true than false more false than true false
26. Males are better at mathematics than females.
true more true than false more false than true false
27. Some ethnic groups are better at mathematics than others.
true more true than false more false than true false
28. To be good at mathematics you must be able to solve problems quickly,
true more true than false more false than true false
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
29. I am very good at learning mathematics.
true more true than false more false than true false
30. I think I will be very good at teaching mathematics.
true more true than false more false than true false
31. Mathematics was one of my least favorite subjects in high school,
true more true than false more false than true false
32. I lack the ability to think mathematically.
true more true than false more false than true false
33. I understand mathematics concepts well enough to be effective in
teaching mathematics.
true more true than false more false than true false
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

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Asset Metadata

Creator York, Darlene R. (author)

Core Title Effects of an integrated content and methods course on preservice teachers' beliefs and efficacy toward mathematics

School Rossier School of Education

Degree Doctor of Education

Degree Program Education

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag education, higher,Education, Mathematics,Education, Teacher Training,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Hocevar, Dennis (committee chair), Cohn, Carl (committee member), Rich, William (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-335686

Unique identifier UC11335882

Identifier 3180327.pdf (filename),usctheses-c16-335686 (legacy record id)

Legacy Identifier 3180327.pdf

Dmrecord 335686

Document Type Dissertation

Rights York, Darlene R.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA