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Teacher's use of socio-constructivist practices to facilitate mathematics learning

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Teacher's use of socio-constructivist practices to facilitate mathematics learning

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TEACHERS’ USE OF SOCIO-CONSTRUCTIVIST PRACTICES TO FACILITATE
MATHEMATICS LEARNING

by

Anna Lillia Arredondo
__________________________________________________________________

A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION

December 2011

Copyright 2011 Anna Lillia Arredondo
ii
DEDICATION
This dissertation is dedicated to my husband Vath and our two amazing boys,
Matthew Tomás and Aiden Neang. You continuously inspire me to be a better person.
I love you.

iii
ACKNOWLEDGEMENTS
I would like to thank my family, friends, and colleagues who continuously
encouraged me throughout this endeavor.
Vath, I don’t know if I could have ever done this without your ongoing support.
From the moment I decided to embark on this journey, you were there to help. Whether
it was a shoulder to lean on, extra help with our boys, or just your editorial input; your
continued support was an integral part of my success. Thank you my love!
A toda mi familia, le agradezco el apoyo y el amor que me brindaron durante
este tiempo. Mami, gracias por inculcar en mí el amor por la escuela. Papi, gracias por
siempre apoyarme en mis estudios. Margarita, thank you for always being there and
listening when I needed to talk and decompress. Ruby, thank you for being there when
I was stressed, overwhelmed and needed an extra set of eyes and hands. Thomas and
Sergio thank you for your interest in my continued success. To my nieces, Antonella,
Noblelynn, and Alani, and my sons, Matthew and Aiden, I hope to instill in you a love
for learning like my own. I hope I can continue to make you all proud.
To my friends, thank you for your continued support throughout this process.
Carmen, my partner throughout this journey, I could have never done this without you
being by my side. Although this journey was much longer than we expected, there is
finally a light at the end of the tunnel for both of us. I am glad we embarked on this
journey together! Elizabeth, you were always there to listen whenever I needed an ear,
thank you for being an academic advisor, a confidant, a sister, and a friend. To the rest
of my NAL sisters, you don’t know how much a simple question: “How’s the
iv
dissertation coming along?” could serve as a motivating factor. Thank you all for your
constant motivation.
I would also like to express my gratitude to the many teachers that helped make
this study a reality. To my colleagues at Magnolia Avenue Elementary School, thank
you for encouraging me when I needed it the most. To the teachers who participated in
this study, thank you for allowing me to contribute to the body of literature for teachers
of mathematics.
Last but not least, I would like to express my deepest gratitude to my
dissertation committee. Dr Reynaldo Baca, thank you for your advice and guidance.
This process would have never culminated without your support. Dr. Linda Fischer,
thank you for taking me under your wing and helping me finish this journey. Your
mentorship was an invaluable component of my success. Dr. Angela Hasan, thank you
for your advice and guidance. Your expertise in the area of mathematics was critical
for my success.

v
TABLE OF CONTENTS
DEDICATION ii

ACKNOWLEDGEMENTS iii

LIST OF TABLES vii

ABSTRACT viii

CHAPTER ONE: INTRODUCTION 1
Background of the Problem 4
Statement of the Problem 7
Purpose of the Study 10
Research Questions 11
Significance of the Study 12
Limitations and Delimitations 14
Operational Definitions 15
Organization of Study 16

CHAPTER TWO: LITERATURE REVIEW 18
Section One: A Historical Analysis 20
Traditional Instruction in Mathematics 21
The Progressive Movement 24
New Math 28
Back to the Basics 32
Section Two: Current Teacher Education and Teaching Practices 35
Learning from History 36
Teaching mathematics in today’s classrooms 37
NCTM influence 38
Standards-Based Reform 40
Integration of Current Reform Efforts in Preservice
Programs 41
Theoretical Perspective 42
Subject Matter Knowledge 44
Pedagogical Practice 46
Teacher Practices 49
Culturally Responsive Pedagogy 54
Summary 56

vi
CHAPTER THREE: METHODOLOGY 58
Sample and Population 60
Instrumentation 60
Data Collection 62
Data Analysis 63

CHAPTER FOUR: RESULTS 65
Participant Description 66
Biographical Sketch of Participants 67
Findings 71
Research Question One 72
Research Question Two 80
Research Question Three 98

CHAPTER FIVE: DISCUSSION 105
Findings 106
Theme 1: Participants seemed to agree on practices to
incorporate and neglect. 107
Theme 2: Only some participants utilized Culturally
Relevant Pedagogy 110
Theme 3: Only some participants claimed their teacher
education programs influenced them. 113
Theme 4: Tracy, a socio-constructivist in disguise 113
Implications for Practice 115
Implications for Research 118
Conclusion 120

REFERENCES 121

APPENDICES 134
Appendix A: Demographic Questionnaire 134
Appendix B: Vignettes 135
Appendix C: Secondary Questionnaire 136
Appendix D: Pre-Observation Questionnaire 140
Appendix E: Lesson Observation 141
Appendix F: Post-Observation Questionnaire 142

vii
LIST OF TABLES
Table 1: Distribution of Teachers Selected 61
Table 2: Number of questionnaire items in each theme 74
Table 3: Instruction benefits from teacher reflection 75
Table 4: Variety or diversity in groupings, strategies, and learning styles 76
Table 5: Students need to reflect and assess their own work 78
Table 6: Instruction needs to build on prior knowledge 79
Table 7: Distribution of teachers selected 81

viii
ABSTRACT
The purpose of this study was to analyze the teaching practices of teachers to
determine if they incorporated socio-constructivist practices in their teaching repertoire.
This study analyzed the use of socio-constructivist practices utilized by a group of ten
teachers, as well as the factors that facilitated their use. The research questions of this
study focused on which socio-constructivist practices were utilized; what factors
facilitated their use; and the impact of teacher education programs for the
implementation of these practices. Data was collected from ten teachers at two different
school sites. Results from the study indicate teachers utilize a variety of socio-
constructivist practices. Teachers with a low preference for mathematics reflected on
their own negative experiences with learning mathematics, and tried to incorporate
strategies that addressed all types of learning modalities in the classroom because they
understood the diverse needs of all students. Teachers with a high preference for
mathematics understood the systematic nature of mathematics and tried to impart that
on their students. Teachers with a preference for traditional instruction utilized more
paper assessments, while teachers who preferred socio-constructivist practices
incorporated more conceptual lessons. Teacher Education programs influenced
teachers’ use of socio-constructivist practices, but only for recent graduates.
Recommendations for research and practice are provided.

1

CHAPTER ONE: INTRODUCTION
The technical demands of modern workplaces call for an increasingly
sophisticated knowledge of mathematics (Burrill, 2001). Besides knowledge of basic
arithmetic skills and procedures, students also need to be adept at applying these skills
and procedures in real world settings. It no longer suffices to teach mathematical
algorithms and procedures as the end product of instruction. Instead students need to
be problem-solvers who are able to apply mathematical concepts to real-world
problems and to their lives.
Technological advances suggest that an environment and experiences which are
as yet unimaginable will shape students’ lives. Thus, students cannot limit their ways
of thinking about mathematics to the basic content that has historically been the focus
of instruction; nor can they learn mathematics in the rote fashion in which most of us
learned (Burrill, 2001). Students must acquire a set of mathematics basics that enable
them to calculate fluently and to solve problems creatively and resourcefully (NCTM,
2007). Our schools, through administrators and teachers, have the responsibility of
endowing students with the power and beauty of mathematics. Teacher preparation
becomes a critical aspect of student success as these programs must adequately prepare
teachers to meet the demands of preparing students to function in an increasingly
complex society.
According to the Third International Mathematics and Science Study (TIMSS),
students in this country are not performing as well as their international counterparts

2

(NCES, 2005). Although some data reveal student improvements, the areas of
improvement are negligible (Tate, 1997). The National Assessment of Education
Progress trend scores show that students are significantly better in computational skills
than they were in 1971 (Reese, Miller, Mazzee & Dossey, 1997). However student
knowledge is fragile; when asked to use mathematical skills in performance tasks to
solve multi-step problems or to explain their reasoning, U.S. students do not do well
(NCES, 2005).
There is also great variation in academic success based on students’ social class
and racial background. In a review of national trend studies, Tate (1997) found that
although the achievement gap between minorities and White students is decreasing,
non-Asian minorities are still performing far below the academic levels of their White
counterparts. Alongside this observation are concerns about a shortage of qualified
teachers in the mathematics field. Students need to be prepared for careers that require
more than basic skills in mathematics, and teacher education programs may not be
doing enough to prepare teachers to serve these needs. Current societal needs call for
problem-solving individuals who are able to use critical thinking skills to analyze and
solve problems more so than when instruction focused on individuals who could
perform the computational procedures that technology can now execute. Also, teacher
education institutions and public schools must adequately prepare teachers for a
workforce that requires people who are problem solvers.

3

In analyzing the patterns of teacher education over time, some researchers have
looked at specific efforts to improve teaching practices whereas other researchers have
analyzed specific periods in history. However, little research has looked at the trends
and patterns in history that have demonstrated how pedagogy and math education have
developed in the last century to the teaching practices that math organizations, like
National Council of Teachers of Mathematics, currently advocate. There is also a lack
of research that analyzes the factors that promote the application of these effective
teaching practices at the classroom level.
Culturally relevant pedagogy has also become critical as the diversification of
schools has made it necessary for teachers to be prepared to teach children of various
backgrounds. According to Gloria Ladson-Billings (1994), culturally relevant
pedagogy is an approach to teaching and learning that empowers students intellectually,
socially, emotionally, and politically by using cultural referents to impart knowledge,
skills, and attitudes. It dictates that teachers focus on students’ academic success while
still helping them maintain a cultural competence (Ladson-Billings, 1995). According
to Howard (2003), the population of students in U.S. schools is rapidly becoming
heterogeneous while the teaching population remains homogenous. He further
elaborates that teacher education must reconceptualize the way it prepares new
teachers. Thus, the integration of culturally relevant pedagogy in teacher education
programs has also become important in the preparation of teachers to educate future
mathematicians.

4

Background of the Problem
Trends and patterns in the last century of teaching mathematics have had
influential effects on current teacher education practices and pedagogy. Historically,
the teaching of students has been related to societal needs. These needs have resulted in
an education pendulum where teaching has alternated between teacher-oriented
instruction and student-centered instruction. In times when teacher-centered practices
predominate, instruction is regimented, prescriptive, standardized and associated with
traditional methodologies. Conversely, student-centered instruction, which is
associated with constructivist teaching, is more open-ended. With this form of
teaching, standardizing student learning is replaced with professionalizing the teaching
profession and teaching is no longer routine (Darling-Hammond, 1988).
Throughout history, researchers and educators have modified existing
mathematics pedagogy to meet the needs of society and the desires of the public.
Efforts at the turn of the century advocated teaching mathematics using traditional
approaches with a focus on skill attainment (O’Brien, 1999). Public dissatisfaction in
the 1930’s led to constructivist applications of mathematics education which was
widely known as New Math (Herrera & Owens, 2001). This New Math era was
characterized by changes to instructional approaches that relied more on discovery and
concrete materials. Teaching was also more prescriptive and individualized with a
majority of instruction done with small groups and with the assistance of emerging
technology (Price, Kelley & Kelley, 1977). This era was short-lived and educators as

5

well as the public at large went “back to the basics” and focused again on traditional
approaches with minimal conceptual understandings. What followed was a critical
analysis of how New Math had failed, and a subsequent interest in constructivism with
a focus on cognition (Herrera & Owens, 2001). Although constructivist approaches to
mathematics education are susceptible to critics who advocate traditional approaches,
literature in the last two decades has centered on incorporating a social aspect to
constructivist approaches. These socio-constructivist approaches have been at the
forefront of efforts to improve education (Maher & Alston, 1990; Wood, Cobb, &
Yackel, 1991).
The latest efforts to improve the teaching of mathematics came from the
involvement of the National Council of Teachers in Mathematics (NCTM), and its 2000
publication, Principles and Standards for School Mathematics. According to NCTM
recommendations, (1) teachers must have stronger qualifications at all levels of
mathematics instruction; (2) more focus must be placed on the content that is being
taught and the pedagogy used to teach it; (3) there should be an emphasis on problem-
solving and critical thinking, without negating the importance of basic skills; (4) there
should be consideration of student motivation; and (5) make the center of instruction
the real-life application and relevance of mathematics It is unclear whether these new
recommendations have made their way into the university setting so that preservice
teachers can use them. Also unknown is whether teaching in this manner is a reality in
today’s classrooms.

6

Teacher preparation is critical for all teachers because it presents an opportunity
for teachers to view mathematics through a different lens than the one through which
they learned mathematics. Many teachers had negative experiences in their own
acquisition of mathematics knowledge. They only know mathematics through the often
unsuccessful attempts of their own schooling. Thus mathematics and mathematics
methods courses become critical in shifting the existing paradigms for learning and
teaching mathematics that many prospective teachers have at the onset of their
education careers. Hence, teacher education programs must prepare new teachers of
mathematics so that they can teach in a socio-constructivist manner. Teacher education
programs can assist prospective teachers by (a) ensuring they have sufficient
knowledge, (b) adequately preparing them with effective pedagogy, and (c) providing
teachers with the appropriate pedagogical content knowledge that enables them to
deliver mathematics instruction. Teacher education program should consider an
integration of culturally relevant pedagogy to address social inequities in teaching
mathematics. By preparing teachers to incorporate aspects of students’ cultural
background in daily instruction, as advocated by Ladson-Billings (1995), teachers will
have greater success in ensuring the academic success of their students. This is
particularly important in teaching of math, where instruction not only has to have
relevance, and thus personal meaning for all students, but it should also be motivating
to students so it can maximize learning.

7

Statement of the Problem
No Child Left Behind is but one of many legislative initiatives that have called
for highly qualified teachers. Researchers stressed the need for teachers with stronger
content-area and pedagogical knowledge (Ball, 1990; Ball & Wilson, 1990; Ma, 1999;
Hill, 2007). They should be knowledgeable about students so they can motivate them
and present information in relevant and meaningful ways (Zaslavsky, 1996).
Instruction should also focus more on problem-solving and critical thinking skills
which are more applicable and relevant to students’ lives, without negating basic skills
instruction. It is also important to allow students to construct their own knowledge,
typically in cooperative working situations. Teachers who facilitate rather than transmit
information enhance this type of learning. These tenets of socio-constructivist learning
theory are critical for the effective delivery of instruction. However, the literature
suggests that this type of instruction is lacking because many prospective teachers do
not have the necessary content knowledge for mathematics instruction, nor do they
have the necessary knowledge about pedagogy (Ball & Wilson, 1990). Just as
important, they are unable to structure effectively mathematics instruction to maximize
learning, what Shulman (1986) coined as pedagogical content knowledge (Lewis,
2005).
For a multitude of reasons, many teachers did not acquire the necessary
knowledge of mathematics to effectively instruct others (Lewis, 2005). Teacher
content knowledge is a critical element of student learning (Hill, Rowan, & Ball, 2005).

8

Teachers need to understand the content they are teaching so they are able to assist
students with their learning and address misconceptions whenever necessary (Hill,
Schilling, & Ball, 2004). Guided investigation, an effective constructivist strategy for
learning mathematics that NCTM advocates, requires a certain degree of teacher
content knowledge. Thus, teachers need to have minimum content based competencies.
Without them, the guided portion of student inquiry suffers. However, the extent to
which teacher education programs focus on preservice teacher’s prior acquisition of
content knowledge is uncertain (Hill, 2007). Although teacher content knowledge is
essential at the secondary level, it is also highly important at the primary level where
teachers are inculcating the foundations of mathematics in today’s youth.
It is also unknown whether teacher education programs are adequately preparing
teachers with pedagogical practices. One such pedagogical practice is the use of
manipulatives to help bridge students’ understandings from concrete to abstract. The
literature suggests that many teachers display a superficial knowledge of curriculum
and manipulative use (Hargreaves, 1994). Teachers who use manipulatives just for the
sake of giving students hands-on experiencing without relating the manipulatives to the
mathematical content in the lesson are prime examples of this surface knowledge
(Windschilt, 2002). An analysis of the pedagogical practices used in the classroom
would reveal the factors associated with teachers’ effective use of these pedagogical
practices.

9

Ball and Bass (2000) defined content knowledge as the subject matter
knowledge or curricular goals that students should master. Likewise, Jones and
Vesilund (1996) characterize pedagogical knowledge as the knowledge of students,
curriculum, planning, instruction and evaluation. It is the pedagogical knowledge of
prospective teachers that is an area of concern. Ball and Bass (2000) elaborate on
Shulman’s (1986) work which originally called attention to the interweaving of
pedagogical knowledge and content knowledge. They define pedagogical content
knowledge as “the representations of particular topics and how students tend to
interpret and use them, or ideas and procedures with which students often have
difficulty” (p.87). This type of knowledge is necessary as teachers are presenting
mathematical content because it informs teachers about effective examples and non-
examples they can use to deliver instruction. It is vital to find out whether prospective
teachers are getting the experiences necessary that will help them structure and
sequence learning opportunities for their students. Thus effective teaching depends on
all three types of knowledge: subject matter knowledge which provides the necessary
knowledge of the curriculum teachers are to deliver; pedagogical knowledge which
includes the teaching elements necessary to deliver effective instruction; and
pedagogical content knowledge which considers difficulties and strengths of particular
teaching methods and lessons with respect to student success and learning.
Finally, an integration of culturally relevant pedagogy is necessary to prepare
teachers to deal with the diversity that comes from students of diverse cultural and

10

experiential backgrounds in the classrooms. Students of diverse backgrounds fill
today’s classrooms and teachers are increasingly teaching students very different from
themselves. Thus it is critical for teachers to understand how to deliver instruction to all
students in their class, not just those that share their same background. In this manner,
an analysis of teaching practices is necessary to determine if teachers are incorporating
culturally relevant practices with teaching mathematics.
When teachers use socio-constructivist methodology they optimize learning
opportunities for students in mathematics. For teachers to teach in a socio-
constructivist manner, teacher preparation programs need to provide them with the
required knowledge of content, pedagogy, pedagogical content knowledge, and
culturally relevant pedagogy. Besides, teacher educators need to ensure that preservice
teachers acquire the necessary skills and knowledge to teach successfully in a
multicultural society. Thus, it is critical to examine teacher practices to ascertain if
teachers are using effective methods. It is also necessary to determine if teacher
preparation programs are doing an adequate job of preparing teachers to teach in this
manner.
Purpose of the Study
The purpose of this study was to analyze teacher practices to determine if they
are using socio-constructivist methodology to enhance student learning. It was also
important to examine the factors that facilitated teachers’ use of this methodology. By
examining the factors that facilitated these existing practices in classrooms, it became

11

evident whether teacher education programs played a role in their inclusion and if they
were demonstrated and implemented at the preservice level so that these teachers could
apply them once they were in the classroom. It was imperative to ascertain if teacher
education programs prepared teachers to provide students with problem-solving
experiences and opportunities for conceptual understanding grounded in the current
research on learning. Just as important, it is necessary to determine if teachers apply
these experiences to maximize mathematics learning in their students.
The effective learning of mathematics is dependant on the use of socio-
constructivist methodology. Thus it was essential to determine the extent of the socio-
constructivist practices applied during the course of a mathematics lesson. By
analyzing the methodology teachers use while teaching mathematics, it became evident
which socio-constructivist practices occurred most often. Various facets of
mathematics lessons needed to be analyzed to determine the extent of socio-
constructivist practices that were integrated into the lessons.
Research Questions
Data analysis focused on questionnaires distributed to faculty of various
elementary schools that are part of a small school district in Southern California. This
data was triangulated with data from a lesson observation and teacher interviews. The
analysis focused on the following overarching questions:
1. What socio-constructivist practices do teachers incorporate in their teaching
of mathematics?

12

2. What kind of past mathematical experiences influence teachers’ use of
socio-constructivist practices when teaching mathematics?
3. What impact do Teacher Education programs have in the effective
implementation of constructivist practices?
Significance of the Study
This study contributes to the literature by providing a critical analysis of the
inclusion of socio-constructivist practices in the teaching of mathematics, and the
factors that facilitate their use. Students benefit from mathematics instruction when
instruction allows them to work with manipulatives to aid their learning, especially if
teachers use them over time to develop conceptual understanding (Riordan & Noyce,
2001; Gearhart et al, 1999). Opportunities for student collaboration, discussion, and
reflection are also a critical component that aid students’ conceptual understanding
(Kazemi & Stipek, 2001; Martin & Schwartz, 2005 Manouchehri & Goodman, 2000).
Furthermore, teachers enhance learning when students can relate to the content,
especially when teachers contextualize the lesson content in students’ real-world
experiences (Semb & Ellis, 1994).
Socio-constructivist practices are a critical component of student understanding.
Findings from this analysis revealed whether teachers integrated these practices into
mathematics lessons, and if so which were more prevalent. These types of learning
opportunities are indispensable, therefore identifying the factors that facilitate a
teachers’ use of them is crucial. The findings from this analysis also indicated whether

13

teacher education programs are doing an adequate job of incorporating these research-
based practices in preparing preservice teachers to deliver instruction in mathematics in
a manner that is consonant with learning research, while employing culturally relevant
pedagogy.
Preliminary findings from this study indicate that teachers integrate some socio-
constructivist practices into mathematics lessons. Teachers who identified with socio-
constructivist methodology, whether high or low in math preparation/preference, were
also more likely to focus on conceptual understanding and make connections to
students’ lives. Most teachers who participated in this study also applied hands-on
experiences to make the content more comprehensible for students. These experiences
allowed students to work together to make meaning out of the math they were learning.
The teachers also considered culturally relevant pedagogy in their planning and
delivery of lessons. Only the two participants who identified with more traditional
methodology and had low math preparation/preference did not consistently use hands-
on experiences, group work, and culturally relevant pedagogy. These findings suggest
that teachers are using socio-constructivist practices in most of today’s mathematics
classrooms. However teachers who have less math preference/preparation may need
more assistance with lessons so that they are able to integrate more socio-constructivist
practices. This can be addressed at the pre-service level as well as the in-service level.
Findings from this study also suggest that teacher education programs have
indeed integrated socio-constructivist practices in their curriculum in recent years.

14

Although at first glance it appeared that teacher education programs did not influence
most participants, further analysis revealed that participants who were influenced were
those that culminated their programs in recent years, while teachers who were not
influenced by their TEP were more influenced by ongoing trainings and workshops.
However these few participants were from the same two teacher education program.
Thus, further analyses can focus on only graduates from recent years from different
programs. This will determine if recent reform based practices have made their way
into multiple institutions, or just a select few.
Limitations and Delimitations
The limitations of this study include time, issues of trust, and access to typical
lessons. Collecting data within such a short time span limits observations to only a
single lesson. This limits the data that can be gathered. The short time spent with each
teacher may also impact trust, which could impact the quality of data collected.
Finally, because each participant will be observed only once, it will be difficult to
ascertain whether the observed lesson is a typical lesson or an unrealistic view of how
each teacher delivers mathematics instruction over the course of time.
Delimitations of the study pertain to the selected schools. Because data will be
focused on schools from a small school district, many institutions will not be part of the
analysis. Furthermore, because many of the teachers at this district come from a select
few universities which are geographically more accessible, the findings may not have
high external validity.

15

Operational Definitions
The following are list of terms that are used throughout the study. These are
accompanied by their corresponding definitions.
Behaviorist theory. Theoretical perspective in which learning and behavior are
described and explained in terms of stimulus-response relationships (Ormrod, 2006)
Culturally Responsive Pedagogy. An approach to teaching and learning that empowers
students intellectually, socially, emotionally, and politically by using cultural referents
to impart knowledge, skills, and attitudes (Ladson-Billings, 1994)
Discovery learning. An approach to instruction in which students develop an
understanding of a topic through firsthand interaction with the environment (Ormrod,
2006)
Expository teaching. An approach to instruction in which information is presented in
more or less the same form in which students are expected to learn it (Ormrod, 2006)
Pedagogical Content Knowledge. The knowledge that goes beyond knowledge of the
subject per se to how the subject matter knowledge is used for teaching. It includes the
most useful representations of those ideas, the most powerful analogies, illustrations,
examples, explanations, and demonstration that make it comprehensible to others
(Shulman,1986)

16

Pedagogy. The art and science of teaching children while focusing on the relationship
between learning and teaching such that one does not exist as separate and distinct from
the other. More so it is about the relationship between teaching and learning and how
together they lead to growth in knowledge and understanding through meaningful
practice (Loughran, 2006)
Project Method. A teaching methodology that advocates student independence.
Students investigate desired topics and engage in desired projects in small groups.
Socio-constructivist practices. Theoretical perspective that focuses on people’s
collective efforts to impose meaning on the world (Ormrod, 2006)
Subject Matter Knowledge. The amount and organization of knowledge per se in the
mind of the teacher (Shulman, 1986)
Organization of Study
Chapter One has provided an overview of the study. Chapter Two presents a
historical overview of mathematics education and provides a literature review of
current practices supported by research on socio-constructivist learning with a focus on
teacher content knowledge, pedagogical knowledge, pedagogical content knowledge,
and culturally relevant pedagogy. Chapter Three outlines the methodology utilized to
select participants and place them in groups according to their responses on
questionnaires. The chapter also provides a summary of the methodology utilized to
interview participants about their lesson and subsequently to observe their lessons.

17

Chapter Four delineates the findings and the methodology used to analyze the data.
The final chapter provides an overview of the findings of the study and the implications
for teacher education and for teaching in a socio-constructivist manner.

18

CHAPTER TWO: LITERATURE REVIEW
Effective teachers exhibit a variety of traits and strategies that allow them to
maximize their students’ learning. Some of these may have been attained by teachers
early in their own educational experiences, since many teachers teach in the same
manner they were taught (Smerdon, Burkam, & Lee, 1999). However, for the most
part, preservice programs attempt to foster these traits and strategies that teachers will
continuously hone throughout their teaching careers.
For more than a century, the focus of these traits and strategies has vacillated
according to societal needs, public opinion, and public policy. Throughout this time,
instruction has alternated between teacher-directed instruction and student-centered
learning. When teacher-centered instruction prevailed, students were compared to raw
products in an assembly line that needed to be processed (Darling-Hammond, 1988).
Teacher-directed instruction during these times has been highly regimented with
instruction influenced by prescriptive curriculum. The assumption is that teachers can
ensure student learning by following a prescribed set of procedures (Darling-
Hammond, 1988). Conversely when student-centered instruction has been the focus,
the needs of every student have been paramount. This view which supports the
professionalization of teaching, advocates for the utilization of teaching techniques that
vary with student needs, and eschews the views of standardized students and routine
teaching (Darling-Hammond, 1988). Supported by recent research on how people learn
(Bransford et al., 2000), this view of student-centered instruction posits that rather than

19

following standardized procedures, teachers must have knowledge of students,
pedagogy, and content, so they are able to make the best curricular decisions for their
diverse students.
These views of best educational practices have been influenced by research and
public policy which are often a result of public opinion (Cuban, 2003). In turn, public
opinion is often fueled by historical events. For more than a century the vacillation of
these practices has resulted in a view of education that has been fine-tuned throughout
time. In mathematics specifically, the teaching practices that are currently advocated,
draw from the success and failure of reform efforts in the past. Although much of the
research supports the use of socio-constructivist practices, it is unclear whether these
practices are fostered in K-12 classrooms. Moreover, the factors that contribute to the
use of socio-constructivist practices are also unknown.
A review of the literature led to the formation of two major sections. In the first
section, a historical analysis of the trends in mathematics education elaborates on the
various eras prior to the more recent reform efforts of the last two decades. The first
four eras are differentiated by their corresponding theoretical frameworks: (a)
traditional instruction in mathematics during the first part of the 20
th
century where
teacher-directed instruction was the focus; (b) the Progressive movement that was
evident from the 1930s to the 1960s where student-centered instruction began to gain
popularity; (c) New Math curriculum that was seen in the 1960s and continued with
student-centered instruction; (d) and Back to the Basics approach in the 1970s where

20

the educational pendulum swung back to teacher-directed instruction. Although the
driving theories of each era apply to learning in general, the focus of this review will be
the application of these theories to teaching mathematics and specifically teaching
mathematics at the primary level. Thus the second section delineates how current
teaching practices came about as a result of the preceding historical trends and how
these have been integrated at the preservice level in an effort to prepare teachers to
maximize student learning in mathematics.
Section One: A Historical Analysis
Throughout the last century, two paradigms of teaching and learning have
prevailed: teacher centered instruction and student centered learning (Darling-
Hammond, 1988). Those who support teacher centered instruction are advocates of
behaviorism and view learning as the transmission of facts. Conversely student-
centered learning advocates follow the teachings of John Dewey and view education as
the construction of meaning; and learning as the uncovering of knowledge (Darling-
Hammond, 1988). These two paradigms are associated with specific eras in history,
and with the learning theories driving instruction during those periods.
Analyzing these trends brings a better understanding of current educational
practices. Evaluating trends from the past stop futile efforts of repeating failed results
via current practices. George Santayana’s quote “those who forget history are doomed
to repeat it” takes relevance as we try not to repeat the same processes that brought
failed results in the past. Understanding the past helps to contextualize current issues in

21

teaching mathematics. Although instruction has swung back and forth between
traditional or teacher-directed instruction and more constructivist approaches which
advocate student-centered learning, current efforts seem to draw from past learning
theories, as new theoretical understandings of student learning arise.
Traditional Instruction in Mathematics
Mathematics instruction has historically been a product of what society values
and needs (Cuban, 2003). Mathematics education in the United States during the last
century has been a reflection of the society at the time (Jones & Coxford, 1970). At the
turn of the 20
th
century, curricular choices and decisions were made to produce good
citizens and functioning members of the workforce (Burrill, 2001). Good citizens were
those that could contribute to society, particularly as members of the workforce.
During this time, farming was the driving force of the economy. Basic skills in
arithmetic then, were necessary to understand purchase agreements, as well as to buy
and sell produce, groceries, or land in this agrarian society (Burrill, 2001).
During these times, education typically culminated in eighth grade and teachers
had little formal education with many teachers not even attending postsecondary
institutions (Burrill, 2001). Although highly desired, postsecondary education was not
required for teaching (Hanus, 1897). Classroom instruction in mathematics focused on
acquiring basic skills with the goal of producing students who knew how to efficiently
add, subtract, divide, and multiply. Students needed an understanding of these basic
operations with whole numbers, fractions, and decimals. Thus teachers typically

22

developed lessons to demonstrate operations and procedures so students would be able
to replicate them. This Direct Instruction method required students to work
independently and practice the modeled procedures so they could apply these
procedures to other situations (Lemlech,1998). Traditional classrooms were replete
with students working independently on algorithmic procedures while the teacher
walked around the classroom to monitor correct application of procedures and answer
questions (Battista, 1999).
Learning was viewed by educators as a direct result of stimulus-response
connections between a subject and his/her environment, an idea the educational
psychologist Edward Thorndike advocated in his principles of strengthening bonds
through stimulus-response activities (Willoughby, 2000). Led by researchers like E.B.
Thorndike, most research recommended that in mathematics, students needed to engage
in plenty of opportunities of drill and practice on correct procedures and facts. This
type of behaviorist instruction led to strengthening correct mental bonds and habits
(Peterson and Knapp, 1993). Behaviorists viewed success in mathematics as that
driven by reinforcement and punishment (Willoughby, 2000). They prescribed plenty of
drill and practice, reinforced by rewarding desirable behavior (i.e. correct answers) and
extinguishing or punishing undesired behaviors (Peterson & Knapp, 1993).
Who is taught, what is taught, and by whom it is taught, is contingent on the
conditions in which people live (Burrill, 2000). Because society viewed schooling as a
means to prepare citizens to function in an agricultural society, most students went to

23

school only until eighth grade and teachers who taught them had little formal education
(Burrill, 2000). Reading, writing and arithmetic were the main focus of instruction.
These were also the focus of the liberal arts courses preservice teachers had to take. To
effectively deliver instruction, mathematics educators had to have procedural
understanding of mathematical processes (Willoughby, 2000). Simply stated, teachers
needed to understand how to do the mathematical steps they would be expecting their
students to master. This would ensure that they would effectively impart that
knowledge on their students. Unlike later eras, where teachers had to understand
students’ thought processes, teachers in this era needed to know only the basics: how to
read, write, and do arithmetic. Thus the curriculum covered in most liberal studies
programs was enough.
The Behaviorist perspectives which led to traditional teaching were regimented
with little room for student creativity. Like many of his behaviorist colleagues,
Thorndike argued that we should reward students for desired behavior. These
behaviorist theories were helpful in explaining animal behavior, but did not consider
the thought processes of students, nor did they acknowledge the experiences each child
brought to the classroom.
Although behaviorist perspectives highly shaped the education of this era, there
were alternative movements that were more student-centered. Thus, around the 1940’s
the pendulum began swinging in the other direction. These programs focused more on
the individual thought processes of each student rather than in engaging students in

24

procedural lessons that focused on the answers students derived. Thorndike’s theories
were criticized as being solely based on objective reasoning, without considering the
experiences students bring to mathematics, or the meaning they make from what is
being learned (Ellis & Berry, 2005). Because of dissatisfaction with behaviorist
teaching, the progressive education gained movement.
The Progressive Movement
Although behaviorism was the dominant teaching methodology until the 1930s,
other nascent theoretical perspectives had already begun to emerge. Early progressive
ideas were being proposed as early as the 19
th
century. However, it was not until
Thorndike’s ideology faded in the early 1900’s that progressive education began to be
more popular (English & Halford, 1995). With societal changes spurring new views of
children and learning, it was becoming clear to many educators that instruction must
change. In his historical analysis of progressive education, Reese (2001) claims that
many educators were tiring of traditional techniques. Educators claimed that traditional
methods denied children the creative venues for their thought processes. Juxtaposed
with that, this country was changing from a rural, agrarian, and mercantile society to
one which was more dependent on commercial and industrial capitalism (Reese, 2001).
Thus a societal need for higher education arose. The eighth grade schooling that most
students were culminating with was no longer enough.
The Progressive Education Association began as a reaction against the
structured rote schooling practices supported by Thorndike’s theories. Three basic

25

principles made up the foundation of progressive education. Progressive educators
argued that children should have the freedom to develop naturally, student interest
should be the inspiration for their work, and teachers should serve as guides, not
taskmasters (Willoughby, 2000; Ellis & Berry, 2005). Such changes to schooling
became possible as middle class families became smaller, which allowed opportunities
for more attention to individual children, a central principle that progressive educators
advocated (Reese 2001). Progressive educators believed there was no value in
knowledge without understanding. They argued for the need to harness and provide
direction to the child’s natural impulses toward learning as well as theorizing that the
best way to ensure learning was to make critical connections to students’ experiences
and interests (English & Halford, 1995). They also criticized the departmentalized
view of learning different disciplines, by stating that real life was not divided into
subjects (Michael, 1991). Thus teaching became less focused on students forming
habits of mind through extensive repetition in distinct disciplines, and more focused on
students’ experiential learning.
Educators like John Dewey, William Heard Kilpatrick, and Harold Rugg were
among the main contributors of early progressive ideology who advocated for
meaningful learning (English & Halford, 1995). Dewey was never satisfied with the
teaching practices of America during the late 19
th
and early 20
th
century. He criticized
the elitism, irrelevance, and triviality of curricula in most schools (Norris, 2004). He
“denounced science educators who emphasized rote memorization and mechanical

26

routine at the expense of inquiry and creativity” (Vandervoort, 1983, p.38). He also
rejected the ideology of the time that equated learning with tediousness and believed
that student learning should begin with curiosity and learning should be a democratic
process that would contribute to the furthering of a democratic society (Norris, 2004).
Kilpatrick, who was often referred to as Dewey’s disciple, contributed to the
Progressive movement by advocating his “project method” of teaching” (Norris, 2004).
The project method is a teaching methodology that advocates student independence.
Students investigate desired topics and engage in desired projects in small groups.
Kilpatrick’s primary belief was that all subjects and subject matter should be real and
relevant to students’ lives (Norris, 2004). He felt that the project method broke down
the artificial barriers between students and the curriculum (Zilversmit, 1993). Rugg
was an advocate of the social reconstructivist movement. In the hope of creating a
more equitable society, he felt that teachers were facilitators who guide students in
using the tools of critical thinking and problem solving to study real-world problems
(Stern and Riley, 2001).
Most of the teaching the Progressive Movement advocated focused on allowing
students to learn through discovery. Student-centered approaches were at the forefront
of education during the progressive education era. Unlike expository instruction, which
predominated in the last era of traditional instruction, discovery learning required
students to interact with their environment and derive information for themselves
(Ormrod, 2006). In the classroom, instruction no longer focused on meaningless drill

27

methods, where speed and accuracy were the considered adequate math knowledge.
Instead, educators were focusing on developing mathematical concepts in meaningful
ways. William Brownell (1935), a prominent math educator of the time argued that
learning was not merely about the mechanical capacity to solve a problem. He argued
for an intelligent grasp of number relations and the ability to deal with mathematical
situations with proper comprehension of the mathematical and practical significance.
Concrete materials and practical applications of the mathematical material being
learned, began playing an important role in instruction (English & Halford, 1995). Drill
and practice activities were utilized only after students understood the ideas and
processes that were being reinforced (English & Halford). Thus a focus on truly
understanding the mathematics behind a problem and understanding its applicability to
one’s life began to have importance. This is something that would continue to
resurface at various points in history.
The social efficiency movement, also a part of the progressive movement,
argued that students could be guided to expressive self-realization and social
integration. This was possible because they were evaluated and trained by experts
according to their natural inclinations and abilities (Holt, 1994). With students having
such diverse needs, inclinations, and abilities, some educators questioned the need for
higher order mathematics for all students. This led them to turn to standardized
assessments as proof that some students were more apt for advanced courses than
others (Ellis & Berry, 2005). The general belief was that teaching advanced

28

mathematic courses should focus on students whose perceived future required such
knowledge (Willoughby, 2000). By the 1940’s, it was standard practice to track
students, with schools steering most students into vocational, consumer, and industrial
mathematics courses (Ellis and Berry, 2005). This change in practices was reflected
statistically as the percentage of students taking algebra sharply declined from 57% in
1905 to 25% in the early 1940’s and 1950’s, while high school student enrollment was
dramatically rising (Jones & Coxford, 1970). Before the New Math movement,
tracking procedures limited students from taking higher-level mathematics courses
which was especially harmful to students in smaller schools where high-level classes
were not even an option (Walmsley, 2003). These tracking practices were to have later
implications nationally.
New Math
While the need for change in the Progressive Era arose from a desire to back
away from traditional methods, the need for reform during the New Math era was due
to a military need (Walmsley, 2003). World War II had a powerful effect on how
people saw mathematics. According to Willoughby (2000) the military was forced to
provide crash courses in mathematics for their recruits. As a result mathematics
instruction becomes the focus of various mathematics commissions. Also, national
security issues (the nation was at war and nuclear weapons which relied on
mathematical formulas were being developed elsewhere) sparked the need for more
rigorous mathematics curriculum to address this national concern. This spurred

29

Congress to create the National Science Foundation in 1950 with the hopes of
promoting education in the sciences. Although the intent to fund mathematics education
was there post World War II, it was not until after 1957 that the federal government
provided substantial support for mathematics and science education. (Willoughby,
2000). Mathematics reform became a major concern amid public dissatisfaction when
the Russians launched the first Sputnik satellite in 1957 (Castle and Aichele, 1994).
Many mathematicians began raising concerns over existing mathematics
programs. This was juxtaposed with the public demands that we educate students who
could put a man on the moon. These concerns led to the creation of the School
Mathematics Study Group (SMSG) and New Math. (Burrill, 2001). New Math
curriculum focused on enriching content; understanding the unifying concepts and
structures of mathematics; and the process of discovery. This was meant to replace the
expository instruction in arithmetic with more meaningful attention to a broader sample
of topics from mathematics (Fey, 1978). Problem solving skills and critical thinking
became a focus of instruction as curriculum materials incorporated activities that
required students to think, reason, and deduce (Nee, 1990). Thus, teaching incorporated
the use of guided discovery with manipulatives, juxtaposed with a focus on problem-
solving so students would comprehend the mathematics they were practicing, rather
than concentrating on problem solutions with minimal understanding (Walmsley,
2003).

30

Jerome Bruner, Zoltan Dienes, and Jean Piaget (English & Halford, 1995)
highly influenced curriculum in this era. The work of these psychologists led to
extensive concern for the use of physical materials in teaching elementary mathematics
through laboratory-like investigations (Fey, 1978). Jerome Bruner’s theory analyzed
how students move through various levels or stages of representation. He was interested
in the cognitive processes that were part of learning and thinking. He proposed that
children went through three stages of representation, with each subsequent stage
building on the prior one. Children first went through the enactive stage where they
had to directly manipulate objects. This was followed by the iconic stage where
students visualized manipulating the objects. The final stage included children who
could manipulate symbols rather than objects or images of these objects.
Zoltan Dienes worked with Bruner and developed concrete materials and games
that teachers could use in structured learning experiences (Moyer, 2001). Dienes used
the phrase “learning cycle” to designate where students progress from concrete
activities to more symbolic formats (Dienes, 1971). Bruner’s stages of representation
and Dienes’ “learning cycle” are often seen in current mathematics lessons as teachers
move students from a physical manipulation of objects to guiding students towards
solving equations where no such representations are available to the student (Fuson,
1975).
Piaget also had an influential role in teaching mathematics with his work on
four stages of cognitive development. His work on cognitive development looked at

31

children as they traversed through various developmental stages. Each of Piaget’s
stages represented a distinct way of interpreting the world, and developed as students
interacted with their environments. Piaget also advocated the use of concrete
manipulatives and opportunities for children to explore with less formalization
(Walmsley, 2003). Piaget’s ideas have continuously resurfaced on the socio-
constructivism end of the curriculum debate.
Mathematics education under New Math required a vision of discovery learning
with small group and class-wide discussions. Teachers needed to possess some level of
minimal competency in pedagogical content knowledge. Walmsley (2003) reports that
teachers needed to be aware of guided discovery approaches that would be used with
students. This meant that teachers questioned students and waited patiently for
responses instead of providing student with answers. According to Woodward (1970)
teachers needed not only an understanding of the formal nature of mathematics, but
they also needed to be able to ask the right questions at the right time. Additionally
teachers were required to follow student thought processes by supporting those that
were correct and rejecting those that were not (NCTM, 1961). A central premise of
teaching during this time was that students were responsible for understanding and
discovering math principles with minimal guidance, rather than teachers verbalizing the
meaning of content matter for students (Walmsley, 2003).
Although New Math worked well for some, it did not suit everybody. The New
Math movement followed as a response to dissatisfaction in preparing American

32

students to compete internationally in Math and Science. Although revolutionary ideas
for mathematics instruction were provided as a result of the New Math movement, it
was never widely accepted and thus was short-lived (Brewer, 1997). The New Math
movement was criticized as being too abstract and having no real-life application to
either children or adults. Besides, critics claimed the use of vocabulary was unfamiliar
to mathematicians and adults and was doing more harm than good in American
education (Willoughby, 2000). What followed was a “Back to the Basics” period
during which publishers stripped most words from mathematics texts and reintroduced
pages of computational problems (Burrill, 2001).
Back to the Basics
After focusing on student-centered approaches and minimal structure in
mathematics, the educational pendulum vacillated back to the highly regimented
methods of traditional methodology. With the failure of New Math curriculum and
increasing frustration levels, many educators began to rely on a “back to the basics”
approach to teaching mathematics. A movement toward accountability exacerbated this
change in the teaching of mathematics. Whereas scientific concerns framed the focus of
instruction during the New Math era, the new concern during the 1970’s was a social
one which advocated school accountability (Walmsley, 2003). During this time, the
easiest way for students to demonstrate the content they had learned was via traditional
standardized testing. Thus by reverting to traditional mathematics and subsequent
traditional testing, the public could hold schools accountable (Walmsley, 2003).

33

During this shift, computational problems and a Madeline Hunter method of mastery
learning based on expository teaching replaced word problems in text books.
Therefore, much like earlier teacher-directed methodologies of teaching, this era of
teaching focused on teachers at the center of instruction and students passively
receiving information.
A return to expository teaching marked this era. Many educators felt that
traditional math was a missing and much needed component of the “fuzzy math” seen
in New Math curriculum. Outcomes-based instruction became the norm and educators
returned to expository approaches of instruction (Ormrod, 2006). These direct
instruction approaches provided students with multiple opportunities to practice skills
and procedures. This movement was also closely tied to the minimum competency
testing movement used in the 1970’s and 80’s. Although standardized assessments saw
a slight increase in scores, critics focused on the inadequate preparation of students for
higher levels of cognition and understanding (Ellis & Berry, 2005).
Just like previous attempts at teaching mathematics in an expository fashion,
this new Back to the Basics movement was no different. Burrill (2002) argues that
traditional curriculum and instructional methods in the United States do not serve our
students well. Traditional strategies go against the way children learn. Zaslavsky
(1996) compares this to children learning to speak; they begin by observing others, and
once they start internalizing some of the rules of language, they begin using their
problem-solving skills to apply what they have observed to speaking. Similarly in

34

mathematics, students need opportunities to learn in meaningful contexts rather than
with routine drills that merely let them practice redundant skills (Zaslavsky, 1996).
Without changes, we will continue to disenfranchise a large portion of the population.
Repetitious practice does not necessarily promote understanding of mathematical
concepts and drills do little to impact learning and transfer of learning to other areas
(Burrill, 2001).
Concurrent with the Back to the Basics approach were new innovative programs
being developed and tested on a much smaller scale. These were precursors to the
NCTM Standards and were based on very different principles from those of the Back to
the Basics movement. They demonstrated that all children could learn a remarkable
amount of mathematics, including higher order-thinking skills, if appropriate
instructional activities were used (Dilworth and Warren, 1980). The complete
abandonment of strategies that New Math advocated had critics jumping to the opposite
end of the pendulum swing. Rather than learning from mistakes in the New Math
curriculum, educators relied on traditional methodology that had already proven
unsuccessful in adequately preparing students. This was counterproductive. NCTM
saw these as futile efforts because as educators we should be learning from the past
(Walmsley, 2003). Thus began an intervention by NCTM that would become the focal
point of current reform.

35

Section Two: Current Teacher Education and Teaching Practices
The preceding section focused on the historical trends of most of the last
century. The following section will analyze how the teaching practices of the last three
decades have been fine-tuned due to historical trends in education, learning theory, and
research-based practices. The section starts with an overview of how history has
impacted current teaching practices. This is followed by an examination of current
mathematics classrooms and the existing needs of students. Finally an analysis of
NCTM’s influence on current teaching practices will lead to a discussion of the latest
standards-based reform movement.
Mathematics reform is as critical now as it has been in the past. However, for
the past three decades it has continued with more caution and educational expertise than
before (Walmsley, 2003). Currently, educators advocate a balanced approach to
teaching mathematics. This includes fluency with basic facts, a traditional approach to
learning; and students working together and constructing their own knowledge, which
reflects more socio-constructivist views of learning. Because of reform efforts in the
past, education has been fine-tuned to reflect those practices that have been successful,
and reject those that have failed. The needs of today’s students have also served to
outline the type of instruction that schools and teachers need to deliver. Standards at
the national level and also at the state level have served a guiding role in determining
what needs to be addressed (Eisenhart, Borko, Underhill, Brown, Jones, & Agard,
1993). NCTM has served as a guiding force in developing standards and assessments,

36

as well as suggesting methodology for implementing math lessons (NCTM, 2006). The
National Mathematics Advisory Panel has also, although to a lesser extent, contributed
to the existing literature (NMAP, 2008).
Learning from History
As predicated in the preceding historical analysis, we have learned much from
our history. Various mathematics researchers compare the evolution of mathematics
education to the swinging of a pendulum where socio-constructivism and behaviorism
lie at opposite ends of the pendulum’s path (Walmsley, 2003; Loveless, 2001; Boaler
2008, Glidden, 1996). In order for educators to truly learn from past reform efforts and
effectively guide future reform efforts, they must be able to choose somewhere within
the pendulum’s path. They cannot be restricted to either of the two extremes
(Walmsley, 2003). Educators should retain ideas from previous curriculum that have
proven effective. They should not be swept aside by the pendulum of reform (Glidden,
1996).
Although the pendulum has continuously swung back and forth, it leaned
towards the more socio-constructivist extreme for the past three decades. Rather than
swinging back to traditional approaches altogether, current reform efforts have found a
way to integrate traditional approaches with mathematics while keeping a mostly socio-
constructivist approach to mathematics instruction. Educators have used the positive
developments from the New Math movement and used them as a guide for future
reform, rather than allowing history to repeat itself (Walmsley, 2003). Current reform

37

draws from the many instructional eras of the past, with continual fine-tuning, it seems
we now have a good indication of what “best practices” are in mathematics
Teaching mathematics in today’s classrooms
Current efforts in mathematics education assert that to prepare students for an
ever changing world, we should teacher mathematics in mostly non-traditional ways.
Thus what has happened in recent years is a view of teaching mathematics in a more
conceptual manner. Alternative lessons are utilized as a means to foster conceptual
understanding. This approach to teaching and learning mathematics emphasizes
student centered learning; students’ collaborative investigations of mathematical ideas;
and the teacher acting as a facilitator rather than disseminating bits of knowledge
among students. Further, teachers create environments where mathematical discourse
takes place and guides learners in the exchange of mathematical ideas.
Current societal needs call for mathematics instruction that highly emphasizes
critical thinking and problem-solving (NCTM, 1989, 2003). After postsecondary
education, success is often predicated on individuals who are able to problem-solve and
think to succeed. In a study of the daily lives of professional engineers, Julie Gainsburg
(2003) found that although they used mathematics extensively throughout their work,
these professional engineers rarely used standard methods and procedures. Instead they
mostly interpreted the problems they were asked to solve; selected and adapted methods
that could be applied to their models; ran calculations; and justified and communicated
their needs. Even though they occasionally were called upon to use standard

38

mathematical formulas, most of their work had minimal structure and was more open-
ended. Thus, traditional K-12 curriculum, which heavily relies on computation, is
inadequate to prepare students for the high-tech workplace and its focus on problem
solving.
Students in today’s society have needs much different from those that have been
at the forefront of education in the past. Boaler (2008) argues that knowing how to do
mathematics is not only one of the most important qualities for workers to possess in
the future, but it is critical for the successful functioning of life. She continues by
affirming that twenty-first-century citizens need mathematics; rather than regurgitating
hundreds of standards methods, they need to know how to reason, problem solve, and
apply methods to new situations (Boaler, 2008). If students are to become powerful
citizens who have full control of their lives when they need to reason mathematically,
think logically, compare numbers, analyze evidence, and reason with numbers. Also,
mathematics classrooms need to catch up to prepare these young people for their lives
(Steen, 1997).
NCTM influence
The National Council of Teachers in Mathematics has had a positive impact on
reform practices since its inception in 1920. However, its first major intervention did
not come about until 1975 with its publication of Overview and Analysis of School
Mathematics, where the authors began discussing the direction of mathematics
education (Glennon, 1976). This was followed by an effort to take control of

39

mathematics reform in 1980, with the publication of Agenda for Action. This report
called for more problem-solving in schools and a focus on critical thinking skills rather
than the “Back to the Basics” approach being taken by many educators. The report gave
eight recommendations that called for a paradigm shift in the way mathematics was
being taught (Walmsley, 2003). These included:
(a) a focus on problem solving
(b) basic skills favored over computational abilities
(c) integration of computers and calculators
(d) teaching math effectively and efficiently
(e) teaching math to all students with options for students with differing needs
(f) teachers exhibiting a high degree of professionalism
(g) public support and understanding of mathematics reform
This was considered to be the precursor to the various versions of Standards
that have been published since then.
The current movement to reform school mathematics in this country
commenced in the 1980’s as a response to the documented failure of traditional
mathematics teaching. It was also an effort to address curriculum changes needed by
the widespread availability of electronic computing devices. This was due to
substantial progress in the scientific study of mathematics learning (Battista, 1999).
Initial attempts at standardizing the national curriculum in mathematics began in 1984
with the publication of a small set of standards for selecting and implementing
instructional materials (Willoughby, 2000). In 1989, NCTM released Curriculum and
Evaluation Standards for School Mathematics which was then followed by the
Professional Teaching Standards in 1991 and the Assessment Standards for School

40

Mathematics in 1995. They formed a foundation for change that was to be
implemented in mathematics curriculum, teaching, and assessment.
Standards-Based Reform
Standards-Based Reform according to NCTM guidelines, initiated a paradigm
shift, where learning was no longer restricted to skills and procedures and students were
asked to use mathematical knowledge to solve problems (Burrill, 2001). This
represented a change in teaching practices that went from teachers disseminating
information to passive students, to a belief that students need to be able to understand
how their own thinking serves as the platform for learning. Assessment practices were
also changing as testing went from end-of-unit tests to using the results of assessments
to inform the teaching and learning process. Thus the focus was on building on student
thinking to inform instruction, and utilizing assessments that measure what students
know, instead of what they do not know (Burrill, 2001). New curricular materials were
also developed that began to give as much importance to developing conceptual
understanding as learning computational skills. They also used concrete materials and
contextual situations to motivate students and began providing scaffolds for
understanding. Furthermore they began to integrate cross curricular content as was
done in other countries and reconfigured the curriculum in different ways so that there
were different emphases at different points in the educational sequence (Burrill, 2001).
Many people misinterpreted the 1989 publication of standards as a guide to
eliminate completely traditional basic skills (Lovin & White, 1999). The belief was that

41

education efforts at the school sites should only focus on teaching critical thinking,
problem-solving, and other higher-order thinking skills (Burrill, 2001). Since this was
never the intent of the original standards, NCTM members revised and published the
standards in 2000 as Principles and Standards for School Mathematics (NCTM, 2000).
The new document included a new set of principles that formed a context for thinking
about teaching and learning mathematics and the effort to identify key mathematical
ideas in each content strand. Furthermore, it included illustrations on how these should
grow and develop throughout each grade band (Burrill, 2001). According to
Willoughby, (2000), the new document makes four things clear. There is a need for
basic and traditional math skills in addition to higher-order thinking skills.
Additionally, equity goals should be embedded in all the previous standards. There is
also a need for technology integration across the curriculum. The NCTM vision also
states that mathematics standards are to be considered as a work in progress that will
encompass continuous future revisions. With the focus on problem solving, critical
thinking, the use of manipulatives and other resources; education once again fluctuated
to socio-constructivist theories of learning. Furthermore a focus on collaborative work
and student discourse which stems from a socio-constructivist theory of learning is also
obvious.
Integration of Current Reform Efforts in Preservice Programs
Currently, educators advocate for a balanced approach to teaching mathematics
which integrates a traditional approach to learning as well as more constructivist views

42

of learning. This includes fluency with basic facts and algorithms and a model of
students working together and constructing their own knowledge. Vygotsky’s theory of
socio-constructivism therefore serves as an important framework for understanding
how teachers are prepared to teach utilizing these research-based practices. To prepare
teachers for this type of learning and teaching, the literature addresses four areas of
preservice education: subject matter knowledge, pedagogical knowledge, teacher
practices; and culturally relevant pedagogy. Efforts need to be made at providing
preservice teachers with adequate subject matter knowledge so that they can focus on
conceptual teaching of topics they are knowledgeable about. Teachers also need the
necessary teaching practices that will enhance student learning and the culturally
relevant pedagogy that will address the needs of diverse learners. The following is an
overview of Vygotsky’s socio-constructivist theory and an analysis of the four areas of
preservice education as addressed by the literature.
Theoretical Perspective
Although some traditional teacher-led approaches to teaching are still evident in
reform efforts, (i.e. fluency with facts and procedures is still a standard), most of the
reform efforts have centered on constructivist student-centered practices and recently
on socio-constructivist approaches. More than two decades of scientific research in
mathematics education have refined the socio-constructivist view of mathematics
learning. These views encompass explanations of how students construct increasingly
sophisticated ideas about particular mathematical topics and what students’

43

mathematical experiences are like. They also provide information of what mental
operations give rise to these experiences, and the socio-cultural factors that affect
students’ construction of mathematical meaning (Battista, 1999). Far from being
synonymous with the discovery learning seen in the New Math era or even simply
integrating manipulatives and cooperative learning opportunities, socio-constructivist
theories have proven to be invaluable in understanding empirical research on student
learning or mathematics.
Stemming from a constructivist theory of learning, Vygotsky’s theory of socio-
constructivism posits that learners construct knowledge but they do so as part of a
social process. Knowledge can be constructed by individuals or it can be socially
constructed. Ormrod (2006) differentiates socially constructed knowledge from
individually constructed knowledge, which may differ considerably from one individual
to another. Socially constructed knowledge, she argues, is shared by two or more
people simultaneously (Ormrod, 2006). Learners go through the process of identifying
the construction of their own knowledge, as well as determining how this fits within a
social context. Thus group work is a critical factor in acquiring mathematics
knowledge. Much of the instruction in recent years advocates socio-constructivist
practices. Social constructivism has a variety of benefits for student learning (Ormrod,
2006).

44

Subject Matter Knowledge
Standards-based instruction advocates an understanding of concepts in
mathematics. Although there is much debate about the importance of subject matter,
there is some indication that teachers need to have a conceptual understanding of the
mathematics that is being done (Ball, 1997; Ma,1999; Manouchehri and Goodman,
2000, Tirosh, 2000). This is not to say that teachers should be able to procedurally
compute a response, instead they should have conceptual understanding about what the
problem is asking.
Ball (1997) found that this type of understanding is not automatic in teachers.
Even secondary teachers who felt they had a mastery of mathematics concepts
struggled with making conceptual connections between procedural algorithms problem
and problem-based learning. Thus teacher education programs need to incorporate a
variety of opportunities for preservice teachers to analyze mathematical concepts and
self-reflect about their mathematical understandings. Similarly teacher education
programs should provide resources for preservice teachers that help them develop a
concise knowledge base, when it comes to mathematics (Morris, Hiebert, & Spitzer,
2009).
Ma (1999) compared the knowledge of teachers in this country against the
knowledge of teachers in China. Her findings indicate that in contrast to Chinese
teachers, American teachers were unable to provide conceptual examples to problems
they were able to solve procedurally. This was specifically evident in the division of

45

fractions where many of the American educators could easily apply algorithmic
procedures; however, they were unsuccessful at providing concrete examples of how
this could apply to their lives. According to Ma, this perpetuates the problem of math
education, as math teachers cannot teach conceptually if they themselves do not have
conceptual understanding of math topics.
Instead of erroneous assumptions about preservice teacher’s mathematical
knowledge teacher preparation programs should provide prospective teachers with
experiences that will help them gain this deep knowledge (Conference Board of the
Mathematical Sciences, 2001, as cited in Newton 2008). Courses that routinely
encourage preservice teachers to make connections between procedural skills and
conceptual knowledge, and regularly assessing them on how and why procedures work,
help bridge this gap (Newton, 2008).
Tirosh (2000) presents findings from a study conducted with 30 preservice
teachers who were enrolled in a math methods class. The study analyzed the
development of preservice teacher’s knowledge as it pertained to dividing fractions.
Findings from the study indicate that prior to taking the methods class, preservice
teachers were able to compute the division of fractions; however, they were unable to
explain the procedure. Implications from the study indicate that preservice programs
should make an effort at familiarizing prospective teachers with common knowledge
that will be addressed with children. Research on mathematical knowledge for teaching
(MKT) has also demonstrated a positive relationship between teacher knowledge and

46

students achievement. In a study of 115 elementary schools, Hill, Rowan, and Ball
(2005) found that teacher knowledge had a positive impact on student learning.
Pedagogical Practice
Teaching in a socio-constructivist manner provides multiple opportunities for
students to work in groups; teachers to build on students’ prior knowledge; and
encouraging students to take multiple paths to solutions. One of the pedagogical
practices that has been recently advocated is group work.
According to Carpenter and Lehrer (1999), learning is also maximized when
teachers build directly on student’s entry knowledge and skills. They assert that
learning must be predicated on prior knowledge because they argue that knowledge is
generative, and builds on itself (Carpenter and Lehrer, 1999). Thus, in order to build
conceptual understanding, teachers need to sequence learning opportunities so that they
allow students to build new knowledge on the prior knowledge they already possess.
Accordingly, Hiebert (1999) argues that when students enter school, many of them are
able to count and perform other mathematical behaviors. Opportunities to teach
mathematics are maximized with this type of learning by gradually increasing the range
of problems and size of numbers.
New alternative forms of teaching mathematics incorporate the provision of
multiple opportunities for student to invent and practice multiple strategies. Rather
than focusing on procedures and rote memorization, allowing students opportunities to

47

invent new strategies encourages them to explain their thought processes and
maximizes conceptual understanding (Hiebert, 1999).
Under the auspices of the National Council of Teachers in Mathematics
(NCTM), teachers need to implement the various tenets of socio-constructivism in their
classrooms. Subsequently, teacher education programs should also incorporate this
methodology in their coursework so that teachers are prepared as they go into the field
of teaching. Preservice teachers need to understand what each of the tenets looks like
and how they can be implemented to effectively deliver mathematics education.
Teacher education courses also need to be designed in such ways that delivery
of instruction is made clear. Feiman-Nemser and Buchmann (1989) argue that
programs need to consider that subject matter and pedagogy are not the only things
being transmitted and that preservice teachers are vicariously learning through their
observations of instructors. Preservice teachers learn many things in their education
courses. Their growth depends not only on the knowledge they encounter, but also on
the way those encounters are structured and the messages that are being conveyed about
teaching and learning to teach, by their instructors (Feimann-Nemser & Buchmann,
1989). In the university setting, the use of socio-constructivist approaches need to be
modeled and preservice teachers need opportunities to analyze the approaches so that
confusions do not arise. Preservice teachers may be unclear as to what is expected of
them. They may attempt to integrate constructivist approaches and find that they are
insufficient. Simply using hands-on activities in problem-solving situations while

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fostering classroom discourses in small and large groups may not garner the desired
effects of student understanding, student ownership of problems is also essential
(Simon, 1995). Burrill (2001) warns preservice teachers of the trappings of reform,
where too much emphasis is placed on cooperative groups, manipulatives, and hands on
activities, with little attention to the mathematics concepts as a focus of instruction.
Teachers are encouraged to use students’ ideas and mathematical knowledge as
the foundation in helping them develop conceptual understanding and relevant
mathematical skills. However, engaging students in this kind of mathematical learning
requires most teachers to think differently about mathematics and the process of
learning it (Remillard, 2000). In order for teachers to alter their practices and go beyond
teaching in the rigid way they themselves were taught, their cognitive schemas need to
be challenged. According to Remillard, (2000) the kind of learning that leads teachers
to fundamental changes occurs over a long period of time, with extensive support and
multiple opportunities to experiment and reflect.
Mathematics teaching is not simply about guiding students to find the
appropriate solutions to the given problems. Mathematics teachers also have to involve
themselves in an “unpacking of mathematics” where much like in literacy instruction,
they decipher the solutions that students give and analyze any misconceptions so that
they may be addressed (Ball, Bass, & Hill, 2004). This is not an easy feat, unlike being
able to procedurally decipher a problem, these types of tasks increase the
“mathematics-in-action problem-solving” demands on the teacher. Thus teacher

49

education programs need to provide preservice teachers with these types of experiences
so that they can make use of them once they are at the novice level (Morris, Hiebert, &
Spitzer, 2009).
Teacher Practices
The literature points to various teacher practices that need to be incorporated
into instruction so that learning is maximized. The use of manipulatives along with
opportunities to collaborate with peers and classroom discourse are central tenets of
socio-constructivism. Thus their integration is necessary to enhance learning
mathematics. Contextualized learning is also important as it makes instruction relevant.
Manipulatives. Concrete manipulatives are a critical element for successful
mathematics instruction, especially if the focus is on conceptual understanding.
Manipulative materials are objects designed to represent explicitly and concretely
mathematical ideas that are often too abstract (Suydam, 1986). Difficult mathematical
content should incorporate the use of concrete manipulation which allows students a
means of appropriate pictorial representations and better conceptual understanding.
When used effectively, they can help concretize abstract concepts for students (Parham,
1983). The active manipulation of these materials allows learners to develop a
repertoire of images that can be used in the mental manipulation of abstract concepts
(Parham, 1983). Manipulatives allow students to visualize and understand basic
concepts. Martin & Schwartz (2005) assert that manipulatives help children learn
because they provide an environment for quantitative activity where children can adapt

50

and reinvent. They prepare children to learn from new resources, perhaps from a
physical demonstration or a lecture and allow students to gain insights into
mathematical concepts (Moyer, 2001). Furthermore, manipulatives allow students to
visualize and better understand concepts, while developing problem solving skills
(Presmeg, 2006). Finally, the use of manipulatives allows teachers and students to
focus more on the reasoning of mathematical actions and concepts rather than focusing
on the procedures (Kazemi & Stipek, 2001).
A caveat about concrete manipulatives is that they do not carry the meaning of
the mathematical ideas behind them. Students must be given opportunities to reflect on
the meaning that is being built with the exploration of manipulatives (Moyer, 2001). In
addition to encouraging the use of manipulatives, teachers should engage their students
in genuine mathematical inquiry and push them to go beyond what is easy for them
(Kazemi & Stipek, 2001). This can be done by pushing students’ conceptual thinking.
In doing so, teachers should ask their students to justify their strategies mathematically,
so that students are not only explaining their thinking but justifying it as well (Kazemi
& Stipek, 2001). Students in particular can benefit from curriculum that incorporates
these reform-oriented practices. Studies (Riordan & Noyce, 2001; Gearhart et al..
1999) indicate that students working in small groups to explore mathematical concepts
using manipulatives outperform students taught with more traditional methodology.
These differences are exacerbated when students are taught with the reform-oriented
curriculum for longer periods of time (Riordan & Noyce, 2001; Gearhart et al.. 1999).

51

However manipulative use is not so easily integrated into the curriculum with
guaranteed success, especially when teachers themselves feel uncomfortable with their
integration into the curriculum. According to Ajose (1999) preservice teachers do not
get sufficient opportunities to work with manipulatives during their preparation.
Therefore teacher education programs need to incorporate opportunities for these
teachers to not only use this methodology but also be experts in their use (Ajose, 1999).
Student Collaboration. Opportunities for collaboration are also critical
elements of effective reform practices. Research indicates that this constructivist
approach is not only essential to mathematical understanding, but a necessary
component for teachers to have mathematically driven classrooms (Kazemi and Stipek,
2001; Martin & Schwartz, 2005). In their observations of video-taped lessons, Kazemi
and Stipek (2001) found that collaboration and the use of mathematics manipulatives
offered students opportunities for conceptual understanding. Discussions within the
class also allowed students to construct their own learning, as well as provided
opportunities for teachers to assess student understanding. In assessing mathematical
understanding, teachers confront students’ under and over-generalizations or
misconceptions in a constructivist manner (Manouchehri & Goodman, 2000).
Collaboration of students allows opportunities for dialogue that encourages
them to clarify their own thinking before they are justified with others (Sfard, Forman,
& Kieran, 2001). Elaboration and justification is also encouraged so that students are
required to clearly conceptualize their ideas. Sharing becomes critical for students so

52

they are exposed to the thinking of other students. Additionally, by forcing students to
work collaboratively, they are able to rationalize their thinking and uncover erroneous
thought processes (Zack & Graves, 2001). Peer interactions also promote higher stages
of cognitive development. Furthermore, discussions and debates force students to
critically analyze the concepts (Mueller & Fleming, 2001). These discussions can also
enhance interpersonal skills. Thus “when students must explain their thinking to
someone else, they usually organize and elaborate on what they have learned” (Ormrod,
2006, pp.228). This promotes a more integrated and thorough understanding of the
material. These practices are critical for learning and can be particularly beneficial
when it comes to the learning of mathematics.
Classroom Discourse. Closely related to collaboration is the necessary
component of classroom discourse. Ball (1993) conducted a self study where she
studied the dilemmas associated with teaching math in a way that is supported by
research. She highlights the importance of discourse in the classroom community so
that teachers can analyze student misconceptions. She asserts that classroom
communities are critical so that students are given opportunities to articulate, refine,
and revise their own thinking. The study also confirms the importance of effective staff
development, so that teachers effectively allow for classroom discourse opportunities in
their classrooms. Although she herself was knowledgeable in the area of teaching
mathematics, she found that there were many lapses between what she was teaching
and what the students were receiving. Although she affirms the importance of

53

constructivist approaches to teaching mathematics, results from her self-study may be
limited to her classroom.
McNair (2000) also describes various purposes for discourse in mathematics
classrooms: subject, purpose, and frame. In order be effective, classroom discourses
should have a subject, or a topic that is being discussed. They should also have a
purpose. The discourse can serve the purpose of solving a problem, elaborating on a
strategy, making comments, or asking questions to continue the discourse patterns.
Finally, the discourse needs to be framed in such a way that the speakers are guided in
their participation. McNair (2000) argues that in order to for classroom discourse to be
effective, these elements need to be considered.
Classroom Environment. In a mathematics classroom, the teacher is
responsible for fostering an atmosphere of respect and trust. This is necessary when
students are expected to share their responses and carry meaningful conversations about
the mathematical content. Furthermore, classroom investigations juxtaposed with high
expectations are the norm. In such an environment, problems are posed and students
wrestle towards solutions. The focus then lies on students actively figuring things out,
testing ideas and making conjectures, developing reasons and offering explanations. In
these environments, it need not matter whether students work in pairs, or individually,
but they are always sharing and discussing. Reasoning is celebrated as students defend
their methods and justify their learning (Van De Walle, 2001).

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Contextualized Learning. Socio-constructivist teachers always find
opportunities to contextualize learning with real-life application. This is in contrast to
many traditional approaches to mathematics where learning is decontextualized and
focused on isolated procedures that have no personal meaning to students (Semb &
Ellis, 1994). In a study of four classes, Brenner (2002), found that teachers who
contextualized mathematical concepts in lessons by relating them to students’ lives,
were more successful than teachers who taught in more traditional ways. Therefore by
contextualizing lessons in students; real-world experiences, teachers can maximize
learning opportunities for their students.
Culturally Responsive Pedagogy
Beyond practicing reform, teachers need to be prepared with culturally
responsive pedagogy. Ladson-Billings (1995) argues that teachers need a culturally
relevant perspective to understand the growing disparity between racial, ethnic, and
cultural characteristics of students and teachers. They also need to address the
continued academic failure of marginalized groups. Although current reform efforts
address diverse learners with NCTM standards, math instruction may still neglect the
needs of diverse learners. By preparing teachers to teach in culturally relevant ways,
teacher education programs can ensure that the needs of diverse populations are being
met. The literature demonstrates how this can be done.
Ladson-Billings (1995) describes culturally responsive pedagogy, as one “that
not only addresses student achievement but also helps students to accept and affirm

55

their cultural identity while developing critical perspectives that challenge inequities
that school (and other institutions) perpetuate” (p.469). She elaborates by stating that to
be effective, teachers must encourage academic success, cultural competence, and assist
students in developing a sociopolitical consciousness which requires them to recognize,
understand, and critique social inequities. In accordance with contextualized learning,
culturally responsive pedagogy advocates that it is crucial to include the knowledge
students bring with them and the skills that they have into the learning activities of the
classroom. Montgomery (2001) asserts that students in culturally responsive
classrooms need to find relevant connections to the subject matter and the tasks they are
being asked to perform. Gaitan (2006) describes classrooms that make use of student
work groups which are organized in such a way that they allow students to engage in
shared inquiry and discovery. She elaborates by stating that teachers need to set high
expectations and include students’ home language whenever necessary. Family and
community contexts need to also be incorporated in the instructional context as much as
possible (Gaitan, 2006).
Zaslavsky (1994) conducted a study of second through fourth grade students of
mostly Mexican heritage. Students learned the same curriculum via different teaching
modalities. The control group’s focus was on rote memory skills and individual seat
work, while the other group used a multicultural program that emphasized cooperative
work and critical learning activities. In the final analysis Zaslavsky reported that
learning with this reform-minded and multicultural ideology enhanced not only

56

students’ computational abilities but also their critical thinking skills. This was
evidenced by the performance of the cooperative learning groups who performed much
better than their counterparts who learned with more traditional approaches.
Although much of the literature focuses on the needs of marginalized groups,
everyone benefits from multicultural approaches to teaching math. Culturally
responsive paradigms are critical in every classroom, not only those comprised of
people of color. Everyone, including non-minority students, benefit from learning
about the contributions of diverse groups of people to the field of mathematics (Gaitan,
2006). While the work of Ladson Billings and others formed the foundation of
culturally relevant pedagogy, the work of Zaslavsky and Gaitan help us understand how
culturally relevant pedagogy specifically applies to the teaching of mathematics.
Summary
Effective instruction in mathematics is critical in today’s society as we prepare
student to compete at a global level, and contribute to society. However, findings at the
national and international level indicated that students are nowhere near where they
should be in terms of conceptual understanding. Nationally there is a wide achievement
gap that places students of color at a disadvantage. Internationally all students are not
competing at an adequate level with international counterparts.
One area where this can be addressed is at the classroom level. However,
before teachers can effectively prepare students to learn mathematics in the conceptual
manner advocated by standards-based reform, they need to have this type of

57

understanding themselves. Reform-oriented approaches also advocate the use of socio-
constructivist teaching practices; however students today are still being taught via
traditional methodology. In an effort to ascertain where the disconnect lies between
research proven methodology and current practices, teacher practices need to be
analyzed. Although NCTM and other reform-oriented organizations advocate socio-
constructivist forms of teaching, it is uncertain what factors facilitate this type of
instruction. Thus it is imperative to determine what these factors are, so that they can
be integrated with teacher preparation, to maximize socio-constructivist teaching
practices across the board.

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CHAPTER THREE: METHODOLOGY
In order to prepare students to be contributing members of this society, teachers
need to prepare them to meet the needs of an evolving world. Mathematics instruction
is one area of education where such preparation can take place. Current data indicates
that students in this country are not prepared for these societal needs. In particular
when compared to their international peers, American students do not perform as well.
These discrepancies are further exacerbated when one looks at students of color, where
there is a wide achievement gap. One of the areas that can be targeted in an effort to
ameliorate this situation is in teacher education, specifically in preparing teachers to
teach mathematics.
Teacher education is replete with concerns over the need for high-quality
teachers and high-caliber education programs. There has been concern over the
effectiveness of teacher education programs and the need for these programs to prepare
qualified teachers. Proponents of alternative programs cite the success of graduates
from their programs who have made a positive impact on student achievement while
criticisms have focused on the failure to apply research-based pedagogy in the
classroom. Similarly, a failure to sustain successful teaching pedagogy, when it comes
to socio-constructivist teaching practices has been an area of concern (Cook,
Smagorinsky, Fry, Konopak & Moore, 2002). This has been particularly evident in the
methodology utilized to teach mathematics.

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Much of current mathematics reforms advocate socio-constructivist learning.
Although the definition can vary widely depending on the researcher, most of the
literature has come to a consensus on some of the major tenets of socio-constructivist
learning and teaching: (a) students constructing their own knowledge at times
independently and at times with social support; (b) teachers facilitating instruction
rather than delivering it; (c) and cooperative working situations (Cook, Smagorinsky,
Fry, Konopak, & Moore, 2002; Phillips, 1995; & Simon, 1995). These constructivist
practices are particularly beneficial and result in student success when it comes to the
delivery of mathematics instruction. Although current reform practices advocate the
use of socio-constructivist practices, and it is believed that prospective teachers obtain
this knowledge in teacher preparation programs, no detailed analysis has been
conducted to determine what factors facilitate some teachers’ use of these constructivist
practices where other teachers struggle and rely on more traditional methodologies.
Thus I am interested in looking at what these concerns are. Specifically I am
interested in addressing the following three questions:
 What socio-constructivist practices do teachers incorporate in their teaching
of mathematics?
 What kind of past mathematical experiences influence teachers’ use of
socio-constructivist practices when teaching mathematics?
 What impact do Teacher Education programs have in the effective
implementation of socio-constructivist practices?

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This chapter begins with a description of the sample and population for the
study. This is followed by a description of the instruments utilized to collect data. An
overview of the data collection steps taken for all three phases of data collection is next.
The last section delineates the steps taken to analyze the data in all four phases of data
analysis.
Sample and Population
The sample was comprised of various elementary teachers from an urban school
district in Southern California. In the first phase, a multitude of elementary school
teachers across the school district were screened via a demographic questionnaire (see
Appendix A) and a vignette (see Appendix B). Ten of these teachers were selected for
the other two phases of the study where they were observed and responded to a survey
that determined the degree to which they considered themselves to be socio-
constructivist educators.
Instrumentation
Data collection for this study occurred in three phases. The first phase focused
on a screening of math teachers to determine the extent of their academic preferences
and mathematics preparation. In addition to gathering demographic information,
Appendix A identified mathematics preparation and subject interest preferences.
Appendix B was also part of the first phase. This instrument includes two vignettes that
describe two teaching styles; one that can be identified as socio-constructivist and one
which is more traditional. With this brief response, participants generally identified

61

with one of two teaching styles, socio-constructivist or traditional. Ten participants
were chosen based on their responses to Appendix A and B.
Table 1: Distribution of Teachers Selected

Teaching Style Preference
Traditional Constructivist
Math
Preparation and
Preference
LOW
2
Teachers
3
Teachers
HIGH
2
Teachers
3
Teachers

Five participants who have a strong preference and preparation for mathematics
were chosen: three who identified with the socio-constructivist approach to teaching,
and two who identified with more traditional methodologies. Five participants whose
preference and preparation was not as high were be chosen, three of these participants
were selected because they identified with socio-constructivist methodologies, and two
who identified with more traditional practices.
The next phase consisted of the distribution of Appendix C to the ten chosen
participants. Appendix C, the secondary questionnaire, was designed to gather more
information about the desired teaching modality each of the participants preferred.
Through the 42-item questionnaire, participant responses better ascertained the teaching
modality each of the ten teachers identified with.

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The last phase of data collection focused on discussions and observations of a
single math lesson. Three meetings were held with each teacher and were comprised of
(1) a pre-observation discussion, (2) lesson observation, and (3) post-observation
discussion. Appendix D was utilized for the pre-observation discussion. The lesson
observation utilized Appendix E. The debriefing process of the post–observation
discussion utilized Appendix F.
Data Collection
The first phase of data collection commenced in April 2010. Screening
questionnaires were distributed at five elementary schools that are part of a small
school district in Southern California. The five selected schools were selected
randomly. Data from these preliminary surveys assisted in the identification of the ten
participants that were involved in the subsequent phases of the study.
The second phase of the data commenced in May 2010, after the identification
of the ten selected participants. In this phase of data collection, participants completed
the secondary questionnaire which provided more information about each participant’s
preferred teaching modality. The scheduling of three subsequent meetings, pre-
observation, lesson observation, and lesson de-briefing also took place at this time.
The third phase of data collection consisted of three meetings and discussions
which focused on a single math lesson chosen by the participant. The first discussion
focused on the planning and goals for the lesson. Appendix D was utilized to guide the
discussion. The next meeting consisted of the lesson observation. The math lesson was

63

recorded to ensure accuracy of data collection. Appendix E was used to guide the
lesson observation. Finally a post-lesson discussion focused on a debriefing of the
lesson. This discussion was guided by Appendix F.
Data Analysis
The first phase of data analysis commenced as soon as all screening
questionnaires were collected in April 2010. Data from the preliminary screening
survey was reviewed and analyzed descriptively to identify the ten participants that
were part of the study. The focus of this analysis was on the identification of the ten
selected participants. The first three participants were selected based on their
association with more socio-constructivist teaching practices and their high preference
of and preparation in mathematics. The next two participants were selected because of
their association with more traditional teaching practices and less preference and
preparation for mathematics. Two more participants were selected who associated with
more traditional teaching practices and had a higher preference of and preparation in
mathematics. The last three participants associated with more constructivist teaching
practices and had less preference and preparation for mathematics.
The second phase of data analysis consisted of reviewing the ten secondary
questionnaires for each identified participant. This analysis focused on identifying
trends and patterns across the 42-item questionnaire. Descriptive statistics were utilized
to provide information across participants and for each individual participant.

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The third data analysis phase focused on the identification of socio-
constructivist and traditional teaching practices across the three meetings with each
participant. Analysis of this data also focused on the corresponding teaching practices
identified by each participant in the pre-observation, across the three meetings.
Discrepancies between stated goals and observations were discussed in the third
debriefing meeting.
A final analysis of the data considered the various data sources in an effort to
triangulate the information. Data from Appendix C, the secondary survey, were
triangulated with data from the pre-observation meetings and the actual observation. In
this manner, the data revealed the degree to which theoretical and pedagogical
orientations remained consistent across the various data sources.

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CHAPTER FOUR: RESULTS
The purpose of this study was to analyze the factors that contributed to teachers’
use of socio-constructivist practices, as well as the factors that hindered their use of
these same practices. Specifically, teachers were interviewed and observed as they
taught a math lesson to determine if they utilized socio-constructivist practices, and if
so to determine which ones they utilized. Some of these teachers associated with socio-
constructivist practices while the rest associated with more traditional methodologies.
By analyzing each teacher’s beliefs about teaching mathematics and their own
educational background, trends were uncovered about some of the factors that
contribute to teaching in a socio-constructivist manner, rather than utilizing more
traditional methodologies. This qualitative study generated descriptive data of factors
that contribute to teaching in a socio-constructivist manner rather than more traditional
methodologies. Teachers from two schools in a small Southern California school
district participated in the study.
This chapter begins with a description of the participants in the study as well as
the methodology utilized for placing them in distinct groups. This is followed by a
second section that delineates the findings of the study for all three research questions,
divided by research question. Some implications are addressed at the conclusion of
each subsection.

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Participant Description
Thirty-five teachers participated in the initial phase, or pre-screening portion of
data collection, which consisted of responses to a demographic questionnaire (see
Appendix A). Among the ten questions, there were two items that identified teachers’
preference for mathematics. In this phase, teachers also responded to a vignette where
participants were asked to identify with one of two teachers (see Appendix B). One of
these, Mr. Jones, exhibited socio-constructivist practices, whereas the other, Mr. Hill,
utilized more traditional practices. Ten teachers were then selected from this group of
35 teachers. Five who had a high preference for mathematics, and five who had a low
preference for mathematics.
These ten teachers then filled out a secondary survey comprised of forty-two
questions (Appendix C). Eight of these questions addressed traditional practices,
whereas the remaining thirty-four items addressed socio-constructivist practices.
Participants were asked to agree or disagree with these forty-two statements. Using
these forty-two statements, in conjunction with the original response to the vignette,
participants were then separated into four groups, see Table 1 below: (a) teachers who
had a low preference for mathematics and agreed with more traditional practices; (b)
teachers who had a high preference for mathematics and agreed with more traditional
practices (c) teachers who had a low preference for mathematics and agreed with more
socio-constructivist practices; (d) teachers who had a high preference for mathematics
and agreed with more socio-constructivist practices.

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These ten participants were also interviewed, and took part in a lesson
observation. The interview was designed to obtain background information about each
participant and as an opportunity to discuss the lesson that was observed at a later time.
The interview and the math lesson observation took place on two separate occasions
and were designed to uncover the factors that facilitated or hindered the delivery of
instruction in a socio-constructivist manner.
Biographical Sketch of Participants
The most experienced teacher in the group of participants was Rae, a 63-year
old teacher who has been in the field of education for twenty-five years. For a majority
of that period she has worked in Special Education and Intervention services. She also
has experience working as a systems analyst, where she spent time developing
computer systems and code. Although she never took Calculus, she has a lot of
confidence when it comes to mathematics and believes she has a strong mathematical
background. A major part of this is due to her experience as a systems analyst where
her analytical skills were honed; because of this she believes she is very analytical in
the way she teaches.
Dana has been worked for the school district for a total of 30 years, but has only
spent twenty-two of those years in the classroom. She was hesitant in participating in
the study due to her low confidence in mathematics. Growing up, she never had
confidence in her mathematical abilities, mostly due to the fact that her father is a
chemical engineer. She has a strong interest in teaching G.A.T.E. (Gifted and Talented

68

Education) classes, since both her children were identified as G.A.T.E. early in their
own education. She is certificated to teach G.A.T.E. classes and has worked with this
population for the past few years.
Tracy is a 47-year old teacher who has worked in the field of education for a
little over twenty years. In addition to a teaching credential, she also has an
administrative credential. Throughout her educational history she has always had high
confidence in mathematics and performed better in mathematics than other academic
subjects. She has taken a variety of mathematics courses, culminating with Calculus at
the university level. She has been inspired by many mathematics experts and has also
attended a multitude of mathematics trainings. Many of these were sponsored by
professional teacher organizations like California Mathematics Council. Utilizing ideas
she has gathered at these trainings, mathematics has been her favorite teaching subject.
Brianna is a 43-year old teacher who has been in the field of education for
thirteen years. In spite of having worked a multitude of jobs prior to teaching, she
considers education her first real career. Although she has attended a multitude of math
trainings and workshops, mathematics is her least favorite subject to teach, which she
ranked 8 out of 8 subjects. This she believes is mostly due to the way in which she was
taught, where manipulatives were never the norm, and intimidation at the middle school
level pushed her away from mathematics. She currently teaches a first grade class, but
has spent most of her teaching career in the upper grades, with most of her experience
in sixth grade.

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Shawna is a 35 year old teacher who has taught for twelve years. She has
always performed well in mathematics, and often with little support. She enjoys the
subject so much that she took advanced courses beyond Calculus at the university level,
for the fun of it. This culminated in her complete understanding of our mathematical
system, how it functions and how it builds on prior knowledge. She constantly tries to
bring in her enthusiasm about mathematics into the classroom and make the material
engaging to all her students. She has attended numerous trainings and has learned how
to bring hands-on, concrete teaching into the classroom, which she believes will help
students move to more abstract concepts in the upper grades. She currently teaches a
Kindergarten class but has spent her teaching career moving up and down throughout
the elementary grades.
Martha, a 49-year old teacher has been teaching for only nine years. Prior to
teaching she spent many years working in the field of Computers, specifically in quality
assurance. Although she never had problems learning mathematics early in her
educational career, she did appreciate it and enjoyed it more as she got older. It was in
High School, that she understood there was a system to Mathematics, which allowed
her to pursue Calculus coursework at the university level. She does not consider herself
an expert, and has no problem re-learning material from the book before teaching it.
This is mostly due to her understanding of its systematic nature. She has also attended
a number of trainings that have focused on teaching mathematics.

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Charlene, a 31-year old teacher has been teaching for eight years. She has
always struggled with mathematics, and never felt she had a teacher of mathematics
who made her feel successful. She feels she was always lectured at, which never
allowed her to feel successful. She appreciates that mathematics is really about
understanding concepts, more than simply doing problems on paper. She did not have
any mathematics courses throughout college and even tested out so she would not have
to take math courses all over again.
Laura is a 28-year old teacher who has been teaching for six years. During her
Master of Arts program where she also earned her teaching credential, she was allowed
the option of choosing a project in any subject area. She chose a Math Pathways
project which allowed her for a week, to develop mathematics games and activities at
the PreKindergarten to 6
th
grade levels. Laura feels that she has never been really good
at mathematics, and reported that she still sometimes struggles with basic fundamental
concepts. This has influenced the way she teaches, and as a result she often tries to
teach subjects in a variety of ways, with the realization that not everyone learns
material one way.
Taylor, another 28-year old teacher has only been teaching for five years.
Although she now teaches first grade, most of her experience has been in the upper
grades. She really enjoys teaching mathematics, especially when it comes to using
manipulatives and getting students involved. She feels this is critical in helping
students understand difficult concepts. She was never very successful with

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mathematics, and often needed tutors to get through coursework. This was despite
having excellent teaching mentors, some of whom she still keeps in contact with. She
attributes her teaching style to some of these teachers, who gave her strategies she now
utilizes with her students. This school year, she earned the Teacher of the Year award
at her school site.
The least experienced teacher was Stephany, a 30-year old who has four years
of teaching experience. She came from the field of Accounting, which is one of the
reasons why she has such a strong mathematical background. She believes her
elementary school experience was much different from the typical public school
experience. She was schooled in the Carden Hall system, which has a strong emphasis
on memorization and basics. Unlike the curriculum in public schools, the Carden Hall
curriculum does less spiraling, and students are required to master a concept before
being allowed to move on to more challenging concepts.
Findings
Much of current mathematics reform advocates socio-constructivist learning.
There is a consensus on some of the major principles of socio-constructivist learning
and teaching: (a) students construct their own knowledge either independently or in
cooperative situations; (b) knowledge is constructed by connecting new information
and knowledge to existing and previous knowledge; (c) authentic learning tasks are
required; (d) the acquisition of new knowledge requires the autonomous and active
restructuring of how one thinks; and (e) teachers facilitate instruction rather than deliver

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it (Cook, Smagorinsky, Fry, Konopak, & Moore, 2002; and Applefield, Huber, &
Moallem, 2001). These socio-constructivist practices are particularly beneficial and
result in student success when it comes to the delivery of mathematics instruction.
The first part of this study focused on the frequency with which participants
agreed with the socio-constructivist practices on the secondary questionnaire. By
analyzing the agreement among teachers, data from this study revealed which socio-
constructivist practices were more prevalent among these participants. This also
identified if any of the traditional practices were also prevalent among this group of
teachers. The subsequent portion of the study focused on an analysis of data obtained
from the interviews and observation. This analysis focused on identifying any factors
that affected teaching practices, whether they were socio-constructivist or more
traditional practices. The final portion of data analysis focused on identifying the
impact of teacher education programs among these ten participants. The following three
sections of this chapter illustrate the findings for each of the three questions this study
was designed to address.
Research Question One
Research Question One asked: What socio-constructivist practices do teachers
incorporate in their teaching of mathematics? The aim of research question one was to
identify which socio-constructivist practices were more prevalent in classrooms. There
are a variety of practices dictated by research that fall within the realm of socio-
constructivist teaching. However the prevalence of them in classrooms is not clear,

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therefore it was critical to uncover which ones were more prevalent and which ones
were less utilized in the classrooms.
The questions in the secondary survey focused on a variety of teaching practices
from the entire spectrum of teaching; from socio-constructivist to traditional. However
the vast majority focused on socio-constructivist practices. There were eight groups of
questions that encompassed the criteria for socio-constructivism delineated above. One
of these groups focused on teacher reflection, and the benefits of instruction with the
inclusion of teacher reflection. A similar feature of socio-constructivist teaching was
incorporated in the second group of questions that addressed student reflection, and its
role in students assessing their own work. The third group of questions dealt with
utilizing diversity while teaching. Diversity was needed for making cooperative
groups, as well as the utilization of teaching strategies. Diversity was also critical in
addressing the variety of learning styles within a classroom. In the fourth group of
questions teachers were asked about building instruction upon prior knowledge. The
fifth group of questions focused on teachers’ use of culturally relevant pedagogy, while
the sixth group focused on teachers making students responsible for their own learning.
In the last two groups of questions, teachers were asked about the need to adjust lessons
for student needs, and the need to treat students as partners in their own learning.
In order to analyze the participant responses, questions from each participant’s
secondary survey were grouped together according to the criteria mentioned above for
teaching practices they addressed. After excluding the eight questions that dealt

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specifically with traditional teaching practices, the thirty-four questions dealing with
socio-constructivist practices fit within the eight criteria (See Table 2). Participant
responses were then analyzed within these criteria to uncover patterns across the sample
of ten participants. This analysis identified consistency of responses across
participants. Questions with significant findings were identified where 7 of the 10
participants agreed.
Table 2: Number of questionnaire items in each theme
Criteria for Questions N
Instruction benefits from teacher reflection 2
Variety or diversity in groupings, strategies, and learning styles 5
Students need to reflect and assess their own work 7
Instruction needs to build on prior knowledge 7
Instruction benefits from culturally relevant pedagogy 4
Students are responsible for their own learning 3
Lessons need to be adjusted for student needs 3
Students are partners in their own learning 3

By focusing the analysis on questions where responses were more consistent, it
was easier to identify which socio-constructivist practices were more prevalent in
classrooms. Questions with higher consistency percentages meant that a large
proportion of teachers agreed with the utilization of that particular teaching practice.
This made it more likely that this particular teaching practice was evident in that
classroom. Conversely for questions that did not have high consistency ratings, this

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likely indicated some participants agreed with the teaching practices and likely utilized
it; while other participants may not have agreed, and therefore neglected to utilize it.
Using the selected criteria, the first group of questions focused on how
Instruction benefits from teacher reflection. The questions within this group exhibited
a very high consistency. Questions grouped under this category had an 80%
consistency rating. This means that 8 of 10 participants agreed with all the statements
within this group of questions, see Table 3 below. Participants strongly agreed that
teachers should (a) engage in dialogue and reflection with colleagues to solve teaching
related problems and (b) reflect on their own instructional success and develop
professional development plans that are based on their reflection and analysis. Thus
despite preference of mathematics and teaching practice, the vast majority of the
participants believed that teacher reflection was a critical component of effective
instruction.
Table 3: Instruction benefits from teacher reflection
Questions Results
Teachers should reflect on their own instructional successes
and develop professional development plans that are based on
their reflection and analysis.
8 of 10 participants
strongly agree
Teachers should engage in dialogue and reflection with
colleagues to solve teaching-related problems.
8 of 10 participants
strongly agree

Questions within the criteria of Variety or diversity in groupings, strategies, and
learning styles also had a high consistency. For the majority of the questions at least 7
of the 10 participants agreed with the practice addressed by the questions. One of the

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questions had a very high consistency rating with 9 of the 10 participants agreeing with
the statement, see Table 4 below. Specifically, the data indicated that 9 of the 10 of
these participants highly agreed with giving students a variety of learning experiences.
Data also indicated that 8 of the 10 participants agreed with both (a) engaging students
in activities that encourage multiple approaches and solutions, and (b) ensuring that
subject matter knowledge encompassed different perspectives. Additional data
suggested that 7 of the 10 participants agreed that variety in teaching strategies and
grouping strategies were important for student learning. Again, despite teaching style
predilection and math preference, most teachers seemed to believe that students
benefited from a variety or diversity of strategies. Both differentiated instruction and
the inclusion of multiple intelligences are correlated with socio-constructivist teachings
(Pelech & Pieper, 2010)
Table 4: Variety or diversity in groupings, strategies, and learning styles
Questions Results
Providing students with a variety of learning experiences
helps accommodate the various learning styles.
9 of 10 participants
strongly agree
Using a variety of strategies to introduce, explain, and restate
subject matter concepts helps all students understand.
7 of 10 participants
strongly agree
Teachers need to provide a variety of grouping structures for
students to engage in collaborative learning.
7 of 10 participants
strongly agree
Students should engage in critical thinking and problem
solving activities and encourage multiple approaches and
solutions.
8 of 10 participants
strongly agree
Teachers should ensure that their knowledge of subject
matter incorporates different perspectives.
8 of 10 participants
strongly agree

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The third criteria, Students need to reflect and asses their own work, also
exhibited high consistency with six of the seven questions having at least 70%
agreement among participants. Thus for these six questions, at least 7 of the 10
participants agreed with the statement. Data from two questions reveal that 8 of the 10
teachers strongly agree that students should be given opportunities to discuss, reflect,
and evaluate information content as well as develop strategies for evaluating and
reflecting on their own learning. Moreover 8 of the 10 participants moderately agreed
that students should also assist with the development of criteria by which their work
will be evaluated. 7 of the 10 participants also moderately agreed that teachers should
develop strategies that help students assess their own work and that students should be
able to understand and monitor their own work. Furthermore 7 of the 10 participants
also strongly agreed that students need time to reflect on their own learning. Therefore,
the majority of the participants in this study agreed that much like teacher reflection,
student reflection is also a critical element of student success.

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Table 5: Students need to reflect and assess their own work
Questions Results
Teachers need to provide opportunities for all students to think,
discuss, interact, reflect, and evaluate information content.
8 of the 10 participants
strongly agree
Students should develop and use strategies for evaluating,
monitoring and reflecting on their own learning.
8 of the 10 participants
strongly agree
Teachers should provide time for students to reflect on their
learning and process of instruction.
7 of the 10 participants
strongly agree
Students should help establish criteria on which their work will be
assessed.
8 of the 10 participants
moderately agree
Teachers should develop and use strategies that help students assess
their own work.
7 of the 10 participants
moderately agree
Students should understand and monitor their own work. 7 of the 10 participants
moderately agree
Questions within the Instruction needs to build on prior knowledge criteria also
had high consistency with five of the seven questions having at least 7 of the 10
participants in agreement. One of these questions, dealing with relating subject matter
to previous lessons and students’ lives, had strong agreement with 9 of the 10
participants agreeing with the statement. 8 of the 10 participants strongly agreed with
the other three questions. These delved into (a) making content relevant and
meaningful by building on student life experiences; (b) connecting new concepts to
what students know; (c) and building on student comments. For the last question,
which addressed connecting new information to students’ life experiences, 7 of the 10
participants agreed with the question. Although there was more diversity of answers in
this group of questions, the majority of participants still felt that instruction needs to
build on prior knowledge, one of the central tenets of socio-constructivist teaching.

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Table 6: Instruction needs to build on prior knowledge
Questions Results
It is important for students to see the connection between what
they already know and new material.
8 of 10 participants
strongly agree
Students need to connect new information and concepts to
their own life experiences.
7 of 10 participants
strongly agree
Building on students’ comments and questions during a lesson
helps extend their understanding.
8 of 10 participants
strongly agree
It is important for students to relate subject matter concepts to
previous lessons and their own lives and to see the
relationships across subject matter.
9 of 10 participants
strongly agree
Teachers should build on student life experiences, prior
knowledge, and interests to make the content relevant and
meaningful.
8 of 10 participants
strongly agree

Although participants in this study agreed with some of the tenets of socio-
constructivism, as a group they were not ready to agree on all tenets of socio-
constructivism. As a group, the participants in this study are consistent in their
agreement with (a) Instruction benefits from teacher reflection; (b) It is necessary to
incorporate variety or diversity in groupings, strategies, and learning styles; (c)
Students need to reflect and assess their work; and (d) Instruction needs to build on
prior knowledge. The four criteria of questions with the most diversity in responses
were (a) Students are responsible for their own learning; (b) Lessons need to be
adjusted for student needs; (c) Students are partners in their own learning; and (d)
Instruction benefits from culturally relevant pedagogy. Therefore, even though the
more traditional participants were willing to agree with building understanding from
prior knowledge, and having diversity in a variety of classroom structures; they were

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unable to agree on the need to make students responsible for their own learning, or
treating them as partners in learning process. They were also unable to agree with the
need to adjust lessons for student needs, or the integration of culturally relevant
pedagogy.
Research Question Two
Research Question Two asked: What factors influence teachers’ use of socio-
constructivist practices when teaching mathematics? Teachers were placed in one of the
four groups using their responses from the initial vignettes (Appendix B) and the
secondary questionnaire (Appendix C), along with math preference information from
the demographic questionnaire (Appendix A). Once the participants were placed in
their respective groups, data from their interview and observation was analyzed to
determine if any trends emerged. The aim was to uncover any trends by level of Math
Preparation and Preference. Thus it was critical to determine whether data differed for
participants who had a high preference for mathematics from participants who had a
low preference for mathematics. Additionally, Teaching Style Preference was also
used as a factor of analysis. This was done by looking specifically for trends in both
groups: teachers who associated more with socio-constructivist practices, and teachers
who advocated more traditional teaching practices. This analysis of data revealed those
factors associated with socio-constructivist teaching styles and those factors associated
with more traditional methodologies.

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The results will be presented by Preference for Mathematics and Teaching
Style. Trends identified in teachers with a Low Math Teaching Preference will be
presented first. These will be followed by any trends uncovered by teachers who have a
High Math Preference. The next set of data will focus on teachers who associate with
more traditional methodologies of instruction. Conversely, results for teachers who
associate with more socio-constructivist practices will be presented next. In the final
section, overall trends across the groups will be presented.
Table 7: Distribution of teachers selected

Teaching Style Preference
Traditional Constructivist
Math
Preparation and
Preference
LOW
Dana
Charlene
Taylor
Brianna
Laura
HIGH
Martha
Tracy
Shawna
Rae
Stephany

Teachers with Low Math Preparation/Preference.
Responses from the five participants with low math preparation and preference
were analyzed to uncover any themes. Three major themes emerged in this group of
teachers. The first theme was that teachers who had lower math preparation and low
math preference reported having difficulty with mathematics. A second theme was that
the negative experiences they had in their own schooling helped them with their

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teaching because it helped them understand the needs of their struggling students. A
third theme was also related to their negative schooling experiences. This group of
participants understood and appreciated that students did not all learn in the same way.
All five participants reported negative experiences with mathematics learning,
with three in particular stating that math was intimidating and tried to avoid it whenever
possible. Brianna spoke of her aversion to mathematics and the source of her
intimidation:
When I was younger I remember really loving math. And then when I
got to middle school, I don’t know if it was the teacher, I’m going to
assume. I was in like the seventh to eighth grade class, it’s like I shut
down. She was very intimidating and I did not want to answer or ask
any questions. And I just remember, like, I don’t want to be in this class
… when I came to high school, since I didn’t really care for it, we didn’t
have to. So, I took the minimum.
Dana similarly related her intimidation with mathematics stemming from her father’s
expertise in the area of Engineering, a field highly dependent on Mathematics. She
also related how her experiences affect her teaching.
I was actually very intimated by math. It wasn’t my strongest suit when
I went through school. I think it was because – I was intimidated
because my dad was a Chemical Engineer and he could dry lab
everything instantaneously, and I didn’t have a – I mean, I am more
right brained then, so, but as I’ve gotten older and into teaching, I
realized that, you know, I can do it. And, I think I am a benefit to a lot of
the students because I realize that it wasn’t the easiest subject for me.
Thus, negative experiences in mathematics were highly correlated with a low
preference for learning and teaching mathematics among these participants. Every
teacher that indicated a low preference for math also indicated negative experiences in

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mathematics in their own schooling, with some of the negative experiences being
recalled as intimidating by these participants.
Teachers within this group also said that their past learning of mathematics
helped them in their teaching. Most of them related negative experiences like Laura,
who spoke about her use of manipulatives and her various explanations resulting from
what she felt, was missing in her own education. Her instruction utilizes manipulatives
and advocates solving problems in various ways because her own learning never
utilized these methodologies:
I was never a really good mathematician and I still sometimes will
struggle with basic fundamental mental math facts, and so I approach
math with that in mind that not everybody is going to learn the
algorithm, and the standard way of doing it, so um I really strive to teach
things two or three different ways, um just because of how I was as a
learner, if maybe if I had used manipulatives for addition, maybe that
would have helped me with mental math, and so I do a lot of that, I do a
lot of manipulatives
Taylor’s experiences were not always negative. She spoke of great teachers who gave
her strategies she utilizes with her students today. Taylor’s own learning included a
focus on solving problems in different ways, which she utilizes with her own students.
I also remember I had the same teacher for fourth and fifth grade and
she was amazing also, and just giving me different ways to do things,
things that I carry on and teach my kids now because it worked for me
so I hope that it will work for them
Thus regardless of whether the experiences were positive or negative, all the teachers
with a lower preference for mathematics said their own educational experiences helped
their own teaching practices now.

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The negative experiences this group related also helped them understand the
needs of different students. Four of these five participants indicated that students have
different learning modalities and that it was important to address these learning
differences. Mostly because of their own struggles with mathematics, these participants
realized that not all students learn in the same manner. Taylor reported the reasons for
why she explains mathematical concepts in more than one way, especially with smaller
groups:
I also really believe in explaining something in more than one way yeah
there’s plenty of kids that get it the first time the first way but um I
know I wasn’t that kid so I know I’m really aware of that and um try and
make sure that I explain things in … at least twice to the whole group
and then other kids obviously need it more than that and that will be in a
small group setting.
Similarly, Laura confirmed the importance of teaching concepts in multiple ways,
especially since this type of instruction was amiss in her own learning.
I approach math with that in mind that not everybody is going to learn
the algorithm, and the standard way of doing it, so um I really strive to
teach things two or three different ways, um just because of how I was
as a learner, if maybe if I had used manipulatives for addition, maybe
that would have helped me with mental math.
Therefore, the negative experiences of these participants benefited their teaching in that
it allowed them to appreciate that different strategies had to be utilized in order to reach
every student.
Teachers with High Math Preparation and Preference
Responses from the five participants with high math preparation and preference
were also analyzed to uncover themes. Three major themes emerged among these

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teachers. Teachers who had higher math preparation and preference indicated that they
appreciated and understood that mathematics worked as a system that could be
analyzed and broken down. They also felt more pressure from district mandates and
pacing plans. Finally this group was more apt to make connections to students’ lives in
their lessons.
Every teacher that had a high preference for math also had an appreciation of
the systematic nature of mathematics. One of the participants, Rae did not take Calculus
at the university level; however, despite not having taken the class, she still felt her
analytical skills were strong. Like every other participant, Rae indicated that she
understood there was a system for mathematics, and felt she had an ease in analyzing
math concepts and problems. Rae also stated that this mathematical knowledge
transferred to her students as she taught:
I have a background - I'm very analytical so I, for a while, I worked as a
systems analyst so I developed computer systems and code so I have a
fairly strong math background, though I don't go into calculus or
anything, you know. So I'm very analytical. So my analytical skills
relate to how I teach because I'm always evaluating and analyzing what's
going on with a learner. And then I break it down, just like you do with
a system, what are the steps? It's that background, you know, that comes
into how I teach.
Another participant went beyond Calculus for the fun of it. Shawna expressed her love
for mathematics. This stemmed from her understanding that mathematical concepts
build upon one another and culminate in an understanding of the entire system:
So for me, math has always been fairly easy, which I know it’s not for a
lot people. And throughout the years I just, I kept taking math all the
way through college, you know, and even differential calculus and I
took it just for fun, didn't have to. I just took it for fun because I just felt

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like, it finally all made sense, like all the things I had learned,
throughout all the years, I was thinking "Oh, so this is why I learned this
and this is why I learned this" and it all kind of culminates at the end, so
it was very interesting. But I love math, I love everything about math
and just try to bring it into the classroom.
Thus regardless of taking courses like Calculus and Differential Calculus, these
participants all understood there was a system for Mathematics. There was an
understanding that there is a set way to do things and that everything followed in a
sequence so that it all culminated with a better understanding of the mathematics in the
end.
Four out of five of these participants were able to connect the mathematics they
were teaching to student’s lives. Stephany helped students make this connection in the
assessment portions of her lesson. She began her fractions lesson with a pre-assessment
that asked students to report on instances where they had seen fractions in their lives.
Later her lesson concluded with a project where students had to apply knowledge of
fraction concepts to a pizza they would like to create. In her interview, Tracy spoke of
the importance of relating mathematics contents to concepts students are familiar with.
Pointing out the relevance of mathematics to students is important for their motivation
and subsequent success.
when I’m introducing it – kind of talking about why this important –
like, you know, relating it to a box of candies – like how many candies
fit in the box, but just so they see kind of how this is relevant to other
things … just like, you know, that hook to get them interested like,
which do you think has more to kind of introduce like why we need to
know

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Similarly, Shawna addressed the importance of making the mathematics content
relevant to the students’ lives. She spoke of her integration of money at home and in
the classroom and how this was relevant to their lives:
Every day when we talk about money, we can talk about, okay, why
would you, you know, why would you need to know this? Okay well, if
you want to buy things at the store. I have the families working on it at
home, I’ve told them to get a coin, you know, keep a coin jar if they
have a coin jar, practice counting money with their kids starting this
week. So really just having…making it relevant in their lives. And for
the most part, money is a pretty…it’s a pretty easy concept for kids to
see the relevance in it, you know.
In addition to addressing pacing plans as the reason certain lessons were taught,
this group of participants also indicated feeling pressure from pacing plans and district
mandates. Three of the five participants elaborated on the difficulty of teaching
conceptually when district mandated pacing plans required them to teach a lesson per
day. Martha expressed her frustration with teaching math in a less than ideal manner,
especially with district expectations in regards to the pacing of lessons:
What I find difficult in the classroom …is they expect us in the
classroom to move through the lessons every day – I mean, to get
through all the lessons the district expects us to – you’d have to do them
one a day. And the timing and the – I kind of extend a lesson with
manipulatives and all that, it would be much better for the kids, but it
doesn’t – it usually doesn’t work with what they expect us to do. So
that’s probably my issue with teaching math …They never – I don’t
think they consider how much time it takes to teach a concept. And they
should be practicing each concept, and then review the concept so, with
the standards we’re suppose to teach and then to get to the testing part –
it’s way too much for – especially our students when they have language
issues too, but I would say for any student – I wouldn’t teach this way
by choice. But it’s what they expect from us.

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Tracy also expressed a concern with teaching in this way, especially when she has
witnessed the benefits of teaching conceptually:
personally, I kind of see even from observations … that it’s not a waste
of time, but we don’t have that luxury of time anymore. So, I get a little
bit frustrated with that total lack of control
So, although these participants would otherwise prefer to teach in a more conceptual
manner, they expressed the difficulty in doing so, when district mandates and pacing
plans were too restrictive.
Teachers who identified with Socio-constructivist Practices
Within the group of teachers who associated with socio-constructivist practices,
five different themes emerged. However three of these themes also emerged for a
majority of the participants. Hence, these three themes will be discussed later as other
findings. Participants in this group were more able to teach conceptually, and they
were also more apt to focus lessons on connections between the mathematics content
and students’ lives.
Although conceptual teaching only emerged in discussions with half of these
participants, this was the highest percentage which included this very important tenet of
socio-constructivist learning. The only three participants in the entire sample to speak
to the role of teaching conceptually were participants who identified with socio-
constructivist practices. Shawna addressed the importance of building the foundation
of concepts in the lower grades for understanding in the upper grades. She realized the
importance of:

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setting the foundation for them for more in-depth concepts....And then
kind of taking that concrete and moving it to more complex things in the
upper grades. So, I am really learning how to help kids understand what
they are doing.
Similarly Laura reported that a good foundation was critical, even though it meant
spending more time on certain mathematical concepts.
we'll take as long as we need, we'll probably take towards the end, till
the end of the school, just to make sure that they have a good
foundation…And so we want to make sure that they have a good
foundation so, you know…at least they have the terminology and the
basic understanding on, you know, how to change and make money.
Lastly Rae indicated that if the foundation is not set, it hinders a student’s ability to
understand the mathematical concepts:
…you go from concrete to semi-concrete to abstract, so we work on
having them not only have further instruction, but also meaning if we
can. So that's why I'm concerned one little girl that, you know, she can
do the calculations if I get her started but she doesn’t fully understand
what she's doing.
Therefore three of these six participants were the only ones to understand that the
building of concepts in mathematics was more important than teaching rote procedures.
Although it was only prevalent across half of these participants, these three participants
were the only ones to declare the importance of teaching mathematics in a conceptual
manner, again one of the major tenets of socio-constructivism.
Like participants who had a high preference for mathematics, these participants
also emphasized the importance of making connections between students’ lives and the
mathematics content they were learning. Taylor spoke about the importance of students
understanding the reasons why they were learning about fractions:

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I think that its great because I know for my sake ... especially in sixth
grade, every time I got up to teach a lesson .. I wanted to say “I’m
teaching you this because” ... so that they had some sort of relation to
the real world… so definitely connecting it to the real world… Just
making sure the kids know you’re not learning this just take it on a test
but to know that you know when your family sits down to eat dinner
how do they make sure everybody gets enough food… because then
they’re li[ke].. you know thinking oh yeah we do all get an equal amount
Laura also spoke about making the connection explicit between students’ lives and
learning about money. She felt that this concept in particular was highly relevant to
student lives.
Well, they’ll have practice using money because I've asked them, you
know, have they…have they been able to go to the store and hand the
money in, you know, to the checker or whatever they are doing? But I
feel like the manipulatives that I'm using are very applicable to just the
entire world because, you need to know how to use money [laughter] to
survive. …It will be the foundation for everything that they do in our
capitalist society. You know, they're going to need to learn how to make
money and make change and they are going to need to recognize that,
you know, a nickel is bigger; it's worth less money than a dime, which
doesn’t makes sense… it would be a real life application starting, you
know, today basically for them. Because if they can start to make
change, then they can help their parents too.
Thus, participants who associated with more socio-constructivist tendencies were very
much like participants who had higher preferences for mathematics in that they were
more likely to make the connections between what they were teaching and their
students’ lives more explicit.
Teachers who identified with Traditional Practices
There were four teachers who associated with traditional methodologies. In the
analysis of data for these four participants, three trends emerged. Participants who
advocated more traditional methodologies utilized a variety of teaching modalities to

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address the various learning styles. They were also likelier to use paper assessments.
Finally, like participants with lower math preference, these participants taught in ways
that were influenced by their own learning.
In the group of four teachers, three of the teachers spoke about using a lot of
different teaching modalities, including modifying their own teaching to address
different learning modalities and ensuring access for all students. Dana claimed that
even though many of her lessons ended with paper and pencil activities, for the most
part she tried to integrate other modalities:
I like to do a lot of modalities. I mean, I like to present it in various
ways, and, but, inevitably, whatever happens is that no matter what I do,
we always end up with paper and pencil work that we still– I’ll use the
smart-board, you know, or I’ll use a lot of chart paper to go over –
sometimes I’ll have them take notes, so that they can, you know, refer to
the notes later, and it just goes from there. It depends on, usually, the
complexity of subject.
Charlene also spoke about the need for hands-on activities and making sure that
students with different learning styles could access the curriculum.
…hands-on I like to be able to make sure that all the kids and all their
learning styles will be able to access it because I know math is hard …
It’s more conceptual than it is something on paper so I like to make sure
the kids can use whatever they need to understand the concept…
Thus although these four participants were mixed in regards to math preference, they all
associated with traditional methodologies. Subsequently three of the four participants
believed that teaching needed to incorporate different modalities so that every student,
regardless of learning style could benefit from instruction.

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Another trend that arose in this group of participants was that they all assessed
student understanding via some type of paper and pencil assessment. Although two of
the participants indicated that walking around the classroom would give them some
indication of which students understood the concepts, and which students would need
additional support. Every single teacher in this group indicated that the final
assessment for the lesson would be a quiz or worksheet that students would need to
complete.
Much like participants who had low math preparation/preference, these
participants were highly influenced by the manner in which they were taught. Martha
spoke about understanding the systematic nature of mathematics, and utilizing this
knowledge during her instruction:
So, I liked the fact that it’s systematic and you can learn it that way – so,
probably that’s probably the way I try to teach it – to break it down into
parts and try to have them see it systematically.
Thus, despite teachers being mixed in their preference for mathematics, these teachers
who preferred more traditional teaching styles were also more influenced by their own
learning, when designing learning opportunities for their students.
Other Findings
In analyzing the factors that contributed to the use of socio-constructivist
practices, data from this study also revealed other findings that were not limited to the
four groups above. Specifically three findings pertained to three of the groups. With
the exception of Dana and Charlene, who both had low math preparation/preference

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and also associated with more traditional methodologies, a vast majority of the
participants utilized (a) hands-on experiences and manipulatives; (b) culturally relevant
pedagogy; and (c) opportunities for partner and group work. Additionally the reasoning
for lesson planning was almost unanimous. All but one of the participants chose the
lesson, because it prepared students for the next school year. Finally, data from one
participant in particular, Tracy, was very informative. Although she associated with
more traditional tendencies according to her initial responses, her interview and lesson
observation indicated that she really found herself equally between both teaching
methodologies.
The group of participants who had low math preference and also associated with
more traditional teaching methodologies was comprised of Dana and Charlene. When
looking at the entire data set, three themes were revealed as significant because the vast
majority of the participants addressed these themes in their interview and observation.
Upon further analysis, Dana and Charlene were the only two participants who
consistently neglected to address these themes in their interview and observation. Thus,
the vast majority of participants addressed the importance of: (a) utilizing
manipulatives and hands-on experiences; (b) integrating culturally relevant pedagogy;
and (c) incorporating group work during instruction.
Six of the ten participants utilized some form of hands-on activities or
manipulatives in the delivery of their lesson, with three of the participants incorporating
games into the lessons, and another incorporating mathematics literature into the

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lessons. Shawna stated that her pull-out instruction used a lot of games, so that students
would not be doing the same paper and pencil activities they had previously struggled
with.
I just now incorporated one of the game activities and to demonstrate
and practice knowledge. I don’t do a lot of focusing on paper and pencil
because they do a lot of that in class.
Similarly Tracy also enhanced her lessons; however, she chose literature connections as
a supplement to the hands-on activities she utilized for many of her mathematics
lessons:
I love to, a lot of times, tie in literature with a math idea and then kind of
hands on. So, in both of those I’d probably start with reading a story and
then go to some sort of hands on group activity, and then go into
something independent
Taylor spoke of using manipulatives, and how this helped student in their
understanding, more than the manipulation of numbers.
Um I really enjoy teaching math, especially when it comes to um using
manipulatives and hand and getting the kids involved … in a sense
…not just sitting up there and crunching numbers but getting kids to
manipulate things I feel that that really helps them in understanding the
concept
Brianna also reported that she used manipulatives whenever possible, especially
because she did not have that opportunity to use them during her own educational
experiences.
I also am a visual learner as, you know, well as I’d like to, you know,
I’d like to be hands on. So, if I do have the manipulatives for them to
use, I do like to use manipulatives where needed. And I don’t feel like,
well in comparison, I don’t feel like I did a lot of that when I was
younger. So if we do [have them] I do like to use them.

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Like other teachers who utilize socio-constructivist practices, the participants in this
study also integrated hands-on activities and manipulatives in their mathematics
lessons. They also incorporated games and literature to enhance the learning of
concepts.
Six of the ten participants in this group mentioned the integration of some form
of Culturally Relevant Pedagogy in their instruction. Of these instances, three
participants focused on the inclusion of students’ culture into lessons. The other three
focused on making instruction relevant to students by integrating their interests and
home lives. One teacher, Laura spoke of how she integrated students’ families into the
classroom community. She elaborated on the importance of integrating students home
lives with their classroom lives.
I speak a lot with the families, I see…because I dismiss directly to the
parent, I am able every day to converse with them. And so just over the
year and most of them are Spanish speaking and I think that that has
helped me create a much stronger bond than the teachers who might not
have the second language. So, through that, you know, I have been
invited to birthday parties and you know, they'll tell me about,
everything that is going on and I ask how the family is, so…having the
daily face time, has been really instrumental in me understanding what
their home life is like. Like, I mean, I feel like I understand the
community and being here for a few years, I understand what the
resources are that they have here. But I often times end up being, you
know, kind of like a counselor myself.
Although this type of family connection was not evident with all six participants, each
participant did indicate the inclusion of some aspect of their students’ lives in order to
enhance instruction.

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Group work was another trend that arose with these participants. With the
exception of Dana and Charlene, every other participant in the study utilized a form of
group work: partners, homogenous groups, and heterogeneous self-selected groups.
Group work also served different purposes. One teacher used groups as a way checking
each other’s work, while two others instructed children to explain concepts to each
other in their groups. Thus despite the organization and purpose of the groups, each
teacher who associated with socio-constructivist practices did incorporate some aspect
of group work in their math lesson.
In analyzing data that pertained to participants’ planning of the lesson, the data
revealed 9 of the 10 participants reported that the lesson they taught was chosen
because it was the next lesson in their sequence of lessons. Of these, 8 of the 10
participants reported that the lesson was specifically designed to get students ready for
the next grade level. Additionally 5 of the 10 participants attributed the choice of
lesson to district mandates, indicating that the lesson was part of the pacing plan and
was chosen because it was what they were supposed to teach.
Another important finding that arose from the data pertained to one participant
in particular. Tracy, who was classified as having higher math preference but also
associated more with teaching via more traditional methodologies, exhibited some
teaching tendencies that were also very socio-constructivist in nature. She confessed
that she often felt the need to incorporate both traditional aspects as well as socio-
constructivist aspects in her instruction:

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I think I try to incorporate … like both sides of that. So I actually have a
daughter …she is not very mathematical and … I know from her just
how I help her – she really likes being told systematically how to do
something…. this is the first step and she latches on to that with her
teachers and sometimes she hears things wrong, or when you try to
show her well, you can do it this way – she’s not about that at all. No,
no, no, this is how we , you know, but, yet I feel like the wave in
education right now is really, you know, with the CGI – it’s trying to
swing away from that and getting kids to discover things on their own
and that idea that, oh, well how did you solve it, oh, you use the
different strategy – so I definitely try to incorporate that into my
classroom, but I still kind of have – I have a foot on both sides because I
do feel like for the kids who are not mathematical – they kind of need at
least something to get them on their path, and I feel like sometimes that
free exploration of like – oh, just give them the paper and see what
they’re going to do – personally, I kind of see even from observations in
CGI classes that it’s not that it’s a waste of time, but we don’t have that
luxury of time anymore. So, I get a little bit frustrated with that total
lack of control – it not’s lack of control, but just handing it totally over
to the kids – I see the value of it and I’m kind of growing on that idea,
but I definitely kind of feel that you need to – you still need to present
some strategies to the kids to help those kids who are maybe not as
mathematical to have a starting place to latch onto.
Due to her experience and tenure as a teacher, her experiences with her children, and
her training she has discovered that it is not always easy to choose one teaching practice
over another.
Overall Findings
Thus in analyzing the factors that influence socio-constructivist instruction, the
data from this study indicates that teachers who had a higher preference for
mathematics were more able to appreciate its systematic nature, whereas teachers who
had a lower preference for mathematics were more likely influenced by the negative
and sometimes intimidating experiences in their own schooling. Also, teachers who
were both lower in math preference and who associated with traditional teaching

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methodologies, were less likely to integrate socio-constructivist practices in their
instruction. The rest of the participants, regardless of teaching modality preference and
math preference utilize some important aspects of socio-constructivist teaching.
Specifically, a majority of the teachers (a) utilized hands-on experiences and
manipulatives; (b) integrated culturally relevant pedagogy, and (c) incorporated some
form of cooperative working situation in the delivery of their lesson.
Research Question Three
Research Question Three asked: What impact do Teacher Education programs
have in the effective implementation of socio-constructivist practices? Teacher
Education programs are critical for teachers because they provide teachers with
opportunities, especially in mathematics to view learning differently from when they
were being taught. Furthermore this preparation is critical for teachers as it (a) presents
an opportunity to ensure they have an adequate knowledge base in mathematics, (b)
adequately prepares them with effective pedagogy; (c) provides them with appropriate
pedagogical knowledge that enables them to teach effectively; and (d) addresses the
integration of culturally relevant pedagogy so that learning is maximized for all
students.
Data from participant interviews and observations revealed some influences
from their teacher education program. Four of the ten participants revealed a positive
influence in their teaching of mathematics, while the other six attributed their teaching
to factors other than their teacher education program. The first participant explained

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that her teacher education program had a focus on culturally relevant pedagogy and the
importance of incorporating students’ families into the classroom community. She also
indicated that her Masters project helped her prepare for mathematics instruction
because it focused on hands-on games and activities for students in PK – 6
th
grades. A
second participant indicated that the master’s project she completed, to culminate from
her teacher education program, focused on group work. Thus her teacher education
program helped with incorporating group work in math instruction. Another participant
elaborated on opportunities to create manipulatives as well as create, practice and
critique mock lessons that focused on conceptual understanding. The final participant
indicated that her teacher education program emphasized the importance of addressing
multiple learning modalities as well as the integration of manipulatives throughout the
math methods courses.
Laura, the first participant to reveal a positive influence from her teacher
education program, spoke about the way her program prepared her to teach
Mathematics in ways that would address the needs of diverse students. She received
her teaching credential from a Teacher Education program highly touted for its social
justice agenda. Many of the courses and course requirement are dedicated to
addressing the needs of diverse students. In addition to being bilingual, Laura spoke of
the many ways she incorporates culturally relevant pedagogy in her instruction:
Well, first of all I speak Spanish and so… Always trying to bring in
their environment so that they can tell me, what their experiences are
with the topic. It's usually how I start everything…..I speak a lot with
the families …because I dismiss directly to the parent, I am able every

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day to converse with them. …I think that that has helped me create a
much stronger bond than the teachers who might not have the second
language. … Like, I mean, I feel like I understand the community and
being here for a few years, I understand what the resources are that they
have here. But I often times end up being, you know, kind of like a
counselor myself. …..Okay. I mean, I really do try to be very sensitive
to the community's needs and whatever…they know that they can
always come in and you know, talk to me
She went on to discuss the content of her lessons and the relevance of the content to her
students’ lives and the importance of teaching this concept:
Well, they’ll have practice using money because I've asked them, you
know, have they…have they been able to go to the store and hand the
money in, you know, to the checker or whatever they are doing? But I
feel like the manipulatives that I'm using are very applicable to just the
entire world because, you need to know how to use money [laughter] to
survive…..It will be the foundation for everything that they do in our
capitalist society. You know, they're going to need to learn how to make
money and make change and they are going to need to recognize that,
you know, a nickel is bigger; it's worth less money than a dime, which
doesn’t makes sense. But, so…so yeah, I mean, it would be a real life
application starting, you know, today basically for them. Because if they
can start to make change, then they can help their parents too.
Additionally, the Math Education department at her institution has faculty well known
across the country for math education. As a result of the coursework at her institution
she completed a mathematics project that has helped her with teaching mathematics.
Hence, this particular teacher education program not only prepared this participant with
appropriate culturally relevant pedagogy, but also provided experiences that enhanced
her subject matter knowledge and her confidence in teaching mathematics.
Cooperative working situations are also an important tenet of socio-
constructivism. The second participant to emphasize a positive impact from her teacher
education program was Stephany. She attended a teacher education program that

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required her to complete a master’s project on a pedagogical practice. She chose to do
her project on collaborative groups and differentiated instruction. She claimed to use a
lot of grouping in her lessons because it allowed her to differentiate her instruction:
I use a lot of grouping. There are certain times when I use cooperative
learning. When I did my master's work that was what my study was on
… So I use some of that. And I try to use the direct instruction model,
but then once it comes to the independent practice that's when I
differentiate the process.
Like Laura, Stephany’s experiences and coursework completed during her credentialing
program assisted her in meeting the needs of her students. She related that she
administered pretests prior to each unit of study in an effort to create homogenous
groups. She also followed a similar practice for her language arts instruction. These
flexible groups allowed her to better address the needs of her students.
Taylor, the third participant to acknowledge the positive influence of her
program attended an institution that gave her multiple opportunities to work with and
analyze mock lessons in mathematics. She stated:
My Teacher Education program made a positive impact on my methods
of teaching math. We did a lot of stuff at the college doing mock
lessons that required you make or create your own manipulative, to get
you ready for having no money to purchase them when you became a
real teacher, and teaching one another different math lessons
She also elaborated on the positive experiences she had during her field experience, and
throughout her own educational experiences:
I learned the most when I worked under Master teacher’s during my
actual student teaching. Mostly I remember how I was not good at math
growing up and my teaching program helped remind me there is always
more than one way to teach something.

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Thus her Teacher Education program was instrumental in allowing her to gain valuable
experiences, in creating, implementing, and critiquing lesson plans as well as in the
field experiences which gave her a valuable resource in her supervising teacher
The final participant to argue the benefits of her teacher education program was
Charlene. She spoke about various tenets of socio-constructivism advocated in her
teacher education program. Specifically she felt strongly about considering multiple
learning modalities in the design of mathematics lessons as well as giving students the
necessary tools to succeed. Because she felt her own learning style was neglected
throughout her own schooling, she argued the importance of addressing various
learning styles as well as the need to include hands-on learning and manipulatives in
mathematics lessons. She stated:
I like to be able to make sure that all the kids and all their learning styles
will be able to access it because I know math is hard … its not … It’s
more conceptual than it is something on paper so I like to make sure the
kids can use whatever they need to understand the concept
When asked specifically about the influence of her particular program she replied:
My teacher education program taught me the importance of teaching to
all of the different learning styles. Using manipulatives and differing my
teaching methods to connect with each of the students’ way of learning
information.
Interestingly, Charlene was the only out of this group to associate with traditional
methodology. Thus, despite being inculcated with some important tenets of socio-
constructivism throughout her teacher education program, she still held onto more
traditional tendencies.

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For the other six participants, Teacher Education programs were not as
instrumental. Five of the six participants indicated that ongoing staff development in
the form of trainings and workshops was the most beneficial in terms of how they
taught mathematics, along with memories of how they were taught. Shawna indicated
that most of her strategies were attained at trainings and meetings with colleagues.
My teacher education program had very little influence on the way I
teach math. For the most part my math teaching has been influenced by
strategies that were used with me in my elementary education in
conjunction with ideas that were brought forth in trainings and meetings
with colleagues
Tracy also spoke of strategies she learned in workshops and gave an account of her
mother’s influence.
I gained more knowledge from attending workshops. I really feel that
attending workshops after I started teaching had a much greater impact
i.e. AIMS, Box it Bag It, Marcy Cook, Math Conference in Palm
Springs, etc… Also, my mom was a teacher, and she used a lot of hands
on techniques because I used to help her make them. At night, we
would make the manipulatives for "Math Their Way" i.e. work jobs,
spraying beans 2 colors, cutting jewels, etc.
Similarly, Dana spoke of her experiences and elaborated on how many of the changes
in education have influenced her teaching:
I feel the workshops I have taken on math have helped me the most. I
have learned the basic techniques, and concepts and the standard
methods to instruct these concepts.... but to then tweak it my own way to
assist my students. This, by the way, changes lessons to lessons, year to
year. Nothing, in my opinion, in education is static...Education needs to
change continuously to assist the needs of the children involved.
Therefore, a majority of the participants in this study felt that the training they received
during their teacher education program was not as beneficial as experiences they had in

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their own educational histories, ongoing learning, and in trainings they attended for
staff development.
Teacher Education programs are critical in providing teachers with the
necessary experiences that will be utilized in classrooms to maximize learning for all
students. However it appears from data that they have not been as beneficial as
expected. One teacher in particular, did not even recall any math preparation:
I don't think my teacher education program had much effect on the way
that I teach math. I don't recall being shown how to teach math
specifically.
Although this seems discouraging at first glance, upon further analysis, the four
participants with positive experiences were also the four least experienced teachers or
those that last attended a teacher education program. Thus it is a possibility that teacher
education programs in the last decade may be doing a more adequate job of preparing
teachers than in previous years.

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CHAPTER FIVE: DISCUSSION
Current reform efforts advocate the use of socio-constructivist practices by
teachers at all levels of teaching. Research also indicates that teachers need to be
knowledgeable about the subject matter they teach, the teaching practices they utilize
and the pedagogical practices that are necessary for effective mathematics instruction
(Hill, Rowan & Ball, 2005). As such, it is critical for teachers to have this knowledge,
and even more important that they are able to use it whenever necessary (Hill, Schilling
& Ball, 2004). It is imperative to determine the extent to which teachers make use of
this knowledge. It is even more critical to uncover the factors that facilitate this type of
instruction, as well as the role of Teacher Education Programs in ensuring this
transmission of information. Thus the purpose of this study was to analyze those factors
that facilitate this type of instruction, as well as analyzing the role of Teacher Education
Programs. Specifically it was imperative to address the following three research
questions:
 What socio-constructivist practices do teachers incorporate in their teaching
of mathematics?
 What factors influence teachers’ use of socio-constructivist practices when
teaching mathematics?
 What impact do Teacher Education programs have in the effective
implementation of constructivist practices?

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These three questions are crucial for the literature. By determining the extent of
socio-constructivist practices found in the classroom and then by identifying those
factors that are associated with their inclusion in the classroom, teacher educators can
identify ways to incorporate them so that preservice teachers will benefit from them in
the future.
Findings
The results from this study are promising in regards to practices being utilized
and the role of Teacher Education programs. Four major themes emerged from the data
which revealed that some of the current teaching practices are aligned to reform-based
practices. The first theme involves the use of socio-constructivist practices being
utilized: teachers consistently used many of the same practices, and neglected to use
many of the same ones. The use of culturally relevant pedagogy constitutes the next
theme. The third theme was that some of the teachers from this study also attributed
their reform-based practices to their teacher education programs, which addresses the
importance of preparation received in these programs. The final theme regards one
teacher in particular who felt she was more of a traditional teacher. However, upon
further analysis it was discovered that she had many socio-constructivist practices in
her pedagogical repertoire.

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Theme 1: Participants seemed to agree on practices to incorporate and
neglect.
Participants from this study created lessons which focused on student
understanding and built on students’ prior knowledge; readily utilized manipulatives in
their lessons; and integrated group work into the lesson design. This indicates that
three of the major tenets of socio-constructivism were integrated as critical components
in instruction regardless of theoretical orientation or math preparation. Thus, it did not
matter whether participants were high or low in mathematics, or whether they identified
with socio-constructivist or traditional methodology, most of the participants in this
study were likely to incorporate prior knowledge, manipulatives, and group work in
their lesson design to promote student understanding.
According to constructivist tenets, building instruction on prior knowledge is
imperative for student understanding. Participants in this study had students draw from
past experiences and lessons as the foundation for new knowledge and lesson content.
Research by Carpenter and Lehrer (1999) posit that conceptual understanding comes
about when students are able to relate new information to prior knowledge. Further,
they contend that when students acquire understanding, they are able to apply existing
knowledge to new ideas. (Carpenter & Lehrer, 1999). Thus, the inclusion of this
important tenet of socio-constructivist methodology ensures students’ conceptual
understanding.

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The use of manipulatives and group work was also evident in every lesson I
observed as part of this study. Although their inclusion was obvious, it is unclear if
over time these tenets of socio-constructivism are integrated in a superficial manner.
According to Windschilt (2002) many teachers adopt some of the more easily
assimilated practices into their pedagogical repertoire without requiring students to
restructure their own learning, a critical component of socio-constructivist learning.
Similar research has found that teachers tend to use socio-constructivist practices and
equate their use with student learning (Prawat 1992; Hargreaves, 1994). Although
findings from this study indicate this is not the case, further analyses would be
necessary to determine the extent and use of these socio-constructivist practices over
time.
Although as a group, these participants were able to use prior knowledge, group
work and manipulatives, other critical tenets of socio-constructivism were missing.
These participants were unwilling to give students the responsibility of their own
learning and adjust lessons for student needs. Both of these are critical for student
understanding. These last two reform-based practices are much more difficult to
integrate into mathematics lessons. Whether teachers feel constrained by the maturity
levels of their students, time constraints, administrative constraints, or district
mandates, these teachers were unable to integrate these other critical elements of socio-
constructivist theory.

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Participants were less likely to give students opportunities to become self-
directed learners. They were also unlikely to support and monitor student autonomy
and choice during their learning experiences. Part of this was due to grade level
assignment with half of the Kindergarten teachers revealing that it was hard to give so
much responsibility to such young students. Furthermore these same participants felt it
inappropriate to make such young students partners in their own learning because they
felt it would be too difficult with such immature students.
Participants also disagreed on the benefits of adjusting lessons for student needs.
Although the consideration of student needs is one of the central features of socio-
constructivist teaching, participants from this study were unable to come to an
agreement on the benefits of incorporating students’ physical, social and emotional
development in lesson. Nor could they agree on the benefits of allowing students to
learn at their own pace, or make the lessons relevant and accessible to each student.
Upon further probing many of the participants indicated that this was due to constraints
placed on them by district mandates. When asked about this, Martha contributed:
I kind of extend a lesson with manipulatives and all that, it would be
much better for the kids, but it doesn’t – it usually doesn’t work with
what they expect us to do. So that’s probably my issue with teaching
math – if you want that information. They never – I don’t think they
consider how much time it takes to teach a concept. And they should be
practicing each concept, and then review the concept so, with the
standards we’re suppose to teach and then to get to the testing part – it’s
way too much for – especially our students when they have language
issues too, but I would say for any student – I wouldn’t teach this way
by choice. But it’s what they expect from us.

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Similarly Brianna insisted that having been deprived the use of manipulatives during
her own educational experiences, has encouraged her to use manipulatives to reach all
her students. However, with district mandates, this becomes very difficult:
if I do have the manipulatives for them to use, I do like to use
manipulatives where needed. And I don’t feel like, well in comparison, I
don’t feel like I did a lot of that when I was younger. So if we do I do
like to use them…I think sometimes we are so concerned about staying
on our pacing plan that we forget to sometimes make it more fun, or skip
some of the more fun activities. And just, you know, we’re obviously
trying to get them to learn our essential power standards and sometimes,
you know, when we’re on a quick, fast pace we’ve got to just keep
going and you know, it’s hard to get everybody
Thus pacing plans and district mandates are very influential in disallowing teachers to
address the needs of diverse learners.
Theme 2: Only some participants utilized Culturally Relevant Pedagogy
The use of culturally relevant pedagogy also surfaced as a theme in the data. In
the last couple of decades, the use of culturally relevant pedagogy has been deemed as
necessary for the success of all students. Brown-Jeffy and Cooper (2011) provide a
conceptual framework for culturally relevant pedagogy that synthesizes the works of
Gay (1994, 2000), Ladson-Billings (1994), and Nieto (1999). They present 5 themes
that encompass the major concepts of culturally relevant pedagogy as discussed in the
literature: (1) Identity and Achievement; (2) Student Teacher Relationships (3) Equity
and Excellence; (4) Developmental Appropriateness; and (5) Teaching Whole Child.
The latter three themes were validated by data from this study.
As part of the equity and excellence theme, all students should benefit from
equity and excellence, including minority students. Brown-Jeffy and Cooper (2011)

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assert that to meet student needs, they need to have equal access to the curriculum,
which might come in the form of differentiated instruction. According to Gay (1994)
when teachers fail to see the diversity in students, they do not see the students; which
then disallows them from meeting students’ social and educational needs. Most study
participants differentiated their lessons to address different learning styles and
modalities. Additionally, many of the participants also grouped students homogenously
so that they could address each group’s needs. Thus there is some indication that
participants from this study are addressing the equity needs of their students.
The theme of developmental appropriateness suggests that it is critical to know
and understand where children are in their cognitive development. Ladson-Billing
asserts that teachers need to acknowledge, explore, and utilize the knowledge that
students bring with them to school. Brown-Jeffy and Cooper (2011) contend that
teachers need to understand that the psychological needs of their students can differ just
as they can have different motivations to learn. Teachers must modify their teaching
styles to address the differences in culturally-based learning styles and preferences of
their students. Teachers in this study not only modified their lessons and targeted the
needs of students by homogenously grouping students, but they also integrated family
life into the classroom so they could better understand their students’ needs. Thus,
findings from this study indicate that some teachers are addressing the developmental
needs of their students, and thus teaching in a culturally relevant manner.

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The last theme, teaching the whole child, refers to teachers remembering the
needs of the whole child, and not just the academic ones in the classroom. Brown-Jeffy
and Cooper (2011) contend that teachers need to incorporate skill development in a
cultural context; engage in a home-school-community collaboration; present learning
outcomes; provide a supportive learning community, and encourage student
empowerment. Besides, teachers need to acknowledge that students come to school
with culturally-based ways of seeing, doing, and knowing; therefore, teachers need to
use these to as foundation for learning (Brown-Jeffy & Cooper, 2011). A minority of
teachers in this study addressed this theme. One teacher in particular spoke at length
about the need to integrate these types of learning experiences. Laura, spoke about the
integration of students’ families and community members into the classroom
environment. Although she does not believe she shares the same cultural background
as her students, her ability to communicate with parents in both Spanish and English
has bridged the barrier she would have otherwise felt between herself and her students’
families.
The data from this study suggest some teachers are utilizing culturally relevant
pedagogy to address the needs of their students; however, those that provided the most
examples of culturally relevant pedagogy were the younger teachers. Laura, a teacher
who addressed most of these themes, was also one of the least experienced teachers in
the group. On further probing, it was revealed that Laura’s teaching credential was
earned at an institution that prides itself as having social justice as major part of its

113

teacher education programs. Thus, teacher education programs provide an important
venue for inculcating teachers with the necessary skills and strategies necessary for
effective instruction in a culturally relevant manner.
Theme 3: Only some participants claimed their teacher education
programs influenced them.
Teacher Education programs have an influential role in the effective delivery of
reform-based practices. Of the four participants to reveal an influence from their
Teacher Education program, half of them reported the influence on their pedagogical
practices. These participants then, felt that their Teacher Education program was
instrumental in preparing them with the necessary pedagogical practices to teach
mathematics effectively. Another teacher referred to the preparation she received in
analyzing lessons and mathematics content, which is more indicative of preparation in
pedagogical content knowledge. Another teacher spoke of her preparation to teach in a
culturally relevant manner. Data from this study then, suggests that participants who
recently attended a teacher education program were more likely to utilize reform-based
practices, like culturally relevant pedagogy and socio-constructivist practices.
Theme 4: Tracy, a socio-constructivist in disguise
Tracy, one of the teachers who associated with more traditional methodologies
also used many of the socio-constructivist practices advocated by current mathematics
reform and the National Council of Teachers in Mathematics. History has demonstrated
that for a long time teaching mathematics was either traditional or constructivist. Not

114

until the last few decades has reform focused on strategies from both ends of the
educational pendulum. Tracy then, is doing what she can to maximize learning in her
students. As one of the most experienced teacher in the group, she has seen the
advantages and disadvantages of different methodologies throughout the years and thus
has been able to draw from the most effective strategies. Tracy has also benefited from
her experience as a mother of two. Through her child rearing experience, she learned
that two very similar students, her children, with similar experiences can learn in very
different ways. This has strengthened her resolve to use various methodologies because
she feels that utilizing a broad spectrum of strategies from both methodologies is the
most effective way to address students’ needs. Data from this study then, indicates that
this may be one of the most effective ways to address the needs of all students.

115

Implications for Practice
Four implications were identified for the preparation of teachers. The first of
these is that Teacher Education programs need to address the subject matter knowledge
needs of pre-service teachers. Additionally Teacher Education programs need to ensure
that pre-service teachers are benefitting from the latest research-based strategies as well
as the other proven strategies. Teacher Education programs also need to prepare
teaching candidates with effective pedagogy that can be utilized with all students
including those that are culturally diverse. Lastly pre-service teachers need to have
experiences which allow them to teach mathematical concepts which encompass a
variety of skills so they are not pressured by district pacing plans that restrict lessons by
focusing on single skills.
Teacher Educators have the responsibility of preparing teachers to meet the
demanding needs of an evolving society. Specifically in preparing students with
adequate mathematics knowledge, teachers need to have adequate math knowledge to
teach students. Teacher Education programs can ensure preservice teachers have this
knowledge prior to admittance into the credential program, as is often the case in
Teacher Education programs. This knowledge can also be fostered by effective
mentoring relationships or it can be cultivated in Teacher Education programs, as
preservice teachers learn to become effective Mathematics instructors.
Findings from this study suggest that the most socio-constructivist practices
were found in classrooms where teachers either had high math preparation/preference

116

or had a predisposition for socio-constructivist teaching style. In the case of teachers
who had a high math preparation/preference, the participants had sufficient subject
matter knowledge. The teachers who were predisposed toward socio-constructivist
practices had either attended a variety of trainings, or had beneficial relationships with
teacher mentors that passed on effective teaching strategies. The common denominator
among these successful teachers was that their subject matter knowledge assisted their
teaching practices. Therefore teachers benefit from Teacher Education programs that
address their subject matter knowledge needs. Whether subject matter is addressed
prior to entry into Teacher Education programs or whether it is addressed throughout
the program, it is a critical component for effective teaching.
Currently research based practices are focusing on socio-constructivist
practices, without neglecting the importance of basic skills advocated by traditional
educators. Teacher Education programs should not be limited to presenting their
teacher candidates with one teaching style, at the expense of excluding others. As
evidenced in Tracy’s testimony, teachers often feel like educational fads advocate a
particular strategy at the expense of other research-based and often successful
strategies. Tracy claimed that having her feet on both sides allowed her to incorporate
innovative research-based socio-constructivist strategies, while still utilizing effective
but yet traditional practices when students needed them for understanding.
Therefore teacher education programs need to ensure that the latest research-
based strategies are implemented when preparing teachers, without negating other

117

proven strategies as well. Fads in education have swung teachers back and forth on a
metaphorical pendulum that has vacillated according to the latest big thing in education.
Teacher Education programs need to be cognizant of these extremes that rarely satisfy
the needs of all students. Rather than forcing preservice teachers to either extreme,
teacher education programs should present the best of both worlds so that preservice
teachers can make use of the strategies that best meet the needs of their students.
Culturally relevant pedagogy is another area that can be addressed at the
Teacher Education level. As evidenced by the data in this study, the younger
participants who also attended Teacher Education programs more recently were more
likely to incorporate culturally relevant pedagogy in their daily instruction. Moreover,
Laura’s interview data indicated she had a strong background in culturally relevant
pedagogy due to her own teacher education experience. Her attendance at an institution
touted for their social justice agenda, adequately prepared her for the needs of the
diverse population she now encounters in her classroom.
This has lasting implications for Teacher Education programs. Evidently,
teachers are incorporating some facets of culturally relevant pedagogy in their
instruction, and some teachers make them the focus of their entire instructional
sequence. Thus Teacher Education programs need to ensure they are adequately
preparing their teaching candidate with effective pedagogy that can be utilized with the
culturally diverse student populations they will undoubtedly encounter in their
classrooms.

118

Lastly, Teacher Education programs must address district mandates in the
preparation of teachers. Most participants in this study indicated that district mandates,
and particularly pacing plans, served as a deterrent of effective instruction. This was
particularly true when it came to addressing the needs of diverse students. Some
participants felt that time constraints did not allow them to do so because they were
expected to teach a lesson a day, with little consideration for the diverse needs of
students.
Teacher Education programs then, need to ensure preservice teachers are given
opportunities to focus on teaching concepts rather than limiting their instruction to
teaching a skill per day, with the scope and sequence dictated by pacing plans. By
preparing teachers to teach mathematical concepts which encompass a variety of skills,
preservice teachers can then focus their instruction on a variety of skills. This will
enable them to not be dependant on pacing plans that delineate the lessons that need to
be taught every day of the week.
Implications for Research
The analysis of data illuminated three areas for further research. A possibility
for further research could focus on the tenets of socio-constructivism that were not as
prevalent, and identifying how they could be more easily integrated into mathematics
lessons. Another area for future research could focus on implementing culturally
relevant pedagogy at the in-service level. Future research could also address the use of
socio-constructivist practices over time.

119

Results from this study revealed that many of the major tenets of socio-
constructivism were being incorporated in these classrooms. However the least
evidenced tenet focused on the use of culturally relevant pedagogy, adjusting lessons
for student needs, and making students responsible for their own learning, as well as
making them partners in their own learning. Future research could identify teachers
who implement these important elements of socio-constructivist teaching, and isolate
those factors that facilitate this type of instruction in these teachers. These tenets are
also critical components of effective instruction, and data from this study indicated that
they were utilized in only a few classrooms. Thus it would be critical to uncover the
factors that allow and encourage teachers to utilize these practices and to keep utilizing
them over the course of time.
Other research opportunities could focus on implementing culturally relevant
pedagogy at the in-service level. Evidence from this study indicated that veteran
teachers who completed their teacher credential programs long ago, were less likely to
incorporate culturally relevant pedagogy. Thus further research could focus on the
effectiveness of in-service opportunities for inculcating veteran teachers with culturally
relevant pedagogy.
Lastly future research opportunities could include multiple observations of
lessons. The data from this study was limited to a single observation of a single lesson.
This snapshot view of each participant can be limited by the circumstances leading to
each lesson, whereas multiple observations would allow the researcher to gain a better

120

understanding of the teaching practices of each participant. Thus future research could
focus on whether these socio-constructivist practices are utilized over time, and on
identifying those factors that facilitate the use of these practices over time.
Conclusion
The latest research on teaching and learning mathematics indicate that in order
to maximize learning, students need to benefit from opportunities to learn in socio-
constructivist ways. This means learning opportunities need to be sequenced in ways
that allow students to learn collaboratively with manipulatives, while ensuring that new
experiences build on prior knowledge. Teachers need to address the needs of all
students. They also need to make students responsible for their own learning, while
also making them partners in their own learning. Lastly, they need to ensure that
culturally relevant pedagogy is incorporated in all lessons so that the needs of all
students are met.
For students to learn in this manner and for teachers to teach in this way, teacher
education programs need to do an adequate job of preparing teachers. This study
sought to identify the factors that allow teachers to teach in this manner. By first
identifying the socio-constructivist practices prevalent in classrooms, and then isolating
the factors that were associated with each of these factors, this study attempted to
uncover the factors that can be addressed at the Teacher Education level.

121

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134

APPENDICES
APPENDIX A: DEMOGRAPHIC QUESTIONNAIRE
Mathematics Instruction Survey -- Demographics Section
Yrs. of teaching experience _____ Age ______ Gender Male Female
Ethnic Background (Choose all that apply)
African-American or Pacific Island Indian/East-Indian
Black American
Filipino Latino or Hispanic Armenian
Asian/Asian-American Native Other
American/Alaskan _______________
Southeast Asian/ Native
Southeast Asian-
American White (Non-Hispanic)
Work Background (Please indicate your prior work experience)
Teaching at a public school was my first full-time job.
Transferred from a job in the Business world, please specify _________________
Transferred from a job in Science/Technology, please specify _________________
Transferred from a job in another Field, if so please specify____________________
Educational Background (Please list all degrees earned)
______________________________________________________________________
Name of institution where you earned your teaching credential ________________
Mathematical Background (Please list all mathematics courses taken after 8
th
grade - i.e.
High School Pre-Calculus, College Calculus)
______________________________________________________________________
Preferred Subject rankings (Please rank 1-8 in order of interest preference)
Reading Mathematics Physical Education
Writing Social Studies/History Foreign Language
Science Visual/Performing Arts Other_____________
____________________________________________________________________________

135

APPENDIX B: VIGNETTES

Mathematics Instruction Survey – Two Case Studies
The following paragraphs describe the manner in which two teachers conduct
their classes, Ms. Hill and Mr. Jones. Answer each question below by
checking the box under the column that best answers that question.

Ms. Hill leads his class in an
animated way, asking questions that
the students can quickly answer;
based on the lesson they have done
the day before. After this review,
Ms. Hill engages the students with
the new class material, again using
simple questions to keep students
attentive and listening to what is
being discussed
Mr. Jones’ class is also having a
class discussion. But the questions
come from the students themselves.
Although Mr. Jones can clarify
student questions and suggest where
students can find relevant
information, he can’t really answer
most of the questions himself.

Definitely
Ms. Hill
Tend toward
Ms. Hill
Tend toward
Mr. Jones
Definitely Mr.
Jones
Which type of class
discussion are you
more comfortable
having in class?

Which type of
discussion do you think
most students prefer to
have?

From which type of
discussion do you think
students gain more
knowledge?

From which type of
discussion do you think
students gain more
useful skills?

136
APPENDIX C: SECONDARY QUESTIONNAIRE
Mathematics Instruction Questionnaire
Strongly
Disagree
Moderately
Disagree
Moderately
Agree
Strongly Agree
1. It is important for students to
see the connection between
what they already know and
new material.

2. Students need to connect new
information and concepts to
their own life experiences.

3. Teachers know much more
than students; they shouldn’t
let students struggle with
concepts when they can just
explain the concepts directly.

4. Building on students
comments and questions
during a lesson helps extend
their understanding.

5. Providing students with a
variety of learning
experiences helps
accommodate the various
learning styles.

6. A quiet classroom is
generally needed for
effective learning.

7. Students are not ready for
“meaningful” learning until
they have acquired basic
reading and math skills.

8. It is better when the teacher,
not the students, decides
what activities are to be
done.

137
9. Using a variety of strategies
to introduce, explain, and
restate subject matter
concepts helps all students
understand.

10. Teachers need to provide a
variety of grouping
structures for students to
engage in collaborative
learning.

11. It is important to support and
monitor student autonomy
and choice during learning
experiences.

12. Teachers need to provide
opportunities for all students
to think, discuss, interact,
reflect, and evaluate
information content.

13. Students should engage in
critical thinking and problem
solving activities and
encourage multiple
approaches and solutions.

14. Students should be
motivated to initiate their
own learning and to strive
for challenging learning
goals.

15. Students should develop and
use strategies for evaluating,
monitoring and reflecting on
their own learning.

16. Student projects often result
in student learning all sorts
of wrong “knowledge”.

138
17. Teachers should help
students accept different
experiences, ideas,
backgrounds, feelings, and
points of view, and create
opportunities for them to
communicate and work with
one another.

18. Homework is a good setting
for having students answer
questions posed in their
textbooks.

19. Students need opportunities
to become self-directed
learners.

20. Teachers should facilitate
student participation in
classroom decision-making.

21. Students need to become
involved in the development
of classroom procedures and
routines.

22. Teachers should provide
time for students to reflect
on their learning and process
of instruction.

23. Students will take more
initiative to learn when they
feel free to move around the
room during class.

24. Students should help
establish criteria on which
their work will be assessed.

25. Teachers should ensure that
their knowledge of subject
matter incorporates different
perspectives.

139
26. It is important for students to
relate subject matter
concepts to previous lessons
and their own lives and to
see the relationships across
subject matter.

27. Teachers should build on
student life experiences,
prior knowledge, and
interests to make the content
relevant and meaningful.

28. Teachers should select and
use learning materials that
reflect the diversity of
students in the classroom.

29. It is important to incorporate
students’ knowledge, as well
as knowledge about their
lives, families, and
communities, in the
planning of curriculum and
instruction.

30. Teachers should incorporate
knowledge of students’
physical, social, and
emotional development in
lesson planning, and design
lessons that challenge
students at their own
development levels.

140
APPENDIX D: PRE-OBSERVATION QUESTIONNAIRE
Pre Observation Discussion
I. Math Background
a. Math Autobiography (Describe yourself as a math educator.)
b. Past Experiences (Does anything stand out in your own educational
history?)
II. Math Lesson
a. Tell me about the lesson you will be teaching. (standards, objectives)
b. What goals do you have for the lesson?
c. What are some of the expectations you have of your students after this
lesson?
d. What are some of the things that you need to do to prepare for this
lesson?
e. What

141
APPENDIX E: LESSON OBSERVATION

PARTICIPANT # ______________________________
Observation Notes
I. Lesson components

II. Focus throughout teaching

III. Strategies used

IV. Student work

142
APPENDIX F: POST-OBSERVATION QUESTIONNAIRE
Post-Observation Discussion
Based on lesson observation, some possible discussion starters:
I. Participant reflection

a. What went well?

b. What might have improved the lesson?

c. Could you elaborate a little on why you …

II. Consistency of statements in pre-observation meeting and actual
observation.

a. I noticed that during the lesson you …

b. I noticed that in the planning you said you would ….

c. Could you elaborate a little on why you …

Abstract (if available)

Abstract The purpose of this study was to analyze the teaching practices of teachers to determine if they incorporated socio-constructivist practices in their teaching repertoire. This study analyzed the use of socio-constructivist practices utilized by a group of ten teachers, as well as the factors that facilitated their use. The research questions of this study focused on which socio-constructivist practices were utilized

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Asset Metadata

Creator Arredondo, Anna Lillia (author)

Core Title Teacher's use of socio-constructivist practices to facilitate mathematics learning

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Publication Date 11/22/2011

Defense Date 10/27/2011

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

Tag OAI-PMH Harvest,socio-constructivist,Teacher education,teaching mathematics

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Baca, Reynaldo R. (committee chair), Fischer, Linda A. (committee member), Hasan, Angela Laila (committee member)

Creator Email anna.arredondo@gmail.com,annalila@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-193716

Unique identifier UC11291570

Identifier usctheses-c3-193716 (legacy record id)

Legacy Identifier etd-ArredondoA-431-0.pdf

Dmrecord 193716

Document Type Dissertation

Rights Arredondo, Anna Lillia

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 a...

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