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INFLUENTIAL FACTORS: A MIXED METHOD STUDY 5

presented by

JI-WON SON

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in CURRICULUM, TEACHING
AND EDUCATIONAL POLICY

Mabr Professor’s Signature
”car (a [my
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MSU is an affirmative—action, equal-opportunity employer

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MAY BE RECALLED with earlier due date i

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ELEMENTARY TEACHERS’ MATHEMATICS TEXTBOOK USE IN TERMS OF
COGNITIVE DEMANDS AND INFLUENTIAL FACTORS:
A MIXED METHOD STUDY

By

J i-Won Son

A DISSERTATION

Submitted to
Michigan State University
In Partial Fulfillment of the Requirement
For the degree of

DOCTOR OF PHILOSOPHY

Curriculum, Teaching, and Educational Policy

2008

ABSTRACT

ELEMENTARY TEACHERS’ MATHEMATICS TEXTBOOK USE IN TERMS OF
COGNITIVE DEMANDS AND IN FLUENTIAL FACTORS:
A MIXED METHOD STUDY

By
Ji-Won Son

The purpose of this study is to examine teachers’ textbook use and its influential
factors from a different angle, that of the cognitive demand of mathematical problems
and teacher questions, and to provide both depth and breadth of analysis on these topics.
A survey was used for the quantitative aspect of the research, along with teachers’ lesson
plan modifications, observations, and interviews for the qualitative part. Among 169
teachers who participated in the survey, eight teachers were observed and interviewed in
order to establish the validity of the survey and to provide additional details.

Textbook use patterns were explored by looking at the relationship between
problems and questions in textbooks and those used by teachers in teaching. Influential
factors were explored in three levels--individual (e.g., teacher knowledge), contextual
(e. g., type of textbook), and teachers’ opportunity-to-learn (e.g., Professional
development [PD] experiences).

Three noticeable findings emerged. First, there exists relatively a simple
relationship between textbook problems and problems used by teachers in teaching.

When teachers use problems, they at least maintain the cognitive demand of textbook

problems in teaching, suggesting that the cognitive demand of textbooks play an
important role in deciding the cognitive demand of problems used by teachers in teaching.

Second, a complicated relationship exists between teacher questions in textbooks
and those in teaching. Although textbooks provide higher cognitive demand teacher
questions, some teachers decrease these levels in their teaching. Even though teachers use
the higher level text problems in teaching, they decrease its level by using lower level
teacher questions. This is because the enactment of mathematical problems is a multi-
dimensional practice intertwined with teachers’ goals, intentions, questions, and
interactions of teachers and students.

Furthermore, the critical factor that influences teachers who decrease higher level
text to lower level cognitive demand or those who increase a lower level text into higher
level cognitive demand is alignment between their teaching goals and those of the
textbooks they use. This study also found other important influential factors, such as type
of textbook (standards-based curriculum), teacher knowledge, teacher perceptions about
student achievement, and PD observation experience. The findings of this study provide
curriculum developers, professional developers, and teacher educators with information
about ways in which they can support teachers in order to help teachers make better use

of their texts.

Copyright by
J i-Won Son
2008

This dissertation is dedicated to my parents,
Joo Yeoul Son and Ae Soon Kim,
For their unconditional love and support,

which has given me the confidence to face the world.

 

 

ACKNOWLEDGMENTS

In looking back at my dissertation study, many memories from the past several
years strike me. In the summer of 2002, I arrived at Michigan State University to study
mathematics teacher education. Around that time, it did not seem to me that it could be
feasible to expect to ever finish a dissertation due to at least several difficulties I had to
face: language, culture, my own na'ive ideas and ability to explore the field I had chosen,
etc. Nevertheless, I have finally arrived here after getting through all of the hardships l
was willing to bear. Thinking back on my time in graduate school at Michigan State, the
biggest thing left in my mind is that I actually experienced joy, happiness, and excitement
at having this unique chance to seriously study education.

I have always realized that I am very lucky not only to have this great opportunity,
but also to get enormous support and warm friendship from others. I would like to
specially thank them all. My co-advisors and dissertation directors, Drs. Sharon Senk and
Sandra Crespo, are my wonderful teachers, mentors, and professors, who guided me to
the current place. They have profoundly shaped the research and writing of this study.
They read numerous drafts, found time to meet with me, and provided insightful

feedback. They offered endless encouragement through my thinking and writing process,

vi

providing the right balance of challenge and support. They are the best teachers, mentors,
and role models for me. I am forever grateful to them.

I am also most appreciative of my dissertation committee, Drs. Helen
Featherstone, Mary Kennedy, and Robert Floden, for their encouragement, guidance,
expertise, and time. Each offered insight and advice relating to his or her areas of
specialization during my doctoral journey. I especially thank Helen F eatherstone for her
warm and consistent support and encouragement to continue my studies.

Thank you also to Drs. Sharon Senk and Michael Battista for my research
assistantship in the project, Teachers for a New Era (TNE) over three years, which
provided me the opportunity to develop my knowledge and skills. I would like to thank
you again to Drs. Sharon Senk and Sandra Crespo for encouraging me to teach several
courses at MSU, which gave me the opportunity to understand American education better.
My experience in teaching and research at MSU has been invaluable in providing me
with a solid foundation for my professional career.

There are also many other people my debts to whom I like to acknowledge. They
are Patricia Edward, Mary Bouck, Jack Smith, Jack Schwille, Lynn Paine, Lynn F endler,
Sandy Willcox, and Nathalie Sinclair. My friends were also always there with me when I

had to solve a variety of problems. They are Marcy Wood, Joy Oslund, Jill Newton,

vii

Khalel Hakim, Sharon Strickland, Ramona Fruja, Irini Papaieronymou, Young Yee Kim,

Dong-Joong Kim, J inYoung Choi, RaeYoung Kim, Nicole Johnson, Jung Eun Kim and

Ga Young Ahn. I specially thank Michael Sherry, Jill Newton, Marcy Wood, Ramona

Fruja, and Irinin Papaieronymou for carefully reading the manuscript of this study and

giving me constructive comments. I appreciate the elementary teachers who gladly

participated in this study and giving me wonderful chances to learn about their

mathematics teaching in relation to their textbook use.

Finally, I deeply appreciate my family’s endless support. I met my husband,

J inWoo Park here. He has always stood by me during my doctoral journey. He was

always with me, when I worked and studied day and night, as my best friend, colleague,

and a thoughtful scholar. In particular, I express my deepest love and acknowledgements

to my parents, J 00 Yeoul Son and Ae Soon Kim. Words could never adequately express

how grateful I am for their unconditional love and support from Day One. This

dissertation is dedicated to them.

viii

 

TABLE OF CONTENTS

LIST OF TABLES .................................................................. xiii
LIST OF FIGURES ................................................................. xvi
CHAPTER 1
INTRODUCTION .......................................................................................... 1
Problem Statement .......................................................................................................... 3
Purpose of the Study ....................................................................................................... 8
Research Questions ......................................................................................................... 9
Definition of the Terms ................................................................................................. 10
Significance of the Study ............................................................................................... 11
Overview of the Dissertation ........................................................................................ 13
CHAPTER 2
LITERATURE REVIEW ............................................................................. 15
Research on Teacher’s Textbook Use and Influential Factors ...................................... 16
Problems of the Current Research Literature on Textbook Use ................................... 25
Conceptual Framework ................................................................................................. 3O
Importance of Cognitive Demands of Problems and Questions ................................ 3O
Framework for Cognitive Demands of Problems ...................................................... 34
Framework for Cognitive Demands of Teachers Questions ...................................... 37
Hypotheses of Transformation Patterns and Influential Factors ................................... 41
Hypothesized Transformation Patterns ...................................................................... 41
Hypothesized Influential Factors ............................................................................... 45
CHAPTER 3
RESEARCH DESIGN AND METHODS .................................................... 48
Research Design Overview ........................................................................................... 48
Quantitative Research Methods .................................................................................... 51

ix

Survey Development and Validation ......................................................................... 51

The Revised Survey ................................................................................................... 54
Survey Participants .................................................................................................... 55
Survey Data Analysis ................................................................................................ 58
Qualitative Research Method ........................................................................................ 59
Case Study Participants ............................................................................................. 59
Case Study Data Collection Procedures .................................................................... 60
Case Study Instruments ............................................................................................. 60
Case Study Analysis .................................................................................................. 63

CHAPTER 4

RESULTS FROM THE SURVEY:

TRANSFORMATION PATTERNS OF COGNITIVE DEMANDS .......................... 68
Preliminary Analysis ..................................................................................................... 69
Transformation Patterns in Terms of Cognitive Demands ........................................... 82

Transformation Patterns in Terms of Cognitive Demand of Problems ..................... 82
Transformation Patterns in Terms of Cognitive Demand of Questions ..................... 88
Transformation Patterns between Problems and Questions in Teaching ................... 93
Validation from Lesson Plan Modification ................................................................... 97
Summary ..................................................................................................................... 109

CHAPTER 5

RESULTS FROM THE SURVEY:

INFLUENTIAL FACTORS ON TRANSFORMATION PATTERNS IN TERMS OF

COGNITIVE DEMANDS ................................................................................ 1 13
Preliminary Analysis of Dependent and Independent Variables ................................. 114
Multinomial Logistic Regression ................................................................................ 126
Factors Influencing Textbook Transformation Patterns in Terms of Cognitive Demands
of Problems ................................................................................................................. 13 1

Factors Distinguishing L-L Pattern from H-H Pattern ............................................ 137
Factors Distinguishing L-H Pattern from H-H Pattern ............................................ 141
Factors Distinguishing L-H Pattern from L-L Pattern ............................................. 143

Factors Influencing Textbook Transformation Patterns in Terms of Cognitive Demands

of Teacher Questions .................................................................................................. 145
Factors Distinguishing L-L Pattern from H-H Pattern ............................................ 149
Factors Distinguishing L-H Pattern from H-H Pattern ............................................ 151
Factors Distinguishing L-H Pattern from L-L Pattern ............................................ 155

Factors Influencing Transition Patterns between Fraction Problems in Teaching and

Teacher Questions about Fractions in Teaching ......................................................... 156
Factors Distinguishing L-L Pattern from H-H Pattern ............................................ 160
Factors Distinguishing H-L Pattern from H-H Pattern ............................................ 162
Factors Distinguishing H-L and L-H from L-L Pattern ........................................... 162

Summary ..................................................................................................................... 165

CHAPTER 6

RESULTS FROM THE CASE STUDIES:
TEACHERS’ TEXTBOOK TRANSFORMATION PATTERNS AND INFLUENTIAL

FACTORS ..................................................................................................... 170
Methods Used to Select and Analyze Teachers for the Case Study ............................ 171
Higher Level Textbook Problems and Questions to Higher Level Problems and
Questions in Teaching: Pattern l ................................................................................ 178

A Vignette of Brad ................................................................................................... 179
Similarities and Differences between the Textbook Lesson and the Lesson Taught in
Terms of Cognitive Demands .................................................................................. 186
Influential Factors on the Textbook Transformation Pattern .................................. 191
Higher Level Textbook Problems and Questions to Higher Level Problems but Lower
Level Questions in Teaching: Pattern 2 ...................................................................... 195
A Vignette of Karen ................................................................................................ 195
Similarities and Differences between the Textbook Lesson and the Lesson Taught in
Terms of Cognitive Demands .................................................................................. 201
Influential Factors on the Textbook Transformation Pattern ................................... 204
Lower Level Textbook Problems and Questions to Lower Level Problems and
Questions in Teaching: Pattern 3 ................................................................................ 207
A Vignette of Randy ................................................................................................ 208
Similarities and Differences between the Textbook Lesson and Lesson Taught in
Terms of Cognitive Demands .................................................................................. 214

xi

.\.—

 

Influential Factors on the Transformation Pattern ................................................... 217

Summary ..................................................................................................................... 220
CHAPTER 7 ............................................................................................... 224
SUMMARY, DISCUSSIONS, AND IMPLICATIONS ............................. 224

Summary ..................................................................................................................... 224

Transformation Patterns in Terms of Cognitive Demands ...................................... 227
Influential Factors Associated with Various Transformation Patterns ..................... 238
Discussion ................................................................................................................... 251
Implications ................................................................................................................. 260
Implication for Policy Makers ................................................................................. 260
Implications for Curriculum Developers ................................................................. 263
Implications for Professional development ............................................................. 264
Implications for Future Study .................................................................................. 266

Concluding Remark .................................................................................................... 269
APPENDICES ............................................................................................ 270

Appendix A. Survey .................................................................................................... 271

Appendix B. Interview Protocol ................................................................................. 282
REFERENCES ........................................................................................... 284

xii

 

LISTS OF TABLES

Table 2. 1 Summary of the Previous Studies on Textbook Use Patterns ......................... 16
Table 2. 2 Summary of the Studies on Influential Factors ............................................... 21
Table 2. 3 Hypothetical Textbook Transformation Patterns ............................................. 47
Table 3. 1 Measures Used in Statistical Analysis ............................................................. 55
Table 3. 2 Characteristics of Survey Participants ............................................................. 57
Table 4. 1 Summary of Descriptive Statistics of the Data for Item 18 ............................. 69
Table 4. 2 Summary of Descriptive Statistics of the Data for Item 19 ............................. 74
Table 4. 3 Results of Factor Analysis and Reliability ...................................................... 79
Table 4. 4 Frequency Distribution of the Transformation Patterns of Problems .............. 87
Table 4. 5 Frequency Distribution of Ttransformation Pattern of Teacher Questions ..... 92 ’
Table 4. 6 Frequency Distribution of Transformation Patterns in Phase 3 ...................... 96

Table 4. 7 Category and Frequency of Each Component in Teachers’ Lesson Plan
Modifications ................................................................................................. 1 01

Table 4. 8 Framework for Categorizing Teachers’ Modifications in Terms of Cognitive
Demands ........................................................................................................ 103

Table 4. 9 Frequencies of Transformation Patterns in Teachers’ Lesson Plan
Modifications ................................................................................................. 105

xiii

Table 4.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

10 Frequency (Percentage) of Different Transformation Patterns Exhibited in
Questions 18 and 19 by Types of Lesson Modification ................................. 107

1 Descriptive Statistics of Dependent and Independent Variables Explored ..... 114

2 Descriptive Statistics of the Independent Variables Explored in Factor Analysis
........................................................................................................................ 1 1 5

3 Frequency of Different Textbook Policies and Student Math Achievement
Level ............................................................................................................... 117

4 Frequency Rating of Agreement on State Curriculum Framework and Tests 117

5 Frequency of Different Textbooks Used by Teachers .................................... 119
6 Descriptive Statistics for Teacher Knowledge Variables ............................... 120
7 Teachers’ Agreement with Contrasting Statements of Teaching Philosophy 121

8 Agreement Frequency about the Textbook Satisfaction ................................ 123

9 Mean Time (in Hours) and Standard Deviation Spent on Professional
Development (PD) Activities ......................................................................... 125

10 Correlations for the Variables Included in the Final Model ........................ 134

11 Parameter Estimates from Logistic Regression Model for Examining Effect
of Seven Independent Variables on the Distinction between Patterns Exhibited
in Transformation Phase 1 ............................................................................. 135

12 Parameter Estimates from Logistic Regression Model for Examining Effect

of Seven Independent Variables on the Distinction between L-H and L-L
Patterns in Phase 1 ......................................................................................... 144

xiv

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 6.

Table 7.

13 Correlations for the Variables Included in the Final Model ........................ 147

14 Parameter Estimates from Logistic Regression Model Examining Effects of
Predictor Variables on the Distinction between Patterns on Transformation
Phase 2 ........................................................................................................... 149

15 Parameter Estimates from Logistic Regression Model for Examining Effect
of Eight Independent Variables on the Distinction between the L-L and the L-
H Patterns ....................................................................................................... 155

16 Correlations for the Variables Included in the Final Model ........................ 158

17 Parameter Estimates from Logistic Regression Model Examining Effects of
Predictor Variables on the Distinction between Patterns in Phase 3 ............. 159

18 Parameter Estimates from Logistic Regression Model Examining Effects of
Predictor Variables on the Distinction Based on the L-L Pattern in
Transformation Phase 3 ................................................................................. 163

1 Information about the Case Study Participants ............................................. 172

1 Frequency of Teachers in Each Pattern ......................................................... 235

XV

LISTS OF FIGURES

Figure 1. 1 Three Phases of Transformation Patterns ...................................................... 10
Figure 2. 1 Transformation of Mathematical Tasks in Terms of Cognitive Demand ....... 32
Figure 2. 2 Hypothetical Textbook Transformation Patterns ........................................... 42
Figure 3. 1 Overview of Study Design ............................................................................. 49
Figure 3. 2 Survey Data Analysis Process for the Study .................................................. 58
Figure 4. 1 Three Analyses of Transformation Patterns ................................................... 68

Figure 4. 2 Mean Frequency Rating for Different Types of Problems in Textbook and
Teaching ......................................................................................................... 70

Figure 4. 3 Average Frequency Differences between High-level and Low-level Problems
in Textbooks and in Teaching ......................................................................... 73

Figure 4. 4 Average Frequency Ratings for Different Types of Questions in Textbook and
in Teaching ..................................................................................................... 75

Figure 4. 5 Average Frequency Differences between Hi gh-level and Low-level Questions
in Textbooks and in Teaching ......................................................................... 77

Figure 4. 6 Scatterplot of Teachers’ Responses about the Average Frequency of Hi gh-
Ievel Problems ................................................................................................ 82

Figure 4. 7 Scatterplot Illustrating the Process of the Categorization of Transformation
Patterns in Terms of the Cognitive Demand of Problems .............................. 86

xvi

Figure 4.

Figure 4.

Figure 4.

Figure 4.

Figure 5.

Figure 5.

Figure 5.

Figure 5.

Figure 6.

Figure 7.

Figure 7.

Figure 7.

8 Scatterplot Illustrating Teachers’ Average Frequency Rating (AFR) of High-
level Teacher Questions in Textbooks against Teachers’ AFR of Hi gh-level
Teacher Questions in Teaching ...................................................................... 89

9 Scatterplot Illustrating the Categorization of the Transformation Patterns in
Terms of the Cognitive Demand of Questions in Textbooks and in Teaching91

10 Scatterplot of Teachers’ Responses about the Average Frequency of High-
level Problems in Teaching against the Average Frequency of High-level
Questions in Teaching .................................................................................... 93

11 Scatterplot Illustrating the Categorization of the Transformation Patterns in
Terms of the Cognitive Demand from Problems to Teacher Questions in
Teaching ......................................................................................................... 95

1 Emphasis Placed on the Different Student Objectives ................................ 122

2 Dependent and Independent Variables in Transformation Phase 1 .............. 131

3 Dependent and Independent Variables in Transformation Phase 2 .............. 145

4 Dependent and Independent Variables in Transformation Phase 3 .............. 157

1 Summary of Lesson Activities in a Lesson .................................................. 180

1 Textbook Transformation Patterns Identified in Terms of Cognitive Demand
of Problems .................................................................................................. 228

2 Textbook Transformation Patterns Identified in Terms of Cognitive Demand
of Teacher Questions .................................................................................... 231

3 Transformation Patterns Identified in Terms of Cognitive Demands of the
Lesson Taught .............................................................................................. 235

xvii

Chapter 1
INTRODUCTION

Historically, curriculum materials or textbooks have been a key agent of policies

to regulate mathematics practice in ways that align instruction with the reformers’ ideas.

Unlike objectives, assessments, and other mechanisms that seek to guide curriculum,

textbooks are concrete, and provide the daily stuff of lessons and units, what teachers and

students do. Textbooks are, therefore, often used as a means to shape what students learn

(Bruner, 1960; Dow, 1991).

However, despite their central role in instruction, there has been controversy over

the role of textbooks in research and practice. There are two different perspectives on the

role of textbooks in practice—( 1) “Textbooks as mechanism of deskilling the

professional work of teaching” and (2) “Textbooks as agents of instructional

improvement” (cf. Russell, 1994).

Some educators argue that curriculum materials that were designed to shape

mathematics instruction “dc-skill” the professional work of teaching, severely limit

teachers’ discretion over curriculum, and therefore limit teachers’ and students’

opportunities to learn. Advocates of this view acclaim teachers who create original

materials and lessons (Apple & Jungck, 1990; Elliott, 1990).

In contrast, other educators argue that curriculum materials can make up for lacks
in teacher knowledge and experience (Ball & F eiman-Nemser, 1988; Ball & Cohen,
1996; Collopy, 2003; Ma, 1999; Remillard, 2000; Russell, 1997). Advocates of this view,
in particular, reformers, often consider a new textbook as a main tool to change
instructional practice (Cohen & Ball, 1990; Stigler & Hiebert, 1999). By providing new
curriculum materials, reformers assume that teachers will follow textbooks, change their
instructional practices in the way they intended, and consequently improve students’
mathematics achievement (Cohen & Ball, 1990; Corcoran, 2003; Rernillard, 2004; Stigler
& Hiebert, 1999).

Do curriculum materials really hamper or help teachers, teacher learning, and
teaching? What are the affordances and limitations of different ways of following a text
book? Ball and Cohen (1996) urge that if textbooks are a key component in most
classrooms, we need to try to understand the relationship between the textbook and
teachers’ practice. In fact, the hostility to textbooks and the idealized image of
professional autonomy have inhibited careful consideration of the constructive role that
textbooks might play (Ball & Cohen, 1996). Between two perspectives on the role of
textbooks described above, this dissertation study prefers the second, which is that

curriculum materials could contribute to professional practice if they were created with

closer attention to processes of curriculum enactment. Indeed, a textbook is the most
commonly used instructional tool, and in some cases it is the only one which the teacher
depends on. Textbooks can have a strong influence or even dominate the nature and
sequence of teaching. For this reason, it is very important that textbooks are used well.
Yet, too little is known about how teachers use textbooks, in particular, about what would
be a good way to use their textbooks.
Problem Statement

Locating this dissertation study at the intersection of curriculum and teaching, I
reviewed the research literature associated with three areas: Reform in mathematics
education, textbook use, and its influential factors. A number of studies have focused on
teachers’ mathematics practice in the context of changes in national and state reforms
(EEPA, 1990; March & Odden, 1991; Spillan, 1995; Spillan & Zeuli, 1999; Wilson,
Peterson, Ball, & Cohen, 1996; Weiss, Pasley, Smith, Banilower, & Heck, 2003). Several
studies have contributed to our understanding of how teachers use their textbooks as an
attempt to revise their practice, and have provided a substantial number of categories of
teachers’ textbook use patterns and influential factors (Freeman & Porter, 1989; Freeman,
Belli, Porter, Floden, Schmidt, & Schwille, 1983; Kauflinan, 2002; Schmidt, Porter,

Floden, Freeman, & Schwille, 1987; Stodolsky, 1989; Sosniak and Stodolsky, 1993).

However, there are limitations associated with the previous research on teachers’
textbook use, in particular, to answer my questions.

First, despite findings about the value and influence of textbooks, the existing
studies on reform and practice do not help us better understand how teachers revise their
practice with textbooks according to reform efforts. Reforrners have gauged teachers’
implementation of reform ideas based on “inclusion of new mathematics topics,” “use of
variety of manipulatives”, “use of calculator,” etc. Some even consider “less use of
textbooks,” as one indicator that reform has permeated the classrooms (e. g., Goertz,
Floden, & O’Day, 1995).

However, textbooks are still the visible curriculum in most classrooms. Teachers
use textbooks to identify topics to be covered, and they select problems and questions
from textbooks to make topics concrete (Ball & Cohen, 1996). In addition, textbooks
provide opportunities for teachers to learn. Textbooks can fimction as an important
Support for teachers, especially beginning teachers, by providing detailed information that

is useful for making decisions (Kauffinan, 2002) and as a professional development tool
for teachers to increase their understanding of content (Ball & Cohen, 1996; Collopy,
2003; Ma, 1999; Remillard, 2000; Russell, 1997). Moreover, textbook change has been

Viewed as a main tool to revise and support teachers’ instructional practice. Therefore, we

need to understand teachers’ practice in connection with the influence of textbooks in
order to better understand teachers’ reform efforts.

Second, while researchers have documented mathematics instructional changes
with different criteria, such as, “new mathematics topics,” “variety of manipulatives,”
“calculator,” etc, in order to understand teachers’ implementation of reform ideas, these
criteria do not suggest what kinds and levels of student learning opportunities are created.

The Professional Standards for Teaching Mathematics (NCTM, 1991) states that
opportunities for student learning are not created simply by putting students into groups,
by placing manipulatives in front of them, or by handing them a calculator. Rather, the
level and kind of thinking in which students engage with mathematical problems, what
Stein and Smith (2000) called, “cognitive demands” of mathematical problems
determines what students will learn (p. 19).

However, most of the previous research on textbook use focused on the maximal
extent of coverage, such as to what extent teachers use textbooks in planning and
teaching school subjects (e. g., Freeman & Porter, 1989; Freeman, Belli, Porter, Floden,
Schmidt, & Schwille, 1983; Sosniak & Stodolsky, 1993). There have been few studies
looking at how teachers transform mathematical problems presented in textbooks when

planning and teaching mathematics in terms of cognitive aspects and what factors support

and constrain these transformations.

Teachers use textbooks to create students’ learning opportunities. According to
NCTM’s (1991) document, the kinds and levels of mathematical problems and questions
teachers use from their textbooks influence kinds and levels of learning opportunities
with which student are provided. To understand what learning opportunities teachers
provide students with, we need to explore how teachers use their textbook in terms of
cognitive demands. Although several studies (e. g., Stein & Smith, 1998; Stein, Grover, &
Henningsen, 1996; 1999, Stein, Smith, Henningsen, & Silver, 2000) have examined
teachers’ practices in terms cognitive demands, they fail to consider the value and
influence of textbooks. They did not consider the teacher-texts relationship (i.e. how the
cognitive demands of mathematical tasks presented in textbooks were changed when
teachers planned and implemented them during instruction). There is a need for an
examination of how mathematics tasks presented in textbooks are transformed by
teachers by using the construct of cognitive demand. This dissertation study therefore
explores whether and how the cognitive demands of the textbook versions of problems
and questions are changed when teachers moved content from text to teaching.

Furthermore, we need to understand teachers’ textbook use in depth and in

breadth by combining quantitative methods and qualitative methods. Most of the previous

 

studies that have examined textbook use patterns employed case studies. There are a few
studies that examined teachers’ textbook use using a quantitative method. Although case
studies on teachers’ textbook use provide detailed descriptions of how teachers use
textbooks and what factors influence their textbook use, the results from case studies are
varying. For instance, some researchers reported that teachers’ beliefs about textbooks are
the most important, but others reported that teachers’ knowledge is important. Patton
(1990) suggests the importance of using quantitative methods and qualitative methods
together, and highlights the advantages of combining survey and case study methods.
Since it is important to understand instructional practice at the micro and macro-level
(Stecher & Borko, 2002), to provide macro-level and micro-level analysis, this study
examined teachers’ textbook transformation patterns and its influential factors by
combining quantitative (survey) and qualitative (case study) methods.

This research focuses on the topic of fractions. Fractions typically represent a
serious excursion into abstract mathematics (Wu, 2005). It is well known that many
students and even adults have difficulties with the meaning of fractions and
understanding operations with fractions (e. g., Behr, Wachsmuth, Post, & Lesh, 1984;
Hiebert & Behr, 1988; Wu, 2001). Nevertheless, mathematicians and educators agree that

instructional programs should enable all students to understand fractions and these

operations meaningfully and conceptually, use efficient and accurate methods for
computing, and use these operations to solve problems (Kilpatrick, Swafford & Findell,
2001; NCTM, 2000). This study therefore focuses on the topic of fractions and examines
elementary teachers’ textbook use.

Purpose of the Study

The purpose of this dissertation study is to examine teachers’ textbook use and its
influential factors from a different angle, that of the cognitive demand of mathematical
problems and teacher questions, and to provide both depth and breadth of analysis with
respect to teachers’ textbook use and its influential factors.

The aim of the study was to extend the previous work, offer a new perspective,
that of the cognitive demand of mathematical problems and teacher questions, and
examine teachers’ textbook use and its influential factors from this perspective in order to
gain insight into the potential for textbooks to contribute to reform in mathematics
teaching. It is hoped that the knowledge about teachers' mathematics textbook
transformation patterns in terms of cognitive demands and the supports and constraints
that influence them gained through this study will help enrich a dialogue among
reformers, curriculum policy makers, and teachers, and lead to more effective ways of

enhancing teachers’ efforts to respond to recent reforms.

Research Questions

This study is guided by the following research questions:

I. What transformation patterns do elementary teachers exhibit as they move content
from text to teaching if the transformation is looked at in terms of cognitive demands

of problems and questions?

2. What factors are associated with various textbook transformation patterns?
0 Contextual-level factors:
(1) Does type of textbook (standards-based vs. conventional) influence teachers’
transformation patterns? In what ways?
(2) Do teachers’ perceptions of the state or district objectives or curriculum
frameworks influence teachers’ transformation patterns? In what ways?
(3) Does teachers’ perception of the state-wide tests (e.g., MEAP) influence
teachers' transformation patterns? In what ways?
(4) Do teachers’ perceptions of students’ achievements or ability levels influence
teachers’ transformation patterns? In what ways?
0 Individual—level factors:
(5) Does teachers’ knowledge about the topic of fractions influence their
transformations? In what ways? i
(6) Do teachers’ views about the textbook influence their transformations? In
what ways?
(7) Does teachers’ fidelity to text influence their transformations? In what ways?
(8) Do teachers’ beliefs about teaching and learning influence their
transformations? In what ways?
(9) Do teachers’ emphases on student objectives (understanding vs. procedural-
fluency) influence their transformations? In what ways?
0 Teachers’ opportunity-to-learn factors:
(10) Does the content of professional development (PD) activities influence
teachers' transformations? In what ways?
(11) Does the pedagogy of PD influence teachers' transformations? In what
ways?
(12) Does the total amount of hours teachers received in PD activities influence
teachers’ transformation patterns? In what ways?

Definition of the Terms

Transformation: The process of moving content from textbook to teaching has
been described as “implementation”, “adaptation”, or “modification”. However, this
study prefers using “transformation” to indicate all these processes. It is because teachers
make a choice when they enact curriculum, even the choice to use the problem exactly as
it is from the textbook involves transformation. Therefore, any pattern of textbook use
can be expressed as a “transformation” in this study. What changes with regard to the
cognitive demands is the nature of the transformation. Transformation patterns are
explored by looking at whether and how the cognitive demand of problems and teacher
questions presented in textbooks were maintained, increased, or decreased.

Transformation patterns were analyzed in three phases as teachers move content from

text to teaching as shown in Figure 1.1.

Textbook Teaching

 

 

Transformation Phase I

 

 

 

Problem % Problem

4-

l Transformation Phase 3

 

 

 

 

 

 

Transformation Phase 2

 

 

A
7

Question Question

 

 

 

 

 

 

Figure 1. 1 Three Phases of Transformation Patterns

Transformation patterns in phase 1 describe the relationship between the cognitive

10

demand of mathematical problems presented in textbooks and that of problems used by
teachers in teaching. Transformation patterns in phase 2 describe the relationship between
the cognitive demand of teacher questions presented in textbooks and the cognitive
demand of teacher questions used by teachers in teaching. Transformation patterns in
phase 3 refer to the relationship between the cognitive demand of problems used in
teaching and the cognitive demand of teacher questions used in teaching.

Significance of the Study

This study attempts to provide policy makers, curriculum developers, and teacher
educators with information about what is needed to help teachers make better use of their
textbooks. First, this study attempts to increase knowledge concerning the texts-teacher
relationship (i.e., how teachers use textbooks and in what ways) and the factors that
influence teachers’ decisions. This understanding is essential when curriculum materials
are considered the primary strategy for improving practice.

Second, this study attempts to demonstrate a different way of analyzing teachers’
use of textbooks (i.e., analysis of textbook use in terms of the cognitive demand of
problems and questions). If all teachers are to have opportunities to promote students’
understanding and foster students’ ability to engage in developing conjectures, and in

justifying their mathematical procedures and solutions, core dimensions of instruction

11

will have to change substantially, especially the mathematical tasks (i.e., the problems,

questions, and exercises students work on) and classroom discourse norms (i.e., the ways

teachers pose questions and the ways teachers interact with students about mathematics).

This study looks at these dimensions in terms of cognitive demands in order to expand

the literature in the areas of teachers’ use of textbook in mathematics education.

Third, this study attempts to contribute to current reform efforts seeking to change

teaching practices in ways that promote students’ understanding and foster students’

ability to engage in ways that promote students’ understanding and foster students’ ability

to engage in developing conjectures, framing and solving mathematical problems, and

justifying their mathematical procedures and solutions. This study identified the factors

(e.g., types of textbooks, teacher knowledge, etc) that influence teachers who revised

their instruction. This understanding will help reformers, school administrators, and

teacher educators to consider in what ways they can support teachers when teachers

attempt to revise their practices according to reform ideas.

12

Overview of the Dissertation

In this chapter, I have established the central questions of the study as well as the
purposes of this study. Chapter 2 synthesizes the findings from the existing literature
about teachers’ textbook use and influential factors. It also points out the limitations of
the previous studies about textbook use and addresses an alternative analytical framework
that was used to understand the texts-teacher relationship in terms of the cognitive
demand of problems and questions.

Chapter 3 presents research design and methods. It includes an overview of the
mixed methods research design and provides a rationale for choosing this research design.
This chapter also describes the data collection instruments, data collection procedures,
details of participants, and methods of data analysis.

Chapters 4 and 5 present the findings from the survey. Chapter 4 presents the
findings from the survey that examined what transformation patterns teachers exhibited
as they moved content from text to teaching. It also presents the findings from the
analyses of teachers’ actual lesson plan modification. Chapter 5 presents the findings
from the survey that examined the factors affecting the distinction between
transformation patterns exhibited in Transformation phase 1, Transformation phase 2, and

Transformation phase 3, respectively that are reported in Chapter 4.

13

Chapter 6 presents the finding fiom the case studies that examined how teachers
transformed problems and teacher questions in terms of cognitive demands as they
moved content from text to teaching in fractions units. The case studies also examined
factors influencing teachers’ textbook use in teaching.

Chapter 7 synthesizes what the findings of the survey and the case study suggest
about the role that textbooks might play in reform in mathematics education and the
factors that we need to consider in order to help teachers make better use of their

textbooks. This chapter also provides implications of the findings from this study.

14

Chapter 2
LITERATURE REVIEW

This chapter synthesizes the findings from the existing literature about teachers’
textbook use and influential factors, and proposes an alternative framework to guide my
dissertation study. This chapter is organized into four sections. The first section
synthesizes the findings from the prior research conducted about teachers’ textbook use
and influential factors to distinguish what has been learned and accomplished in this area
of study and what still needs to be explored. This section not only includes the claims
made in the existing literature but also examines the research methods used to better
understand whether the claims are warranted.

The second section points out the problems of the current research literature, in
particular in terms of the analytical framework as well as in terms of the research method
used, which, in turn, permits me to take a new perspective and to propose a new
combination of methods for the analysis of teachers’ textbook use in this study.

The third section presents the theoretical framework this dissertation study builds
on, that of the cognitive demand of mathematical problems and teacher questions. The
fourth section addresses hypothesized transformation patterns as teachers move content

from text to teaching and hypothesized factors influencing teachers’ transformations.

15

Research on Teacher’s Textbook Use and Influential Factors

Research on teachers’ textbook use and influential factors in practice goes back
for decades (Freeman & Porter, 1989). Several key studies have contributed to our
understanding of teachers’ textbook use and influential factors in practice (Freeman &
Porter, 1989; Freeman, Belli, Porter, Floden, Schmidt, & Schwille, 1983; Kauffman,
2002; Kon, 1993; Schmidt, Porter, Floden, Freeman, & Schwille, 1987; Stodolsky, 1989;
Sosniak and Stodolsky, 1993; Weiss, Pasley, Smith, Banilower, & Heck, 2003). Although
most of the studies on teacher’s textbook use have accompanied the analysis of factors
that affected teacher’s different styles, I review them separately to provide a clear view of
textbook use and influential factors.
Research on Teacher ’s Textbook Use

Table 2.1 shows a summary of the findings from the previous studies and presents

different trends about teachers’ textbook use that this study needed to consider

Table 2. 1 Summary of the Previous Studies on Textbook Use Patterns

 

 

 

 

Edy Focus Method Findings
1. Bagley The extent to Analysis of the results of Texts were followed almost
(1931) which the previous surveys on the slavishly; the curiosity of the
textbook matter, from as early as students was seldom aroused;
dominates 1898, collected from state rarely was an inquiring spirit
instruction and local inspectors and stimulated by the teacher
x supervisors.

 

 

 

 

16

Table 2.1 (Continued)

 

 

 

 

 

 

1\

 

 

 

2. Freeman The extent to Three aspects of (a) T extbook-bound teachers
& Porter which a teacher’s instruction: (b) F ocus-on-the-basic style of
(1989) instruction (a) The total number of teachers
matched the minutes of mathematics (c) F0cus-on-district-objectives
topics and instruction, (b) a list of style of teachers
sequencing topics, activities, and
presented in a problems each students
mathematics studied, (c) estimates of
textbooks the total amount of time
spent on each topic.
3. Freeman, Exploring what is Match between the (1) the textbook-bound teacher
Belli, Porter, covered by the content of material who omits nothing
Floden, aggregate of presented in fourth-grade (2) the textbook—bound teacher
Schmidt, & nationally used mathematics textbooks those who selectively omits
Schwille fourth-grade and the content of items sections
(1983) standardized tests on standardized tests for (3) the focus—on-the—basic
that grade level teacher who includes a unit on
measurement
(4) the focus-on-the-basic
teacher who does not include a
unit on measurement
(5) the focus-on-district-
objectives teacher.
4. Schmidt, To look for Short questionnaires and (1) the classic textbook-follower
Porter, patterns of effects interview and probes that without alteration
Floden, across the request a description of (2) textbook follower/strong
Freeman, & influential factors teachers’ lessons over a student influence
Schwille and define year and effect on that (3) follower of district objectives
(1987) teachers’ patterns content of nine factors, (4) follower of conception and
of content such as teachers’ past past experience
decision-making grperience
5. Stodolsky To what extent Three components of (1) Teachers covered only the
(1989) and in what ways textbooks: topics in the books, though not
teachers adhere to (a) topic necessary all of those topics or in
textbook content (b) actual material the order presented.
and suggestions contained on the pages in (2) At the level of content,
of teacher’s guide the books Teachers were very autonomous
in order to be (c) Activities suggested in their textbook use and only a
teaching by the in the teacher’s editions minority of Ts really followed
book in social that accompany student the text in the page-by-page
studies texts. manner.
6. Sosniak To see how Four fourth grade Textbooks were not blueprints;
& Stodolsky robust the teachers they were simply material
(1993) textbook use was Observation and available, “tools, props,
across subjects interview curriculum embodiments” (p.
270)

 

l7

 

Table 2.1 (Continued)

 

 

 

 

 

 

 

 

7. Kon The role of Analysis of the (a) Textbook as a primary
(1993) textbook in instructional activities resource
planning and listed in the lesson plan in (b) Textbook as an active
influential factors terms of its (a) duration, resource
(b) instructional (c) Textbook as a limited source.
grouping, (0) use of text
or other instructional
materials and interview.
8. Lambdin To see how 34 teachers using sixth (1) The frustrated methodologist
& Preston teachers adopt grade Connected (2) The teacher on the grow
(1995) new curriculum Mathematics (3) The standards bearer
materials
9. Remillard To examine the Observation and Three main categories of use:
&Bryans ways in which interview of eight (1) Intermittent
(2003) teachers use elementary school (2) Adopting and adapting
Investigations teachers (3LThrough piloting
10. Drake & To describe Use a fi'amework of The teachers developed a vision
Sherin teachers’ models of curriculum use of reform instruction through the
(2002) practices when (1) reading cuniculum use of the curriculum materials
using standard- materials and consequently the teachers
based textbooks (2) evaluating cuniculum began to trust the cuniculum
materials
(3) adapting cuniculum
materials

 

First, the previous studies revealed that the notion that teachers’ content decisions

are dictated solely by the textbook cannot be supported. That is, teachers do not strictly

follow the textbooks. One of the first studies regarding textbook use (Bagley, 1931) had

as its primary concern “the extent to which the textbook still dominates instruction in

American Schools” and reported teachers’ excessive dependence on textbooks.

However, the rest of the studies revealed that there are distinct styles of textbook

use. For example, Freeman and Porter (1989) focused on overlap between textbook

content and content taught in instruction. They found three categories of teachers’

textbook use: (a) textbook-bound, in which the teacher followed the textbook page by

18

 

page; (b) focus-on-the-basics, in which the teacher taught lessons directly related to basic
mathematical concepts and skills and skipped lessons viewed as being unrelated to these
concepts or skills; and (c)focus-on-district-objectives, in which teachers followed closely
their district’s recommendations on the topics to be taught. Indeed, most of the previous
studies on textbook use reported that what teachers teach is in the books, but they do not
teach everything that is in the textbook (e.g., Stodolsky, 1989). This finding suggests that
textbooks do not control the elementary curriculum to the extent ordinarily assumed, and
teachers regularly modify textbooks in their classrooms (Freeman & Porter, 1987;
Stodolsky, 1989; Schwille et al., 1982; Schmidt, et al., 1987).

Second, several studies questioned the long held conviction that teachers who
closely follow the textbook teach more traditionally than in reform ways, and revealed
contradicting results. For example, Freeman and Porter (1989) showed that the teachers
who deviated most from the textbook placed a greater emphasis on drill and practice of
computational skills, while those who followed the text more closely emphasized
applications and conceptual understanding. Moreover, studies on teachers’ use of the
standard-based textbooks revealed that the teachers “developed a vision of reform
instruction” through the use of the curriculum materials (Drake & Sherin, 2002; Lloyd &

Behm, 2002; Remillard & Bryans, 2004), suggesting that “thorough piloting” of the

19

standards-based textbooks by the teachers, while not a panacea, can play important roles
in fostering refonn-based practices (Remillard & Bryans, 2004).

Third and lastly, the previous studies suggest the necessity of studying what
influences the decisions that teachers make regarding content and pedagogy when using
textbooks. The previous studies on textbook use revealed that various factors lead
teachers to use textbooks differently. For example, in Freeman and Porter’s (1989) study,
the focus-on-district—objectives teacher category indicates that district objectives have
much more influence than the textbook itself when selecting content. The focus-on-the-
basics teacher category shows that teacher’s beliefs about what to teach and how to teach
have stronger effect on content decisions than the textbook itself. These findings imply a
need for understanding teachers’ larger curricular agendas and the role the textbook play
in them, suggesting that researchers who study textbooks and their use in classrooms also
have to pay attention to a connection between teachers’ thinking and beliefs about
instructional plans and activities, and their actual use of materials (Sosniak & Stodolsky,
1993, p. 272). Built on previous studies, my dissertation study involves studying teachers’

enacted curriculum and the role that texts play in it.

20

Research on Influential Factors on Teachers’ Textbook Use

A number of studies exarrrined what factors influence the decisions that teachers

make regarding content and pedagogy. Table 2.2 shows the summary of the previous

studies that examined the various influential factors on teachers’ decision making. This

shows different findings and tendencies in the previous studies that examined influential

factors on teachers’ decisions and that this study needed to consider.

Table 2. 2 Summary of the Studies on Influential Factors

 

 

 

 

 

 

Study Factors examined Findings
1. Freeman & Teacher view on text 0 Teachers’ adherence to textbook
Porter (1989) Teacher view on master basics topics is a function of the degree to
which teachers see the textbook as a
legitimate content authority and of
their own convictions about the
content for S3 to master.
2. Schmidt, 0 Teachers’ beliefs about math 0 Textbooks have the greatest influence
Porter, FIOdCD, 0 Educational experiences on content selection decision,
Freeman, & 0 District Objectives/ Tests followed by teachers’ conceptions
Schwille o Textbook and then students.
(1987) . Different subject-matters o The other factors had little overall
. Parents/ Student impact on content decisions.
. Other teachers 0 Parents especially had no noticeable
impact.
3. Ball & 0 Pre-service training experience 0 Due to teacher education programs
Feiman- that emphasized that textbooks can
Nemser (1983) only be used as a limited resource,
most of the students from both
programs were unprepared to use the
given textbooks in their practice
4- Kaffeman 0 Teachers’ skill & knowledge 0 Beliefs about mathematics and
(1939) 0 Teacher beliefs & attitude mathematics instruction (particularly
toward textbooks how those beliefs do or do not match
0 School context/ students the available curriculum materials) is
0 District objectives dominant individual-level influence.
. Subject matter being taught - Most prominent contextual factors
. Nature of textbook are institutional norms and

 

 

expectation for textbook use

 

21

 

Table 2.2 (Continued)

 

5. Kon (1993)

0 Educational experiences
0 Teacher views on the goals of
social education

The factors are quite varied and
interrelated.
No single factor consistently stands

 

 

 

 

- Teacher view on how to teach it out.
0 School characteristics
0 Classrooms characteristics
6. Remillard 0 Teacher beliefs about Teacher beliefs about the nature of
(1996;1997;19 mathematics mathematics and about teaching and
99; &2000) o Beliefs about teaching and learning influenced how teachers read
learning the textbook
0 Teacher perception about Within the construction area, where
students and their learning the cuniculum is constructed by
adapting tasks according to the
students’ performance and reactions
to the planned tasks, the teachers
examined and analyzed the tasks
guided by their ideas about learning
and about what students need to
know
7. Remillard & 0 Teacher view about mathematics Engagement with the curriculum
Bryans (2004) and the teaching of mathematics materials is affected by the teachers’
0 Teacher view of a particular orientation toward using a
curriculum curriculum, and in turn it influences
the teacher’s views and ideas of and
about mathematics
3. Weiss, 0 Teacher knowledge State/district cuniculum standards
P331655 Smith, 0 Teacher beliefs has the greatest influence on content
Banilower, & 0 Teaching experience SCIGCI dCCiSiOD, fOIIOWCd by
Heck (2003) . State/district curriculum textbooks, tests, teacher knowledge,
standards beliefs, & experience, student
. Test accountability characteristic, teacher collegiality,
, Students and PD experience
0 Professional Development Teacher knowledge, beliefs, &
. Teacher collegiality experience has the major influence on
. Textbooks mstructronal strategy, followed by

 

 

textbooks, student characteristic, PD
experience, teacher collegiality, tests,
and state/district curriculum standards

 

First, some researchers place the teacher-text relationship at the center of analyses

of teaching (e.g., Freeman & Porter, 1989), whereas other researchers consider teachers’

larger curricular agendas and the role the textbook plays in them (e.g., Schmidt, Porter,

Floden, Freeman, & Schwille, 1987; Weiss, Pasley, Smith, Banilower, & Heck, 2003),

22

 

focusing less on the teacher-textbook relationship and more on the teacher-curriculum

relationship (Remillard, 2005). Although the second type of research did not explain why

some teachers use their textbooks in a particular way, these studies revealed the crucial

role of textbooks in deciding content selections and instructional strategies, and suggested

the necessity of study examining the factors influencing teachers’ textbooks use. For

example, Schmidt, Porter, Floden, Freeman, and Schwille (1987) examined nine factors

affecting teachers’ content decisions as shown in Table 2.2 and revealed that textbooks

had the largest impact on most of the teachers’ content decisions, followed by teachers’

conceptions, and then students. Along the same line, in a more recent report of a large

national observation study on K-12 mathematics and science education, Weiss, Pasley,

Smith, Banilower, & Heck (2003) also reported that the textbook designated for a class is

the second most important factor that influences both the selection of content and

instructional strategies.

Second, the previous studies suggest the importance of examination of the factors

from multiple aspects. Although the existing research has identified many influences over

what teachers actually teach, findings vary among the studies. One possible reason is that

while some researchers have focused on only individual-level influences, such as

teachers’ beliefs about textbooks (e.g., Sosniak & Stodolsky, 1993), others explored both

23

individual-level and contextual-level influences (e.g., Kauffrnan, 2003). Knapp (2000)

suggests the importance of studying the connection between individual teachers and their

workplace environments. He noted the difficulty of conducting this kind of research. Yet,

since individual teachers reside in the contexts in which they work, a better approach

would be to look at the influential factors on teachers’ textbook use considering both

individual factors and contextual factors.

Furthermore, the previous studies suggest studying influential factors on content

selection and instructional strategies separately. Recently, Weiss, et a1. (2003) explored

what led teachers to select the content in the lessons, and why they chose the pedagogy

and the materials used in the lessons. With 364 mathematics and science teachers in K-1 2

grade, Weiss, et a1. conducted interviews and classroom observations and found that

different factor influence content selection and instructional strategies differently. They

reported that state and district curriculum standards had the most influence on their

content decisions and next was the textbook/program designated for the classroom.

Teachers’ knowledge, beliefs, and experience had the most influence on their

instructional strategies. This finding suggests that we need to examine influential factors

differently by separating content selection and instructional strategy.

24

Problems of the Current Research Literature on Textbook Use

Several studies have contributed to our understanding of teachers’ attempts to
revise their practice with textbooks and have provided us with a substantial number of
categories of teachers’ textbook use patterns and influential factors. However, there are
limitations associated with the previous research on teachers’ practice concerning
textbooks.

First, despite findings about the value and influence of textbooks, the existing
studies on reform and practice do not help us better understand how teachers revise their
practice with textbooks according to reform efforts. Reforrners have gauged teachers’

9, 6‘

implementation of reform ideas based on “inclusion of new mathematics topics, use of
variety of manipulatives”, “use of calculator,” etc, However, they often consider “less
reliance on textbooks,” as one indicator that reform has permeated the classrooms (e.g.,
Goertz, Floden, & O’Day, 1995).

However, textbooks are the visible curriculum in most classrooms. Teachers use
textbooks to identify topics to be covered, and they select problems and questions from
textbooks to make topics concrete (Ball & Cohen, 1996). In addition, textbook change

has been viewed as a main tool to revise and support teachers’ instructional practice.

Therefore, we need to understand teachers’ practice in connection with the influence of

25

textbooks in order to better understand teachers’ reform efforts.

Second, the criteria that researchers have used to understand the teacher-text
relationship do not suggest what kinds and levels of student learning opportunities are
created.

The Professional Standards for Teaching Mathematics (N CTM, 1991) states that
opportunities for student learning are not created simply by putting students into groups,
by placing manipulatives in front of them, or by handing them a calculator. Rather, the
level and kind of thinking in which students engage with mathematical tasks, what Stein
and Smith (2000) called, “cognitive demands of student thinking” determines what they
will Ieam (p. 19).

However, most of the previous research on textbook use focused on the maximal
extent of coverage, such as to what extent teachers use textbooks in planning and
teaching school subjects (e.g., Sosniak & Stodolsky, 1993). There is little research on
how teachers use their textbook to provide different kinds and levels of student thinking
that shape students’ experience in the classroom.

The QUASAR project (Quantitative Understanding: Amplifying Student
Achievement and Reasoning) examined how mathematical tasks are transformed by

teachers from the standpoint this study investigated, that is, in terms of cognitive

26

'.u I“

 

a:
-.L' l

luau...‘

'V'fi‘;

flu. .-
4‘

 

5‘}...
‘HL,

 

demands (Henningsen & Stein, 1997; Silver & Stein, 1996; Stein, Grover, & Henningsen,

1996; Stein & Smith, 2000). Although the QUASAR project examined how the cognitive

demands of mathematical tasks planned by teachers were changed during instruction,

they did not consider the teacher-texts relationship (i.e. how the cognitive demands of

mathematical tasks presented in textbooks were changed when teachers planned and

implemented them during instruction). There is a need for an examination of how

mathematics tasks presented in textbooks are transformed by teachers by using the

construct of cognitive demand. This dissertation study therefore explored whether and

how the cognitive demands of the textbook versions of problems and questions are

changed when teachers moved content from text to teaching.

Furthermore, most of the previous studies that have examined textbook use

patterns and the influential factors employed case study, and there are a few studies that

examined teachers’ textbook use using a quantitative method. Although case studies on

teachers’ textbook use provide detailed descriptions of how teachers use textbooks and

what factors influence their textbook use, the results from case studies are varying.

For instance, some researchers reported that teachers’ beliefs about textbooks are

the most important, but others reported that teachers’ knowledge is important. Recently,

with 364 mathematics and science teachers in grades K-12, Weiss, and his colleagues

27

(2003) provided more general quantitative findings regarding influential factors on
content decisions and instructional strategies. Unlike case studies, Weiss et al’s study
does not provide detailed information about how such factors are related to teachers’
content selection and instructional strategies.

Patton (1990) suggests the importance of using quantitative methods and
qualitative methods together and highlights the advantages of survey and case study
methods.

The advantage of a quantitative method is that it is possible to measure the reactions

of a great many people to a limited set of questions, thus facilitating comparison and

statistical aggregation of the data. This gives a broad, generalizable set of findings
presented succinctly and parsimoniously. By contrast, qualitative methods typically
produce a wealth of detailed information about a much smaller number of people and

cases. This increases understanding of the cases and situations studied but reduces
generalizability (p. 14).

It is important to understand instructional practice at the micro and macro—level (Stecher
& Borko, 2002). The methodological weakness of the previous studies I pointed out
above suggests the need for a combination of quantitative and qualitative methods in
order to compensate for those weaknesses. In order to provide macro-level and micro-
level analysis, this dissertation study examined teachers’ textbook transformation patterns
and influential factors by combining quantitative (survey) and qualitative (case study,
combination of observation and interview) methods.

In summary, the existing research literature on teachers’ textbook use has

28

limitations in its analytical framework, as well as its methodological aspects. Therefore,

this dissertation study extends the previous work and offers a new perspective, that of the

cognitive demand of mathematical problems and teacher questions, to examine teachers’

textbook use and its influential factors. In addition, this study used a mixed method

approach to provide both depth and breadth of analysis with respect to teachers’ textbook

use and its influential factors. In the next section, I will address an alternative analytical

framework that was used to understand the texts-teacher relationship in terms of the

cognitive demand of problems and questions.

29

 

Conceptual Framework

This section presents the theoretical framework this dissertation study builds on,
that of the cognitive demand of mathematical problems and teacher questions, and the
hypothesized transformation patterns and influential factors. In this section, I first
describe the importance of the cognitive demands of mathematical problems in forming
students’ opportunity to learn in mathematics lesson. I also emphasize the importance of
the cognitive demand of teacher questions in supporting students’ learning. Second, I
address the analytical framework used to classify various types of mathematical problems
and teacher questions in terms of cognitive demands. Third and finally, I describe the
hypothesized transformation patterns and influential factors derived from the previous
studies as associated with teachers’ use of textbooks.
Importance of Cognitive Demands of Problems and Questions

Mathematical problems used in instruction form the basis of students’
opportunities to learn mathematics (Doyle, 1983, 1979; 1980; Stein & Smith, 2000).
The Professional Standards for Teaching Mathematics (NCTM, 1991) articulated that
opportunities for student learning are not created simply by placing manipulatives in front
of students, or by handing them a calculator. Rather, the level and kind of thinking in

which students engage with mathematical problems, what Stein and Smith (2000) called,

30

“cognitive demands of problems” determines what they will learn (p. 19).

Indeed, mathematical problems draw students’ attention to particular ways of
thinking about and doing mathematics. For example, if the problems students work on
present the mathematical ideas as finished products, students will not need to think
through the concepts and engage in using them to reason about mathematics, and thereby
they may perceive mathematics as a statement of end products—definitions, rules, and
procedures--for memorization. Conversely, if a problem students work on demands
engagement with concepts through reasoning and argument, students will learn
mathematics by engaging in mathematical thinking, offering conjectures, responding to
one another’s ideas (and the teacher’s), and defending and justifying their ideas, as
opposed to mainly knowing computational procedures and following predetermined steps
to compute correct answers (NCTM, 1989, 1991, 2000). Therefore, being aware of the
cognitive demand of problems is central in the selection of mathematical tasks from
textbooks or in the creation of mathematical tasks.

The cognitive demands of mathematical tasks are maintained or changed as
teachers move content from textbook to teaching. Figure 2.1 describes the processes of
how the cognitive demand of mathematical tasks unfolds from textbook to teaching by

teachers. The framework developed from the QUASAR project (Quantitative

31

Understanding: Amplifying Student Achievement and reasoning) was adapted (Silver &

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stein, 1996).
Cognitive Cognitive Cognitive
Demands of Demands of Demands of
Problems __, Problems _1_? Problems _’
As presented As set up by As Student
in textbooks teachers implemented Learning
by teachers
v
v
First Transformation Second Transformation
(Design Arena) (Implementation Arena)

 

 

 

 

 

Source: Adapted from Stein, M.K., Grover,B.W., Henningsen, M. (1996). “Building Student Capacity for
Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms.”
American Educational Research Journal, v(33), pp.455-488.

Figure 2. 1 Transformation of Mathematical Tasks in Terms of Cognitive Demand

First, mathematical tasks or (problems) are presented in textbooks. These
mathematical tasks reflect curriculum developer’s intention of what mathematics is and
how students should learn mathematics. The cognitive demands of problems and
questions presented in textbooks are decided by curriculum developers.

These cognitive demands of textbook problems and questions are transformed
first by teachers when teachers plan lessons. Teachers use, adapt, or alter textbook
problems and questions to present their students. In this process, various factors influence

teachers’ selection of problems. The tasks that a teacher selects, regardless of the extent

32

to which they differ from those described in the textbook, represent the teacher’s
assumptions about content (what and how students should learn). Depending on the
cognitive demands of problems selected by teachers, the cognitive demands of problems
presented in textbooks are maintained or changed into different levels.

The cognitive demand levels of the mathematics tasks planned and set up by
teachers are again changed during instruction. Teachers enact the planned tasks in the
classroom by posing questions to guide students work on mathematical tasks and
responding to students’ interactions with them. In the second transforming process, the
ways the teacher and students talk, what they talk about, and how they agree and disagree
are especially important here because they can fundamentally transform the manner in
which a particular academic task is enacted by teacher and students (Spillane & Zeuli,
1999). In particular, teacher questioning has the potential to greatly facilitate the learning
process in a way that transfers factual knowledge to conceptual understanding (Brown &
Campione, 1990; Brown, Collins, & Duguid, 1989; Simon, 1986; Spillane & Zeuli,
1999). For example, even though teachers select the tasks that concentrate on factual
knowledge, teachers pose questions that require students to use higher order thinking or
reasoning. By engaging in this kind of teacher questioning, students do not remember

only factual knowledge. Instead, they can use their knowledge to problem solve, to

33

analyze, and to evaluate. Depending on the cognitive demands of teacher questions

teachers use during instruction, the cognitive demand of mathematical tasks set up by

teachers can be decreased, maintained, or increased, which eventually contribute to the

kinds and levels of student opportunity to learn. The following section describes the

framework that was used to classify the cognitive demand of problems and teacher

questions presented in textbooks and used by teachers in teaching.

Framework for Cognitive Demands of Problems

This study employed the Stein and Smith’s (2000) framework. A mathematical

problem in this study refers to a mathematical object to be solved by students.

Mathematical problems typically differ with respect to the level of cognitive demand they

place on student leaming. Mathematical problems presented in textbooks and in teaching

are categorized into two levels with respect to cognitive demands—(l) low-level and (2)

hi gh—level.

Low-level cognitive demand problems. This type of problems asks students to

perform a demonstrated procedure in a routinized way and hence place low-level, mostly

proceduralized demands on student learning. Low-level cognitive demand problems

would involve reproducing facts, rules, or formulas and consist of two types of tasks--

memorization tasks and procedures without connection tasks. Memorization tasks involve

34

exact reproductions of what students learned previously. Mathematical problems in this
type of task are clearly and directly stated. There is little ambiguity about what needs to
be done and how to do it. No connection is made to the concepts, meanings, or
understandings that underlie the procedure being used. Problems in this type of task
require no explanations or explanations that focus solely on describing the procedure that
was used. For instance, in the topic of fi'actions, this type of task could consist of

problems requiring students to memorize the equivalent forms of specific fractional

l = 0.5 = 50%).
2

quantities (e. g.,
Another type of low-level problem is procedures without connections tasks,

which also do not require students to make connections to the concepts or meanings that
underlie the procedure being used Procedures without connections tasks require limited
cognitive demand for successful completion. Problems in this type of task are focused on
producing correct answers rather than developing mathematical understanding. A typical
example is a problem that asks students to convert fractions to percents or decimals using
standard conversion algorithms in the absence of additional context or meaning (e.g.,
convert the fraction 3- to a decimal by dividing the numerator by the denominator to get

0.375; change 0.375 to a percent by moving the decimal point two places to the right to

get 37.5%).

35

High-level cognitive demand problems. High-level problems ask students to
make conceptual connections and to think and reason in sustained and thoughtful ways.
Like low—level cognitive demand problems, hi gh-level cognitive demand problems could
involve using procedures, but must do so in a way that builds connections to underlying
concepts and meaning. High-level problems also consist of two types of tasks--
procedures with connection tasks and doing mathematics tasks. Problems in procedures
with connection tasks require students to make connections between ideas and
procedures, possibly by using multiple representations. The purpose of addressing
procedures in these tasks is to develop deeper levels of understanding of mathematics
concepts and ideas. Procedures usually are represented in multiple ways, such as visual
diagams, manipulatives, symbols, and problem situations. For instance, problems might
ask students to use a diagam to illustrate how the fraction % represents the same
quantity as the decimal 0.6 or 60%. Students could also be asked to record their results in
a table containing the decimal, fiaction, percent, and pictorial representations, thereby
allowing them to make connections among the various representations and attach
meaning to their work by referring to the pictorial representation of the quantity every
step of the way.

Another hi gh-level problem is doing mathematics tasks, which would entail

36

.‘K’

I
u

asking students to explore the relationships and invent ways to solve problems.

Students would not be provided with the conventional procedures to solve the problems.

Rather, students could be asked to use a visual diagam to solve the problem, or be

allowed to choose a method for solving the problem. In this type of task, students are

challenged to apply their understanding of mathematics concepts in novel ways.

Framework for Cognitive Demands of Teachers Questions

A question in this study refers to a pedagogical object suggested in the textbook

and used by teachers in teaching that directs the students to think in certain ways and to

reflect on their math work. As in the case of the cognitive demand of problems, the

cognitive demand of questions here refers to as the kind and level of student thinking

required when (students) engage with “teacher questions”.

Six categories were adapted by blending the revised Bloom’s taxonomy

(Anderson & Krathwohl, 2001), Stein and Smith’s framework (2000), and the TIMSS

Video study’s (2003) mathematics cognitive domains, in order to describe the cognitive

demands of teacher questions. These six categories lie in continuum from remembering

through evaluating, which are categorized into three different cognitive levels—(1) low-

level, (2) medium level, and (3) high-level. The first two categories—remembering and

knowing procedures—are low-level teacher question. The middle category—

37

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understanding—is medium level. The rest three categories—applying, reasoning, and

evaluating—are hi gh-level teacher questions.

Low-level cognitive demand questions Low-level cognitive demand of teacher

questions concentrate on factual information that can be memorized or carrying out

algorithmic procedures. This type of question can limit students by not helping them to

acquire a deep, elaborate understanding of the subject matter. Low-level cognitive

demand of teacher question consists of two categories-~remembering and knowing

procedure. Remembering type of question requires students to draw up previously learned

knowledge. This kind of question is used for students to tell teachers what they already

know because of what teachers have taught them and what they have perceived and/or

experienced for themselves. Students are asked to recognize or recall information.

Another type of low-level question is knowing procedures. This type of a question

asks students to carry out algorithmic procedures for +, -, X, +, or a combination of these

with whole numbers, fractions, and decimals. This kind of question is used for students to

compute given problems based on an algorithm.

Medium level cognitive demand questions Understanding category is the

medium level cognitive demand, which means that, in this type of question, it is hard to

tell whether questions require hi gh-level of student thinking or low-level of student

38

thinking. Typically, this type of questions tests students’ comprehension. This level of
question is used for students to demonstrate that they understand what they know.
Students are asked to reword, rephrase, or describe what they know.

High-level cognitive demand questions This level of questions can be defined as
questions that require students to use higher order tlninking or reasoning skills. By using
these skills, students do not remember only factual knowledge. Instead, they use their
knowledge to problem solve, to analyze, and to evaluate. Applying, reasoning and
evaluating categories are high-level questions. Applying category question is used for
students to be able to select and use knowledge to complete a task by taking what they
have already learned and applying it to other situations. Students are asked to apply
known facts, principles, and/or generalizations to solve a problem.

Another type of hi gh-level question is Reasoning category, which encourages
analysis. Teachers use this type of questions for students to be able to support their
arguments and opinions by organizing ideas into logical patterns of understanding.
Students are asked to analyze and provide a justification for the truth or falsity of a
statement by reference to mathematical results or properties. Students are also asked to
apply mathematical procedures in unfamiliar or complex contexts and support their

arguments and opinions by organizing ideas into local patterns of understanding.

39

Evaluating question is the highest level that involves making a judgnent based
upon the application of a set of standards or criteria. Teachers want students to consider
the values implicit in their thinking by looking at evidence and establishing criteria.
Students are asked to synthesize and integate to evaluate results and determine how
closely a concept or idea is consistent with standards or value. Some of the thinking skills
are: synthesizing and integating, summarizing, judging, criticizing, or arguing.

In my dissertation, I believe that teachers should ask a combination of all three
levels of teacher questions in order to foster student understanding and achievement. For
example, it is possible that teachers have to break high level cognitive demand problems
into lower level questions in order to help students to understand, thus using low level
questions as a scaffold to high level learning opportunities. However, evidence from
previous studies reported that teachers spend most of their time asking low-level
cognitive questions (Hiebert & Weame, 1993; Klinzing, Klinzing-Eurich, & Tisher,
1985; Wilen, 1991). Teachers rarely ask ‘higher order’ questions, even though these have
been identified as important tools in developing student understanding (Hiebert &
Weame, 1993; Klinzing, Klinzing—Eurich, & Tisher, 1985).

Previous studies reported that student learning gains were geatest in classrooms

in which instructional tasks consistently encouraged hi gh-level student thinking and

40

reasoning and least in classroom in which instructional tasks were consistently procedural
in nature (Silver & Stein, 1996; Stein & Smith, 2000). I argue that teachers in both
instruction and evaluation should devote “a minimum of one-thir ” of the time allotted to
questioning to levels above memory (Sanders, 1966, p, 156).

In this study, referring to the ratio Sanders articulated, I classified the cognitive
demand of mathematics problems and teacher questions presented in mathematics and
that of mathematical problems and teacher questions used by teachers into two levels--
(1) higher level and (2) lower level. If more than 1/3 of problems are high-level problems
and questions, these textbooks (or teachers) were considered as providing higher level of
problems and teacher questions. Otherwise, these textbooks (or teachers) were considered
as providing lower level of problems and teacher question.

Hypotheses of 'Ikansformation Patterns and Influential Factors
Hypothesized Transformation Patterns

Based on the frameworks for the cognitive demand of mathematical problems and
teacher questions described above, four hypothetical transformation patterns from
textbook to teaching were expected to appear with respect to the cognitive demand of

problems and the cognitive demand of teacher questions as shown in Figure 2.2.

41

'3!
.,

5:,
11,
“1

"‘~.
..

'1‘

Transformation

 

 

 

 

 

 

Textbook Teaching Transformation
Patterns
I Problems/QuestionsI I Problems/Questions I
Lower level L-L 1
Lower level .
............ 1‘ .
Higher level L-H 2
Lower level H-L 3
Higher level .
................ ’ Higher level H-H 4

Figure 2. 2 Hypothetical Textbook Transformation Patterns

The first transformation pattem describes the situation where teachers have
textbooks that require lower level problems and questions, and they use lower level
problems and questions in their teaching. The second transformation pattern describes the
case where teachers transform lower level of textbook problems and questions into higher
level cognitive demand problems and questions in teaching. Conversely, the third
transformation pattern illustrates the situation where teachers have textbooks that provide
higher level problems and questions, but they lower the higher level in their teaching. The
fourth transformation pattern describes the situation where teachers have textbooks that
provide higher level problems and questions, and they maintain the cognitive demand in
their teaching. Among four transformation patterns, some reformers and researchers may
favor’the second and fourth transformation patterns (e. g., Silver & Stein, 1996; NCTM,

1989, 1991, & 2000), since problems and questions in these teachers’ classes will

42

demonstrate that mathematics is comprised of key ideas and concepts that can be used to

construct procedures for solving problems as opposed to a statement of end products for

memorization.

However, several very well-known studies (e. g., Cohen, 1990; Doyle, 1986;

Eisenhart, Borko, Underhill, Brown, Jones, & Agard, 1993; Stigler & Hiebert, 2004)

have found that many teachers reduce the cognitive level of problems and questions when

they use them in their teaching. For example, The Third International Mathematics and

Science Study included a video component whose examination suggests that teachers in

the highest-achieving counties maintained the high level of tasks whereas teachers in the

US implemented none of the high level problems in the way in which they were intended

(Stigler & Hiebert, 2006). According to them, the US. teachers turned most of the high

level problems into low level problems such as procedural exercises or just supplied

students with the answers to the problem.

Consistent with the results from Stigler and Hiebert’ study, Cohen (1990) reported

that due to teachers’ lack of knowledge, teachers decreased the high level of tasks

presented in curriculum materials as they conducted in their instruction. Cohen examined

one teacher, Mrs. 0, who believed that she had changed her mathematics teaching in

ways that provide students with high level of student thinking by using the high level

43

problems and questions presented in the new curriculum materials. However, his analyses
revealed that although Mrs. 0 used the new materials (high-level problems) in her
teaching, she conducted the class in ways that discouraged exploration of students’
understanding, which will be categorized into H-L pattern in this study. Thus, according
to previous studies, one might expect to see a large number of teachers in the H-L pattern
or the L-L pattern and a smaller number of teachers who follow the H-H pattern. In
particular, if one did an observational study, it would not be surprising to find that many
who identified themselves as H-H would be classified as H-L by an observer.

As a consequence of these studies, much more emphasis has been put on teacher
education and professional development progams in order to help teachers use problems
and questions in ways that improve students’ understanding. In addition, with the
availability of reform-inspired curriculum materials that provide high-level problems and
questions, there is considerable emphasis on the wide-spread adoption of new curriculum
materials in current education. School districts increasingly regulate mathematics
teaching practices by mandating the use of a “single” curriculum (Remillard, 2005).

However, as Ball and Cohen (1996) pointed out, too little is known about how
teachers use textbooks. In keeping with the findings from previous studies and current

emphasis on teacher education and curriculum policies, this study examined what

44

—

 

 

patterns elementary teachers exhibit as they move content from text to teaching in terms
of cognitive demands of problems and questions.
H ypothesized Influential Factors

Teachers do not work in a vacuum. Various factors influence teachers’ decision.
Existing research has identified two distinct influences—( l) Individual-level influence
and (2) contextual level influence. Some researchers reported individual-level influences
on teachers’ textbook use. For examples, teachers’ knowledge (Ball & Feiman-Nemser,
1988; Brophy, 1982; Cohen, 1990; Kauffman, 2002), teachers’ beliefs about mathematics
teaching and learning (Barr, 1988; Cohen, 1990; Stodolsky, 1989; Schmidt, et a1, 1987),
and teachers’ view about textbooks (Ball & Feiman-Nemser, 1988; Freeman, & Porter,
1989) are reported as all associated with teachers’ use of textbooks.

Other researchers reported contextual-level influences on teachers’ decision. For
example, teacher perception of state/district curriculum framework (Freeman, & Porter,
1989; Schmidt et al., 1987; Weiss, et al. 2003), state-wide tests (Freeman, & Porter, 1989;
Schmidt et al., 1987; Weiss, et a1. 2003), and school climate (e.g., limited time and
resource) (e.g., Wilson, 1990) all influence teachers’ use of cuniculum materials.

Although these studies did not examined influential factors on teachers’ textbook

use in terms of the cognitive demands, this study assumes that identified factors described

45

above would also influence teachers’ use of textbook in terms of the cognitive demand
since individual teachers reside in the contexts in which they work and influences of
contextual factors are mediated by influences of individual factors (Knapp, 2002).
Builds on the previous studies, this study hypothesizes that both individual-level and
contextual-level influence over teachers’ use of textbooks in terms of cognitive demands,
However, this study extends these two-level influences to three-level--individual-
level factors, contextual-level factors, and teachers ’ opportunity-to-learn factors. The
previous research categorized “professional development opportunities” into either
individual-level factors or contextual-level factors (see Kauffman (2002)’s study and Ball
& Feiman—Nemser (1993)’s study). Yet, professional development participation is the
mixture of individual-level influence and contextual-level influence. In some districts,
participating in professional development opportunities is mandatory, but in other district
it is voluntary. Profession development participation is mandatory to some teachers but
not other teachers. In addition, regardless of whether it is from individual-level or
contextual-level influence, professional development participation provides teachers with
opportunities to learn. As such, this study separated and added one more category,
teachers ’ opportunity-to-learn factors to examine the effect of professional development

opportunities over teachers’ use of textbooks. Table 2.3 presents the hypothesized factors

46

associated with teachers’ textbook transformation patterns in terms of cognitive demands.

Table 2. 3 Hypothetical Textbook Transformation Patterns

 

Factor-level

Sub-category

 

Contextual-level

District textbook policy

Type of textbook (standards-based vs. conventional)
Teacher perception of students’ mathematics achievement
Teacher perception of state/district curriculum frameworks
Teacher perception of state-wide tests (e.g., MEAP test)

 

Individual-level

Teacher knowledge

Teacher beliefs about teaching and learning
Teacher view on textbooks

Teacher emphasis of student objectives
Teacher fidelity to textbook

 

Teachers’
opportunity to learn

Professional development participation hours
Types of content areas studied on PD activity
Types of pedagogy participated in during PD activity

 

47

Chapter 3

RESEARCH DESIGN AND METHODS

This chapter consists of three sections. The first section presents an overview of
the mixed methods research design and a rationale for choosing this research design. The
second and third sections describe the quantitative research method and qualitative
research method, respectively, describing the data collection instrument and procedures,
details of participants, as well as methods of data analysis.

Research Design Overview

This dissertation study employed a mixed method design that combines both
quantitative and qualitative research methods to explore elementary teachers’ textbook
transformation patterns in terms of cognitive demands and influential factors. Figure 3.1
shows an overview of the study design.

A survey was developed to collect data for the quantitative aspect of the research.
An interview protocol and observation protocol were developed for the qualitative part. I
used the survey in order to explore the general ideas of the ways in which teachers
transform the cognitive demand of problems and teacher questions when they move
mathematics content from text to teaching. The factors that are associated with textbook

transformation patterns were also explored.

48

 

 

1. Quantitative method 2. Qualitative method
(Survey) (Case Studies)

 

V

 

 

 

 

 

 

 

 

Individual-level Study Focus

 
  

 

 

 

 

 

 

     
   
 
    
    
 
   

Teachers
with standards-
based curricula

 
 
  
   
     
 

  

Transformation
patterns in terms of
cognitive demands of
problem/teacher

question &

 

Contextual-level

 

 

 

 
   

Influential factors
Teachers

with
conventional
curricula

 

Teachers’

opportunity-to-learn

 

 

 

Figure 3. I Overview of Study Design

The purpose of the survey was to address two research questions: (1) What
transformation patterns do elementary teachers exhibit as they move content fi'om text to
teaching if the transformation is looked at in terms of cognitive demands of problems and
questions?; and (2) What factors are associated with various transformation patterns?

Next, I planned to select about 8 teachers from among the survey participants

purposefully for the case studies in order to gain more in-depth understanding of teachers’

49

textbook transformations and influential factors. I planned to observe each teacher’s

lesson once (in some cases, twice) and interview each teacher. During the interview(s), I

sought to answer the same research questions as those in the survey mentioned in the

previous paragaph.

The quantitative (a survey) and qualitative (observation & interview) methods

complement each other. For instance, studies based on qualitative methods can provide a

detailed description of a small number of teachers’ use of textbooks in terms of cognitive

demands, but they do not provide a general picture of what factors influence their use of

textbooks. In contrast, quantitative methods can miss information that the qualitative

methods can provide (i.e., detailed descriptions). Therefore, using two methods

sequentially, this study intended to provide both breadth and depth of analysis on

teachers’ use of textbooks in terms of cognitive demands and influential factors.

50

Quantitative Research Methods
Survey Development and Validation

The survey was developed by employing items from Weiss, Pasley, Smith,

Banilower, and Heck, (2003), Ravitz, Becker, and Wong (2000), Desimone, Porter,
Birman, and Garet (2002), and Choi (2005). Several survey items were modified based
on the Third International Mathematics and Science Study (TIMSS) questionnaires,
Freeman and Porter (1989), Schmidt, Porter, Floden, Freeman, and Schwille (1987), and

Stein and Smith’s (2000) framework.

After developing the survey, survey validations were conducted in three different
processes. The surveys must offer an accurate portrait of what teachers do in their classrooms
and how teachers use their textbook in their classroom. However, there are several reasons to
be concerned that they might not. First of all, in the survey much of the data are self-reported
by teachers. Teachers might provide biased responses to a survey because they feel that they
should (for a variety of reaons) respond to the questions in “acceptable” or “socially
desirable” ways; or teachers might unknowingly provide misleading responses to the survey
questions (Mayer, 1999). In particular, research suggests that teachers sometimes truly
believe they are embracing pedagogical reforems, but in pratices, their teaching comes

nowhere near the vision of the reformers (Cohen, 1990). In addition, the teaching process

51

consists of complex interactions between students and teachers. Surveys are limited in their
abilities to portray a valid picture of the schooling process, and in particular, aspects of
curricular practices, such as the interactions between teachers and students and their role in
the learning process (Burstein et al., 1995; Mayer, 1999).

To minimize these methodological limitation and use more valid surveys, the pilot
studies were conducted in three ways. First, I examined content validity, which can be
obtained from literature and experts (Carmines & Zeller, 1979). I asked a Ph.D. gaduate
student who had eight years of teaching experience in elementary schools in the US in
mathematics education to check if the survey made sense to her as a gaduate student
studying mathematics education, as well as a teacher who understands teaching practice
well. Although I have been studying mathematics education in Michigan, my limited
experience as an international student in the US. context might have potentially
generated some misunderstanding among my targeted participants.

After that, I asked five other Ph.D. students who were studying mathematics
education to indicate their opinions about the degee of relevance of each item in my
survey by marking one of three statements (Good, So S0, or Not Good) about each
survey item and providing specific explanations about their opinions when they chose

“So So” or “Not Good”. Overall, most items were perceived as “Good”. For items

52

perceived as “So So” or “Not Good,” I revised the statements based on their comments.

Lastly, the survey items were validated by five in-service teachers. The key issue

in this validation process is whether the intended participants make sense of the questions

asked of them. I tried to understand whether teachers’ understanding of the question

matched what I had in mind.

This pilot study was conducted through think-aloud protocols, which involve

participants verbally describing their thinking as they are performing a set of given tasks

(Swanson, O’Connor, & Cooney, 1990). This validity test enabled me to see how

participants made sense of the survey questions asked of them and how they interpreted

and justified their choices. Based on teachers’ comments, I revised the survey instrument

and reformatted the survey to facilitate easier reading. The wording of some questions

was changed for clarity, the scales of items were changed and some items were removed.

In addition, this validity test with in-service teachers enabled me to compare their

estimates of the cognitive demands of their texts and my own classification of them. Using

the survey, I intended to examine the relationship between the cognitive demands of

problems and questions presented in textbooks and the cognitive demands of problems and

questions used by them in teaching. My own classification of the cognitive demands of texts

provided one potential external standard against which to assess the validty of teacher self-

53

reports. All five participants in the pilot studies used standards-based curricula, which were

categorized as providing higher level cognitive demand of problems and questions by my

analyses (Stein & Kim, 2007); and their perceptions of the cognitive demand were similar to

mine although one teacher’s estimates of her own practice were a little bit upwardly biased.

Overall, these survey validation processes helped me to revise the survey in a way

that would capture a more valid and accurate portrait of how teachers use their textbooks in

terms of the cognitive demands of problems and questions and what factors influnece these

transformation patterns.

The Revised Survey

The revised survey consisted of 25 questions orgarnized in tlnree parts: (1)

backgound information, (2) textbook use in fraction units, (3) three levels of factors that

influence teachers’ use of textbooks including contextual—level factors, individual-level

factors, and teachers’ opportunity-to-leam factors. The survey is attached in Appendix A.

Questions 1-8 collect demogaphic data. Among the remaining questions, 16 are

used to answer the research questions. These are summarized in Table 3.1.

54

Table 3. 1 Measures Used in Statistical Analysis

 

 

 

 

Measures Item
number

A. Textbook transformation patterns

Cognitive demands of problems in textbooks and in teaching Q 18

Cognitive demands of questions in textbooks and in teaching Q 19
B. Factors influencing textbook use

Student math achievement Q 8

District textbook policy Q 9
Contextual-level Type of textbook Q 12

District curriculum framework Q 23

State-wide tests Q 23

Teacher knowledge Q 21

Teacher view on textbook Q 13
Individual-level Teacher beliefs on teaching Q 22

Teacher emphasis on student learning Q 16

Teacher fidelity to textbook Q 14
Teachers’ Professional development (PD) participation hours Q 24
opportunity-to- Content type in PD Q 24
learn Pedagogy Type during PD Q 25
C. Textbook transformation pattern validation

Cognitive demands of lesson plan modification Q20

 

Among the 16 main measures are two measures for teachers’ transformation

patterns, five measures for contextual-level factors, five measures for individual-level

factors, three measures for teachers’ opportunity-to-learn factors, and one measure for

validation of identified textbook transformation patterns.

Survey Participants

Prior to data collection, a description of this study and the data collection

instruments were reviewed and approved by the University Comnnittee on Research

Involving Human Subjects (UCRIHS). The survey data were collected from summer

2006 through fall semester 2007. For survey participant recruitment, the letter

introducing this study was sent out to instructors and elementary teachers who were

55

taking Master’s courses at a large suburban Midwestern University or those who were
registered for professional development progams.

Because the study aimed to include teachers who taught fractions, targeted
participants were originally teachers from second to sixth gade. According to Michigan’s
Mathematics Grade Level Content Expectations (GLCE), these are the gades at which
fractions are taught. However, different districts develop different content frameworks
and pacing guides, thus teachers in 1St gade were also included if they said they taught
fraction concepts. Participants were recruited among teachers who were taking Master’s
courses at a large suburban Midwestern University or those who were registered for
professional development progams. Thus, these participants are convenient samples
meaning that the participants do not represent the entire population. In particular, given
their participations in either Master progams or profession development progams, these
teachers might be very familiar with reform ideas (e. g., NCTM standards) or might be
influenced by these progams and might thereby show different transformation patterns
from those identified by other researchers (e. g., Cohen, 1990; Stigler & Hiebert, 2004) or
previous studies (e. g., Stein & Smith, 1998).

A total of 169 teachers from first to sixth gade participated in this study. Table

3.2 presents a summary of the characteristics of the survey participants as well as brief

56

information on teachers’ use of the textbook assigned by their districts or schools.

Table 3. 2 Characteristics of Survey Participants

 

 

 

 

 

 

 

may may
Type of school Public 161 (96%)
Private, religious 4 (2%)
Other (e.gLChart) 4 (2%)
Gender Male 33 (20%)
Female 1 36 (80%)
Grade taught 1St 18 (11%)
2“d 28 (17%)
3rd 30 (18%)
4‘h 26 (15%)
5‘h 35 (21%)
6‘1‘ 33 (1 8%L
Teaching experience 0 to 4 years 45 (26%)
5 to 9 years 52 (31%)
10 to 14 years 21 (12%)
15 to 19 years 16 (10%)
20 years or above 35 (21%)
. . . Yes 162 (96%)
Use the textbook assrgned by dnstrrct No 7 (4%)
Yes 53 (32%)
Use other textbooks No 1 1 5 (6 8% I

 

Almost all survey participants were working in public schools and were female.

The distributions for the gade-level were similar among six different gade levels, even

there were a few more fifth gade teachers. Approximately 74% of teachers had teaching

experience of five years or longer. Vnrtually all reported using the textbooks assigned to

them by their district or school. About one-third of the teachers used other textbooks

additionally, suggesting that the assigned textbooks are the primary source for teachers to

plan and teach mathematics in their practice.

57

Survey Data Analysis

Figure 3.2 describes the processes of survey data analyses

 

 

A. Transformation Pattern Analysis

 

 

1.Descriptive statistics

 

2. Factor analysis & reliability

 

Step 1: Preliminary analysis 3. Composite scores

 

 

 

 

 

 

 

1. Scatter plots & best-fit line

2. Categorization of teachers’

 

. responses on survey item 18 &
Step 2: Transforrnatron pattern . .
19 wrth a rating scale of 3

 

 

analysis

 

3. Interpretations of patterns
4. Analysis of teachers’ responses

of survey item 20 (validation)

 

 

 

 

B. Influential Factor Analysis

 

 

 

 

1.Descriptive statistics

 

Step 1: Preliminary analysis 2. Factor analysis & reliability

 

 

 

3. Composite scores

 

 

Multinomial logistic regression

 

Step 2: Influential factor analysis . 1. Test significance of a model

 

 

2. Select independent variables (p-
value $1.25)

3. Examine factors distinguishing

 

reference patterns to other
patterns

4. Interpretation

5. Multivariate regression and

interpretation (validation)

 

 

 

 

Figure 3. 2 Survey Data Analysis Process for the Study

58

Data analyses of survey items 1-19 and 21-25 was done using the Statistical
Package of the Social Science (SPSS) version 15.0. Details are provided in the following
chapters. To validate the findings from survey items 18 and 19, survey item 20 asks the
teachers comment on and modify a lesson plan reproduced fiom a lesson about
equivalent fractions. Each teacher’s response on survey item 20 was analyzed with two
questions in mind: (1) What did the teachers actually modify from the daily lesson plan in
terms of student objectives, classroom activity, mathematical problems, and teacher
questions? and (2) How did teachers modify the lesson in terms of the cognitive demand
of problem and questions? Once these data were coded, frequencies and of the
description statements were also obtained by using SPSS version 15.0.

Qualitative Research Method
Case Study Participants

Among the 169 survey participants, a total of eight teachers were chosen to
participate in the second part of the study, the case studies. These teachers were selected
purposively from 52 survey participants who volunteered to participate in the case study.
Two criteria were used in the selection of participants. The first criterion was gade level.
Although fi'action concepts are introduced from first or second gade, this topic is

developed intensively from fourth to sixth gade at the elementary level. Therefore, I only

59

selected teachers who were teaching in those gades.

The second criterion used in the selection of case study participants was the type

of textbooks teachers used. One suggestion from the findings based on teachers’ self-

reports was that types of textbooks (standards-based versus conventional curriculum)

play an important role in deciding the cognitive demand of problems and teacher

questions used by teachers in teaching. So I selected teachers who used standards-based

and conventional textbooks.

Case Study Data Collection Procedures

Each participant was observed at least once when they were teaching units about

fractions and a post-observation interview was conducted that lasted approximately 45-60

minutes. Prior to starting the interview, participants consented to being tape recorded.

Although each participant was observed and interviewed once or twice, case study data

were collected over the period of a year, since teachers taught fraction units at different

times.

Case Study Instruments

Three sets of data were collected from the eight elementary teachers—(l)

teachers’ planned lessons, (2) my observation and observation notes, and (3) structured

interviews.

60

Teachers ’ planed lessons using teachers’manuals Before teaching the lessons,

teachers were asked to provide their lesson plan. However, rather than keep a detailed

lesson plan, teachers were asked to use problems and teacher questions presented in the

teacher’s guide of their textbook to create their lesson plan. Teachers circled problems

and teacher questions they planned to use from the teacher’s guide and gave a copy to me.

With this lesson plan from the teacher’s guide, I observed each teacher’s lesson once or

twice. These planning lessons from the teacher’s guide helped me to see the main

activities that would take place during the lesson. This also allowed me to examine what

were the similarities and differences between the textbook lesson and the lesson planned

by teachers.

Classroom observation (and notes) The classroom observations focused on

exannining the teachers’ use of textbooks in practice. While I was observing each

teacher’s lesson, I took field notes, especially regarding what the teachers or students

recorded on the boards. Classroom observations permitted me to examine the changes

made between their planned lesson and their enacted lesson. I looked to see in what ways

the teachers enacted the planned lesson from their textbook and examined what kinds of

teacher questions they used.

61

Post-observation Interview The post-observation interview was designed to
elicit data on teachers’ use of textbooks and factors influencing their practices. This made
it possible for me to examine how teachers reflected on their use of textbooks from the
planning stage to teaching practice and explore what teachers recognized as constraints or
support in their use of textbook. The interview protocol consists of two parts——(1) Use of
their textbooks in planning and teaching fractions and (2) Influential factors on their
textbook use.

(1) Use of textbooks in planning and teaching lessons This set of questions was
designed to explore the process of teachers’ transformation from textbook lessons to
planned lesson and lesson taught. Teachers were asked to explain resources they used in
planning lessons and similarities or differences between the existing textbooks, their
planned lessons, and the lesson taught with respect to student objectives, classroom
activities, problems, and teacher questions. Teachers were also asked to provide a
rationale for why they added or deleted particular activities, problems, and questions
from the textbook. F urtherrnore, teachers were asked to explain their plan for the next
lesson using textbooks. Questions about teachers’ use of textbooks in planning and

teaching lessons were adopted from the interview protocol employed by Kauffman

(2002)

62

(2) Influential factors on the use of textbooks Questions related to teacher
knowledge, teacher learning goals, teacher view of textbook were asked in order to
examine whether these factors function as possible constraints on or supports to teachers’
use of textbooks. To examine teachers’ knowledge about fractions, teachers were asked to
describe the big ideas in the fraction unit. To examine whether learning goals align with
problems and questions used in teaching in terms of cognitive demands, teachers were
asked to provide learning goals for the specific lesson I observed as well as overall goals
for students’ mathematics learning. Teachers were also asked to answer whether there
were any parts of the book they found particularly helpfirl and in what ways their
textbook was not helpful.

Lastly, teachers were asked to consider the factors which influenced their use of
textbooks when teaching fractions in order to explore what teachers recognized as
constraints or support in their use of textbook. Appendix B describes the specific
questions of the interview protocol.

Case Study Analysis

The case study data analysis involves: (1) analyses of textbook lessons, (2)

analyses of teachers’ taught lesson, and (3) interview data. Because textbook lessons and

teachers’ taught lessons were analyzed in the same way, I describe them together.

63

Textbook Lesson Analysis Textbook lessons and teachers’ taught lesson were

analyzed in light of two aspects—(l) mathematics content covered and (2) cognitive

demands of learning goals, problems and teacher questions.

(1) Mathematics content covered This study first examined the similarities

and differences in the mathematics content of the lessons presented in their textbook and

the lesson teachers taught, in particular with regard to the lessons’ goals and choice of

problems, activities and teacher questions.

(2) Cognitive demands of mathematics content covered This study next

examined the similarities and differences between the textbook and taught lesson in terms

of cognitive demands of these lesson’s goals, problems and questions. First, learning

goals presented in the textbook lesson and articulated by teachers were classified as either

higher level or lower level based on the revised Bloom’s taxonomy (Anderson &

Krathwohl, 2001).

Second, each problem presented in the textbook lesson was classified as either

high-level or low-level based on Stein and Smith’s (2000) framework. Next, referring to

teachers’ planned lesson and my observation notes, each problem used by teachers was

classified into either high-level or low-level.

After that, the ratio of “1/3” is used to determine the overall cognitive demand of

64

problems presented in textbooks and the overall cognitive demand of problems used by
teachers in teaching. For example, if a textbook provides problems that" require high-level
cognitive demand more than 1/3 of the total problems, it was considered as providing
higher level cognitive demand problems. Conversely, if a textbook provides them as less
than 1/3 of the total problems, it was considered as providing lower level cognitive
demand problems. The ratio of 1/3 was selected based on previous studies. For example,
Sanders (1966) articulated that teachers in both instruction and evaluation should devote
“a minimum of one-third” of the time allotted to questioning to levels above memory (p.
156). Referring to ratio of cognitive domain in TIMSS 1999 video study, Kadijevic’

(2002) stated that:

The chosen target percentage to the cognitive domains in gade 8-—
knowing facts and procedures (15%), using concepts (20%), solving
routine problems (40%) and reasoning (25%)—are quite appropriate and
well balanced (p. 98).

In the TIMSS cognitive domain, only reasoning is matched with the high
cognitive levels in teachers’ questions and its percent (25%) is less than the ratio of one-
third that Sanders articulated. Therefore, the ratio of 1/3 is considered an appropriate
criterion to decide the overall cognitive demand of problems and questions in textbooks
and in teaching.

Third, teacher questions presented in textbook lesson were categorized into three

65

cognitive levels—high, medium, and low level--based on the revised Bloom’s Taxonomy
(Anderson & Krathwohl, 2001 ). As in the analysis of problems in terms of the cognitive
demands, if a textbook (a teacher’s guide of the textbook) provided teacher questions that
require hi gh-level cognitive demand more than 1/3 of the total questions, it was
considered as providing higher level cognitive demand teacher questions. Otherwise, it
was considered as lower level.

For the cognitive demand of teacher questions in the lesson taught, interview data
and my observations were used. One interview question was: “What questions did you
pose to have students get involved with these activities?” and “What are typical questions
you frequently use in your teaching?” By using teachers’ reports and my observations, the
cognitive demand of teacher questions used in the lesson taught was decided.

Interview data analysis The interview data were analyzed via the following
steps. First, interviews were transcribed into printed text. I then read the transcripts
carefully. While I obtained a general sense of the information through that reading, I
made a list of themes related to the findings from survey and emergent ideas from the
reading of the data (Cresswell, Tashakkori, Jensen & Shapley, 2003; Glaser & Strauss,
1967; Miles & Huberman, 1994). A third step was to code the content of transcripts in

relationship to these themes. The N-vivo computer software allowed me to categorize the

66

responses. Next, analytic case study narratives were written for each teacher (within-case

analysis) (Miles & Huberman, 1994). Finally, cases were compared to one another based

on the textbook transformation patterns identified from the analysis of each teacher’s

textbook lesson and observed lesson in order to explore possible reasons why teachers

used the same textbook differently or used different textbooks in a similar way (across-

case analysis).

Interview data were triangulated for each case with the survey, the textbook

lesson, and the observation (and notes). For example, teachers were asked to describe

their use of the textbook, and the survey had similar questions, including the percentage

of teaching time or lessons when the teacher would use the textbook. At the same time,

observation data would corroborate what was stated during the interview and in the

survey. To ensure that the narrative of each case and the claims made therein were

accurate and trustworthy descriptions of each case and lesson tapes selected were

reviewed again and compared with the narrative.

67

Chapter 4

RESULTS FROM THE SURVEY:
TRANSFORMATION PATTERNS OF COGNITIVE DEMANDS

This chapter is organized into four sections. The first section presents preliminary

information about the cognitive demand of problems and questions presented in

textbooks and the cognitive demand of problems and questions used by teachers in

teaching. These results are based on the analysis of teachers’ responses on survey items

18 and 19.

The second section presents the results of transformation patterns in terms of the

cognitive demands. Transformation patterns were explored in three phases as shown in

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1.
Textbook Teaching
Transformation phase 1
Survey
Item 18 Problem ; Problem
I Transformation phase 3
Transformation phase 2 ,
Survey . .
Item 19 Questron 1. Questron

 

 

 

 

 

 

Figure 4. 1 Three Analyses of Transformation Patterns

The third section presents the findings from the teachers’ transformations of an
actual lesson plan given in survey item 20. The fourth section summarizes and

synthesizes the findings regarding transformation patterns from the survey, as well as

68

from the lesson plan modifications.
Preliminary Analysis
Descriptive Analysis

Based on an initial summary of teachers’ responses on survey items 18 and 19, a
descriptive analysis was performed to provide preliminary information about the
problems and questions presented in textbooks and those used by teachers in teaching.

In survey item 18, to measure the cognitive demand of problems in textbooks and
the cognitive demand of problems used by teachers in teaching, teachers were asked to
indicate how often various types of problems were presented in their textbooks and how
often teachers used corresponding types of problems in their teaching. A five-point Likert
scale was used, which was coded l= Never, 2 = Rarely (once a unit), 3 = Sometimes (2-3
times a unit), 4 = Often (4-5 times a unit), 5 = Almost all lessons. Table 4.1 displays the

basic statistics obtained from item 18, including the mean and standard deviation.

Table 4. 1 Summary of Descriptive Statistics of the Data for Item 18

 

 

 

Textbooks Teaching
Item # Types of problems M SD M SD
18 a Problems requiring methods of inquiry 3.15 0.98 3.42 0.87
18 b Problems requiring explanation & justification 3.61 1.02 3.95 0.80
18 c Problems requiring computations with speed 2.63 1.07 3.21 1.12
18 (1 Problems involving communication 3.51 1.04 3.87 0.96
18 e Problems requiring developing own methods 3.19 1.11 3.50 0.94
18 f Problems emphasizing the relationships 3.45 0.93 3.70 0.83
18 g Problems requiring the use of representations 3.53 0.93 3.75 0.89
18 h Problems requiring real world application. 3.60 0.88 3.87 0.78

18 i Problems requiringrecalling facts and formulas 2.89 1.14 3.07 1.04

69

Table 4.1 shows that the mean frequency ratings of all types of problems used in teaching

are greater than those presented irn textbooks. It suggests that teachers reported that they

used all types of problems in their teaching more frequently than the textbook did.

Each type problem mentioned in Question 18 was categorized into either high-

level cognitive demand (HCD) or low-level cognitive demand (LCD), based on the Steirn

and Srrnith’s framework (2000). Among nine problems, only two—problems requiring

computations with speed and problems recalling facts and formulas—were classified as

requiring low-level cognitive demand. The rest were categorized as requiring high-level

cognitive demand. Figure 4.2 illustrates the data presented in Table 4.1.

 

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HCD1 HCDZ HCD3 HCD4 HCD5 HCDS HCD7 LCD1 LCDZ

 

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Note: HCD1: Use method of inquiry HCD2=Use explanation/Justification, HCD3= Communication,
HCD4=Develop own method, HCD5=Understand relationships, HCD6=Use representation,
HCD7=Application, LCDl=Compute with speed & accuracy. LCD2=Recall facts

Figure 4.2 Mean Frequency Rating for Different Types of Problems in Textbook and Teaching

70

First, the frequencies in all types of problems used in teaching are slightly higher
than those presented in textbooks, indicating that teachers reported that they used each
type of problems more ofien than the textbook did. In particular, teachers reported that
problems that require computation with speed and accuracy—which demand low-level of
student thinking (LCD1)——were used in teaching much more often than those presented
in the textbooks (note that the difference in frequency is geatest for this question).

Second, among problems categorized into the high-level cognitive demand
category (HCD1 to HCD7), problems that require students to explain and justify their
ideas were used most often (HCD2), followed by problems requiring communication
with peers and teachers (HCD4), and problems that require students to apply concepts to
a real-world problem (HCD7).

To know which level of the cognitive demand of problems was presented more
often in textbooks and which level of the cognitive demand of problems the teachers used
more often in teaching, the difference between the average score of problems requiring
high-level cognitive demand and that of problems requiring low-level cognitive demand
both in textbooks and teaching was calculated for each teacher. A difference of 0 signifies
that teachers reported using hi gh-level problems with the same frequency as low-level

cognitive demand problems. If the difference is less than 0, it means that teachers

71

reported using low-level cognitive demand problems more often than hi gh-level cognitive

demand problems. If the difference is geater than 0, it signifies that teachers reported

using high-level cognitive demand problems more often than low-level cognitive demand

problems. Figure 4.3 shows the results of this analysis.

While many teachers had scores close to zero, there is a noticeable skew towards

the positive end of the continuum (i.e. difference is geater than zero) in both textbook

problems and problems used in teaching. Sixty eight percent of teachers had scores

geater than 0 with regard to textbooks and the same percentage of teachers had scores

geater than 0 with regard to teaching. This indicates that on average, teachers reported

that high-level cognitive demand problems appeared more fiequently in their textbooks

than did low-level cognitive demand problems, and they also reported that, in their

teaching, they used high-level cognitive demand problems more often than low-level

01168.

In addition, although the same percentages of teachers had scores geater than 0 in

textbooks and in teaching, a comparison of the percentage of teachers who scored less

than 0 in textbooks (26.6%) and the percentage of teachers who scored less than 0 in

teaching (27.2%) shows that more teachers reported that they used low-level problems in

teaching more often than such problems presented in their textbooks.

72

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In survey item 19, to measure the cognitive demand of teacher questions

presented in textbooks and that used by teachers in teaching, teachers were asked to

indicate the frequency of the presence of various types of teacher questions botln in their

textbooks and in their teaching. The same five-point Likert scale was used as in survey

item 18, ranging from 1 (Never) to 5 (Almost all lessons). Table 4.2 displays the basic

data for item 19.

Table 4. 2 Summary of Descriptive Statistics of the Data for Item 19

 

Textbook Teaching__

 

 

Item # Types of Teacher Questrons M SD M SD
19 a Pose open-ended questions 3.16 1.07 3.61 0.95
19 b Require students to explain reasoning when giving an answer 3.73 1.02 4.18 0.74
19 c Ask students to explain concepts to one another 3.16 1.17 3.71 0.90
19 (I Ask students to consider alternative solutions 3.38 1.19 3.85 0.95
19 e Ask students to read from a textbook in class. 2.73 1.31 2.65 1.25
19 f Ask students to answer questions about what they have read. 2.73 1.29 2.84 1.25
19 g Ask students to evaluate each other’s ideas 2.56 1.21 3.02 1.12

 

Among the seven types of teacher questions, only the frequency of asking

students to read from a textbook in class while teaching is less than the corresponding

questions found in the textbooks. This suggests that teachers reported that they used this

type of question less frequently than the textbook did.

To explore tendencies of teacher questions presented in textbooks and used by

teachers in terms of the cognitive demand, applying similar cognitive demand categories

as in item 18, these teacher questions were categorized into two levels——(1) high-level

cognitive demand (HCD) and (2) low-level cognitive demand (LCD). Two types of

74

teacher questions—teacher questions that ask students to read fi'om a textbook in class

and teacher questions that ask students to answer questions about what they have read—

were classified as requiring low-level student thinking. The rest were classified as

requiring high-level student thinking. Figure 4.4 presents the results of this analysis.

 

 

HCD1 HCD2 HCD3 HCD4 HCDS LCDl LCD2

 

El Mean frequency of questions in text I Mean frequency of questions in teaching

 

 

 

 

Note: HCD1= pose open-ended questions, HCD2= require students reasoning, HCD3= Ask students
alternative methods, HCD4=Ask students peer-evaluation, HCD5=Ask student explanation, DCDl= Read
from textbook. LCD2= Answer from textbook

Figure 4. 4 Average Frequency Ratings for Different Types of Questions in Textbook and in
Teaching

Note that teachers indicated that they used questions categorized as high-level cognitive

demand (HCD1 to HCD5) more often than those presented in textbooks, whereas they

indicated that they used questions categorized as the low-level (LCD1 & LCD2) in a

similar frequency to their textbooks. Among the questions categorized as high-level

75

cognitive demand, teachers reported that they most often used questions that required that
students explain their reasoning when giving an answer (HCD2).

As was done to analyze results of question 18, the differences between the
average scores of questions requiring high-level of student thinking and that of problems
requiring low-level were obtained. Figure 4.4 illustrates the results.

There is a noticeable skew towards the positive end of the continuum (i.e. geater
than zero equals higher cognitive demand). Sixty percent of teachers had scores geater
than 0 with regard to textbooks and 73.4% of teachers had scores geater than 0 in
teaching, indicating that on average, teachers believed that questions requiring hi gh-level
cognitive demand are presented more often in textbooks and are used more often in
teaching than questions involving low-level cognitive demand.

However, different from the result of Question 18 concerning the cognitive
demand of problems, as the percentages of teachers who had scores geater than 0 in
textbooks and in teaching show, teachers reported that they more often used high-level
cognitive demand questions in teaching than those presented in their textbooks. In
particular, a comparison of the percentages of teachers who scored less than 0 in
textbooks (33.3%) and in teaching (19.5%) shows that teachers reported that they used

low-level questions in teaching less than such questions presented in their textbooks.

76

 

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77

F actorAnalysis

Afier the descriptive analysis was completed, factor analysis was conducted to
test the theoretical categories of this study. Factor analysis is a statistical procedure used
to uncover relationships among many variables. This allows numerous inter-correlated
variables to be condensed into fewer dimensions, called factors (Thompson, 2004).
Although all the survey questions were designed based on theory, factor analysis was
conducted in order to examine whether the sub-items measure each factor cluster together.
For example, problems used in survey item 18 were classified into either high-level or
low-level. Using factor analysis, this study attempted to investigate whether problems
and questions classified as high—level or low-level clustered together.

Exploratory factor analysis was used since the nature of the research itself is
exploratory. The principal component analysis method for initial factor extraction, with
the criterion eigenvalue geater than 1 and the Varimax method of rotation was applied.
Coefficient display formats suppress the absolute value less than .05 in the factor analysis.

Factor analysis results revealed that without any manipulation, two factors were
extracted from the responses to each survey item 18 and 19, which are—(1) the factor
measuring high-level cognitive demand and (2) the factor measuring low-level cognitive

demand. The reliability of each factor was calculated to verify whether a scale

78

consistently reflected the construct it was measuring. Table 4.3 presents the resulting

factors, the survey item numbers comprised of the factor, and its reliability.

Table 4. 3 Results of Factor Analysis and Reliability

 

 

 

Factors measuring Survey item # Textbook Teaching
Cronbach a Cronbach 0:
Problems
High-level cognitive demand Q 18 a,b,d,e,f,g,h 0.89 0.83
Low-level cognitive demand Q 18 c,i 0.61 0.50
Teacher questions
High-level cognitive demand Q 19 a, b,c,d, g 0.88 0.80
Low-level cognitive demand Q 19 e,f 0.78 0.79

 

The Cronbach’s alpha in all factors was geater than or equal to 0.5. In particular, factors
measuring high-level cognitive demand in both textbooks and in teaching and in both
problems and questions are highly reliable (0L 2 0-8 ).

For the exploration of transformation patterns in terms of the cognitive demand of
problems and questions, the factor measuring high-level cognitive demand in botln
problems and questions was selected for several reasons. First of all, factors measuring
low-level cognitive demand had lower reliability than the general acceptance level of
reliability for research instruments, 0.70 (Cheung & Lee, 2001; Lee & Turban, 2001; Teo,
2001), whereas the reliability of all factors measuring high-level cognitive demand
exceeded 0.8. This indicates that the factors measuring high-level cognitive demand were
very reliable (or, at least, teachers understood survey item 19 in the same way).

In addition to this technical reason, there has been much more emphasis on high-

79

level mathematical thinking in mathematics education. For example, Professional

Standards for Teaching Mathematics (NCTM, 1991) stressed the importance of providing

higher level cognitive demand in student learning opportunities. Along the same lines,

Principles and Standards for School Mathematics (2000) described the kinds of tasks that

teachers should play in promoting higher level cognitive demand. In correspondence to

such needs and emphasis, the factor measuring higher level cognitive demand in both

problems and questions was selected and analyzed in exploring transformation patterns in

terms of cognitive demand.

Composite Scores and Average Composite Scores

In the third and final step of the preliminary analysis, individual teacher’s

responses on survey items that measure hi gh-level cognitive demand in both problems

and teacher questions were combined into a single composite measure, called a

composite score, and the composite score was divided by the total number of survey

items. This new score, the average composite score, was utilized in exploring

transformation patterns in terms of cognitive demand of problems and questions. For

example, for the analysis of transformation patterns, teachers’ ratings for the presence of

high-level problems in textbook and in teaching were added togetlner, and then the sum

was divided by the total number of sub—items (i.e.,[ Q19a+Q19b+Q19c+ Q19d+Q19e]/5).

80

These resulting average frequency ratings of high-level problems and questions in

textbooks and those of high-level problems and questions in teaching were utilized for

the analysis of transformation patterns.

81

Transformation Patterns in Terms of Cognitive Demands

Transformation Patterns in Terms of Cognitive Demand of Problems

Figure 4.6 shows a scatterplot displaying the relation between the cognitive
demand of fraction problems presented in textbook and the cognitive demand of fraction
problems used by teachers in teaching. If we assume that teachers’ reports about the level
of problems in the textbooks are predictive of the textbook problems they are using, the
associations shown in the scatterplot in Figure 4.6 are an indication of differences in
teachers’ responses about the frequency of hi gh-level textbook problems and those about

the corresponding high-level problems used in teaching.

 

   

 

 

 

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Figure 4.6 Scatterplot of Teachers’ Responses about the Average Frequency of High-level

Problems

82

In Figure 4.6, we notice that, in general, as the value of the average frequency rating of
high-level problems presented in textbook increases, so does the value of the average
frequency rating of hi gh-level problems used by teachers in teaching. This scatterplot
clearly indicates that, per individual teacher, there is a positive association between the
frequency of hi gh-level problems presented in textbooks and the frequency of high-level
problems used by teachers in teaching. This also suggests that there is a tendency for low
frequencies of the presence of hi gh—level problems in textbooks to be associated with low
frequencies of the presence of hi gh-level problems in teaching.

To find the line that best represents the relationship between the frequency rating
of high-level problems in textbooks and that in teaching, correlation and regession were
used. The best-fit line shown in the scatterplot summarizes the linear relationship
between the presence of high-level problems in textbooks and that in teaching. The
strength of this relation, r2 70.53, indicates that according to teachers’ reports, about 53
percent of the variance in the use of hi gh-level problems in teaching was accounted for
by the frequency of hi gh-level problems in textbooks.

This positive relationship shows two different transformation patterns. One
pattern is what I will refer to as an H—H pattern. This pattern occurs when, teachers have

textbooks that provide higher level problems, and they use the higher level problems in

83

their teaching. The other pattern is what I will refer to as an L—L pattern in which teachers
have textbooks that demand lower level cognitive demand problems and they use
problems in teaching that demand lower level of student thinking.

Other possible patterns exist in addition to these two patterns, since there are
variations around the linear relationship. For example, teachers represented on the dashed
vertical line shown in the scatterplot above, ageed on the frequency of high-level
problems in their textbooks, whereas none of them indicated the same frequency of high-
level problems used by them in teaching. Notice that there is much variation in the
frequency of hi gh-level problems used by these teachers in teaching, ranging from
approximately 2.5 to 4.5. These variations show that other transformation patterns exist
as well.

To explore the transformation patterns in terms of the cognitive demand of
problems, the frequency rating scale 3 out of a 5-point Likert scale ( 2-3 times a unit) was
selected based on previous studies (e. g., Sanders, 1966; Kadijevié, 2002; Martin, Mullis,
& Chrostowski, 2004) so that the cognitive demand of textbook problems and the
cognitive demand of problems used in teaching were categorized into requiring either
higher level or lower level. For example, Sanders (1966) articulated that teachers in both

instruction and evaluation should devote “a minimum of one-thir ” of the time allotted to

84

higher level cognitive demand above in both instruction and evaluation (p. 156). Since

one unit in a textbook ranges typically from a minimum of 4 lessons to a maximum of 10

lessons, a scale of 3 (2-3 times a unit) was considered to adequately correspond to the

suggested ratio of one-third of the time allotted.

Using a scale point of 3 as an average frequency rating on both axes, I divided the

scatterplot into 4 quadrants representing the four possible transformation patterns, H-H,

H-L, L-H, and L-L. For example, if an average fi'equency rating of high-level problems

presented in textbooks was geater than or equal to 3, textbooks were considered as

requiring higher level cognitive demand of student thinking. Otherwise, the textbooks

were categorized as providing lower level cognitive demand. For the cognitive demand of

problems used by teachers in teaching, the same procedure was employed.

Figure 4.7 illustrates the resulting transformation patterns in terms of the

cognitive demand of problems.

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Average frequency rating (AFR) of hlgh-level problems In Text

Figure 4. 7 Scatterplot Illustrating the Process of the Categorization of Transformation Patterns in

Terms of the Cognitive Demand of Problems

Three transformation patterns were identified in the data in terms of the cognitive
demand of problems: (1) L-L, (2) L-H, and (3) H—H. One pattern, H-L, was not
represented by any individual teacher in this study. Within the L-L pattern, since teachers’
average fi'equency rating of high-level problems is lower than 3, the cognitive demand of
problems both in textbooks and in teaching was categorized as lower level. In H-H,
since teachers’ average frequency rating of hi gh-level problems is geater than or equal to
3, the cognitive demand of problems both in textbooks and in teaching were categorized
as higher level. In the L-H pattern, teachers’ responses about the cognitive demand of

problems in their textbooks were categorized as lower level and those about the cognitive

86

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demand of problems in teaching were as higher level. Table 4.4 shows a frequency

distribution of the transformation patterns.

Table 4.4 Frequency Distribution of the Transformation Patterns of Problems

 

 

Pattern Cognitive Demand of Cognitive Demand of Frequency Percent
Problems in Textbook Problems in Teaching (n=l 69) (100%)
1 Low Low 12 8
2 Low . , High 32(4) 18
'3' ' . High ‘ f [ Low , .0 ' _0_.
4 High High 125 (7) 74

 

Note: The number in ( ) represents the numbers of teachers who had 3 as average composite score in each

pattern

These frequencies reveal interesting trends. First, a relatively large percentage of
teachers (74%, 125 out of 169) were categorized as having textbooks that require higher
level of student tlninking. Second, there are different transformation trends between
teachers categorized as having textbook problems that require lower level cognitive
demand and those categorized as having textbook problems with higher level cognitive
demand. According to teachers’ self-reports, when teachers have textbook problems that
require a lower level of student thinking, 30% teachers (13 out of 44) were categorized as
maintaining its cognitive demand level (L-L pattern), whereas 70% of the teachers (31
out of 44) were categorized as increasing the cognitive demand of problems in teaching
(L-H pattern).

Interestingly, all teachers categorized as having textbooks that require higher level

cognitive demand were categorized as using problems that require higher level cognitive

87

demand of student tlninking. These results revealed that when teachers perceived their

textbook problems as requiring higher level of student thinking, these teachers reported

that they at least maintained the same level of cognitive demand in teaching. Although

there exists the possibility of the effect of various other factors on this (e.g., student needs,

lack of supplementary materials, curricular mandates, etc.), these findings suggest that

teachers in this study reported that they either maintained or increased the cognitive

demand of problems presented in textbook when they used problems in their teaching.

This implies that the cognitive demand of textbook problems plays an important role in

maintaining the cognitive demand of problems in teaching.

Transformation Patterns in Terms of Cognitive Demand of Questions

Teachers’ average frequency ratings of hi gh-level questions in textbooks and

those of high-level questions in teaching were obtained from responses to Question 19.

Figure 4.8 shows the resulting scatterplot that displays the association between the

variables. Again, if we assume that teachers’ reports about the level of teacher questions

in the textbooks are predictive of the textbook questions they are using, the associations

shown in the scatterplot in Figure 4.8 are an indication of differences in teachers’

responses about the frequency of high-level textbook questions and those about the

corresponding hi gh-level questions used in teaching.

88

 

   

 

 

 

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Average frequency rating (AFR) of high-level questions In Text
Figure 4. 8 Scatterplot Illustrating Teachers’ Average Frequency Rating (AFR) of High-level

Teacher Questions in Textbooks against Teachers’ AFR of High-level Teacher Questions in
Teaching

This scatterplot indicates that, in general, as the value of the average frequency rating of
the presence of hi gh-level teacher questions in textbooks increases, so does the value of
the average fiequency rating of the use of hi gh-level teacher questions in teaching.

The best-fit line shown in the scatterplot summarizes the linear relationship
between the presence of hi gh-level teacher questions in textbooks and the use of hi gh-
level teacher questions in teaching. The strength of this relation, 1'2 =0.457, indicates that,
according to teachers’ reports, about 46 percent of the variance in the use of high-level

teacher questions in teaching was accounted for by the presence of hi gh-level teacher

89

questions in textbooks. This positive relationship indicates that there are two prominent

transformation patterns in terms of the cognitive demand of questions. One pattern is the

H-H, where teachers have textbook questions that require higher level of student thinking

and they use questions that also require higher level of student thinking. The other pattern

is the L-L pattern, in which teachers have textbook questions that require lower level of

student thinking and they, in turn, use questions in teaching that require lower level of

student thinking.

In addition to these two patterns, other patterns exist since there are variations

around the linear relationship, as the dashed vertical line shown in Figure 4.8 illustrates.

Teachers represented on that line ageed on the presence of high-level teacher questions

in their textbooks. However, none of these teachers reported the use of hi gh-level teacher

questions in their teaching at the smne level. This suggests that, although these teachers

ageed on the same level of cognitive demand in textbook questions, they responded

differently about the cognitive demand of questions they used in teaching. These

variations show that other transformation patterns exist as well.

The same criteria (the rating scale of 3) and the same procedures were used to

determine the cognitive demand of teacher questions in textbooks and that of teacher

questions in teaching. Based on a rating scale of 3, teachers’ average frequency rating of

90

high-level teacher questions in textbooks and that in teaching were categorized into either
requiring higher level or lower level cognitive demand. Figure 4.9 shows the resulting

categories of the transformation patterns in terms of the cognitive demand of teacher

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Average frequency rating (AFR) of high-level questions In Text

Figure 4.9 Scatterplot Illustrating the Categorization of the Transformation Patterns in Terms of

the Cognitive Demand of Questions in Textbooks and in Teaching

The same types of transformation patterns as in the cognitive demand of problems
were also identified in terms of the cognitive demand of teacher questions: (1) the L-L
pattern, (2) the L-H pattern, and (3) the H-H pattern. The H-L patterrn was once again

empty. Table 4.5 shows the frequency distribution of each transformation pattern in terms

91

of cognitive demand of teacher questions from textbook to teaching.

Table 4.5 Frequency Distribution of T transformation Pattern of Teacher Questions

 

 

Cognitive Demand of Cognitive Demand of Frequency Percent

Questions in Textbook Questions in Teaching (n=1 69) (100%)
Low Low 27 l 5
Low High 43 26
High Low 0 0
High High 99 59

 

Similar to the findings in transformation phase 1, more than half of teachers (59%, 99 out

of 169) were categorized as having textbook questions that require higher level of student

tlninking. In addition, these teachers reported using teacher questions in their teaching that

demand higher level of student tlninking. These results revealed that when teachers

reported the cognitive demand of teacher questions in textbooks as higher level, they

declared that they at least maintained their cognitive demand in teaching (notice that the

H-L quadrant is empty).

Interestingly, when teachers perceived their textbook questions as demanding

lower level of student thinking, more than half (63%, 44 out of 70 teachers) reported that

they increased the cognitive demand of teacher questions in teaching. Taken together with

the findings in phase 1, this finding suggests that the cognitive demand of textbook

problems and questions associates with the cognitive demand of problems and questions

used by teachers in teaching.

92

Transformation Patterns between Problems and Questions in Teaching

This section presents the results of transformation patterns in phase 3, the
relationship between the cognitive demand of fraction problems in teaching and the
cognitive demand of teacher questions in teaching (see Figure 4.1). Results of responses
to survey item 18, measuring the frequency of high-level problems in teaching and survey
item 19, measuring the frequency of high-level questions in teaching were analyzed.

Figure 4.10 shows the resulting scatterplot.

 

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Average frequency rating (AFR) of high-level problems in Teaching

Figure 4.10 Scatterplot of Teachers’ Responses about the Average Frequency of High-level
Problems in Teaching against the Average Frequency of High-level Questions in Teaching

It is noticeable that the distribution is negatively skewed, indicating that most teachers

93

reported using hi gh-level problems frequently in their teaching, as well as high-level
questions in teaching.

In addition, Figure 4.10 shows that as the value of the frequency rating of high-
level problems in teaching increases, so does the value of the frequency rating of high-
level questions in teaching. It indicates that, for an individual teacher, there is a positive
association between the frequency of hi gh-level problems used in teaching and the
frequency of hi gh-level teacher questions used in teaching. This also suggests that there is
a tendency for low frequencies of hi gh-level problems used in teaching to be associated
with low fiequencies of high-level teacher questions in teaching. These relationships
suggest two prominent transformation patterns. One pattern is the H-H pattern in which
teachers’ responses on the cognitive demand of problems in teaching and those of teacher
questions in teaching are both categorized as requiring higher level of student thinking.
Also present is the L-L pattern, requiring lower level cognitive demand problems in
teaching, and lower level cognitive demand questions in teaching.

The best-fit line shown in the scatterplot summarizes the strength of the linear
relationship, r2 20.597, between the frequency of hi gh-level problems and the fi'equency
of high-level teacher questions in teaching. This relationship is the strongest among the

three transformations (notice that the strengths of the linear relationship in phase 1 and

94

phase 2 are 0.53 and 0.456 respectively). This might be expected, since both variables in
Transformation 3 were involved in teaching. However, there are also variations around
the linear relationship, indicating that other transformation patterns exist.

To examine the possible transformation patterns, the average frequency ratings
of problems and teacher questions in teaching were categorized into either higher level
cognitive demand or lower level, using a scale point of 3 as the boundary between the

two categories. Figure 4.11 shows the result of the categorization.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Average frequency rating (AFR) of high-level problems in Teaching

Figure 4.11 Scatterplot Illustrating the Categorization of the Transformation Patterns in Terms of
the Cognitive Demand from Problems to Teacher Questions in Teaching

Four transformation patterns were identified fiom the figure. In addition to the H-H, the

L-L, and the L-H patterns, the H-L pattern, which did not appear in transformation phase

95

l and 2, was exhibited in this transformation that presents the relationship between the
cognitive demand of fraction problems in teaching and the cognitive demand of teacher
questions in teaching. Using average frequency ratings in the H-L pattern, teachers’
responses regarding the cognitive demand of the problems used in teaching were
categorized as requiring higher level of student thinking, and those describing the
cognitive demand of the questions used in teaching were categorized as requiring lower
level of student thinking. In the L—H pattern, teachers’ responses about the cognitive
demand of problems in teaching were classified as lower level cognitive demand, but the
cognitive demand of teacher questions in teaching were categorized as demanding higher

level of student thinking. Table 4.6 shows the frequency of each transformation pattern.

Table 4. 6 Frequency Distribution of Transformation Patterns in Phase 3

 

 

Pattern Cognitive demand of Cognitive demand of Frequency Percent
Jroblems in teaching questions in teaching (Total=l 69L (100%)

1 Low Low 7 4

2 Low High 6 (2) 4

3 High Low 19 (2) ll

4 High High 137 (5) 81

 

Note: the number in ( ) represents the numbers of teachers who had 3 as average composite score

in each pattern

All four possible patterns occur (L-L, L-H, H-L, and, H-H), with H-H being the most
prevalent. The H-L pattern and the L-H pattern suggest that the cognitive demand of
problems can be firndamentally changed by the cognitive demand of questions teachers

use. This finding indicates that it is necessary to explore the factors that influence

96

teachers; discussion to change the cognitive demand of questions.
Validation from Lesson Plan Modification

This section presents the findings from the analyses of teachers’ lesson plan
modifications to the survey presented in Question 20. In this item, teachers are provided a
lesson planning scenario for teaching the concept of equivalent fractions to 4th gaders,
and they were asked to describe the modification(s) they would make to the lesson
components—student objectives, classroom activities, problems, and questions—all of
which were lower level in the original plan, itself categorized as lower level overall. The
daily lesson plan utilized in survey item 20 appears in Appendix A.

This analysis was conducted to validate one common transformation pattern, the
L-H pattern, in which teachers who have textbook problems and questions that require
lower level of student thinking, increase their cognitive level to higher level when they
use problems or questions in their teaching. This pattern appeared consistently in the
analyses of transformation patterns in the three phases. However, this finding is rather
inconsistent with the findings from previous studies. For example, Kennedy (2005)
reported that teachers decreased the cognitive demand of textbooks by modifying difficult
problems into easy ones. While little is known, in general, about how teachers use

textbooks (Ball, 2000), still less research exists that examines how teachers use textbooks

97

that provide lower level of problems and questions, which is still prevalent.

In addition, as described earlier, transformation patterns from textbook to teaching

were identified from survey data—which relies on teachers’ self-reports. One limitation

of self-report for this study is the potential for teachers to provide biased responses to the

survey because they feel that they should respond to the questions in “acceptable” or

“socially desirable” ways; or teachers might unknowingly provide misleading responses

to the survey questions. For these reasons, to validate this finding and to know more

about what and how teachers modify the lower level problems and teacher questions in a

lesson plan, survey question 20 was used.

Among 169 survey participants, 51 teachers (30%) answered that either they

would use the lesson plan exactly as it is. The rest, 118 teachers (70%), responded that

they would modify the lesson plan. These 118 teachers’ modification(s) of the original

lesson plan were explored with the following three questions: (1) How many teachers

provided their modification(s) in a way that survey item 20 required, such as including

student objectives, classroom activities, problems, and question? (2) What did teachers

actually modify from the daily lesson plan with regards student objectives, classroom

activities, problems, and questions? (3) Did teachers increase the cognitive demand of the

given lesson with respect to problems and questions and how did they do so?

98

How Many Componenm Teachers Provided in Their Lesson Plan

Before exploring transformation patterns in teachers’ lesson plan modifications,

the lesson components--student objectives, classroom activities, problems, and

questions—were defined based on previous studies. (1) In this study, a student objective

refers to a statement describing what students should learn or be able to do as a result of

student engagement in classroom activities, problems, or teacher questions. Some

teachers stated student objectives explicitly and other teachers did it implicitly. For

example, only five teachers out of 118 teachers clearly labeled student objectives in their

modification. The rest of the teachers stated them within classroom activities, problems,

or teacher questions. As a clue for identifying student objectives stated implicitly, “to”

was used. Some examples of student objectives stated implicitly are: “To find the

discovery on their own,” “To understand the concept of equivalent fractions,” etc.

(2) In this study, a classroom activity refers to a request that students use physical

objects, or to participate in games related to the lesson. As a cue for identifying classroom

activities, a statement requiring students to do sometlning physically was used. For

example, in a following statement--“Have students use manipulatives,” “use

manipulatives” indicates a classroom activity.

(3) In this study, a problem is defined to be a mathematical situation or object for

99

the students to solve or figure out. For example, in the statement below, “to find the

fraction cards equivalent to V2” is a mathematical object for students to figure out.

“I would first have them find '/2 then they would look through the rest of
their cards to find ones that matched up exactly with 1/2”

(4) In this study, a question is a statement indicated by the verb “to ask”. Urnlike

the cases of student objectives, teacher questions stated implicitly were not counted. For

example, in the statement above, “have them find V2” could involve using questions.

However, the statement could also involve showing students how to find 1/2; and because

of this ambiguity, such statements were not counted as a “question”.

The analysis revealed that among 118 teachers who answered that they would

modify the lesson, 10 teachers (9%) included the four components in their lesson

modification; 17 (14%) teachers included three components; 37 (31%) included two

components; 21 (18%) included only one component. A relatively large proportion of

teachers (33 out of 118, 28%) indicated that they would modify the given lesson in

survey item 20, but did not provide any modification.

Among the four components, teachers attempted to modify (or supplement) the

classroom activities most often (76 teachers out of 118), followed by problems (56

teachers), student objectives (43 teachers), and then teacher questions (15 teachers).

100

What Teachers Modified in Each Component

Table 4.7 presents categories of teachers’ modifications in student objectives,

classroom activities, problems and questions and the frequencies of each category.

Table 4.7 Category and Frequency of Each Component in Teachers ’Lesson Plan Modifications

 

 

 

 

 

 

Components Category Examples *N 0f
Teachers
0 Set up content-specific Understand the concept of equivalent 3 5
Student student objectives fractions
objectives 0 Set up non-content-specific My students need to be an active part 8
student ob'gctives of the activity
0 Add or supplement Have students draw pictures of 36
additional problems different equivalent fractions
0 Modify the given problems I would change the denominators to
to make them easier for 261 when first introducing this 2
Problems students concept
0 Add the real world Attempt to make the fractional work
situation problems more “real” putting it into a story 12
problem format
0 Select and use some of I may still use some of the sample
problems in a given lesson problems to have them build and 5
solve.
0 Add concrete materials or I would give the students an
Activity hands-on activities opportunity to use manipulatives when 76
first solvrng the problems.
0 Use group works or games I would add on a mathematical game 8
to reinforce ideas taught
0 Use different questioning I would also include a writing portion
under each segnent of slightly
Questions different questions having students 7
explain how they know their answer is
correct or why it is.
0 Modify pacing Divide the given one lesson into 7
several lessons
Others . Change teaching approach Have students explore or discover the
(teacher-centered to student- relationship between two fractions and 13

centered)

find the patterns on their own

 

Note: N of teachers in each category is out of 118 teachers who described their modification.

101

How Teachers Increased the Cognitive Demand of the Lesson

Table 4.7 shows that 35 teachers set up different student objectives from the daily

lesson, 36 teachers added additional problems, 76 teachers included hands-on activities or

manipulatives for classroom activities, and 7 teachers planned to use different

questioning. These teachers’ modification(s) were explored to determine whether and

how teachers maintained or increased the cognitive level of the given lesson. To do so,

the framework that describes teachers’ modification associated with maintenance or

change in the level of cognitive demand of each component was developed based on the

revised Bloom’s Taxonomy (Anderson & Krathwohl, 2001) and Stein and Smith (2000).

Although the factors increasing or decreasing cognitive demands for all four components

were developed, student objectives were not used to determine the change of the

cognitive demand level in a lesson plan. This is because student objectives differ from the

rest of the components in a lesson plan in that student objectives present one’s goals for

student learning, whereas the rest of the components are means selected by teachers to

achieve these goals. Table 4.8 presents the framework that illustrates teachers’

modifications in each component of a lesson plan, which is associated with maintenance

or change in the cognitive demand.

102

 

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104

With this framework, each individual teacher’s modification in classroom

activities, problems and questions was classified in three categories—(l) Lower level,

(2) Higher level and (3) Hard to tell. In determining the cognitive demand level of each

individual teacher’s modification, the category of “hard to tell” was collapsed into the

other two categories. For example, if a teacher’s modifications in some components were

categorized as “hard to tell” and those in other components were categorized as lower

level, this teacher’s modifications were categorized as “decrease the level of cognitive

demand of the lesson.” On the contrary, if a teacher’s modifications in some components

were categorized into “hard to tell” and those in other components were categorized as

higher level, the overall modifications were then categorized as “increase the level of

cognitive demand of the lesson”. None of the teachers were categorized as both lower

level and higher level.

Table 4.9 presents the frequencies of teachers who were identified as increasing

the cognitive level of the given lesson plan and those identified as maintaining or

decreasing to a lower level.

Table 4. 9 Frequencies of Transformation Patterns in Teachers ’Lesson Plan Modifications

 

 

 

Transformation patterns N of
The daily lesson plan Modified lesson plan teachers
Decrease the cognitive demand L L 13 (11%)
Increase the cognitive demand L H 19 (16%)
Hard to tell L Hard to tell 86 (73%)

 

105

A large proportion of teachers were classified in the category “hard to tell.” However,

consistent with the findings from teachers’ self-reports, these results indicate that some

teachers maintained the lower cognitive levels of the given textbook lesson, whereas

other teachers increased the lower cognitive demand level toward a higher level. These

findings validate the trends from teachers’ self-reports that, when teachers have textbook

problems and questions that require lower level of student thinking, some teachers tend

to increase their level, at least in their planning.

After this analysis was completed, I further explored to what extent teachers’

responses in question 20 (see table 4.9) were consistent with those in survey questions 18

and 19. In particular, since a large number of teachers were categorized as the “Hard to

tell” in question 20, I compared teachers’ responses across these three different survey

questions and explored whether and their responses were similar in light of the cognitive

demand.

Among 169 survey participants, 51 teachers answered that they would use the

lower level problems and questions presented in the lesson plan exactly as it is. Among

118 teachers who attempted to modify the lesson plan, 13 teachers were categorized as

maintaining or decreasing the given lesson plan (see Table 4.9). Therefore, the total

number of teachers decreasing the given lesson plan are 64 teachers. Table 4.10 presents

106

the fi'equency of teachers categorized into three different categories by lesson plan
modification.

Table 4. 10 Frequency (Percentage) of Dijferent Transformation Patterns Exhibited in Questions
18 and I 9 by Types of Lesson Modification

 

Type of Lesson modification

 

 

 

Decrease the Increased the Frequency

Pattern cognitive demand cognitive demand Hard to tell (%)

(L-L) (L-H)

Q18' H-H 39 (30%) 14 (11%) 72 (58%) 125 (100%)
Problems L-H 12 (38%) 3 (6%) 17 (53%) 32 (100%)
L-L 3 (25%) 2 (17%) 7 (58%) 12 QOO‘VQ
Q19' H-H 28 (28%) 13 (13%) 58 (59%) 99 (100%)
Questions L-H 14 (33%) 3 (7%) 26 (60%) 43 (100%)
L-L 12 (41%) 3 (11%) 12 (48%) 27 (100%)

64 (13+51) 19 86 169

 

A large number of teachers categorized by questions 18 and 19 as increasing the
cognitive demand (i.e., H-H & L-H) were classified as “Hard to tell” in lesson plan
modification; some of them were even classified as decreasing the cognitive demand in
lesson plan modification. In contrast, some teachers categorized by questions 18 and 19
as decreasing the cognitive level (i.e., L-L) were found to be increasing the cognitive
demand in lesson plan modification.

Using teachers’ lesson plan modification, this study attempted to minimize issues
of accuracy of self-reports. Although only a small number of teachers were categorized

as increasing the cognitive demand of the lesson, these results validate the existence of

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the L-H pattern. However, a large number of teachers, in particular, ones identified as

increasing the cognitive demands, were categorized as "Hard to tell".

One possibility is methodological limitation. Participants were asked to respond

to 25 survey items in a limited time. The number of items and time might act as

constraints for teachers to respond to the survey. In a situation where teachers were asked

to describe their modification in detail as in question 20, teachers are likely to provide a

brief lesson plan, and therefore their modification can easily be categorized as “Hard to

tell.”

Another possibility is that teachers might unknowingly provide misleading responses

to the survey questions. Research suggests that teachers sometimes truly believe they are

embracing pedagogical reforms, but in pratice, their teaching is a mixture of the vision of

traditional methods and reformers (Cohen, 1990). As table 4.7 shows, a large number of

teachers modified the lesson by adding the use of manipulatives/hands-on activities or real-

world problems. However, these modifications do not mean that teachers increase the

cognitive demands of the given lesson. Without more information about how teachers use

these manipulatives or real-world problems in their lesson plan modification, and/or without

observing their actual lesson, it is hard to decide whether they would increase the cognitive

demands of the lesson. Through the lesson plan modification I attempted to provide more

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accurate results, but these methodological limitations provide the necessity of the case
studies.
Summary

This chapter presents the findings from the survey that examined transformation
patterns when teachers moved content from text to teaching in terms of cognitive
demands. Transformation patterns were explored in two ways using a survey: (1)
teachers’ self-reports and (2) teachers’ lesson plan modifications.

First, transformation patterns were explored by looking at the relationship
between the presence of hi gh-level problems and teacher questions in their textbooks and
the presence of high-level problems and teacher questions in teaching. In survey item 18
(and item 19), teachers were asked to indicate on a five-point Likert scale the frequency
of the various types of problems (and teacher questions) in textbooks and then in their
teaching. Using a scale of 3 out of 5 Likert scale, teachers’ responses about the frequency
of problems in textbooks and in teaching were classified into either lower level or higher
level. For example, if a teacher’s average frequency rating of problems in textbooks is
greater than or equal to 3, textbook problems is classified as requiring higher level
cognitive demand. Otherwise, it is classified as requiring lower level. Transformation

patterns from textbook to teaching were analyzed in three phases: (1) problems to

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problems, (2) questions to questions, and (3) problems in teaching to teacher questions in

teaching.

Three transformation patterns from textbook to teaching were exhibited in terms

of the cognitive demand of problems and teacher questions (i.e., L-L, L-H, & H-H). Four

transformation patterns appeared between the cognitive demand of problems in teaching

and that of teacher questions in teaching (i.e., L-L, L-H, H-L, & H-H).

Next, to validate these findings from survey items 18 and 19, in particular the L-

H pattern, whether and how teachers modify the lesson plan classified as lower level

cognitive demand of student thinking. Two patterns appeared: L-L & L-H.

Three common findings emerged fiom teachers’ self-reports and lesson plan

modifications. First, teachers claimed to at least maintain the cognitive demand of

textbook problems and questions in their teaching (i.e., L-L & H-H pattern). This finding

resonates with those of previous studies (e. g., Remillard & Bryans, 2003; Drake &

Sherien, 2002), suggesting that the cognitive demand of textbooks plays an important

role in deciding the cognitive demand of problems and teacher questions used in teaching.

Another common finding was that when teachers perceived their textbook

problems and questions as requiring lower level cognitive demand, some teachers tried to

increase its cognitive demand in their teaching (i.e., L-H) whereas others maintained the

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lower level cognitive demand in teaching. This finding is inconsistent with the findings

from previous studies (e. g., Cohen, 1990; Kennedy, 2005; Stigler & Hiebert, 2006).

However, the analysis of teachers’ lesson plan modifications validated L-H pattern.

Furthermore, the H-L pattern appeared in describing the relationship between the

cognitive demand of problems in teaching and that of teacher questions in teaching. This

pattern has been ofien reported from previous studies (e. g., Cohen, 1990; Kennedy, 2005;

Henningsen & Stein, 1997; Stein et al., 1996; Stigler & Hiebert, 2006). Although

teachers in this pattern reported using problems that required a higher level of student

thinking, they eventually decreased the cogritive demand of the problems by using a-

lower level of teacher questions in teaching. This pattern urges us to examine influential

factors on teachers’ decision-making.

There are several aspects to consider. First, in contrast with the previous studies

(e. g., Stigler & Hiebert, 2006), a large number of participants were categorized in the H-

H pattern. In addition, none of teachers were categorized as the H-L pattern in describing

the relationship between textbook problems and questions and problems and questions

used by teachers in teaching. This discrepancy may be related to the use of the code of 3

as the cut off score in classifying the cognitive demand of problems and questions in

textbook and in teaching into either higher level cognitive demand or lower level

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cognitive demand. Although this study chose this cut off score based on the previous
studies (e. g., Sanders, 1966), perhaps the use of code of 3 as the cut off score might be
too generous. We may see the H-L pattern and less teachers in H-H pattern if a code of 4
was used.

In addition, this study used high-level cognitive demand problems and questions
in exploring textbook transformation patterns. What would happen if one compared
teachers on their frequency of using the low-demand options in questions 18 and 19?
This is an alternative way of asking the same research question that may avoid teachers'
embracing of theoretically fashionable terms. However, due to the small number of
survey items measuring the low-demand options, in particular, relatively low reliability,
high-level cognitive demand problems and questions were used. Future research should
explore transformation patterns using both hi gh-and low-level problems and questions in
survey in order to capture more accurate portrait of what teachers do in their classrooms
and how they trasforrn their textbook in their classroom. In the next chapter, I describe

factors associated with these different transformation patterns.

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Chapter 5

RESULTS FROM THE SURVEY:

INFLUENTIAL FACTORS ON TRANSFORMATION PATTERNS
IN TERMS OF COGNITIVE DEMANDS

Data reported in the previous chapter revealed three transformation patterns from
textbook to teaching in terms of the cognitive demand of problems and questions, and
four transformation patterns in terms of the cognitive demand between problems in
teaching and teacher questions in teaching. This chapter presents the findings from the
survey that examined which factors were associated with the various textbook
transformation patterns exhibited. The research question I sought to answer was: What
factors are associated with various textbook transformation patterns?--that is, I wondered
what factors distinguish teachers exhibiting one transformation pattern from teachers in
the other patterns in phase 1, phase 2, and phase 3?

This chapter is organized into four sections. The first section begins with
preliminary information about the dependent variables and the independent variables.
Descriptive statistics were utilized and interpreted to provide initial summary of teachers’
responses on each sub-item of the independent variables. The second section briefly
introduces a multinomial logistic regression used for the analysis and provides the

rationale of why multinomial logistic regressions were used. The third section presents

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the findings from multinomial logistic regressions that illustrate the factors affecting on
the distinction between transformation patterns exhibited in problem-problem text phase,
question-question text phase, and problem-question enactment phase, respectively.
Preliminary Analysis of Dependent and Independent Variables
Dependent variables in this study are transformation patterns exhibited in each
phase. For example, in the problem-problem text phase, teachers are categorized into
three pattems—L-L pattern, L-H pattern, and H-H pattern. The dependent variables are
comprised of these three groups of the transformation patterns. Table 5.1 presents the

number of teachers categorized into each category of dependent variables.

Table 5. 1 Descriptive Statistics of Dependent and Independent Variables Explored

 

 

 

 

Transformation phases Category of Dependent variables N 83 :6? 213:3

L-L pattern 12

Problem-problem text phase L-H pattern 32
H-H pattern 125

L-L pattern 27

Question-Question text phase L-H pattern 43
H-H pattern 99

L-L pattern 7

. L-H pattern 6
Problem-Questron enactment phase H-L pattern 19
H-H pattern 137

 

In order to examine the factors influencing group membership of transformation

patterns, survey items measuring three levels of factors—contextual-level, individual-

level, and teachers’ opportunity-to-learn factors—were developed. Factor analyses were

performed in order to examine whether various survey items really measure what they

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intend to measure. Teachers’ responses for each variable were factor analyzed to

determine which item should be discarded and which should be used to create summated

scales. Table 5.2 presents the results. The survey is attached in Appendix A.

Table 5. 2 Descriptive Statistics of the Independent Variables Explored in Factor Analysis

 

 

Factor Survey Cron
Level N b Name of Factors N Item b h
um er ac
Number
Context- 1 District Curriculum policy 9
ual Same textbooks across grade level 123
Other textbook Policy 46
2 Teacher perception of Student math 8
achievement
Mostly high achieving 8
Other achievement level 161
3 District cuniculum Framework 23
Alignment of district curriculum 1 69
framework with NCTM
4 Teacher perception of State-wide Tests 23
Alignment of state-wide tests with l 69
NCTM
5 Type of textbook 1 58 12
Standards-based 103
Conventional 55
Individu Teacher Knowledge
'31 6 Content Knowledge 169 21 a,b,c,d,e 0.85
7 Pedagogical Knowledge 169 21 g,h,i,j,k 0.85
8 Curriculum Knowledge 169 21 l,m,n, o 0.81
Teacher beliefs about teaching
9 Constructivist approach 169 16 a,d,g,h,j 0.82
10 Traditional approach 169 16, b,c, f,i 0.79
11 Teacher View on Textbook 169 13 a,b,c,d,e 0.93
Teacher Emphasis of Student Objectives
12 Understanding-oriented 169 22 b,c 0.76
13 Procedural Fluency oriented 169 22 a,d 0.84
14 Teacher Fidelity of Textbook 14 b,c,d,e 0.82
Teacher- Type of Content in PD
Learnin- 15 PD Content Knowledge 169 24 a,b,c,d,e 0.90
8' l6 PD Pedagogical Content Knowledge 168 24 g,h,i,j 0.84
0pm“ 17 PD Curriculum Knowledge 169 24 l,m,n 0.78
Type of pedagogy in PD
18 Presentation 169 25 a (-b) 0.57
19 Observation 169 25 c, d 0.75
20 Active Participation 169 25 e,f,g,h,i 0.76
21 Total PD hour 157 25

 

A total of 21 factors in Table 5.2 were utilized as independent variables in order

to explore influential factors on the distinction between the transformation patterns. After

factor analyses, a composite score and an average composite score were calculated. The

composite score was created by simply adding individual item scores and the composite

score was divided by the total number of survey items. This new score, the average

composite score, was utilized in exploring the factors influencing transformation patterns.

For example, for “teacher view on text”, individual teacher’s responses on survey item

13 (a, b, c, d, & e) were added together and then divided by 5. This new score was used

in exploring influential factors on textbook transformation patterns. Descriptive statistics

were utilized to provide initial summary of teachers’ responses on each sub-item used to

measure three levels of factors.

Descriptive Findings from the C ontextual-Level Factors

Five factors were derived from previous studies on textbook at the contextual-

level--( 1) “District policy regarding the textbook series”, (2) “Teacher perception of

overall student mathematics achievement”, (3) “Teacher perception of state curriculum

framewor ”, (4) “Teacher perception of state-wide tests”, and (5) “Type of textbook”

(see Appendix A for the description of each survey item).

In survey item 9, teachers were given five options for district policies regarding

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textbook use and were asked to indicate the one that applied them. In survey item 8,

teachers were asked to indicate the mathematics achievement level of students in their

classroom. Table 5.3 presents the resulting frequency distributions.

Table 5. 3 Frequency of Dzflerent Textbook Policies and Student Math Achievement Level

 

 

 

Factors Sub-category Frequency (n=169)
a. Same textbook series in the building 123 (73%)
District b. Different series across grade level in the building 26 (15%)
textbook c. Own textbooks per schools 10 (6%)
policy (I. No policy 3 (2%)
e. Different policy from those listed above 7 (4%)
a. Mostly high achieving 8 (5%)
Student b. Mostly average achieving 98 (5 8%)
achievement c. Mostly low achieving 23 (14%)
(1. Students at a range of achievement level 40 (24%)

 

According to Remillard (2005), school districts increasingly regulate

mathematics teaching practice by mandating the use of a single curriculum. The

frequency of this policy shown in Table 5.1 corresponds to this trend.

In survey item 23, teachers were asked to indicate the degree of agreement or

disagreement about state-wide tests and state curriculum framework. Table 5.4 presents

the frequencies of agreement and disagreement on them.

Table 5. 4 Frequency Rating of Agreement on State Curriculum Framework and Tests

 

 

 

Factor Agreement orhdglgoriZtSlstlczllle Disagreement
State a. Obligation 161 (95%) 5 (3%) 3 (2%)
curriculum b. Alignment with their beliefs 97 (57%) 49 (29%) 23 (14%)
framework c. Alignment with NCTM ideas 11] (66%) 48 (28%) 10 (6%)
State-wide d. Obligation 135 (80%) 29 (17%) 5 (3%)
tests e. Alignment with their beliefs 37 (22%) 73 (43%) 59 (35%)
f. Alignment with NCTM ideas 40 (24%) 96 (56%) 33 (20%)

 

There are common tendencies in teachers’ perception of the state curriculum framework

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and the state-wide tests. First, a large percentage of the teachers agreed that they had an

obligation to follow the district curriculum fiamework and to teach the content of state-

wide tests (95% and 80%, respectively). However, a relatively lower percentage of

teachers agreed that the district curriculum framework and state-wide tests were aligned

with NCTM reform ideas (66% and 24%), and an even smaller percent of teachers

reported that the district curriculum framework and state-wide tests matched with their

beliefs about what is important in mathematics.

According to a body of research about the effect of tests, teachers gear their

teaching methods and strategies to the type of performance elicited by standardized tests,

particularly when the tests are the basis for important decisions about students or schools

(Haney & Madaus, 1986; Darling-Hammond, 1990; Darling-Hammond & Wise, 1985;

Goodlad, 1984; Bussis, 1982). Supporting this finding, the data suggests that although

teachers think that state curriculum framework and state-wide tests mismatched with

their beliefs of what is important in mathematics or with NCTM reform ideas, a large

percentage of teachers feel an obligation towards implementing the state curriculum

framework and teaching the content of state-wide tests.

In survey item 12, teachers were asked to list the title of textbooks students and

teachers used in their class. For the analysis, these listed textbooks were classified into

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two types based on Stein and Kim’s (In press) framework——(l) standards-based
curriculum and (2) conventional curriculum.

Among 169 participants, 158 teachers (93% out of 169) listed the title of
textbooks they were using. In total, 12 different textbooks were identified. Table 5.5

presents the frequencies of the textbooks listed by teachers in this study.

Table 5.5 Frequency of Diflerent Textbooks Used by Teachers

 

 

 

of :fibzok The title of textbooks Frequency (£32518)
Investigations in Number, Data & Space 53 (31%)

Standards- Connected Mathematics 11 (7%) 103 (65%)

based Everyday Mathematics 20 (12%)
Math Trailblazers 19 (11%)
Scott Foresman Addison-Wesley 22 (14%)
Health Math 5 (3%)
Harcourt l7 (1 1%)
0
Conventional 18331:; Hall 3 853 55 (35%)

Silver, Burdett & Ginn 6 (4%)
Math to Know 1 (1%)
Think Math 1 (1%)

 

Descriptive Findings from the Individual-Level Factors

Five variables derived from previous studies (e. g., Ball & Feiman-Nemser, 1988;
Brophy, 1982; Cohen, 1990; Kauflinan, 2002; Barr, 1988; Stodolsky, 1989; Schmidt,
Porter, Floden, Freeman, & Schwille, 1987) were included at the individual-level: (1)
“Teacher knowledge”, (2) “Teacher beliefs about teaching approach”, (3) “Teacher view

on textbook”, (4) “Teacher emphasis of student objectives”, and (5) “Teacher fidelity to

textbook”.

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In survey item 21, teachers were asked to rate on a five-point Likert scale various

types of knowledge identified as important for teaching mathematics fiom the literature.

Table 5.6 presents minimum values, maximum values, mean, and standard deviation of

teachers’ average rating about each type of teacher knowledge.

Table 5. 6 Descriptive Statistics for Teacher Knowledge Variables

 

 

 

 

Factors Variables Min. Max. M SD

Whole numbers 3 5 4.38 .62

Content Rational numbers 1 5 3.92 .84

Knowledge Geometry 2 5 3.96 .72

Measurement 2 5 4.09 .78

Data AnalLsis & Probability 2 5 3.76 .84

Knowledge of how students learn 2 5 4.07 .61

Ways of teaching strategies 2 5 4.05 .65

Pedagogical Ways of improving student basic skills 2 5 3.78 .70

Content Ways of improving understanding 2 5 3.76 .67
Knowledge

Ways of improving problem solving 2 5 3.81 .65

Ways of using curriculum material 2 5 3.84 .72

Curriculum Ways of modifying difficult problem 2 5 3.94 .61

Knowledge Ways of modifying easy problem 2 5 3.82 .7 2

Ways of developing high-order thian 2 5 3.70 .69

 

Teachers in this study rated their knowledge on whole number the highest not .

only in content knowledge area but also across all three types of knowledgeh—content

knowledge, pedagogical content knowledge, and curriculum. This suggests that teachers

reported that they had stronger knowledge about whole number than any other types of

content knowledge.

In addition, among the five content knowledge areas, teachers reported that they

have relatively lower content knowledge on rational numbers than other areas. A

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minimum rate of 1 (Very poor) that appeared only in this area supports this finding. Six

teachers out of 169 (4%) indicated that they had very poor or poor knowledge on rational

numbers.

Furthermore, the means ratings suggest that among three types of knowledge—

content knowledge, pedagogical content knowledge, and curriculum knowledge--

curriculum knowledge was reported as the weakest one.

In survey item 22, teachers were asked to rate several pairs of statements about

teaching philosophy or beliefs on a 4-point Likert-type scale in such a way as to indicate

which statement between constructivist perspective and traditional transmission

perspective came closer to their own point of view. Table 5.7 presents the frequency of

teachers’ agreement with contrasting statements of teaching philosophy.

Table 5. 7 Teachers ’Agreement with Contrasting Statements of Teaching Philosophy

 

 

Favored the more Favored the more
constructivist statement Traditional Position

Facilitator vs. Explainer 112(66%) 57 (34%)
Sense-making vs. Curriculum 119 (70%) 50 (30%)
coverage

Breadth vs. Depth 88 (52%) 81 (48%)
Multiple actrvrtres vs. Short-term 93 (55%) 76 (45%)
whole-class assrgnment

Student-oriented vs. Teacher-directed 140 (83% 29 Q 7%)

 

Overall, teachers substantially preferred the constructivist philosophy to the

traditional-transmission philosophy in their response to each of these items. The data

suggest that given an argument between support for a philosophical position consistent

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with constructivist approaches and one reflecting a more traditional viewpoint, many
more teachers in this study will select agreement with constructivist approaches than
with traditional teaching practices.

In survey item 16, teachers were asked to indicate the degree of emphasis on the
various student objectives for teaching fractions. For the analysis, these student
objectives were classified into two types based on Adding It Up (NRC, 2001): (1)
procedural fluency-oriented student objectives (PS0) and (2) understanding-oriented

student objectives (USO). Figure 5.1 shows the average rating of each student objective.

 

Eemphasis Mean Rating
to

     

U801 U802 U803 U804 U805 P801 P802 P803 P804

 

 

 

Note: USOl= Lean mathematical concepts, USO 2=Learn to reason, USO3=Learn to explain ideas,
US04=Learn to connect ideas, USO 5=Learn to how to solve problems, PSOl=Learn algorithms,
PSOZ=Develop computational skills, PSO3=Spwd and accuracy, PSO4=Standardized test.

Figure 5. 1 Emphasis Placed on the Difi‘erent Student Objectives

Teachers reported using understanding-oriented student objectives (U S01 to

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USO6) much more than procedural fluency—oriented student objectives (PSOl to PSO4).

In particular, the student objective regarding learning mathematical

algorithms/procedures received moderate emphasis.

For “teacher view of textbook” teachers were asked to indicate how much they

agree or disagree (on a five point Likert-scale) with five statements about the textbook

they were using in survey item 13. For “teacher fidelity to textbook,” teachers were

asked to indicate how much they agreed or disagreed with the six statements in survey

item 14. Table 5.8 presents the frequency of agreement and disagreement to the given

statements.

Table 5. 8 Agreement Frequency about the Textbook Satisfaction

 

 

 

. Middle .

Factors Variables Agree position Disagree
3. Satisfaction with the textbook 107 (63%) 35 (21%) 27 (16%)
b. Alignment of text with state-tests 92 (55%) 21 (12%) 56 (33%)

View c. Alignment of text with what teachers think 0 o o
of text was important in math. 95 (56 A) 41 (24 /°) 33 (20 /°)
(1. Alignment of textbook with NCT M ideas 113 (66%) 33 (20%) 23 (14%)
e. Satisfaction with the teacher’s manual 107(63%) 35 Q1%) 27 (16%)
a. Use the textbook as a primary resource 123 (73%) 10 (6%) 36 (21%)
b. In-class activities came directly from text 99 (59%) 28 (17%) 42 (24 %)
Fidelity c. Planned problems came directly from text 99 (59%) 36 (21%) 34 (20%)

d. Homework assignments were typically 0 o o
to text drawn from the textbook 98 (58 A) 28 (17 Ar) 43 (25 /o)
e. Most of questions planned to use came 86 (51%) 26 (15%) 57 (33%)

directly from teachers’ manual of the text

 

A relatively larger percentage of the teachers agreed that they used their textbooks as a

primary resource than that of teachers who agreed that they satisfied with their textbooks.

In addition, more than half of teachers in this study have fidelity to their textbooks.

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They reported intensively using their textbooks in selecting problems, activities and

homework assignments. Teachers’ fidelity to their textbook or teachers’ reliance on their

textbook has been often criticized from the public or educators or reformers. However,

without consideration of consequent result of teachers’ textbooks use (i.e., the kinds and

levels of student learning opportunities), it is hard to tell whether teachers’ modifications

of their textbooks need to be encouraged or whether teachers’ reliance on their textbooks

would be criticized. Therefore, this study examined whether teachers’ fidelity to their

textbooks (teachers’ reliance on their textbooks) has a significant effect on teachers’

transformation patterns.

Descriptive Findings from Teachers 'Opportunity-to—Learn factors

Three sub-variables derived from the previous studies (e. g., Ball & F eiman-

Nemser, 1988; Ma, 1999) were included under the teachers 'opportunity-to-learn

variables: (1) “Total participation hours in professional development (PD) activities”, (2)

“Type of content areas teachers studied during PD activities”, and (3) “Type of pedagogy

teachers participated in during PD activities”.

For “total participation hours in PD”, teachers were asked to write down

approximately how many total hours they had spent on professional development (PD)

activities in the last 24 months including attendance at professional meetings, workshops,

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conferences, and formal courses for which teachers received college credit in survey item

24. Among 169 participants in this study, 157 teachers wrote down their approximation

of PD hours received, which ranged from 1 hour to 360 hours.

For “content types teachers studied in PD” and “pedagogy types during PD”,

teachers were asked to indicate how much time they spent on PD activities relative to

various types of knowledge identified as important for teaching in survey item 24 and to

indicate the frequency of various activities teachers participated in during PD in survey

item 25. Table 5.9 presents descriptive statistics of this factor

Table 5. 9 Mean Time (in Hours) and Standard Deviation Spent on Professional Development
(PD) Activities

 

 

 

 

 

 

Factors Variables M SD

Whole number 2.15 .96

Rational number 2.05 .96

Content knowledge Geometry 2.07 .95
Measurement 1 .95 .84

Data Analysis & Probability 1.95 .92

Knowledge of how students learn 2.53 1.01

PD . Specific ways of teaching strategies 2.59 .98
Content Pedalgcrilgrceilgontent Specific ways of improving basic skill 2.20 .95
ow e ge Specific ways of improving understanding 2.40 1.02

Specific ways of improving problem solving 2.34 1.01

C . Specific ways of using cuniculum material 2.33 1.01

umculum . . .

Knowledge Specific ways of modrfymg problems 1.79 .90

Specrfic ways for hgher order thmkrng 2.33 1.07

Passive participation Listening to a lecture 3.41 1.07
Presentation 2.21 1 . 15

Being observed 2.04 .97

PD Observe demonstration 2.68 1.08
Pedagogy Active participation Observe a lesson of expert 2.05 .98
Design lessons 2.56 1.11

Analyze student work 2.68 1.02

Lead discussion 2.39 1.12

Demonstrate lessons 2.07 1.03

 

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The means ratings show that among three types of knowledge—content knowledge,
pedagogical content knowledge, and curriculum knowledge, teachers on average spent
more time studying pedagogical content knowledge than any other types of knowledge
during PD activities. In addition, teachers in this study spent the least amount of time
studying specific ways of modifying problems followed by measurement and data
analysis and probability. Furthermore, the means ratings show that teachers in this study
participated in passive learning more often than active learning during PD activities

Based on these understanding of teachers’ overall responses on each factor
explored, this study examined the research question, whether and how teachers’
responses in each factor differ according to the transformation patterns. The following
section presents the statistical methods utilized in order to explore influential factors on
these transformation patterns.

Multinomial Logistic Regression

A multinomial logistic regression (MLR) is an extension of the binary logistic
regression technique that predicts a two-categorical dependent variable. A multinomial
logistic regression is used when the dependent variable has more than two groups.

Like multiple linear regression, logistic regression develops equations that

describe the relationship between multiple independent variables and a single dependent

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variable. In order to develop a regression equation, the categorical dependent variable in

logistic regression is transformed into the natural log of the odds ratio. The frequency of

categories in the dependent variable is first transformed into a probability that an event

will occur. Then, the dependent variable is transformed into an odds ratio, which is ratio

of the probability that an event will occur to the probability that an event will not occur.

After that, this odds ratio of the dependent variable is transformed to be the natural log of

the odds, which is called a logit.

For example, suppose these are two textbook transformation patterns of a

dependent variable, which are H-H pattern coded 1 and L-L pattern coded 0. Among 100

teachers, 70 teachers were categorized as the H-H pattern and 30 teachers were in the L-

L pattern. Research interest is to model the factors characterizing teachers in the H-H

pattern compared to teachers in the L-L pattern. Then, using the frequencies of two

patterns, logistic regression develops the probability of being categorized into the H-H

pattern (P=O.7) and the probability of not being categorized into the H-H (i.e., the

probability of being categorized into the L-L pattern, P =O.3).

The probability of being categorized into the H-H pattern is used as reference

category. Although teachers in the L-H pattern are equally interesting for their ability to

transform lower level cognitive demand into higher level, a small number of teachers are

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categorized into the L-H pattern compared to teachers in the H-H pattern. In addition to
this sample size problem, since logits (logistic regression contrasts) are formed based on
the probability of being categorized into the L-H pattern compared to the probability of
being categorized into the H-H pattern, we do not lose information about L-H patterns.
For these reasons, H-H pattern was chosen as the standard.

In logistic regression, the dependent variable is expressed as ratio of the
probability of being categorized into the L-L pattern to the probability of not being

categorized into the H-H pattern (= 3—3 =0.428), which is called odds ratio

 

(= Prob “FL pattern) . Then, the odds ratio of the dependent variable is again
Prob (H—H pattern)

Prob (L—L pattern)

Prob (H—H pattern)

 

transformed into natural log of the odds, which is log[ ] . After that,

the natural log of logits as dependent variable is predicted from a weighted combination
of the independent variables.

As a result of these transformations (probability—» odds —->then natural log of the
odds ratio), the dependent variable ranges from minus to plus infinity, as a continuous

variable. The logistic regression model can be written as:

 

logl: Prob (event)

= a + b1*X| + b2*X2 + + bk*Xk
Prob (no event)]

Where a = Intercept (constant)

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k = Number of independent variables
b; to bk = Coefficients estimated for the k predictor variables
X; to X k = Values of the k independent variables

Positive coefficients in the logistic regression increase the probability that an

event will occur, which is the probability of being categorized into the L-L pattern.

Negative coefficients decrease the probability that an event will occur. In other words,

negative coefficients increase the probability that event will not occur, which is the

probability of being categorized into the H-H pattern.

Although logistic regression finds a "best fitting" equation just as linear

regression does, it uses a maximum likelihood method, which maximizes the probability

of getting the observed results given the fitted regression coefficients. Therefore, logistic

regression does not assume linearity of relationship between the independent variables

and the dependent variable, does not require normally distributed variables, does not

assume homoscedasticity, and in general has less stringent requirements (Polit, 1996).

This makes it especially suited to the present study because some independent variables

did not fulfill the assumption of homoscedasticity.

This study employed multinomial logistic regressions since the dependent

variable is comprised of three or four categories (e. g., L-L, L-H, H-L or, H-H patterns).

In Transformation phases 1 and 2 where the dependent variable is comprised of three

129

categorical indexes, two logistic regression contrasts were developed based on: (1) the

probability of being categorized into the L-L pattern compared to the probability of being

categorized into the H-H pattern and (2) the probability of being categorized into the L-H

pattern compared to the probability of being categorized into the H-H pattern.

In the cases of Transformation phase 3 where the dependent variable is comprised

of four categorical indexes, three logistic regression contrasts were developed based on

the natural log of : (1) the probability of being categorized into the L-L pattern compared

to the probability of being categorized into the H-H pattern; (2) the probability of being

categorized into the L-H pattern compared to the probability of being categorized into

the H-H pattern; (3) the probability of being categorized into the H-L pattern compared

to the probability of being categorized into the H-H pattern. The following sections

present the findings from these multinomial logistic regressions that describe influential

factors on the distinction between transformation patterns. Using a SPSS 15.0 version,

multiple multinomial logistic regressions were explored.

130

Factors Influencing Textbook Transformation Patterns in Terms of
Cognitive Demands of Problems
This section presents the findings from multinomial logistic regression used to
answer the following research questions: What factors are associated with the
transformation patterns in Transformation phase 1. Figure 5.2 presents the dependent

variable and the independent variables explored in the analysis.

Independent Variable Dependent Variable

 

Contextual-level
District Curriculum policy
Type of textbook

 

 

 

 

 

 

 

Teacher perception of student achievement
Teacher perception of district framework \ Textbook Teachigg_
Teacher perception of state-wide tests
1 Problem l—r
Individual-level L L
Teacher Knowledge [A L H

Teacher Beliefs H H
Teacher View on Textbook
Teacher Use of Student Objectives
Teacher Fidelity to Textbook

 

 

 

 

Opportunity-to-learn
PD content area studied
PD pedagogy type participated
PD total hour

 

 

 

Figure 5. 2 Dependent and Independent Variables in Transformation Phase 1

Because the dependent variable consists of three categorical indexes, two
nonredundant logits (logistic regression contrasts) are formed; (1) based on the
probability of being categorized into the L-L pattern compared to the probability of being

categorized into the H-H pattern and (2) based on the probability of being categorized

131

into the L-H pattern compared to the probability of being categorized into the H-H

pattern. The last category (the H-H pattern=4) was set up as the baseline or reference

group to which the other two groups (L-L pattern and L-H pattern) were compared based

on the independent variables.

One more analysis was conducted after setting up the L-L pattern as the reference

group in order to explore the factors that influence teachers to raise the level of cognitive

demand of problems. One logistic regression contrast was formed based on the

probability of being categorized into the L-H pattern compared to the probability of

being categorized into the L-L pattern.

All independent variables shown in Figure 6.1 were included in the original

multinomial logistic regression model. A total of 21 independent variables presented in

Table 5.2 were entered in the original multinomial logistic regression model.

The goal of the analysis is to select variables that result in a “best” model that

represent the factors distinguish teachers in the H-H pattern compared to teachers in the

L-L pattern and the L-H pattern, or vice versa. According to Hosmer & lemeshow (2000),

when the number of variables in the model is large relative to the number of participants,

the model may be “overfit” and produce numerically unstable estimates (e. g.,

unrealistically large estimated coefficients and/or estimated standard errors). Since the

132

sample size of this study, 169 participants is small relative to the number of variables

explored in the model (i.e. 21 independent variables), to find a “best” model, several

predictive variables were dropped from subsequent analyses.

In the selection of variables for the final model, the significance of the overall

model and goodness-of-fit was considered first. With the condition that the overall model

is statistically significant (p < 0.05) and the model adequately fits the data (the p-value of

goodness-of-fit is greater than 0.05), any variable whose variable test has a p-value <0.25

was considered a candidate for the model along with all variables of known importance.

Hosmer and lemeshow (2000) recommend that a 0.25 level be used as a screening

criterion for variable selection based on the work by Bendel and Afifi (1977) on linear

regression and on the work by Mickey and Greenland (1989) on logistic regression.

These authors show that use of a more traditional level (such as 0.05) often fails to

identify variables known to be important. According to them, use of the higher level has

the disadvantage of including variables that are of questionable importance at the model-

building stage. Indeed, it is possible for individual variables not to exhibit strong

confounding, but when taken collectively, considerable confounding can be present in the

data. Therefore, for these reasons, this study used a p-value <0.25 as a criterion in the

selection of the variable for a logistic regression model. Based on this criterion, only

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seven variables were included in the final model. The final model includes the seven

individual variables: (1) “ Type of textbook;”” (2) “Teacher fidelity to textbook”; (3)

“Teacher view on Textbook”; (4) Teacher use of understanding-oriented student

objectives (USO); (5) “Teacher use of procedural fluency-oriented student objectives”

(PSO); (6) “Teacher pedagogical content knowledge” (PCK) ; (7) “Teacher presentation

experience during PD”. Table 5.10 shows correlations for the variables used in the

subsequent analysis. “Type of textbook” variable is excluded since it is categorical.

Table 5. 10 Correlations for the Variables Included in the Final Model

 

 

1 2 3 4 5 6

l. Fidelity to Textbook l
2. View on Text .543(**) 1
3. Understanding-oriented student .1 6 6 0,) .184(*) 1

objectives
4. Procedural fluency-oriented H

student objectives .110 .057 .316( ) 1
5. Pedagogical Content Knowledge -.080 .013 .1 16 .045 l
6. PD Presentation -.054 .016 .129 -.041 .063 l

 

Note. * p<.05, **p<.001

For the multinomial logistic regression examining the effects of the seven
independent variables, the likelihood ratio test for the overall model revealed that the
overall model was significantly better than the intercept-only model (x2 =72.101, p
<0.000). In other words, the null hypothesis that the regression coefficients of the
independent variables are zero was rejected. In addition, the likelihood ratio test for

individual effects reveals that among the 7 variables included in the final model, 5

134

variables are significantly related to the categories of the dependent variable (“Type of
textbook” 2x2 [2]=12.099, p <0.000; “Teacher use of understanding-oriented student
objective”: x2 [2]=15.727, p <0.000; “Teacher fidelity to textbook”, x2 [2]= 13.855, p
<0.000;“Teacher view on textbook”: )8 [2]=7.324, p <0.05; “teachers’ pedagogical
content knowledge”: x2 [2]=9.629, p <0.001).

Table 5.11 reports the parameter estimates (regression coefficients) fi'om the
logistic regression model examining the effects of the independent variables on three

transformation patterns exhibited in Transformation phase 1.

Table 5.11 Parameter Estimates from Logistic Regression Model for Examining Eflect of Seven

Independent Variables on the Distinction between Patterns Exhibited in Transformation Phase I

 

Transformation patterns
in terms of the cognitive demand of problems

 

 

L-L L-H

Variables [3 Odds [3 Odds
Intercept 6.775 9.487
Type of Textbook

[Conventional texts=1] 2159* 8 .664 1.465“ 4.326

[Standards-based =0] - - - -
Teacher Fidelity to Textbook. -l .662" .190 -.687* .503
Teacher View on Textbook 1.137 3.1 l 8 -.296 .744
Teacher use of understanding-
oriented student objective ~2.546** .078 -l .990“ .137
Teacher use of procedural fluency- *
oriented student objective 1.660 5.257 .235 1.265
Pedagogical Content Knowledge -1.964* .140 -l .103* .332
Presentation experience during PD .661 1.936 .685 1.983

 

Note. The reference category is the H-H pattern. R2 = 0.479 (Nagelkerke), N=158. * p<.05, **p<.01

Estimates of the independent variables are provided for the distinction between

teachers in the L-L pattern compared to teachers in the H-H pattern and on the distinction

135

of teachers in the L-H pattern compared to teachers in the H-H pattern. Table 5.11

presents logistic regression coefficients ([3) in the second and fourth column, which

correspond to b coefficients in multiple regression. These logistic regression coefficients

construct prediction equations and generate predicted values, which are called logistic

scores in logistic regression. If regression coefficients of the factors are positive, this

factor increases the odds ratio of the probability of being categorized into the L-L pattern

or the probability of being categorized into the L-H pattern compared to the probability

of being categorized into the H-H pattern. If regression coefficients of the factors are

negative, this factor decreases the odds ratio of the probability of being categorized into

the L-L pattern or the probability of being categorized into the L-H pattern compared to

the probability of being categorized into the H-H pattern.

The odds ratios in Table 5.11 present a measure of effect size. The odds ratios of

the independents indicate relative importance of the independent variables with respect

to chance of being categorized into the L-L pattern compared to chance of being

categorized into the H-H pattern, or chance of being categorized into the L-H pattern

compared to chance of being categorized into the H-H pattern.

Table 5.11 presents effect of sizes among the seven factors. Five factors—“type

of textbook”, “teacher fidelity to textbook”, “teacher use of understanding-oriented

136

9, ‘6

student objective , teacher use of procedural—oriented student objective”, and

“pedagogical content knowledge”--are found to be significantly related to characterizing

teachers in the L-L pattern and teachers in the H-H pattern. Four factors—“type of

textbook”, “teacher fidelity to textbook”, “teacher use of understanding-oriented student

objective”, and “pedagogical content knowledge”--are found to be significantly related to

characterizing teachers in the L-H pattern and teachers in the H-H pattern.

Factors Distinguishing L-L Pattern from H-H Pattern

The coefficients of “type of textbook” and “teacher use of procedural fluency-

oriented student objectives” are significant and positive, which means that these factors

increase the probability of being categorized into the L-L pattern compared to the

probability of being categorized into the H-H pattern. In contrast, the coefficients of

“teacher fidelity to textbook”, “teachers use of understanding-oriented student

objectives”, and “teacher pedagogical content knowledge” are significant and negative,

which means that these factors increase the probability of teachers being categorized into

the H-H pattern. A closer look at the coefficients (13) suggests more interesting trends

with respect to the factors that affect the probability of teachers being categorized into

the L-L pattern compared to the probability of teachers being categorized into the H-H

pattern.

137

First, in survey item 12, teachers were asked to list the title, year, and edition of

textbooks (e. g., Everyday Mathematics, Grade 3, 2001, 2nd edition.) students and

teachers used in their class. Teachers’ responses to this survey item were first recoded

using the titles teachers provided, and then these textbooks were classified into two

types——(l) standards-based curriculum and (2) conventional curriculum—based on Stein

& Kim’s definitions (2006). Investigations in Number; Data, and Space, Connected

Mathematics, Everyday Mathematics, and Math Trailblazers were categorized into

“standards-base ” curriculum (code 0). The rest of curricula were categorized into

“conventional” curriculum (code 1).

Table 5.11 shows that the logistic regression coefficient of “type of textbook”

coded the value 1, which is “conventional textbook”, is significant and positive

([3 =2.309, p <0.05). This result indicates that teachers in the L-L pattern are more likely

to use conventional textbooks, than teachers in the H-H pattern. Conversely, teachers in

the H-H pattern are more likely to use “standards-based textbooks” than teachers in the

L-L pattern.

Second, the coeflicient of “teacher use of procedural fluency-oriented student

objectives” factor is also significant and positive (0 =1 .622, p <0.01). To measure

“teacher use of procedural fluency-oriented student objective”, teachers were asked to

138

indicate the degree of emphasis on several student objectives categorized as procedural

fluency-oriented (e.g., develop students’ computational skills; learn to perform

computations with speed and accuracy). The positive coefficient of this factor means the

increases of the chance of being in the L-L pattern compared to the chance of being in

the H-H pattern. It suggests that teachers who reported frequent use of student objectives

categorized as procedural fluency-oriented student objectives are more likely to be

categorized into the L-L pattern than into the H-H pattern.

Unlike “type of textbook” variable and “teacher use of procedural-oriented

student objectives” factors, the coefficients of the rest of three variables identified as

significant are negative: “Teacher fidelity to textbook” ([3 =-1.852, p <0.01), “Teacher

use of understanding-oriented student objectives” ([3 =-2.558, p <0.05), and “Teacher

pedagogical content knowledge” (PCK) (B =-2.032, p <0.05).

First, to measure “teacher fidelity to textbook”, teachers were asked to indicate on

a five-point Likert scale how much they agree with the given statements that asked

whether their in-class activities, problems, questions, and homework assignment came

directly from the textbooks. The negative coefficient of this variable ([3 =-1.852, p <0.01)

indicates that this factor decreases the chance of being categorized into the L-L pattern

compared to that of the H-H pattern. This result implies that teachers in the L-L pattern

139

were less likely to report that their lessons came directly from the textbook they were
using than teachers in the H-H pattern. In other words, teachers in the H-H pattern
reported heavier reliance on their textbook than teachers in the L—L pattern. Therefore,
teachers who report the faithful use of their textbooks are likely to be categorized as the
H-H pattern rather than as the L-L pattern.

Second, the coefficient of “teacher use of understanding-oriented student
objectives” variable is also significant and negative (13 =-2.558, p <0.05). To measure
“teacher use of understanding-oriented student objective”, teachers were asked to
indicate the degree of emphasis on several student objectives categorized as
understanding-oriented (e. g., learn to reason mathematically; learn to explain ideas
effectively). The negative coefficient of this variable suggests that teachers who reported
that they fiequently used student objectives categorized as understanding-oriented
student objectives were more likely to be in the H-H pattern than in the L-L pattern.

Finally, to measure “teacher pedagogical content knowledge”, teachers were
asked to rate the following types of knowledge: knowledge of how students learn (e.g.,
common mistakes/confusion), specific ways of teaching strategies, specific ways of
improving student basic skills, specific ways of improving understanding, and specific

ways of improving problem solving. The negative coefficient of this value ([3 =-2.032, p

140

<0.05) suggests that teachers in the L-L pattern were less likely to rate their knowledge
as higher level than teachers in the H-H pattern. In other words, according to teachers’
self-reports, teachers who reported higher level of PCK are likely to be in the H-H
pattern.

The odds ratio in Table 5.11 presents a measure of effect size, relative importance
of the independent variables in terms of effect on the dependent variable's odds (Menard,
2001). Among five variables identified as significant, “type of textbook” (8.664) has the
strongest effect on distinguishing teachers in the L—L pattern from teachers in the H-H,
followed by “teacher use of procedural fluency-oriented student objectives” (5.257),
“teacher fidelity to textbook” (0.190), “teacher pedagogical content knowledge” (0.140),
and “teacher use of understanding-oriented student objectives” (0.078).

Factors Distinguishing L-H Pattern from H—H Pattern

Table 5.11 indicates that four factors--“type of textbook” , “teacher fidelity to
textbook” , “teachers’ use of understanding-oriented student objectives”, and “teacher
PCK” are significantly related to the distinction between teachers in the L-H pattern and
those in the H-H pattern. Among four factors, only the coefficient of type of textbook
variable is significant and positive, whereas the coefficients of “teachers’ fidelity to

textbook”, “teachers’ use of understanding-oriented student objectives”, and “teacher

141

PCK” factors are negative. These results indicate that “type of textbook” (conventional

textbook) factor increases the probability of being categorized into the L-H pattern

compared to the probability of being categorized into the H-H pattern, whereas “teacher

fidelity to textbook”, “teacher use of understanding-oriented student objectives”, and

“teacher pedagogical content knowledge” factors increases the probability of being

categorized into the L-H pattern compared to the probability of being categorized into

the H-H pattern. A closer look at the coefficients ([3) of these variables reveals

interesting results.

First, the logistic regression coefficient of “type of textbook” coded

“conventional textbooks”, is significant and positive ([3 =1 .509, p <0.01). This result

suggests that teachers in the L-H pattern are more likely to use conventional textbooks,

than are those in the H-H pattern. Conversely, teachers in the H-H pattern are more likely

to use standards-based textbooks than teachers in the L-H pattern.

Second, the coefficients of the rest of the factors--“teacher fidelity to textbook”,

“teacher use of understanding-oriented student objectives”, “teacher PCK”--were found

to be significant and negative. The results suggest that for “teacher fidelity to textbook”,

teachers in the L-H pattern were less likely to report that their lesson came directly from

the textbook than teachers in the H-H pattern. In other words, teachers in the H-H pattern

142

are more likely to agree that they rely on their textbook more strongly than teachers in

the L-H pattern; for “teachers’ use of understanding-oriented student objectives” factor

(B =-1.967, p <0.01), teachers who reported that they frequently used student objectives

categorized as understanding-oriented student objectives were more likely to be in the H-

H pattern than in the L-H pattern; for “teacher pedagogical content knowledge”, teachers

in the L-L pattern were less likely to rate their knowledge level as higher than teachers in

the H-H pattern. In other words, teachers who rate higher level of PCK are likely to be in

the H-H pattern.

The odds ratio in Table 5.11 indicates that “types of textbook” (4.326) has the

strongest effect on distinguishing teachers in the L-H pattern from teachers in the H-H,

followed by “teacher fidelity to textbook” (.503), “teacher PCK” (0.332), and then

“teacher use of understanding-oriented student objectives” (0.137).

Factors Distinguishing L-H Pattern from L-L Pattern

To explore the factors that influence teachers to raise the level of cognitive

demand of problems, one more analysis were conducted after setting up the L-L pattern

as the reference group. Table 5.12 reports the parameter estimates (regression

coefficients) from the logistic regression model examining the effects of the independent

variables for the distinction between teachers in the L-H pattern compared to teachers in

143

the L-L pattern.

Table 5. 12 Parameter Estimates from Logistic Regression Model for Examining Eflect of Seven
Independent Variables on the Distinction between L-H and L-L Patterns in Phase I

 

 

L-H

Variables [3 Odds
Intercept 2.712
Type of Textbook

[Conventional texts=1] -0.694 .499

[Standards-based =0] - -
Teacher Fidelity to Textbook .975 2.650
Teacher View on Textbook -l .433* .239
Teacher use of understanding-oriented student objective .557 1.745
Teacher use of procedural fluency-oriented student objective -1 .425 .241
Pedagogical Content Knowledge .861 2.366
Presentation experience during PD .024 1.024

 

Note. The reference category is the L-L pattern. R2 = 0.479 (Nagelkerke), N=158. * p<.05, **p<.01

Table 5.12 indicates that only the parameter estimates for “teacher view on
textbook” (B =-1.433, p <0.01) is significantly related to the distinction between teachers
in the L-H pattern and those in the L-L pattern. In particular, the coefficient of “teacher
view on textbook” is negative, indicating that this factor increases the probability of
being categorized into the L-H pattern compared to the probability of being categorized
into the L—L pattern. This suggests that teachers in the L-H pattern were less likely to
report that they were pleased with the textbook than teachers in the L-L pattern. In other
words, teachers in the L-L pattern are more likely to agree that they were pleased with
their textbook more strongly than teachers in the L-H pattern.

The type of textbook is found to be insignificant in distinguishing teachers in the

L-H pattern fiom teachers in the L-L pattern, indicating that teachers in the L-H pattern

144

and in the L—L pattern are using the same type of textbook (i.e., conventional textbooks).
Factors Influencing Textbook Transformation Patterns in Terms of
Cognitive Demands of Teacher Questions
This section presents the findings from multinomial logistic regressions used to
answer the following research question: What factors are associated with the
transformation patterns in Transformation phase 2? Figure 5.3 presents categorical

dependent variables and independent variables explored in this analysis.

Independent Variable DQendent Variable

 

Contextual-level
District Curriculum policy
Student math achievement
District curriculum Framework

 

 

 

 

 

 

 

 

 

 

 

 

 

State-wide Tests Textbook Teachipg_
Type of textbook
Teacher Teacher
Questions —-—-> Questions
Individual-level L L

Teacher Knowledge / L H
Teacher beliefs H H
Teacher View on Textbook

Use of Student Objectives
Teacher Fidelity to Textbook

 

 

Opportunity-to-Iearn
PD content area studied
PD pedagogy type participated
PD hour

 

 

 

Figure 5. 3 Dependent and Independent Variables in Transformation Phase 2

All independent variables shown in Figure 5.3 were included in the original multinomial

logistic regression model. A total of 21 independent variables were entered in the

145

original multinomial logistic regression model. The same procedures were used to

identify the variables that were associated with teachers’ transformation patterns in terms

of the cognitive demand of teacher questions. With the condition that the overall model

is statistically significant (p < 0.05) and the model adequately fits the data (the p-value of

goodness-of-fit is greater than 0.05), any variable whose variable test has a p—value <0.25

was considered a candidate for the final model along with all variables of known

importance.

The final model in Transformation phase 2 includes the eight variables: (1) “Type

of textbook” (Conventional vs. Standards-based); (2) “Teacher perception about students’

achievement” (High achieving vs. the other levels); (3) “Teacher fidelity to textbook”;

(4) “Teacher view on textbook”; (5) “Teacher beliefs about traditional approach”; (6)

“Teacher beliefs about constructivist approach”; (7) “Teacher use of understanding-

oriented student objectives”; (8) “teachers’ observation experience during PD activity”.

Since the first two variables--(1) “Type of textbook”; (2) “Teacher perception about

students’ achievement”—are categorical, they were excluded in exploring correlations

between identified factors. Table 5.13 presents correlations for the variables used in the

subsequent analysis.

146

Table 5.13 Correlations for the Variables Included in the Final Model

 

 

l 2 3 4 5 6
1. Teacher Fidelity to Text 1
2. Teacher View on Text .543(**) 1
3. Teacher beliefs about traditional approach -.023 -.049 l

4. Teacher beliefs about constructivist approach .153(*) .017 -.O33 l

5. Teacher use of understanding-oriented .1 6 6 (t) ;184( ”023 -.031 1
objectrves )
6. Teacher Observation during PD .136 -.017 -.082 .057 -.009 l

 

Note. "' p<.05, ”p<.001

For the multinomial logistic regression examining the effects of the independent
variables, the likelihood ratio test for the overall model revealed that the overall model
was significantly better than the intercept-only model (x2 =80.099, p <0.000). In other
words, the null hypothesis that the regression coefficients of the independent variables
are zero was rejected. In addition, the likelihood ratio test for individual effects reveals
that among eight variables, five variables are found to be statistically significant in
characterizing teachers exhibiting transformation patterns in terms of the cognitive
demand of teacher questions (“type of the textbook”: x2 [2] = 22.470, p <0.000; “Teacher
perception about students’ achievement”: x2 [2] = 6.616, p <0.05; “teacher fidelity to
textbook”: 12 [2] = 11.741 , p <0.004; “teacher use of understanding-oriented student
objective”: x2 [2] = 7.927, p <0.01; “teacher beliefs about traditional approach”: x2 [2]
= 6.404, p <0.05 (emphasis of sense-making and depth). Although, the rest of variables
included in the final model were found to be insignificant, it was expected that when

taken collectively, considerable confounding could be present in the data.

147

Among three categories of the dependent variable (i.e., L-L pattern, L-H pattern

and the H-H pattern), the H-H pattern was set up as the baseline or reference group to

which the other two groups (L-L pattern and L-H pattern) were compared based on the

independent variables. Therefore, multinomial logistic regressions form two logistic

regression models (contrasts): one logistic regression model for examining the factors

that affect the probability of being categorized into L-L pattern compared to the

probability of being categorized into the H-H pattern and the other for examining the

factors that affect the probability of being categorized into L-H pattern compared to the

probability of being categorized into the H-H pattern.

Table 5.14 reports the parameter estimates from the logistic regression model

examining the effects of the independent variables on the distinction between three

different transformation patterns exhibited on Transformation phase 2. Estimates of the

independent variables are provided for the distinction between teachers in the L-L pattern

compared to teachers in the H-H pattern and on the distinction of teachers in the L-H

pattern compared to teachers in the H-H pattern.

148

Table 5.14 Parameter Estimates from Logistic Regression Model Examining Effects of Predictor

Variables on the Distinction between Patterns on Transformation Phase 2

 

Transformation patterns of teacher questions

 

 

L-L L-H

Variable (B ) Odds (B ) Odds
Intercept 5.346 10.338
Type of Textbook

[Conventional texts =1] 1.526M .574 2.491 ** 12.072

[Standards-based texts =0] - - - -
Teacher perception of student achievement

[High achieving=l] -.099 1.408 -2.l69* .114

[Other levels=0]
Teacher Fidelity to Textbook -.938** .347 -.835** .434
Teacher View on Textbook .730 .410 -. 145 .865
Teacher use of understanding-oriented _1 .820" .616 _.490 .613
student objectives
Teacher beliefs about constructrvrst .006 .432 -1.084* .338
approach
Teacher beliefs about traditional approach -.469 .453 -.802 .448
Observation of Teaching in PD .180 .281 -.304 .735

 

Note. The reference category is H-H pattern. R2 = 0.470 (Nagelkerke), N=156. * p<.05, "p<.01

Table 5.14 shows that three factors—“type of textbook”, “teacher fidelity to

textbook”, and “teacher use of understanding-oriented student objectives” are

significantly related to distinguishing teachers in the L—L pattern from teachers in the H-

H pattern. Four factors—“type of textbook”, “teacher perception of student mathematics

achievement” “teacher fidelity to textbook”, and “teacher beliefs about constructivist

approach” are significantly related to distinguishing teachers in the L—H pattern from

teachers in the H-H pattern.

Factors Distinguishing L-L Pattern from H-H Pattern

The coefficients of “type of textbook” variable are positive whereas the

coefficients of “teacher fidelity to textbook” and “teacher use of understanding-oriented

149

student objectives” variable are negative. A closer look at the coefficients (B) of these

three factors found to be significant reveals interesting findings.

First, the logistic regression coefficient of “type of textbook” coded as

“conventional textbook”, is significant and positive (B =l.l.45, p <0.05). This indicates

that the parameter of textbook type coded the value 1 increases the odds of the given

category of the dependent variable (i.e. the L-L pattern) compared to the reference

category response (the H-H pattern). This result suggests that teachers in the L-L pattern

are more likely to use conventional textbooks than are those in the H-H pattern.

Conversely, teachers in the H-H pattern are more likely to “standards-based textbooks”

than teachers in the L-L pattern in Transformation phase 2. This finding is consistent

with the finding in Transformation phase 1, suggesting that type of textbooks is a

significant indicator characterizing teachers in the H-H pattern and in the L-L pattern in

terms of cognitive demands of problems as well as in terms of cognitive demands of

teacher questions.

Second, the logistic regression coefficient for “teacher fidelity to textbook” (B =-

.93 8, p <0.001), and “teacher use of understanding-oriented student objectives” (B =-

l.802, p <0.01) are significant and negative. This result implies that for “teacher fidelity

to textbook” variable, teachers in the L-L pattern were less likely to report that their

150

lesson came directly from the textbook than teachers the H-H pattern. For “teacher use of

understanding-oriented student objectives” variable, teachers in the L-L pattern were less

likely to report the frequent use of student objectives categorized as understanding-

oriented than teachers in the H-H pattern. That is, teachers who report the frequent use of

student objectives categorized as understanding-oriented are more likely to be in the H-H

pattern than in the L-L pattern.

The odds ratio in Table 5.14 presents a measure of effect size, which is relative

importance of the independent variables in terms of effect on the dependent variable's

odds. Among 3 variables identified as significant, “type of textbook” (4.723) has the

strongest effect on distinguishing teachers in the L-L pattern fi'om teachers in the H-H

pattern, followed by “teacher use of understanding-oriented student objectives” (0.347)

and “teacher fidelity on textbook” (0.616).

Factors Distinguishing L-H Pattern from H-H Pattern

Table 5.14 also indicates that four factors--“type of textbook” variable, “teacher

perception about student mathematics achievement”, “teacher fidelity to textbook”, and

“teacher beliefs about constructivist approach” factors are found to be significant in

distinguishing teachers in the L-H pattern from teachers in the H-H pattern in

transformation phase 2. The coefficients of “type of textbook” and “teacher perception of

151

student achievement” are positive whereas the coefficients of “teacher fidelity to
textbook” and “teacher beliefs about constructivist approach” variable are negative. A
closer look at the coefficients (B) of these three factors found to be significant
demonstrate important trends, exhibit certain patterns.

First, the logistic regression coefficient of textbook type coded the value lis
significant and positive (B =2.490, p <0.001). This result indicates that teachers in the L-
H pattern are more likely to use textbook type coded as “conventional textbooks” than
those in the H-H pattern. As in the comparison between L-L and H-H, variable, teachers
in the H-H pattern are more likely to use “standards-based textbooks” than teachers in
the L-H pattern.

Second, for “teacher perception of student mathematics achievement”, in survey
item 8, teachers were asked to indicate mathematics achievement level of students in
their classroom. Teachers had the option of indicating that their students were mostly
high achieving, mostly average achieving, mostly low achieving, or at a range of
achievement level. Teachers who perceive their student mathematics achievement as
'mostly high are likely to provide more high cognitive demand problems and questions
rather than teachers who perceive their students’ mathematics achievement as being at

other levels (Saracho, 2003). For “teacher perception of student mathematics

152

achievement”, mostly high achieving was coded the value I and the other achievement

levels (that is, mostly average achieving, mostly low achieving, and students at a range

of achievement level) were coded the value 0. The logistic regression coefficient of

student achievement level coded the value 1 is significant and positive (B =-2.155, p

<0.01). This result suggests that teachers in the L-H pattern are less likely to indicate

their students’ mathematics achievement level as “other achievement levels” than

teachers in the H—H pattern. In other words, teachers in the L—H pattern are more likely to

indicate their students’ mathematics achievement level as mostly high achieving than

teachers in the H—H pattern. This result implies one possible reason why teachers in the

L-H pattern transform cognitive level of teacher questions presented in their textbook

into higher level when using teacher questions in their teaching. That is, since teachers

perceive their students’ mathematics achievement level as mostly high, they may change

lower level cognitive demand of textbook teacher questions into higher level.

Third, the negative coefficient of “teacher fidelity to textbook” factor (B =-l .084,

p <0.05) indicates that teachers in the L-L pattem were less likely to report that their

lesson came directly from the textbook than teachers the H-H pattern.

Fourth, for “teacher beliefs about constructivist approach”, questions which

presented two pairs of contrasting statements about teaching philosophy—(1) curriculum

153

coverage vs. sense-making and (2) breadth vs. depth-- were used. Teachers were asked to

rate these two pairs of statements on a 4-point Likert-type scale in such a way so as to

indicate which statement among the pair came closer to their own point of view.

“Teacher beliefs about constructivist approach” means importance of sense-making over

content coverage and importance of depth vs. breadth. The negative coefficient for

“teacher beliefs about constructivist approach” variable (B =-1.084, p <0.05) are

significant and negative, suggesting that teachers in the L—H pattern more strongly agreed

on the importance of cuniculum content-coverage over sense-making and importance of

depth over breadth in student learning than teachers in the H-H pattern. That is, teachers

in the H-H pattern are more likely to agree to sense-making and breadth with regard to

student leanring than curriculum content-coverage and depth.

The odds ratio in Table 5.14 indicates that “type of textbook” (12.072) has the

strongest effect on distinguishing teachers in the L-H pattern from teachers in the H-H,

followed by “teacher fidelity to textbook” (0.434), “teacher beliefs about constructivist

approac ” (importance of sense-making over curriculum coverage and importance of

depth over breadth) (0.338), and then “teacher perception of student mathematics

achievement” (.1 14).

154

Factors Distinguishing L-H Pattern from L-L Pattern

Table 5.15 reports the parameter estimates fiom the logistic regression model

examining the effects of the independent variables for the distinction between teachers in

the L-H pattern compared to teachers in the L-L pattern in transformation phase 2.

Table 5. 15 Parameter Estimates from Logistic Regression Model for Examining Eflect of Eight
Independent Variables on the Distinction between the L-L and the L-H Patterns

 

 

L-H

Variable (B ) Odds
Intercept 4.992
Type of Textbook

[Conventional texts =1] .965 2.264

[Standards-based texts =0] - -
Teacher perception of student achievement

[High achieving=l] -2.070 1.126

[Other levels=0]
Teacher Fidelity to Textbook .103 1.108
Teacher View on Textbook -.875* .417
Teacher use of understanding-oriented 1.331 g 3.784
student objectives
Teacher beliefs about constructrvrst _. 625 . 53 5
approach
Teacher beliefs about traditional approach -.808 .446
Observation of Teaching in PD -.488 .614

 

Note. The reference category is the L-L pattern. R2 = 0.479 (Nagelkerke), N=158. "' p<.05, "p<.01

Table 5.15 indicates that the parameter estimates for “teacher view on textbook”
variable (B =-.875, p <0.05) and “teacher use of understanding-oriented student
objectives” factors (B =1.33 l, p <0.05) are found to be significant in distinguishing
teachers in the L-H pattern fiom teachers in the L-L pattern in transformation phase 2.
The coefficient of “teacher view on textbook” is negative, indicating that teachers in the

L-H pattern are less likely to report that they were pleased with their textbook than

teachers in the L—L pattern. In other words, teachers in the L-L pattern are more likely to
agree that they were pleased with their textbook more strongly than teachers in the L-H
pattern.

In contrast, the coefficient of “teacher use of understanding-oriented student
objective” is positive, indicating that teachers in the L—H pattern are more likely to report
the frequent use of student objectives categorized as understanding-oriented than
teachers in the L-L pattern, That is, teachers who report the frequent use of student
objectives categorized as understanding-oriented are more likely to be in the L-H pattern
than in the L-L pattern.

The odds ratio in Table 5.15 indicates that “teacher use of understanding-oriented
student objective” (3.784) has the strongest effect on distinguishing teachers in the L-H
pattern from teachers in the L-L and followed by “teacher view on textbook” (0.417).

Factors Influencing Transition Patterns between Fraction Problems in

Teaching and Teacher Questions about Fractions in Teaching

This section presents the findings from multinomial logistic regression used to
answer the following research question: What factors are associated with the
transformation patterns in Transformation phase 3? Figure 5.4 presents categorical

dependent variables and independent variables explored in this analysis.

156

Independent Variable Dependent Variable

 

 

Contextual-level
District Curriculum policy
Student math achievement
District curriculum Framework

 

 

 

 

 

 

 

 

 

 

 

 

 

State-wide Tests Teaching Teaching
The type ofthe textbooks
Teacher
Problem .____, .
Questron
Individual-level L L
Teacher Knowledge / L H
Teacher beliefs H L
View on Textbook H L
Student Objectives
Fidelity to Textbook

 

 

Opportunity-to—learn
PD content area studied
PD pedagogy type participated
PD hour

 

 

 

Figure 5. 4 Dependent and Independent Variables in Transformation Phase 3

All independent variables shown in Figure 5.4 were included in the original multinomial

logistic regression model. A total of 21 independent variables were entered in the

original multinomial logistic regression model. The same procedures were used to

identify the variables associated with teachers’ transformation patterns. With the

condition that the overall model is statistically significant (p < 0.05) and the model

adequately fits the data (the p-value of goodness-of-fit is greater than 0.05), any variable

whose variable test has a p-value <0.25 was considered a candidate for the model along

with all variables of known importance.

The final model included the seven individual variables: (1) “Type of textbook”,

157

(2) “Teacher fidelity to textbook”, (3) “Teacher view on textbook”, (4) “Teacher use of

understanding-oriented student objectives”, (5) “Teacher use of procedural fluency-

oriented student objectives”, (6) “Teacher beliefs about traditional approach”, and (7)

“Teachers’ PD observation experience”. The categorical variable, “type of textbook”

factor, was excluded in correlations for the variables used in the subsequent analysis.

Table 5.16 reports correlations between the variables explored.

Table 5. l6 Correlations for the Variables Included in the Final Model

 

 

l 2 3 4 5
1. Teacher fidelity to textbook 1
2. Teacher view on textbooks .543("‘*) 1
3. Teacher use of understanding-oriented .1 6 6 C“) .18 4 0,) 1
objectives
4. Teacher use of procedural fluency-oriented .110 .057 316C") 1
objectives
5. Teacher beliefs about traditional approach
6. PD observation .136 -.017 -.009 -.055 1

 

Note. * p<.05, **p<.001

For the multinomial logistic regression examining the effects of the seven
independent variables, the likelihood ratio test for the overall model revealed that the
overall model was significantly better than the intercept-only model (x2 [18] =43.978, p
<0.000). In other words, the null hypothesis that the regression coefficients of the
independent variables are zero was rejected. In addition, the likelihood ratio test for
individual effects reveals that among 3 variables, 3 variables are found to be statistically

significant in characterizing teaching exhibiting these transformation patterns (“teacher

158

fidelity to textbook”: x2 [3]= 174.844, p <0.01; “type of textbook”: x2 [3]= 173.382, p
<0.01; “teacher use of understanding-oriented student objective” variable:
x2 [3]=1 74.623, p <0.01). The rest of variables included in the final model were found to
be insignificant. However, since the p-value of these variables are smaller than 0.25, it
was expected that when taken collectively, considerable confounding can be present in
the data.

Table 5.17 reports the parameter estimates from the logistic regression model
examining the effects of the independent variables on the distinction of teachers in the L-

L pattern and teachers in the L-H pattern from teachers in the H-H pattern.

Table 5. 17 Parameter Estimates fiom Logistic Regression Model Examining Effects of Predictor

Variables on the Distinction between Patterns in Transformation Phase 3

 

Patterns in Transformation phase 3

 

 

L-L L-H H-L

Independent variable (B ) Odds (B) Odds (B) Odds
Intercept -9.793 -1.275 2.236
Type of textbook

[Conventional texts =1] 2.814 16.684 2.063 7.870 .839 2.306

[Standards-based texts=0] - - - - - -
Teacher fidelity to textbook -3.448** .032 -.121 .886 -.286 .752
Teacher view on textbooks 1.642 5.165 .716 2.047 .578 1.782

Teacher use of understanding-
oriented objectives

Teacher use of procedural fluency-
oriented objectives

-2.526 .080 -l.513 .220 -l.539"'* .215

2.130 8.419 1.502 4.490 -.416 .660

Teacher beliefs about traditional 2.973 * 19.548 _.959 .383 _. 026 .974
approach
PD observation experience 1.156 3.177 -1 .419 .242 .196 1 .217

 

Note. The reference category is H-H pattern. R2 = 0.332 (Nagelkerke), N=156. * p<.05, **p<.01

Table 5.17 shows that two factors—“teacher fidelity to textbook” and “teacher

159

beliefs on traditional approach”—are found to be significant in characterizing teachers in

the L-L pattern compared to teachers in the H-H pattern. None of factors are significant

in characterizing teachers in the L-H pattern compared to teachers in the H-H pattern.

Only one factor—”teacher use of understanding-oriented student objective” --is

significant in characterizing teachers in the H-L pattern compared to teachers in the H—H

pattern. A closer look at the coefficients (B) of these factors found to be significant

reveals interesting findings in each contrast.

Factors Distinguishing L-L Pattern from H-H Pattern

Two factors—“teacher fidelity to textbook” (B =-3.448, p <0.01) and “teacher

beliefs about traditional approach” (B =2.973, p <0.01)—are found to be significant in

characterizing teachers in the L-L pattern compared to teachers in the H-H pattern. First,

the logistic regression coefficient for “teacher fidelity to textbook” is significant and

negative. This result implies that for “teacher fidelity to textbook” variable, teachers in

the L-L pattern were less likely to report that their lesson came directly from the textbook

than teachers the H-H pattern.

Second, the coefficient of “teacher beliefs about traditional approach” variable is

significant and positive. For “teacher beliefs about traditional approach”, questions

which presented the categories of contrasting statements about teaching philosophy—(1)

160

Facilitator vs. Explainer and (2) Multiple activities vs. Short-term whole-class

assignment -- were used. Teachers were asked to rate these two pair of statements on a 4-

point Likert-type scale in such a way so as to indicate which statement among the pair

came closer to their own point of view. Although previous research conducted on this

topic in various ways, I used this contrasting statement to categorize teacher beliefs about

traditional approach. One is called traditional approach and the other is constructivist

approach. “Teacher beliefs about traditional approach” means importance of a role of the

teachers’ as explainer over facilitator and importance of short-term whole-class

assignment over multiple activities (see Chapter 4 or Appendix for more specific

explanation). The positive coeflicient of this factor indicates that, in this study, “teacher

beliefs about traditional approach” factor increases chance of teachers being categorized

into the L-L pattern rather than into the H-H pattern. This means that if teachers report

strong agreement with the role of a teacher as an explainer and use of short-term whole-

class assignment, they are more likely to be categorized into the L-L pattern than into the

H-H pattern.

The odds ratio in Table 5.17 presents a measure of effect size, which is relative

importance of the independent variables in terms of effect on the dependent variable's

odds. “Teacher beliefs about traditional approach” (19.548) has the strongest effect on

161

characterizing teachers in the L-L pattern from teachers in the H-H pattern, followed by

“teacher fidelity to textbook” (0.032).

Factors Distinguishing H-L Pattern from H-H Pattern

Only one factor—“teacher use of understanding-oriented student objective”

factor (B =-1.539, p <0.01)--is significantly related to distinguishing teachers in the H-L

pattern from teachers in the H-H pattern. The regression coefficient for “teacher use of

understanding-oriented student objectives” is negative, suggesting that frequent use of

understanding-oriented student objectives decreases chance of teachers being categorized

into the H-L pattern. Teachers who report the frequent use of student objectives

categorized as understanding-oriented student objectives are likely to be categorized into

the H—H pattern. That is, teachers in the H-L pattern reported less frequent use of

understanding-oriented student objectives. This result explains why teachers in the H-L

pattern decrease the cognitive level of teacher questions in their teaching or at least it

contributes to explaining this trend.

Factors Distinguishing H-L and L-H from L-L Pattern

Table 5.18 reports the parameter estimates from the logistic regression model

examining the effects of the independent variables on the distinction of teachers in the L-

H pattern and teachers in the H-L pattern fi'om teachers in the L-L pattern.

162

Table 5.18 Parameter Estimates from Logistic Regression Model Examining Eflects of Predictor

Variables on the Distinction Based on the L-L Pattern in Transformation Phase 3

 

Patterns in Transformation phase 3

 

 

L-H H-L

Independent variable (B) Odds (B) Odds
Intercept 8.518 12.029
Type of textbook

[Conventional texts =1] -.751 .472 .839 2.306

[Standards-based texts=0] - - - -
Teacher fidelity to textbook -3.327* 27.860 3.163* 23.635
Teacher view on textbooks -.926 .396 -1.064 .345
Timber “3". 0f l‘nde’standmg' 1.102 2.752 .987 2.683
oriented objectives
Teacher use of procedural fluency- _. 629 .533 _2. 5 46 .078
oriented objectives
Teacher beliefs about traditional 4.932,, .020 4.999,, .050
approach
PD observation experience 2575* .076 -.959 .383

 

Note. The reference category is L—L pattern. R2 = 0.332 (Nagelkerke), N=156. * p<.05, "p<.01

Three factors are found to be significant in distinguishing teachers in the L-H pattern

from teachers in the L-L pattern in transformation phase 2: “teacher fidelity to textbook”

variable (B =-3.327, p <0.05), “teacher beliefs about traditional approach (B =-3.932, p

<0.05), and “PD observation experience” factors (B =2.575, p <0.05). The coefficients of

“teacher fidelity to textbook” variable and “teacher beliefs about traditional approach”

are significant and negative, suggesting that teachers in the L-H pattern were less likely

to report that their lesson came directly from the textbook than teachers in the L-L

pattern; and they were also less likely to report agreement with the role of a teacher as an

explainer than teachers in the L-L pattern.

The coefficient of “PD observation experience” is significant and positive,

163

indicating that teachers in the L-H pattern are more likely to report their frequent

participation of observation experience in their professional development programs than

teachers in the L-L pattern.

Two factors-“teacher fidelity to textbook” variable (B =3.163, p <0.05) and

“teacher beliefs about traditional approach (B =-2.999, p <0.05)—are also found to be

significant in distinguishing teachers in the H-L pattern from teachers in the L-L pattern

in transformation phase 2. The coefficients of “teacher fidelity to textbook” variable is

positive whereas the coefficient of “teacher beliefs about traditional approach” is

negative, suggesting that teachers in the H-L pattern were more likely to report that their

lesson came directly from the textbook than teachers in the L-L pattern; and they were

less likely to report agreement with the role of a teacher as an explainer than teachers in

the L-L pattern.

The odds ratio in Table 5.18 indicates that “teacher fidelity to textbook” (27.860)

has the strongest effect on distinguishing teachers in the L-H pattern from teachers in the

L-L, followed by “PD observation experience” (0.076) and “teacher beliefs about

traditional approach” (0.20). For distinction between teachers in the H-L pattern from

teachers in the LL, that “teacher fidelity to textbook” (23.63 5) has the strongest and

followed by “PD observation experience” (0.050).

164

Summary

This chapter presented the findings from multinomial logistic regressions that
examined which factors were associated with various textbook transformation patterns
exhibited in Transformation phase 1, Transformation phase 2, and Transformation phase
3, respectively.

The factors that distinguish teachers in the H-H pattern from teachers in the other
patterns were found to be “type of textbook”, “teacher fidelity to textbook”, “teacher use
of understanding-oriented student objectives” and “teacher pedagogical content
knowledge” across Transformation phase 1, Transformation phase 2, and Transformation
phase 3. Compared to teachers in the L-L pattern and teachers in the L-H pattern,
teachers in the H-I-l pattern are more likely to use “standards-based” curriculum; report
heavier reliance on their textbook in the selection of problems, questions, and homework
assignment; report the frequent use of student objectives categorized as understanding-
oriented; rate their level of pedagogical content knowledge higher than teachers in other
transformation patterns. In other words, teachers in the L-L pattern and the L-H pattern
are more likely to use conventional textbooks; report less reliance on their textbook; less
use of understanding-oriented student objectives; and report lower level of pedagogical

content knowledge compared to teachers in the H-H pattern.

165

In addition to the factors described above, several factors were found to important
indicators that characterize teachers in the L-L pattern, teachers in the L-H pattern, or
teachers in the H-L pattern compared to teachers in the H-H pattern. “Teacher use of
procedural fluency-oriented student objectives” was found to increase chance of teachers
being categorized into the L-L pattern compared to teachers in the H-H pattern. Teachers
in the L-L pattern are likely to report more frequent useof procedural fluency-oriented
student objectives than teachers in the H-H pattern.

“Teacher perception of student mathematics achievement (high achieving vs.
other achievement levels)” is a factor characterizing teachers in the L-H pattern
compared to teachers in the H-H pattern. Teachers in the L-H pattern are more likely to
report their students’ mathematics achievement level as mostly high achieving than
teachers in the H-H pattern.

Furthermore, “teacher use of understanding-oriented student objectives” was
found to be an important factor that explains the differences between teachers in the H-H
pattern and teachers in the H-L pattern. Teachers in the H-L pattern are more likely to
report less fiequent use of understanding-oriented student objectives than teachers in the
H-H pattern.

These factors identified from this chapter provide a possible reason for why

166

different textbook transformation patterns appeared in terms of cognitive demands.

Based on the analysis of teachers’ self-reports, the strongest factor that distinguishes

teachers in the H-H pattern from other transformation patterns (i.e., the L-L pattern and

the L-H pattern) is the type of textbooks teachers use. When teachers use standards-based

curriculum, they are more likely to provide higher cognitive level of problems and

questions in their teaching. These teachers at the same time are likely to use their

textbooks closely and put more emphasis on understanding of mathematical concept than

procedural-fluency.

However, teachers in the H-L pattern showed that even when teachers use

standards-based curriculum, they can provide lower level of teacher questions since they

put more emphasis on procedural-fluency than understanding of mathematical concepts.

When teachers use conventional textbooks, some teachers are likely to provide

the lower level of mathematical problems and teacher questions (L-L pattern) whereas

others provide the higher level of mathematical problems and teacher questions in

teaching (L-H pattern). Teachers in the L-L pattern, teachers who provide lower level of

mathematical problems and teacher questions with conventional textbook, are more

likely to put more emphasis on procedural-fluency than understanding of mathematical

concepts and to perceive the role of a teacher as an explainer rather than a facilitator.

167

However, teachers in the L—H pattern, teachers who provide higher level of

mathematical problems and questions in their teaching are more likely to perceive their

students’ mathematics achievement as mostly high achieving. Because of different

perception of student mathematics achievement, some teachers are likely to transform the

lower cognitive demand of mathematical problems and teacher questions presented in

textbook into higher level in their teaching.

The factors that distinguish teachers in the L—H pattern fiom teachers in the L-L

pattern were found to be “teacher views on conventional textbook”, “teacher use of

understanding-oriented teaching goals”, and “teacher observation of experts’ teaching

during professional development programs.” Compared to teachers in the L-L pattern,

teachers in the L—H pattern are less likely to agree that they were pleased with their

textbook than teachers in the L-L pattern; these teachers moreover are more likely to

report heavier emphasis on understanding-oriented teaching goals and to report their

frequent opportunities to observe expert teachers and to be observed teaching during

their professional development programs than teachers in the L-L pattern.

Interestingly, teachers’ fidelity to their textbook--teachers’ faithful use of their

textbook—-was found to be an indicator that distinguishes teachers in the H-H pattern

from teachers in other patterns. According to teachers’ reports, the more teachers closely

168

follow their textbook, the more they tended to be categorized into H-H pattern. Although

teachers’ fidelity to their textbook is often criticized by reformers or educators (e. g.,

Apple & Jungck, 1990), the finding of this study suggests that teachers’ fidelity to their

textbook itself should not be considered the indicator or criteria to gauge teachers’

practice; teachers’ fidelity to their textbook should be considered in conjunction with the

type of textbook teachers’ use.

However, these findings are based on teachers’ self—reports. Despite these insights

about teachers’ use of textbooks, previous studies identified significant discrepancies

between teachers’ reports and the practices observed by researchers (Cohen et al., 1997).

To minimize this kind of criticism drawn from methodological limitation and to examine

how the findings from teachers’ reports work in actual practice, this study further

examined teachers’ textbook use and influential factor through case studies. Next chapter

will present the findings from the case studies.

169

Chapter 6

RESULTS FROM THE CASE STUDIES:

TEACHERS’ TEXTBOOK TRANSFORMATION PATTERNS AND
INFLUENTIAL FACTORS

This chapter presents the findings from the case studies that examined how

teachers transformed problems and teacher questions in terms of cognitive demands as

they moved content from text to teaching when teaching units about fractions. The case

studies also examined factors influencing teachers’ textbook use in teaching. I sought to

answer the same research questions in the case studies as those in the survey: (1) What

transformation patterns do elementary teachers exhibit as they move fraction content

from text to teaching if the transformation is looked at in terms of the cognitive demands

of problems and questions? (2) What factors are associated with these transformation

patterns? The case studies were conducted to establish the validity of the written survey

and to provide additional details about teachers’ transformation patterns that can not be

obtained from the survey.

This chapter is organized into four sections. The first section summarizes how the

case study participants were chosen, the method used to collect and analyze data, and the

overview of patterns identified from the case studies. In most cases, the patterns

observed in the case studies were consistent with the patterns of teachers’ responses on

170

the survey. However, this chapter also examines cases in which the results of the survey
did not correspond to the patterns observed in the case studies.

Three transformation patterns from textbook to teaching were exhibited in the
case studies: The second section describes a representative case of teachers who use
higher level problems and questions in their teaching. The third section describes a
representative case of teachers who use higher level of problems fiom their textbooks but
use lower level of questions in their teaching. The fourth section describes a
representative case of teachers who use lower level problems and questions in their
teaching. In each section, I attempt to infer possible reasons why each pattern appears. In
addition, by cross-case comparison and analysis, I also attempt to explore why
discrepancies occur between teachers’ responses on the survey and the case studies.

Methods Used to Select and Analyze Teachers for the Case Study

Among 169 survey participants, 52 teachers volunteered to participate in the case
study. None of the teachers whose survey responses were categorized as the H-L pattern
or the L-H pattern volunteered to participate in the case study. The grade level and types
of textbook teachers use were considered to select the case study participants. Since
understanding of fractions is developed intensively from fourth grade to sixth grade,

teachers who were teaching these grade levels were considered first. Among 52 teachers,

171

27 teachers taught the targeted grade levels--fourth (6), fifth (15), and sixth (6) grade. I

then tried to select teachers who used different types of textbooks (standards-based vs.

conventional textbooks) since the finding from the survey showed that the type of

textbook is an important factor in deciding the cognitive demand of problems and teacher

questions in teaching.

Ultimately, eight teachers were selected to be observed and interviewed. More

teachers classified as the H-H pattern in the survey participated in the case study than

teachers classified as the L-L pattern since most of the teachers who were willing to

participate in the case study were classified as the H-H pattern in the survey and a few

teachers were classified as the L-L pattern. Table 6.1 shows a summary of the

characteristics of the case study participants including transformation patterns identified

from their responses on the survey (all names are pseudonyms).

Table 6. 1 Information about the Case Study Participants

 

 

Teaching Transformation pattern
Teacher Grade Year Textbook used in Survey
Problems Questions
Catherine 6 9 Connected Mathematics HI-I HH
Tom 5 33 Everyday Mathematics HH HH
Michelle 5 l 9 Investigations HH I-IH
Brad 4 7 Math Trailblazers HH HH
Karen 4 3 Math Trailblazers HH HH
Teri 5 13 Harcourt LL LL
JoAnne 5 14 *Investigations & Scott Foresman HH HH
Randy 5 33 Investigations & *Scott Foresman LL LL

 

Note: "‘ signifies the main textbook

Each teacher was observed and interviewed at least once (and some twice) while

172

they were teaching a fraction unit. During the observation, I took field notes focusing on

problems and questions the teachers or students worked on. Interviews were tape-

recorded. In the analysis of transformation patterns, audio-taped interview data, the

transcripts of interviews, documents related to teachers’ use of textbooks (teachers’

lesson plan from the teacher’s manual and my observation notes), and my observation

were used.

As in the survey, transformation patterns in the case studies were explored by

examining the relationship between the cognitive demand of problems and questions in

textbooks and that of problems and questions used by teachers in their lesson(s). The first

step of the analysis was to classify each problem and question presented in textbook

lesson and used in the taught lesson as either high-level or low-level. Each problem in

the textbook and in the taught lesson was classified into either high-level or low-level

based on Stein and Smith’s (2000) framework as described in Chapter Two. Similarly,

teacher questions in the textbook lesson were categorized into three cognitive levels—

high, medium, and low level--based on the revised Bloom’s Taxonomy (Anderson &

Krathwohl, 2001). For the cognitive demand of teacher questions in taught lesson(s),

teachers’ reports on typical questions fi'equently used in their teaching and my

observations were analyzed and classified as either high, medium, or low-level.

The second step of the analysis was to decide the overall cognitive level of
problems and questions in the textbook lesson and taught lesson based on the selected the
ratio of 1/3 (Kadijevié, 2002; Sanders, 1966). For instance, if more than 1/3 of the total
problems were high-level in textbooks and/or in teaching, textbook problems and
problems used in teaching were considered as requiring higher level of student thinking.
Otherwise, they were considered as requiring lower level of student thinking (see
Chapter two for more detailed explanation).

Three transformation patterns were exhibited in the case studies. One pattern
occurred when teachers provide higher level problems and questions from a textbook,
and they use higher level problems and questions in their teaching. For example, a

teacher used the following problem presented in Math Trailblazers (Grade 4, Wagreich,

=i,and
8

Alix.)
NI—

9

et a1, 2004, p. 922), “Look for patterns in the number sentences-:—

-1— = 1—50 and find another equivalent fractions.” This problem requires students to focus

2
on the relationship between numerators and denominators and use the patterns, as
opposed to simply following the rule, in order to find other fractions equivalent to one
half. This teacher also used questions presented in the teacher’s manual of Math

Trailblazers that require students to engage in thinking, reasoning, problem-solving,

justifying, and communicating about mathematics. These questions include the language

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,9 9,

like “look for”, “explain , justify”. Problems and questions presented in this pattern
require students to use procedures with connection to concepts, meaning or
understanding and doing mathematics (Hiebert, 1999; Stein & Kim, 2007). Four teachers
(Catherine, Tom, Michelle, and Brad) were identified as having taught lessons which
correspond to this pattern. All four used standards-bases mathematics curricula.

Another pattern occurred when teachers use higher level problems from a
textbook, but use lower level of teacher questions in teaching. Karen and J oAnne were
classified as following this pattern. Although these teachers used standards-based
curricula and presented the suggested examples and problems from the lesson guide, they
shifted the focus from finding relationships to following rules, used teacher questions
that require only a correct answer, and thereby provided lower level student learning
opportunity.

The third pattern occurred when teachers use lower level problems and questions
from a textbook in their teaching. For example, one teacher used the following problem
presented in Scott F oresman — Addison Wesley Mathematics (Randall, et al., 2005, pp.
372), “Find the sum. Simplify, 3% + 2% .” This problem requires students to use a well-
established procedure for finding the sum, and not to connect the procedure to meaning

or the underlying concepts—the need for the same whole unit in order to add two

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fractions with unlike denominators. In keeping with the cognitive demand of the problem,

this teacher used questions that focus on merely getting the one answer. Most of the

classes observed that were taught the problems and questions by teachers (Teri, J oAnne,

and Randy) in this pattern required the use of procedures without connection to concepts,

meaning or understanding or memorization (reproducing previously learned facts, rules,

formulae, or definitions or committing them to memory) (Romberg, 1983; Stein & Kim,

2007)

Teri used a conventional textbook. J oAnne and Randy used a combination of

standards-based and conventional textbooks in their classrooms. Depending on the type

of textbook she used, JoAnne’s teaching was categorized into either the L—L pattern (with

conventional textbook) or the H-L pattern (with standards-based textbook). Although

JoAnne indicated in the survey that the standards-based curricula as her main textbook,

she used the conventional textbook on the first day I observed her lesson. Although she

selectively used problems and teacher questions fi'om the conventional textbook, she

ended up with providing lower level cognitive demand of problems and teacher questions.

When I observed a second lesson, she used the standards-based cuniculum and provided

the problems that require multiple representations for fractions in her teaching. But like

the case of Karen, she used teacher questions that require only correct answers rather

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than engaging in thinking and reasoning.

Among the eight teachers, two teachers--Karen and J oAnne--showed

discrepancies between their responses on the survey and actual teaching. These two

teachers commonly reported using higher level teacher questions in their teaching with

textbooks presenting higher level teacher questions (the H-H pattern) in the survey.

However, in my observation, they were categorized as using lower level teacher

questions with standards-based curriculum, which is the H-L pattern in terms of the

cognitive demand of teacher questions (notice J oAnne also was categorized as the L-L

pattern with conventional textbook).

The following sections will describe each transformation pattern in detail

combined with the analysis of the cognitive demand of problems and questions presented

in textbook lessons and that of problems and questions used in lesson(s). To describe

each of the three main transformation patterns, one case was selected as representative.

Each case functions as an anchor to make comparisons among and generalizations about

other teachers in the same pattern. For the H-H pattern, the case of Brad was selected and

for the L-L pattern, the case of Randy was selected. For the H-L pattern, Karen’s case is

used. Although the case of JoAnne is interesting, her use of standards-based textbook can

be explained from the case of Karen and her use of conventional textbook can be

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explained from the case of Randy. In addition, since Karen and Brad were categorized
into different transformation patterns even though they taught the same lesson with same
textbooks, it was interesting to see why such different transformation patterns appeared.

Within-case analysis and cross-case comparisons between the transformation
patterns allowed me to infer possible explanations of why teachers use the same textbook
differently or use a different textbook in a similar way as well as why discrepancies occur.
Although one selected case was used to describe each transformation pattern, interview
data from other teachers in the same pattern were also used to describe and elicit the
possible reason why teachers used their textbooks in such a way and why the
discrepancies happened between teachers’ response on the survey and actual teaching.

Higher Level Textbook Problems and Questions to Higher Level
Problems and Questions in Teaching: Pattern 1

Four teachers use tasks in ways that can be described as the H-H pattern, in which
teachers have textbook problems that provide higher level problems and questions, and
they use higher level questions in their teaching. All used “standards-based textbooks”,
which were developed based on a philosophy for learning and teaching mathematics that
is consistent with that of NCTM standards (1991, 1999, & 2000). However, it is not the

use of standards-based textbook alone that sets these four teachers apart fiom other

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teachers. It was also the conception of knowing and doing mathematics represented in

higher level of problems and supported by higher level of teachers’ questions in his

practice set him apart from teachers in other patterns. The features of the following

vignette from Brad’s teaching exemplify practices of other teachers in this pattern.

A Vignette of Brad

Brad has 7 years of teaching experience at the elementary school. The school in

which he taught serves students from both middle and working-class backgrounds. Brad

described his class as “average achieving” in mathematics achievement. On average, a

slightly lower percentage of students in the school (82%) met or exceeded standards for

mathematics of the 2007 Michigan Educational Assessment Program (MEAP) than that

of the state average (85%).

On the day I observed Brad’s lesson, he taught the topic of equivalent fractions

using Math Trailblazers (Wagreich et al., 2004). Before observing his class, Brad gave

me a copy of his lesson plan, reproduced as Figure 6.1, which came directly from the

teachers’ manual of his textbook.

179

 

At a Glance

 

1. Ask students to use their fraction chart from Lesson 3 to find all of the fractions that
are equivalent to 1/2. List these on the board or overhead.
2. Ask students to compare the numerators and the denominators of the equivalent

fractions in order to look for patterns.

3. Ask students to suggest other factions that are equivalent to 1/2

4. Write number sentences on the board or overhead showing the equivalencies

5. Students look for patterns in the number sentences

6. Students use the patterns (multiplying or dividing the numerator and the denominator
by the same number) to find fractions equivalent to 3/4, 1/3, and 2/5.

7. Students use the patterns to complete number sentences involving equivalent fractions

Students complete Qtestions 1-5 on the Equivalent Fractions Activity Pages in the
Student Guide as independent practice

9. Assign Homework @stions 1-15 on the Equivalent Fractions Activity Pages in the

 

Student Guide. Students will need their fiaction charts to complete this assignment

 

 

Source: From Math Trailblazers Fourth Grade Teacher Is Lesson Guide (p. 924), by Wagreich, P.,
Goldberg, H., Bieler, J .L., Beissinger, J.S., Cirulis, A., Gartzrnan, M., Inzerillo, C., Isaacs, A., Kelso, C.R.,
& Peters, L(2004), TIMS Project, Dubuque, IAzKendall/Hunt.

Figure 6. 1 Summary of Lesson Activities in a Lesson

Brad started out his lesson by reviewing what students learned in the previous

lesson, such as finding the missing factor in a number sentence and the multiples of 23.

Then he began the “teacher-led” activity suggested in the lesson guide. He asked students

to find equivalent fractions to one half using a fraction chart that they used in the

previous lesson. By looking at fraction chart, students found all fractions equivalent to

one half, for example, “two fourths, three sixths, four eights, five tenths, and six

twelfths.” When students said that there were no more equivalent fractions, Brad said

“those are the equivalent fractions to one half on our charts. Are there more equivalent

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fractions to one half?” Students agreed that there were more equivalent fiactions. From
there, Brad introduced the topic saying “today we are going to find equivalent fractions”.
As suggested, Brad listed all equivalent fractions to one half on the board, such as
-— = -— = -— = —~ = 120-. He then asked students to explore the patterns among denominators
“two, four, six, eight, and, ten”. Students found that “they are all even, skipping
counting, multiples of two”. Brad continued to have students explore the relationship
between the numerator and denominator of each equivalent fraction. Students found the
patterns, which are “Double the numerator and you get a denominator”, “two times the
numerator is the denominator”, and “the numerator is half of the denominator.”

As students figured out the relationship, Brad then put a fraction 31% and l:

30
on the board and asked students to see whether these fractions were equal to one half. He
then said to his students “you know me, right? If you say “yes”, I am going to say, “how
do you know”? If you say “not”, [I say] “why not?” When Brad called on one student, he
answered “because if we take way the zeros it’s just the same as two fourth.” Another
student added that “because twenty plus twenty equals forty and that (20) is half of 40”.
In watching Brad interact with his students, it was obvious that a teacher-led activity in

his class involves discussion with the teacher and students rather than recitation or

lecture in which only the teacher talks. In particular, by using teacher questions that

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require reasoning and explanation, Brad seemed to try to elicit opinions and ideas--not

just “right” answers-- from students.

As students were able to recognize the relationship between the numerator and

the denominator of fractions equivalent to one half (e.g., the numerator is half of the

denominator), Brad set equivalent fractions separately, for example,

NI—
OOH:-

Z
49

N|-—-

and% = % , and asked students to explore the relationship in the number sentences. For
example, pointing out the first number sentence, he asked students to look at two
equivalent fractions and find any pattern by comparing the numerators of the fractions
and then the denominators.

Although most of students seemed to see the relationship at the end of the class,
which is “one times two is two” and “two times two equals four,” several students
struggled with finding this relationship and provided incorrect answers. One common
error that repeatedly came up in his class was comparing the denominator of the first
fraction and the numerator of the second factions to find the pattern. For example, in the
first number sentence % = 3, one selected student answered, “If you take four and time
it two you get eight.” This gave me an Opportunity to see how Brad reacted to those

students and what kinds of questions he used.

One of the challenges that many teachers face when they try to conduct lessons

182

that take account of and productively build on student responses is how to respond to
students’ incorrect response (e. g., Ball, 2001; Brown & Campione, 1994; Chazen & Ball,
2001; Lampert, 2001; Leinhardt & Steele, 2005; Schoenfeld, 1998; Sherin, 2002). It is
often reported that teachers take over students’ thinking and reasoning, tell students how
to do the problem, and thereby reduce the cognitive demands of student thinking (Stein
& Smith, 2000).

However, in this instance, and many other similar instances, Brad accepted the
student’s incorrect reasoning and had students discuss it. Sometimes, he redirected and
let students know what he wanted them to look at in order to find the relation between
the number sentences. For example, Brad responded back to the student “Okay, but you
are looking at the numerator and denominator four eights. I want you to compare the
numerator of one half to the numerator of four eights”. Then he asked another student to
share her/his ideas. The selected student answered “You multiple it one times four equals
four”. These strategies seemed to work well in his classroom. When Brad asked students
to find the relationship in the rest of the number sentences, students responded correctly
and were able to explain why. Indeed, this strategy gives students an opportunity to listen
to each other, and not just take the teacher’s word.

When students were able to find the pattern in number sentences, Brad asked

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students to generate the rule for equivalent fractions. He first went over the process they

= -5— l = f— , and asked to students,

1
210’2 12

1:1
28’

th

’

did in each number sentence é-
“How could we use what we did here to help us find another equivalent fraction for one
half?” Several students tried out their conjecture, but they could not articulate the rule for
finding equivalent fractions. In response, Brad gave the rule saying “you multiply the
numerators and the denominator by the same number to get an equivalent fraction.”
Several students shouted out “cool.” After letting students know the rule, Brad asked
students to find equivalent fraction to one half by multiplying the number “eight” and
“nine.” Students came up with “eight sixteenths” and “nine eighteenths.” These students’
correct responses made me think that students were capable of using the rule that Brad
told without much difficulty in order to find equivalent fractions.

Like teachers in other patterns, Brad did not continue to push students to generate
the rule for finding equivalent fractions when students could not create the rule. However,
one feature that distinguishes his practice from teachers in other patterns was that Brad
did not just ask students to use the rule but also asked them to verify the rule. After
letting students know the rule for finding equivalent fractions, Brad asked students to

apply this rule to find equivalent fi'actions, for example, to three fourths, and also check

that answer from a fraction chart. He said “let’s see if we can prove it with one that is on

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your fraction chart. Now, three fourths is equivalent to six eights. Look at three fourths.
Look at six eights. Is it true?” Students said “yes.” Brad not only used this questioning
technique, but was intentional in his use of it. For example, in an interview, he said,
On the fraction chart, it shows fraction one through twelfth. I want them to recognize
and physically see it that three fourths is equal to six eighths. They can look at that from
the fraction chart and say “I can see it, three fourths is equal to six eighths”. Or if I say,

they are going to take my word for it that those are equivalent. It’s going to be less
connection because they could not see that for themselves (interview, 4/24/2007).

Brad said that he wanted to have his students recognize how the rule works for

finding equivalent fractions. It was interesting to see that Brad transformed a lower level

question, which is “apply the rule to find equivalent fractions”, into a higher level one,

“prove it”, using the context. With additional examples, Brad and his student repeated the

validation process---applying the rule and then checking the results from the fraction

chart. This lesson took 45 minutes. Brad only covered an introductory “teacher-led”

activity up to statement five in Figure 6.1. From the perspective of someone who didn’t

see the lesson, it might appear that Brad did only the “teacher-centered” part of the

lesson, which might lead one to assume that his teaching was more conventional, less

constructivist, and lower level in its cognitive demand. Instead, he turned the “teacher-

led” activity into one which was facilitated by the teacher but depended on students’

discovery of patterns and reasoning concerning the rule for finding equivalent fractions.

On the following day, he continued to cover the remaining parts in the lesson

185

guide. He used problems and examples presented in the teacher’s manual in the same

manner he did on the first day. He also assigned Questions 1-5 from the Activity Pages,

as_suggested in the lesson guide, to his students. Students worked individually on that

assignment. He worked around and gave some help. When I went around, I noticed all

but a few students in his class solved the problems correctly.

Similarities and Differences between the Textbook Lesson and the Lesson Taught in

Terms of Cognitive Demands

Although Brad spent more than the one period the textbook recommended for this

lesson, he closely followed suggestions presented in the lesson guide. Most of the

examples, problems, and teacher questions came directly fiom the teachers’ manual of

Math Trailblazers. Therefore, he (at least) maintained the cognitive demand of the

student objectives, the problems, and the teacher questions presented in textbooks.

Lesson goals As suggested in Figure 6.1, the students are expected to find

patterns and use these patterns to find other equivalent fraction. This objective asks

students to develop a conjecture, which correspond to a higher level student objective

category, reasoning, in the Revised Bloom’s Taxonomy (Anderson & Krathwohl, 2001).

During the interview, Brad said that his goal was the same as the goal presented

in the lesson guide. During the interview, Brad said,

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Today I tried to get them discover for themselves. “Look for the pattern and tell me what
you see.” And then through class discussion, different kids were talking about it. I hope
they would draw the conclusion and they came up with discovery to me (interview,
4/24/2007).

His goal is not to use the algorithm to find equivalent fractions. Brad set the goal

for students to develop an understanding of equivalent fractions and use that

understanding to develop a conceptually based algorithm. Brad’s goals are thus aligned

with the textbook, and in keeping with higher level cognitive demand.

Problems The cognitive demands of problems were analyzed in both an

introductory teacher-led activity as well as student-exercise. In Brad’s class,

mathematical problems in the teacher-led activity include: (1) Find all of the fractions

from a fraction chart that are equivalent to 1/2, (2) Look for patterns by comparing the

numerators and the denominators of the equivalent fractions to one-half, (3) Look for

patterns in the number sentences involving fractions equivalent to one half, and (4) Find

fractions equivalent to 3/4, 1/3, and 2/5 using the patterns (multiplying or dividing the

numerator and the denominator by the same number). Based on Stein and Smith’s

framework (2000), these problems require students to use procedures with connection to

concepts, meaning or understanding, which demand complex thinking and reasoning and

a considerable amount of cognitive effort.

Brad assigned a total of sixteen problems as a student exercise in the Equivalent

187

Fraction Activity of the student textbook. Some problems are presented with contexts
whereas other problems are without context. For example, the first problem is addressed
in the real-world context where Elam wants to know how many eighths of a cup of sugar
is the same as three fourths of a cup to bake some cookies. In this problem, students are
asked to explain a strategy that Elam can use and find the missing numerator. This
problem requires student not just to use the procedure for equivalence between two
. 3 . . . .

fractions, 4 = £8) , but also to do the reasoning and explain it, which demands a
considerable amount of cognitive effort.

However, there are also problems that require the recall of the previously learned
rule for finding equivalent fiactions in “student exercise”, for example, “complete the

3 ? . . .

number sentence, 4 = 8 and check your work usrng your fraction chart (Wagreich et al.,
2004, p. 224).” Without understanding Brad’s class, this problem can be categorized as
requiring lower level cognitive demand. However, as the following comment indicates,
from his viewpoint, this problem was very meaningful to his students.

Seeing the equivalent fiactions when they multiply numerator and denominator

times the same number, they give me equivalent fractions. Looking at the

fraction chart and seeing that, because I think that’s the part they are going to

thinking, that’s going to give them most understanding as opposed to just

knowing the rule. Yeah, I want them to know the rule, follow the equivalent

fraction rule. That’s important. . .. But, as a teacher I want those kids to

understand exactly what that means when they are making equivalent fractions.

That means that kids know that they [equivalent fractions] are equal and they can
see it (interview, 4/24/2007).

188

Through the discussion in Brad’s class, students attempted first to find patterns,

then to derive the rule, and finally to verify the rule from a fraction chart. Therefore the

above problem that could have involved just reproducing the equivalent fraction rules

instead involved justification and verification of procedures to find equivalent fiactions,

which lead to deeper levels of understanding of the idea of equivalent fractions. Indeed,

the cognitive demand of the problem depends on how it was used in the context of the

lesson. What might have seemed like a lower level cognitive demand problem, in the

context of Brad’s lesson, became part of a higher level cognitive demand process. This

problem therefore required students to use procedures with connection to concepts,

meaning or understanding.

Although general procedures may be followed to find equivalent fractions,

students do not need to follow the procedures mindlessly. Brad’s teaching suggests that a

teacher can help students use procedures with connections to the conceptual ideas that

underlie the procedures and develop understanding. Overall, most of problems presented

in the textbook and used by Brad require higher level student thinking.

Teacher Questions The question most frequently used in Brad’s class was

“What are the patterns between the numbers and the denominators?” which is suggested

in the lesson guide of Math Trailblazers. Figure 6.1 indicates specific teacher questions

189

suggested in the lesson guide. Many use verbs which require higher level thinking of
students, such as asking students to “compare” fi'actions and “look for patterns”. In my
observation, I noticed Brad using these questions frequently. In addition, he also
frequently used teacher questions that require students to explain, justify, and verify their
thinking, for example, “How do you know?”, “Is it true?”, “Prove it.” These questions
led students to see the need to explain their ideas, make conjectures, test them out, and
verify them, which require higher level thinking of students. In an interview, he provided
his rationale for using this type of questions,
One point I said, “You know me. I need you to explain why”. Because that’s never
enough for them to just say “here is the answer”. I always want them to explain why. For
this, I want them to explain why that works. I need to explain their thinking and give
them a deeper understanding. And a lot of time, they are right in their math and in their
explanation I have to write out how they solve the problem. They get used to talking like

that. That’s going to easier for them to write like that too. I don’t ask “yes or no” type of
questions. I am always pushing them to explain their thinking (interview, 4/24/2007).

The cognitive demand of problems alone does not guarantee students’ higher

level of learning opportunities. Classroom discourse——e.g., the ways the teacher uses

questions, the way teacher and students talk, what they talk about, how they agree and

disagree——are important to the cognitive demands because they can fundamentally

transform the cognitive demand of a particular task as a teacher enacts them in the

classroom (Spillane & Zeuli, 1999). Overall, Brad constantly used teacher questions that

require students to see relationships between fractions, explain and justify their thinking,

190

verify it, and thereby maintained the complexity of the mathematical problems. Similarly,

the other three teachers categorized in this pattern closely followed their textbook

lesson(s) and maintained the higher level cognitive demand in their teaching.

Influential Factors on the Textbook Transformation Pattern

One noticeable feature that sets teachers in this pattern apart from teachers in

other patterns is their faithful use of textbooks that provide higher level problems and

teacher questions. For example, Brad claimed to be a “follower” of the textbook. In the

survey, he strongly agreed that he was pleased with the textbooks and that the textbook

series matches what he thought was important in mathematics. He also strongly agreed

that he used the textbook as a primary resource. In keeping with this, he reported rarely

omitting questions or activities presented in his textbook and almost every day using the

problems and questions in the textbook exactly as they are phrased. Therefore, survey

responses confirm this conclusion about his faithful use of the textbooks.

However, textbook use is not the sole factor that determines the content of

mathematics instruction in his classroom. During interviews, he said that although he

used their textbook as a primary resource, the Grade Level Content Expectations (GLCE)

from the State of Michigan decide what gets taught in his classroom.

First thing is we got the Grade Level Content Expectations from the State of Michigan
and So I look at that. If a lesson doesn’t necessarily fit with what the content expectation

191

is to be taught, we might omit it or omit part of it. Because what we found was that we
are given all these grade level content expectations (GLCE), so we got all these GLCEs
needed to be covered. But if we taught everything in Trailblazers in the order that it is in
there, we aren’t going to get to the GLCEs. So we have to start omitting something that
were in the Trailblazers some lessons that weren’t going to help to teach those GLCEs.
And also for everything that wasn’t in Trailblazers but was expected to teach, then we
have to supplement. In the fractions unit, in particular, we‘teach it all [lessons presented
in Dailblazers] because there are so many GLCEs related to fractions. In the case of
Trailblazers there things aren’t in Trailblazers, we need to supplement (interview,
4/24/2007).

Indeed, this practice of maintaining higher level problems and teacher questions
in teaching is important because it has the potential to suggest a different idea of what
counts as mathematical knowledge and doing mathematics in school. For teachers in this
pattern, mathematics was not presented as a statement of end products—definitions, rules,
procedures-for memorization. Rather, the problems used by teachers in this pattern
exposed concepts and ideas that can be used to construct procedures for solving
problems (Lampert, 1986).

Moreover, along with questions requiring higher level of student thinking, doing
mathematics in these teachers’ classroom does not mean mainly following predetermined
steps to compute correct answers (Greeno, 1990; Romberg, 1983; Spillane & Zeuli,
1999). As showed in Brad’s classroom, doing mathematics involves engaging in
mathematical thinking, offering conjectures, responding to one another’s ideas (and the
teacher’s), and solving problems as well as explaining, justifying, and defending their

ideas and solutions. This contrasts sharply with the manner of doing mathematics in

192

other classrooms where mathematics is typically presented as the statement of

established mathematical truths. Such a different notion of doing mathematics and

knowing mathematics sets teachers in this pattern apart from teachers in other patterns.

Analysis of interview data revealed two additional factors that are associated with

maintaining a high level of cognitive demand in problems and questions used in teaching.

Understanding and sense-making as a teaching goal One common factor that

influences these teachers’ use of the textbook is their notion about how students learn.

During interviews and in teaching practice, all four teachers--Brad, Tom, Catherine, and

Michelle put priority on understanding and sense-making in learning mathematics. None

of the teachers put priority on tests (e.g., MEAP) in teaching mathematics. As described

earlier, Brad articulated his goal in teaching mathematics, in particular, knowing “why,”

not only knowing “how”. Likewise, the other teachers in this pattern pointed out the

importance of understanding and reasoning in learning mathematics. For instance, Tom,

who has 33 years of teaching experience, closely followed Everyday Mathematics in my

observation He used textbook problems that require students to explore relations

between fiactions and decimals using representations (e. g., concrete, pictorial, symbolic)

in teaching. He also used teacher questions that require students to explain, justify, and

defend their answers using pictures and words. During an interview, he articulated his

193

goals as follows:

I want them to kind of think mathematically and be able to justify their answers and I use
that same philosophy in science. So it’s not whether or not they get the right answer.
Whether they can defend their answers is the most important. The right answer is
important but whether they have some thinking to backup their answers so that’s the
most important. So that’s what I’m trying to get them to that point (interview,
12/05/2006).

Teachers in this pattern commonly said that students needed to learn not only
mathematics concepts, procedures, and rules, but also mathematical ways of thinking,
such as explanation, justification, and verification, which require higher level of thinking.

Alignment of Teachers’ goals with the Standards-based curriculum Another
common factor in this pattern is that all four teachers--Brad, Tom, Catherine, and
Michelle perceived that the textbook they used matched well with what they wanted to
teach. During the interview, all four teachers commonly reported that they really liked
the textbook they were using because the conception of doing mathematics and knowing
mathematics presented in the textbook matched with their notion about how students

learn mathematics. For example, Tom said,

I think if this is much better than the old method that we had, the old method was
just practice memorization. This one is making them think more about
mathematics and more creative. But it’s a lot of regimentation. But at least it’s

making them think a little bit higher mathematically and down here (interview,
12/05/2006).

All four teachers remarked that they were pleased with their textbooks since the

curriculum materials they used emphasized exploration and student understanding. Such

194

emphasis presented in cuniculum materials matched with their notion about how
students learn mathematics. This factor accounts for us why these teachers closely follow
their textbooks.
Higher Level Textbook Problems and Questions to Higher Level

Problems but Lower Level Questions in Teaching: Pattern 2

Two teachers—Karen and JoAnne use tasks in ways that can be described as the
H-L pattern in which teachers use textbooks that provide higher level problems and
questions, but they use lower level teacher questions in their teaching. The case of Karen
is used to illustrate his pattern although she used higher level problems, she regularly
lowered the cognitive demand of problems by using lower level teacher questions, and
thereby she eventually provided lower level student learning opportunities. Karen
infrequently used teacher questions that require justifying and explaining their answers.
The features of the following vignette exemplify practices of both teachers observed who
followed this pattern.

A Vignette of Karen

Karen has four years of teaching experience. She works together with Brad at the

same elementary school and in the same grade. Like Brad, Karen also perceived her

students as mostly average achieving. I observed the same lesson “equivalent fractions”

195

in Karen’s class, which I observed in Brad’s class.

On the day I observed Karen’s lesson, the following terms were written in vertical
order on the left side of the boar , ‘2;- , denominator, numerator, if): , other” were
written in vertical order. On the opposite side of the board, the word “equivalent
fraction” was written.

Like Brad, Karen began the lesson by reviewing what students learned previously.
She asked students to “tell me one thing that you know about fractions.” As students
came up with their ideas about fractions, for example, “the top number is called a
numerator and the bottom is called a denominator,” or “the numerator tells you how
many pieces we’re looking at and the denominator says how many pieces make up the
whole,” Karen continued to connect students’ ideas to other ideas by asking students
what is the numerator and what is the denominator in g- .

Karen then addressed the topic they were going to work with together. She said to
her students “Today, we are going to talk about equivalent fraction” and asked students to
write the words “equivalent fraction” down in their notebook. In Karen’s class, “writing
down” is one of the routines students use while studying mathematics. For the rest of the
class, Karen frequently had students write down what she dictated.

As students finished writing it down, Karen started to use activities suggested in

196

the lesson guide. Karen asked students to look at the fraction chart they used in the
previous lessons and find all of the fractions that were equivalent to one half. Like Brad’s
class, students in Karen’s class easily came up with the equivalent fractions. As
suggested in the lesson guide, Karen asked students to arrange these equivalent fractions
in an order such that a denominator gets bigger. As in Brad’s class, students arranged
them and Karen wrote them on the board, such as -1— = —2- = 2 = i = 3. Karen again

2 4 6 8 10
asked students to write this arrangement of these equivalent fractions down underneath
the word “equivalent fractions” in students’ notebook. In the meantime, she told them
that all of these fractions are the same size.

As students finished writing down equivalent fraction sentence, Karen asked
students to find patterns by comparing the denominators. Students noticed two patterns,
“counting by twos” and “even”, but they did not come up with the answer that Karen
expected, which is multiples of two. In this instance, Karen interacted differently from
Brad. While Brad gave hints, waited until students volunteered to answer, and had
students discuss the incorrect answer, Karen told her students the expected answer. This
interaction with students exemplifies how teachers decrease the cognitive demand of

problems during instruction. Henningsen and Stein (2000) reported that teachers often

decrease the cognitive demand of student thinking by taking over student reasoning and

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telling students how to do the problems.

For the rest of the lesson, I repeatedly observed this pattern. For example, after
having students find the relationship between the numerators and between the
denominators of fractions equivalent to one half, Karen asked students to look at the
numerators and denominators together of each of equivalent fractions and find patterns.
Several students responded, but none provided her expected answer. Karen again said to
students “the numerator is half of the denominator, isn’t it?” Indeed, Karen’s telling was
more obvious when students did not figure out patterns in the number sentences. She
wrote four number sentences on the board and said;

I have on the board four number sentences. Four number sentences. The first one, one

half equals two fourths. The second one, one half equals four eighths. The third one, one

half equals five tenths and the fourth one is one half equals six twelfths. Look at that

pattern in these number sentences, first by looking at the numerators and then by looking
at the denominators. What do you see?

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One student answered but his answer did not show the relationship between
numbers and denominators. The same mistake occurred in Karen’s class as in Brad’s
class. In both Karen’s and Brad’s classes, when students were asked to find the
relationship in the number sentences as above, students tended to compare the

denominator of the first fraction and the numerator of the second fraction, and find the

relationship. For example, in Karen’s class, one student found the relationship for the

l‘t8

first number sentence, such as “If you take two and times it two you get four”. Karen
responded to this study as follows: “Is this what you were saying? One times two is?”
Karen paused and students said “two”. Karen again said “two times two is?” and paused.
Students said “four.”

Although Karen set up problems finding the relationship and used it to find other
equivalent fractions, as students became confused, Karen gave the pattern as a rule that
students needed to follow. Karen said, “Here is the rule. Get your pens ready. If you
multiply both the numerator and the denominator of a fraction by the same number, the
result will be an equivalent fraction”. Karen asked students to copy out the rule in their
notebook, and repeatedly stated the rule while students were writing the rule down.
During an interview, Karen mentioned that “students need to memorize it to apply it,”
contrast with Brad who said that “students need to make a discovery for themselves.”

By using the rule, Karen tried to have students see patterns in the rest of number
sentences. For example, Karen asked students to take a look at the third number sentence
é— : % and see whether the rule works. Some students nodded their heads to say yes
and some kids shook their heads for no. She said that

That’s what the rule says. Multiply both the numerator, that’s my numerator and the

denominator by the same number. The result will be an equivalent fraction and I said
over here that one half equals five tenths (interview, 04/27/2007).

Karen led the students through four number sentences involving equivalent

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fractions, all the while focusing on the procedure. For example, for the fourth number

sentence, Karen repeatedly asked, “Three times what equals six?” Karen paused and

students said “two.” Karen asked again, “Four times what equals twelfths?” and paused.

Students said “two.” Karen’s questions became procedure-oriented, required only

multiplication facts. The conversations that Karen had with students during this portion

of class revealed what her goal for the lesson was: writing equivalent fiactions by

applying the rule.

Like Brad, Karen asked students to explain and justify their answers. For example,

Karen also asked students to see if it is true that one third equals two sixths. However, in

her class, it was sufficient and acceptable for students to rely on the rule. Consider the

following Karen’ remark: “We said it’s true if we could multiply both the numerator and

the denominator by the same number to come up with the answer. What number do I

multiply one times two is equal?”

During the final portion of the class, Karen had students work on the problems

from the textbook, Questions 1-5 on the Equivalent Fractions Activity Pages in the

Student Guide. Students occasionally raised their hands with questions. Karen answered

individual questions. Karen ended up the lesson by asking students to create the number

sentences involving equivalent fractions such as “one times two is two and six times two

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is twelve”, which is expressed as 2x: = —2— in a numerical form. Karen called out one
X

 

student. But the selected student said “I don’t know how to do this”. Karen called out
other students. A few students could make the number sentences described above.
Indeed, when I walked around while students worked on the student exercise problems, I
noticed that a majority of students struggled with finding equivalent fractions.

On the following day, Karen continued to have her students work on problems
that she did not cover in the previous lesson. She even had students work on homework
problems in class, however, in the same manner she did on the first day. Karen kept
emphasizing knowing procedures and applying it on the second day. For example, she

repeatedly said “that what the rule said. Apply the rule” to find fiactions equivalent to %,

MIN

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3
Similarities and Difi'erences between the Textbook Lesson and the Lesson Taught in
Terms of Cognitive Demands

There were similarities between Karen and Brad, chiefly on the dimension of the
mathematical problems they set up. The same mathematical problem was used in both
classes and was set up in essentially the same manner. The ways in which students
actually went about working on the problems differed, however, in the two classes.

During instruction, Karen shifted the emphasis from meaning, concepts, or

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understanding to using procedures. She took over students’ thinking and reasoning and
specified explicit procedures for finding equivalent fiactions. As class went on, Karen’s
questions became much narrower, asking students to fill in the blank rather than
construct an answer (e.g., “two times two is?). Karen attempted, and rarely managed, to
elicit students’ mathematical thinking using her questions.

Learning goal Indeed, Karen’s goals for learning are not the same as Brad’s.
Brad’s learning goals are for students to discover the patterns between equivalent
fractions and use that pattern to find equivalent fractions. His priority is on meaning and
sense-making. In contrast, Karen’s learning goal is more procedural. During the
interview, Karen articulated her learning goal as “proficiency in writing equivalent
fractions” for the lesson I observed, and “mastering the concepts and applying it” in
general. Like Brad, Karen remarked that “I want kids to be able to do that and explain
themselves”. However, the nature of “explaining” in Karen’s class differs from the nature
of explaining in Brad’s. In her class, explaining is a tool for restating the process of
how to apply the rule to get the correct answer. Karen changed the learning goal
suggested in the lesson guide from developing relationship between equivalent fractions
to using the procedure to find finding equivalent fractions, which is a lower level of

student thinking.

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Problems Karen used the same mathematical problems as those used in Brad’s
class for an introductory teacher activity and student exercise. Therefore, the problems
set up in Karen’s class focus students’ attention on the discovery of patterns and the use
of the patterns for finding equivalent fractions, which demands complex thinking and a
considerable amount of cognitive effort.

Teacher Questions The questions most frequently used in Karen’s class were
“what number do we multiply?” which require only multiplication facts. In Karen’s class,
the procedure of “how to do it” was stressed above all else. In an interview, she
confirmed her typical questions as below:

I think probably the question that I asked over and over again and maybe not in this exact

word is “What number should I multiply both the numerator and the denominator by to

find the equivalent fraction?” I probably said that a hundred times (interview,
04/27/2007).

Her remark “a hundred times” shows not only how frequently she used this type

of questions during the class but also how she was aware of her fi'equent use of this type

of questions in her class. Students in Karen’s classroom were rarely pushed to elaborate

on their answers. If students’ responses reflected the correct answer, the teacher did not

raise follow-up questions to make students’ mathematical thinking explicit. If students’

responses reflected the incorrect answer, teachers paraphrased the answer, changing it to

make it more “accurate.” Although Trailblazers provide a lot of suggestions in the lesson

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guide for how teachers should approach questioning or approach discussing concepts,
Karen did not use those questions. She shifted the emphasis of their work from meaning
or understanding to the use of procedure without connections, and decreased the
cognitive demands of teacher questions in ways that require lower level of student
thinking.

Indeed, Karen used the similar questions that Brad used, such as “is that true?”
However, in her class, explaining seems to means for describing how to get the answer
rather than justifying and defending their ideas. It was sufficient and acceptable for
students to rely on the rule in her class, which signifies that the nature of explaining
requires different cognitive level of student thinking from that in Brad’s. Notice that
Karen used a higher level technique (explaining) in a lower level way in context whereas
Brad used a lower level problem as higher level in the context of the lesson. Students’
opportunities to appreciate doing mathematics in Karen’s class are curtailed by questions
that require lower level of student thinking. Students in her class might learn and do
mathematics as involving merely following predetermined steps to compute correct
answer as opposed to involving mathematical thinking and sense-making.

Influential Factors on the Textbook Transformation Pattern

Karen supplements the lessons from Trailblazers with practice problems. While

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Brad claimed to be a “follower” of Math Trailblazers, Karen claimed to be “modifier” of
the textbook. She reported that, in total, 50% of lessons came directly from Math
Trailblazers and the rest 50% came from other resources such as other textbooks. She
said that she supplements the textbook with more practice problems from other textbooks,
in particular, an old textbook, Houghton Mifllin Mathematics. During an interview, Karen
said,
I’ve used old textbooks. I find that in the Trailblazers series there is a lot of introducing
the concepts but not a lot of mastering the concepts... We have some old texts [Houghton
Mrfllin] like that has more practice than the Trailblazers does....Houghton Mtfilin has
some from when I was in school so 1980’s. The Trailblazers series provide a lot of

explore, look at, manipulate but it’s very shallow in the practice areas as you can see

from the lesson that there’s I don’t know ten problems or seven problems (interview,
04/27/2007).

Together with her use of lower level teacher questions, frequent supplementing of
the standards-based textbook with more practice problems reduces the cognitive demand
of student thinking.

Why did Karen transform her textbook in that way? What factors account for
Karen’s transformation? During the interview, Karen provided the rationales for why she
added practice problems out of her obligation to meet the Grade Level Content
Expectations (GLCE) for Michigan. In addition, Karen’s notion of how students learn

also is a factor that influences her use of the textbook and her teaching practice.

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Conflict between Teachers ’ Goal and Perceived Goals of the Textbook Karen
believes that she should cover all the content presented in the GLCEs. Karen described

her use of the textbook as follows:

When I start out at the beginning of the year, I lay out the content expectations that the
state has mandated then I try to match up the text with the GLCEs. So when I get to each
unit I have to first make sure that I’ve covered all of those content expectations.

Because that’s what the state says we have to do. The district has said, this is the text
we’re using and that the Trailblazers series. But the district also has said that the
Trailblazers series does not cover all of the Michigan grade level content expectations.
So therefore you must supplement. So when I’m planning I take a look at the lessons that
are in the Trailblazer series and I see which ones of those cover the grade level content
expectations and then I look at all the holes that are left over and start using other
resources (interview, 04/27/2007).

Karen’s obligation to the contents of the GLCEs influences her pedagogy. During

an interview, Karen said,
When I was at a math meeting a year and half ago we were talking about the new content
expectations, what 4‘” graders are required to know. . ..The curriculum director at the time
said, these things are suppose to be taught to mastery and here was this huge group of
experts saying, we’re introducing the concepts but the children are not practicing it
enough to say that we’ve mastered it. . ..So I’m working really hard on trying to make
sure that I can pull from anywhere that I can find practice stuff to make sure my kids are
going home and working on it on their own, working on it with friends here, that their
parents are informed about what we’re doing and also working with them at on it
(interview, 04/27/2007).

Indeed, “mastery of the concepts” can be interpreted in various ways. Some may
think it only from procedural aspects, such as proficiency in computation, whereas others
consider it from both conceptual aspects and procedural aspects of the mathematics
contents. Considering the recommendations in the GLCEs, definition of mastery requires
both conceptual understanding and computational skill (Mcthighe & Wiggins, 2005;

Wormeli, 2006). However, Karen seems to recognize the definition of mastery from the

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procedural aspects. This understanding may cause her to lead instruction more
procedure-oriented and emphasize application of the rule in her class.

Procedural fluency as a Teaching Goal Indeed, Karen’s obligation to meet the
GLCEs is in keeping with her notion of how students learn, as evidenced by the focus of
her questions on procedures—on how rather than why. Karen put more emphasis on
application of the rule than on sense-making or meaning in learning mathematics. Karen
articulated her goals as “proficiency in writing equivalent fractions” and “mastering the
concepts and applying it” in general. This differs fiom the evidenced by teachers in the
first pattern which matches the category ‘firnderstanding-oriented”. Together with
Karen’s view on the emphasis of Grade Level Content Expectations, her learning goal
seems to push her to supplement the textbook with more practice problems and change
the emphases from understanding to following rules to solve problems in her classroom.

Lower Level Textbook Problems and Questions to Lower Level
Problems and Questions in Teaching: Pattern 3

Three teachers use tasks in ways that can be described as the L-L pattern, in
which teachers have textbook problems that provide lower level problems and questions,
and they use lower level questions in their teaching. All used “conventional textbooks.”

Although teachers in this pattern selectively used problems and examples fiom their

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textbooks, most of the problems focused on mathematical facts and procedures, as

distinct from mathematical knowledge that required higher level of student thinking.

Moreover, they spent most of their time asking low-level cognitive questions to elicit

correct answers. As a result, the conception of doing mathematics and knowing

mathematics presented in the mathematical problems and teacher questions in their

classroom differs substantially fiom that of teachers in other patterns, in particular,

teachers in the first pattern. The features of the following vignette from Randy’s teaching

exemplify practices of other teachers in this pattern.

A Vignette of Randy

Randy has 33 years of teaching experience. He is currently teaching fifth grade.

He perceived his students as mostly average achieving as Brad and Karen did. In his

district, teachers have two different textbook series—one is Investigations and the other

is Scott F oresman - Addison Wesley Mathematics. Which textbook teachers use is up to

the teachers. He indicated Scott F oresman - Addison Wesley Mathematics (Randall, et al.,

2005 as his main textbook. 1 observed two lessons in Randy’s class. Like other teachers,

lessons were organized into two activities: an introductory “teacher-led” activity and

“student exercise”. However, different from teachers in other patterns, a “teacher—led”

activity in his class is more of lecture and demonstration rather than discussion. Randy

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guided the students through example problems, with the problem broken down into

simple steps. Then the students were given practice problems to accomplish on their own.

On the first day I observed, Randy taught “adding and subtracting mixed

numbers” and the second day, he taught “adding mixed numbers”. Like other case study

participants, before observing his class, Randy gave me his lesson plan using the

teacher’s manual of Scott F oresman - Addison Wesley Mathematics, Grade 5 (Randall, et

al., 2005, pp. 372—374).

The lesson guide states a learning goal as “Add and subtract mixed numbers”.

This lesson is organized into three big parts; (1) Introduce, (2) Teach, and (3) Close and

Assess. In an introduction, students review adding and subtracting fractions with unlike

denominators. 1n Teach part, teachers are recommended to address Example 1, which

illustrates procedure of subtracting mixed numbers, and to ask students to work on

practice problems. Two more example problems were presented in the textbook lesson.

In Close and Assess, teachers are suggested to have students demonstrate how students

would find the sum with two mixed numbers.

Like teachers in other patterns, Randy started out his lesson with reviewing what

they learned previously. Randy asked students “what is mixed numerals?” Several

students tried out, such as “whole number” and “mixed with a fraction.” When one

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student said “a whole number in a fraction,” Randy said “Yes, I like that definition. It’s a

whole number okay complimented with a fraction.” He again asked students to find the

definition of mixed numerals in the glossary presented in their textbooks. As one student

read aloud from the textbook, Randy said, “Okay, it is a whole number and part of the

next whole number.”

He addressed the topic they were going to work with together, such as “that is

what we are going to be adding and subtracting today.” Randy then began the “example

problem” suggested in the lesson guide of his textbook. He took the suggested example

below from his textbook. He first wrote it on the board and then asked students to copy

down the problem as Karen did. Randy asked students “what do you think we need to

do?”

Indeed, this example problem is presented as a word problem in the textbook.

However, without introducing a story or problem contexts, Randy asked students to tell

him how to solve the problem. Word problems often provide contexts that allow students

to connect the problem to a familiar context in their own lives. Problem contexts or word

problems can provide practice with real-life problem situations, motivate students to

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understand the importance of math concepts, and help students to develop their creative,

critical and problem solving abilities (NCTM, 1991; Verschaffel, 2002).

However, Randy spent no time discussing the textual aspects of the problem as it

was presented. Rather, Randy quickly reduced the word problem to a mechanical fraction

subtraction problem. He provided the following rationale for studying this topic: “ It’s on

our assessment and that’s why we’re doing it.” None of the other teachers in other

patterns put priority on assessment.

Textbook lesson provides four steps to solve this problem: (1) change mixed

number into improper fiactions, (2) find equivalent fractions, (3) subtract the fractions,

and (4) subtract the whole number. He led the students through this problem at each stage.

Randy was calling on individual students to answer questions. While observing his

lesson, I paid attention to how Randy reacted to students’ incorrect answer. I noticed that

like the teachers in the H-L pattern, Karen, Randy barely paused, moving quickly on to

the correct answer. Even Randy did not raise follow-up questions if students’ responses

did not reflect the correct answer. If students’ responses reflected the correct answer, he

paraphrased the answer, changed it slightly to make it more accurate and explicit. For

example, as students finished writing it down, Randy asked, “What do you think the first

thing we have to do?” One selected student said “We have to simplify the numbers still.”

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Randy responded back to the student by saying “Not yet. At the end yes, we will have to
simplify at the end if we get to the end.” Another selected student said “I just want to say
that we change the denominator and numerator.” Randy responded back to students
“Which means we are finding what?” As none of students did answer, Randy said to
students “The equivalent fiactions. We’re finding equivalent fi'actions. We have to find
equivalent fiactions here at least for one of them because what do we need before we can
add.” One student asked back to Randy “We need to have them have the same
denominator?” Randy answered “yes, they have to have the same denominator”.

In watching Randy interact with his students, it was obvious that a teacher-led
activity in his class involves recitation or lecture in which the teachers talk most of the
time. Considering Randy’s questions that asked students to fill in the blank rather than
construct an answer. Randy seemed to try to elicit just “right” answer from students.

After working one example with his students, Randy assigned the first part of
the “practice” problems to students. All practice problems asked students to find each
sum or difference and then simply the answer, which required only procedure. While
students were working on practice problems from the textbook, Randy answered
individual questions, prompted the next step in the procedure, pointed out mistakes.

As students finished working on the given tasks, Randy demonstrated the

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following example of “adding mixed number” in the same manner he did in the first

 

example.
41 = 43
2 6
+5- = + 53
3 6
9.5.
6

Like other problems Randy and his students worked on, this problem also

requires low cognitive demand of student thinking. After demonstrating how to do this

problem, Randy asked students to work on the rest of practice problems.

Clearly, students had learned a set of steps to go through in order to solve these

problems, for example, change the mixed number into improper fractions, subtract, and

change the fractions into mixed number, etc. However, his class is procedure-oriented.

For example, in the classes I observed, there was no drawing and no use of manipulatives.

During the interview, Randy described his class as “very much traditionally lesson from

the textbook.” The conversations that Randy had with students during this portion of

class reveal what his goal for the lesson was: mastery of the “procedure” for adding and

subtracting mixed numbers.

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Similarities and Difl'eren ces between the Textbook Lesson and Lesson Taught in Terms

of Cognitive Demands

Like teachers in the H-H pattern, Randy maintained the cognitive demand of

problems and teacher questions in his teaching. However, the problems and questions

that Randy and Teri used commonly require the straightforward application of a formula,

which demand students to remember and apply a series of rules and procedures to arrive

at the right answer. The conception of mathematical knowledge that dominated in the

problems and teacher questions used by teachers in this pattern differed substantially

from teachers in the first pattern.

Learning goal Randy used the same goal presented in the conventional

textbook. He put priority on knowing procedures over understanding the concepts and

sense-making. For example, Randy articulated his goal as “to get students to understand

this one specific algorithm in adding and subtracting mixed numerals.” When he was

asked to explain what he meant by “understanding”, he said, “know how to apply the

algorithm in adding and subtracting mixed numbers” (interview, 01/23/2007). Even

though he used the verb “understand,” for Randy, “understand” seems to mean know the

process of how to “apply” the rule to add and subtract mixed numbers, which requires

students to remember and memorize rules and procedures.

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Problem Randy also used problems fi'om the textbooks that afford lower
cognitive demand of student thinking. A typical problem presented in the lesson is “Find
each sum or difference. Simplify, 3% + 2%” (Randall, et al., 2005, pp. 372). This type of
problem exclusively focuses on procedural mathematical knowledge that requires the
recall of previously learned information. There is no connection to what underlying this
procedure. Many of the problems found in conventional textbooks in this study ask
students to perform a demonstrated procedure in a routinized way and hence place low-
level, mostly proceduralized demands on student learning as opposed to the problems
found in standards-based curricula asking students to make conceptual connections and
to think and reason in sustained and thoughtful ways (Stein & Kim, 2007). Randy used
lower level problems as an example and student practice in his teaching.

Indeed, the conventional textbook Randy uses also provides several word
problems which contain “real-world” context in “problem solving” section. During an
interview, Randy mentioned that he would use “problem solving” section later to prepare
their students for assessments since word problems are one type of problems in
assessments. However, even word problems presented in the textbook require only the

use of a well-established procedure for finding sum or difference between two mixed

numbers, which can be categorized as one lower level task, procedure without

215

connections, based on Stein and Smith’s fi'amework (2000).

Teacher Questions Randy asked almost all the questions, and students

typically articulated the procedures that led to an answer and/or the answer itself by way

of responding. Further discussion was rare. The interview confirmed this pattern. During

an interview, I asked Randy about typical questions he would often use in his lessons.

Randy said “I’ll pose questions that have detailed answer and sometimes students will do

part of that answer” (interview, 01/23/2007). His remark “detailed answer” implies lower

level of student thinking required. Randy neither asked students to generate their own

algorithms nor to explain how or why procedure worked. Randy’s questions were

procedure-oriented. There was little or no questions that asked students to conceptualize

or think about the content.

Indeed, the conventional textbook Randy uses does not provide many teacher

questions he could use. One teacher question suggested in the lesson guide was “ask

students which steps could be skipped if the fractional parts of each mixed number had

like denominators” (p. 328). This question requires students to compare different

problem situation between mixed numbers with like denominator and those with unlike

denominator, which at least require more cognitive efforts than following the procedure.

Yet, Randy did not use this question. Overall, the textbook provides lower level problems

216

and teacher questions and Randy maintained the cognitive level when using problems

and teacher questions.

Influential Factors on the Transformation Pattern

One noticeable feature is that Randy used “a conventional textbook”. However, it

is not the use of conventional textbook alone that sets Randy apart from teachers in other

patterns. The conception of knowing and doing mathematics represented in lower level

problems and maintained by lower level teacher questions also set him apart fi'om

teachers in other patterns. Similarly, the other two teachers in this pattern used problems

and questions in their teaching in ways that matched with the notion of doing

mathematics embedded in their conventional textbooks, which is mathematics as a set of

procedures that students needed to master in order to solve exercises involving addition,

subtraction, multiplication, and division of fractions.

What factors account for these teachers’ textbook transformation pattern? Open

about their choice, teachers in this pattern named thee causes: assessment, their students,

time. In addition to these factors, teachers’ notion of how students learn can be an

indicator of this pattern.

The pressure of assessment from the GLCEs Teachers in this pattern commonly

remarked that the press to get high test scores on tests also limited their ability to teach

for deeper understandings. Similar to teachers in other patterns, teachers in this pattern
felt obligation to the Grade Level Content Expectations (GLCE). However, teachers in
this pattern seemed more concerned about assessment than the GLCEs. During an
interview, Randy said that

Assessment drives the curriculum. That’s what our administration expects that we are

doing. The curriculum and the assessments because of the amount of work that we have
to get exposed to the kids it drives it and it is too much (interview, 01/23/2007).

Randy mentioned repeatedly how much assessment constrained his use of
problems. Even when he answered why he did not include certain problems in his lesson,
he simply said that “Just it’s not on the assessment so I skipped it”. Indeed, In Randy’s
case, assessment dictates what is taught. Randy was working on a district that bought an
assessment program, which is now the curriculum of a small county. He described his
use of the textbook as follows:

I use the book based on whatever my assessment is. If the book meets the criteria for the
student to do in the assessment then I use the book. If I can do Investigation and it meets
those same criteria then I’ll do it. If it means drawing pictures then I’ll use that. If it
means coming up with constructive responses in small groups then that’s what gets you.
It just depends; I’m not totally textbook driven. It just depends on the resources that I

have and how that it fits the assessment that I am trying to teach or that I’m trying to get
the kids ready for (interview, 01/23/2007).

He, therefore, would skip some contents that are not on the assessment and
delete examples and problems that do not fit with assessment.
The Pressure of Time Teachers in this pattern also pointed out their limited time

for covering all contents to be taught. Randy believes that with his limited time, he can

218

only work on the basic foundation—the rules and procedures. During the interview, I
asked him why he skipped certain problems presented in his textbook. Randy simply
found his decision from “time”. He said, “When do I have the time to teach? Definitely 1
will cover. But when?” (interview, 01/23/2007). In addition to time constraint, he
provided the rationale for why he did not include certain type of problem in that those
problems are not tested in assessment. He said that,

I skipped it because there wasn’t, there wasn’t on assignment where the students had to

round off the mix numeral to the nearest whole and then add. . . .It’s not on the assessment
(01/23/2007).

Taken together with Randy’s obligation to assessment, time factor seems to
constrain his use of textbook problems that require higher level.

Students’Diverse Ability Students’ diversity in terms of mathematics ability is
another commonly perceived factor by teachers in this pattern. Teachers in this pattern
commonly mentioned that their students’ low economic level and achievement in
mathematics influenced their use of textbooks. For example, during the interview, Randy
mentioned that:

What comes into play is the ability of the students and some students are going

to get done a lot faster than others which then allow me more time to do with
students that need more help. That comes into play (interview, 01/23/2007).

Indeed, student ability is one commonly referred factor by teachers to describe

their students’ low achievement (Wilson, 1990). Although analyses of overall MEAP test

results in these schools show that a slightly higher percentage of students in the school

3219

(78 or 79%) met standards for mathematics of the 2007 Michigan Educational
Assessment Program (MEAP) that that of the state average (76%), teachers in this
pattern commonly mentioned that because of their students’ diversity in mathematics
achievement, they focused on practice problems.

Procedural fluency as a Teaching Goal Furthermore, even though teachers in
this pattern did not point out their view on fiactions and his goal as influence on his
textbooks, their teaching goal is possible factor that influences their use of textbooks in
terms of cognitive demands. In particular, Randy’s notion of how students learn, as
evidenced by the focus of his questions on procedures—on how rather than why—
matches the category ‘procedural-fluency oriented’ developed by the present study to
describe teaching which favored the more traditional transmission perspective than
constructivist approach.

Summary

This chapter presents the findings fi'om the case studies that exploral what
textbook transformation patterns exhibit when teachers move content from text to
teaching and what factors influence these transformation patterns. More specifically, this
study explored: (1) What are the similarities and differences between the textbook and

taught lesson in terms of cognitive demands of lesson’s goals, problems, and questions?

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and (2) What factors teachers perceived influence their use of textbooks?

Three transformation patterns were identified in terms of the cognitive demand of

problems and teacher questions—(1) H-H pattern, (2) H-L pattern, and (3) L-L pattern.

First, when teachers used standards-based curriculum that provides higher level cognitive

demand of problems and teacher questions, some teachers closely follow their textbooks

and thereby maintain the higher level cognitive demand of student thinking in their

teaching. This pattern is called the H-H pattern. In this pattern, mathematical problems

were set up to help students explore concepts, meaning, and underlying ideas of the

procedure. Teacher questions sustained the cognitive level of problems by demanding

students conjecturing, problem-solving, and justification of ideas.

However, some teachers with standards-based curriculum lowered the cognitive

demand level of the textbook lessons by using teacher questions that focus on procedure

and finding the answer. This pattern is called as the H-L pattern. In this pattern, while

mathematics problems in these classrooms were presented in ways that highlighted

meaning and sense-making, teacher questions focused students’ attention on procedures

rather than meaning.

One of the big differences between teachers in the H—H pattern and those in the

H-L pattern was different emphasis of student learning. Teachers in the H-H pattern

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emphasize the importance of understanding, meaning, and sense-making over procedure

and computation, whereas teachers in the H-L put more emphasis on proficiency (or

knowing procedure). This finding is consistent with the findings from the survey.

Two teachers categorized as the H-L pattern in the case studies were categorized

as the H-H pattern in the survey. For example, Karen and JoAnne both reported that they

often used teacher questions that require explanation, multiple representations, and

students’ justification. They were, therefore, categorized as the H-H, representing

teachers who have textbooks presenting higher level teacher questions and use higher

level teacher questions in their teaching. Yet, in the case analysis, these teachers were

identified as using lower level cognitive demand of teacher questions.

Indeed, Karen and JoAnne asked students to use representations, explain their

answer, and justify their answer in their classroom. However, interestingly, “explaining,”

in the context of her lesson was a tool to restate the processes of how to get the answer,

which involves a lower level application of rule and procedure. This may explain why

discrepancies occur between teachers’ responses on the survey and the case studies.

Second, all teachers who used conventional textbooks were considered as

presenting lower level of problems and using lower level of teacher questions, which

were firmly grounded in procedural mathematical knowledge and computational skills.

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This pattern was called the L-L pattern. Although teachers in this category selectively

used problems and questions presented in the textbooks, most of problems and questions

presented in the lesson plan lend themselves requiring lower level of thinking, which

require students to memorize and follow the rule to solve problems. Teachers in this

category commonly put more emphasis on knowing procedures and algorithm than

meaning and sense-making. Even though these teachers also recognize the importance of

having students opportunity to experiences related to underlying ideas (e. g., using

manipulative, game, etc), these teachers put more emphases on the ability required on the

assessment when selecting problems and using teacher questions. These teachers

commonly reported their rationales for why they did not include problems requiring

higher level cognitive demand from the assessment, time, and their students’ low

mathematics ability.

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Chapter 7
SUMMARY, DISCUSSIONS, AND IMPLICATIONS

This chapter is comprised of four sections. In the first section I summarize the
findings from the survey and the case studies, and in the second section I discuss the
theoretical and practical contributions of this work and clarify its limitations. In the third
section, I provide implications of the findings from this study by considering what they
illuminate about teachers’ textbook use and the role that textbooks and practical and
curricular support for teachers might play. In the fourth and last section, I conclude this
study by providing final thoughts about changing teachers’ practices using textbooks.

Summary

This study was prompted by the current cuniculum policies that use textbooks as
an important strategy to regulate mathematics practice and align instruction with the
reformers’ ideas. The purpose of this mixed method study was to examine teachers’
textbook use and its influential factors from a different angle, that of the cognitive
demand of mathematical problems and teacher questions, and to provide both depth and
breadth of analysis with respect to teachers’ textbook use and its influential factors.

I sought to answer the following research questions: (1) What transformation

patterns do elementary teachers exhibit as they move content from text to teaching if the

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transformation is looked at in terms of cognitive demands of problems and questions? (2)

What factors are associated with various textbook transformation patterns?

This study began with four hypothetical textbook transformation patterns that

describe the relationship between the cognitive demand of problems and questions

presented in textbooks and that of problems and questions used by teachers in teaching—

(1) H-H, (2) H-L, (3) L-L, and (4) L-H. The H-H pattern refers to teachers who had a

textbook categorized as providing “higher level” problems and teacher questions (e. g.,

problems and questions requiring students’ mathematical thinking or justification) and

who maintained this higher level in their teaching. The H-L pattern means teachers had

textbook problems and teacher questions that required higher level student thinking but

decreased the cognitive level in their teaching. The L-L pattern refers to teachers who

had textbook problems and teacher questions requiring a lower level of student thinking

(e.g., problems and questions requiring students merely to follow the rules and

procedures) and who maintained the low cognitive level in their teaching, whereas the L-

H pattern means that teachers increased the low cognitive level textbook problems and

teacher question in their teaching.

I used the term “high” and “low” not only to imply cognitive demands for student

thinking required when working on problems and questions, but also to suggest the

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extent to which problems and questions represented the conception of knowing and

doing mathematics in ways that corresponded to reform ideas. At the core of the

mathematics reforms (N CTM, 1989, 1991, 2000) is an attempt to revise what counts as

mathematical knowledge and doing mathematics in school. Reformers and educators

argue that students should come to understand mathematics as involving more than

manipulating numbers to compute right answers and should develop mathematical

thinking, reasoning and problem-solving (Cobb, 1988; Lampert, 1990; NCTM, 1989,

1999, & 2000; Schifer & Fosnot, 1993; Simon, 1986). In this study, problems and

questions categorized as “high level” were perceived as more likely to foster the

conception of doing mathematics and knowledge mathematics supported by the NCTM

reform ideas than problems and teacher questions categorized as “low level.” By

exploring changes between the cognitive demand of textbook problems and teacher

questions and the co griitive demand of problems and teacher questions used by teachers

in their lessons, this study attempted to understand whether and how teachers use their

textbooks to provide students’ learning opportunities supported by the NCTM (1989,

1991, & 2000) ideas. This study also sought to understand whether the type of textbook

(e. g., standards-based vs. conventional textbook) influenced these transformations and

what other factors supported or constrained teachers’ use of textbooks. Thus, this study

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was concerned with exploring how teachers’ use of textbooks relates to students’

opportunities to learn mathematics in ways considered meaningful by the

field/professionals in this domain.

A survey was used for the quantitative aspect of the research, along with

interview and observation. for the qualitative part. Among 169 teachers who participated

in the survey, eight teachers were observed and interviewed in order to establish the

validity of the written survey and to provide additional details about teachers’

transformation patterns that can’t be obtained fiom the survey. In the following section, I

summarize the findings from the survey and the case studies.

Transformation Patterns in Terms of Cognitive Demands

Transformation patterns in terms of cognitive demands were explored in relation

to both the text phase and the enactment phase of instruction. Within the text phases, I

examined transformations of textbook problems in teaching and transformations of

textbook teacher questions in teaching. Within the enactment phase, I examined the

enactment of problems along with teacher questions. It is important to note that, while all

transformation might be considered ‘enactment,’ I use this word here to describe how the

problems in teaching were repeatedly enacted by teacher questions in teaching.

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Transformation Patterns of Textbook Problem

First, in the problem-problem phase, this study explored how problems presented

in textbooks were changed in terms of the cognitive demand when teachers selected

problems and used them in their teaching. Figure 7.1 presents the transformation patterns

from text to teaching in terms of the cognitive demand of problems identified after

combining the findings from the survey and the case study.

 

 

 

 

 

 

 

 

 

 

 

 

Textbook Lesson taught——
Problems Problems

Hi h level I ,IJ High level I

............... Hi h level I

I Low level "W I ....... =I Low level I

 

 

Figure 7. I Textbook Transformation Patterns Identified in Terms of Cognitive Demand of

Problems

The solid line represents the pattern identified from both the survey and the case study.

The dashed line represents the patterns identified from either only the survey or only the

case study. The H-H pattern and the L-L pattern appeared both in the survey and the case

study. When teachers used textbooks that presented higher level problems (standards-

based curriculum), they used higher level problems in their teaching. When teachers used

textbooks presenting lower cognitive demand problems (conventional curriculum),

teachers used lower cognitive demand problems in their teaching even when they

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selectively used textbook problems.

This finding indicates that teachers report that they maintain the cognitive

demand of textbook problems when they use problems in their teaching. This implies that

the cognitive demand of problems presented in textbook plays an important role in

deciding the cognitive demand of problems used by teachers in teaching. In particular,

because standards-based curricula portray mathematics as a process of exploring

relations, which requires students’ high cognitive demand (Stein & Kim, 2007), teachers’

use of standards-based cuniculum might increase the likelihood that students would

engage with these mathematical concepts and do mathematics involving developing

conjectures, explaining and justifying their solutions rather than following rules.

This inference can also be supported by previous studies. For example, Star and

Hoffrnann (2005) examined whether and how students who worked with standards-based

curricula perceived the conception of mathematics and doing mathematics differently

from students who worked with non-standards-based curricula (in this study,

conventional curricula). They reported that students at the standards-based site expressed

more sophisticated conceptions of knowing and doing mathematics than the students

from the non-standards-based site. Since teachers in this study who used standards-based

curricula were found to be maintaining the cognitive demand of textbook problems,

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which means that teachers used problems that required student thinking and reasoning, it

is plausible that students in these teachers’ classrooms had more opportunities to work on

mathematical problems that required exploring mathematical concepts and underlying

ideas through representations, communications, and reasoning and were able to do so

more than students whose teachers used conventional curricula.

Due to the selection of teachers in the case study, the L-H pattern in the problem-

problem phase did not appear in the case studies (all participants in the case studies were

either the H-H pattern or the L-L pattern in the survey). However, the analysis of a lesson

plan and the survey suggest the possibility of teachers who fit into the L-H pattern when

they use problems in their teaching. The fact that this possibility was not represented here

suggests the need for firture studies that investigate this pattern and teaching practice in

relation to textbooks.

Transformation Patterns of Textbook Teacher Questions

In the teacher question-teacher question phase, this study examined how teacher

questions presented in textbooks were changed in terms of cognitive demands when

teachers selected them from textbooks and used them in their teaching. Figure 7.2

presents the transformation patterns in this phase exhibited in the survey and the case

study.

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Textbook Lesson tau ht
Teacher Questions Teacher Questions

I High level ., I J High level

 

 

 

 

 

 

J
~L I Low level J
l

.......... 7 Hi h level

I Low level ‘+—"I Low level I

Figure 7.2 Textbook Transformation Patterns Identified in Terms of Cognitive Demand of

 

 

Teacher Questions

Four patterns appear when combining the findings from the survey and the case studies

in the question-question phase. The H-H pattern and the L-L pattern were exhibited both

in the survey and the case study. The L-H pattern emerged from the survey while the H-L

pattern emerged only from the case study. These patterns show interesting trends.

First, the confirmation of the H-H pattern and the L-L pattern from the case study

suggests that, as in the problems-problem transformation phase, the cognitive demand of

textbook questions plays an important role in deciding the cognitive demand of questions

used by teachers in teaching. If teachers use textbooks that present higher level questions,

then some teachers use higher level questions in their teaching. This relationship between

text and teaching suggests the possibility that textbooks that require higher cognitive

demand of problems may contribute to change teaching in ways that reformers intend to.

However, the appearance of the H-L pattern and the L-H pattern, in particular the

H-L pattern, reveal that there are other factors that influence teachers’ decision making

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when teachers use questions in their teaching. In terms of the cognitive demand of

problems, some teachers at least maintained the cognitive level of textbooks in their

teaching. However, as far as teacher questions are concerned, some teachers decreased in

their teaching the cognitive level of teacher questions presented in textbooks. In the case

studies, two teachers, Karen and J oAnne, used standards-based curriculum that presented

higher level questions, but they lowered the cognitive level by changing the emphasis

from exploring mathematical concepts and meaning to applying the rules to solve the

problems and using the questions that focused on following the procedures.

This finding provides important suggestions to understand better the text-teacher

relationship, in particular in terms of teacher questions. For example, the implication is

that although the cognitive demand of questions presented in textbooks is important,

teachers are influenced by various factors when using questions in their teaching. This

recognition suggests that we need to consider other factors, including teachers’ individual

characteristics, to understand teachers’ textbook use.

Transformation Patterns in Problem-Questions Enactment Phase

In the problem-teacher question enactment phase, this study explored whether

and how the cognitive demand of problems used in teaching was maintained or

decreased by the questions used by teachers in teaching. Even though the transformation

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pattems explored in this phase did not explain directly the text-teacher relationship,

without understanding the change in this phase we do not know which level of learning

opportunities are provided for students. Since teacher questions can fundamentally

transform the manner in which a particular problem is enacted by the teacher and

students, this study explored transformation patterns in this phase. Transformation

pattems that appeared in the problem-teacher enactment phase can be used as an

indicator that implies the students’ level of cognitive engagement.

Four transformation patterns were observed—(l) H-H, (2) H-L, (3) L—H, and (4)

L-L pattern. Among teachers who reported using mathematical problems that required

higher cognitive demand, some teachers reported using teacher questions that sustained

the cognitive demand of the mathematical problems. However, other teachers like Karen

and JoAnne in the case study, set up problems that required students’ mathematical

thinking and relationship, but used teacher questions that lowered the cognitive demand

of the problems by changing the emphasis from understanding to knowing procedures.

Some students in the H-L pattern may know and do mathematics as involving more than

manipulating numbers since they worked on higher level problems that required

mathematical thinking, reasoning, and explanation. However, many students in this

pattern may perceive mathematics as just involving following predetermined steps to

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compute correct answers because of the way the teacher posed the questions.

In contrast, among the teachers who used mathematical problems that required

lower cognitive demand (e. g., following the rule to compute correct answers), some

teachers, like Randy in the case study, used teacher questions that sustained the low

cognitive demand of the problems. Yet, other teachers might increase the cognitive

demand of the mathematical problems by using higher level questions requiring that

students develop conjecture, explain, justify, and defend their solution and thereby

provide higher level of students’ cognitive engagement. Although the L-H pattern did not

appear in the case study, this pattern might be possible (e. g., Martin, 1997). For example,

Deborah Ball began her class with a low level problem (e. g., compare two fractions, '/2

and 1A). However, by using questions that require students to explain and justify their

answer, or defend their solutions with various representations, she ultimately increased

the cognitive demand of the problem and led her students to learn mathematics in ways

that not only manipulate numbers but also develop mathematical concepts and

relationships.

To understand teachers’ enactment of problems and teacher questions in relation

to textbooks, I connected transformation patterns that appeared in the problem-problem

phase with those in the problem-teacher questions phase, according to teachers’ response

234

on the survey. Figure 7.3 presents the transformations from text problems to problems

used in the lesson to questions used by teachers in teaching in terms of the cognitive

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

demand.
Textbook Lesson taught Lesson taught
Problems Problems Teacher Questions
L High level I 1 High level I Hi h level 1
”J High level I.:::: ....
I Low level H Low level P ..... Low level I

 

 

 

 

Figure 7. 3 Transformation Patterns Identified in Terms of Cognitive Demands of the Lesson

Taught

When teachers used problems in their lesson from their textbooks, there was not much

variation between the cognitive demand of textbook problems and the cognitive demand

of problems used by teachers in teaching. However, when the problems were enacted by

teachers with students in teaching, there was much more variation between the cognitive

demand of problems and that of questions used by teachers in teaching. Table 7.1 shows

the frequencies of teachers in each pattern.

Table 7. 1 Frequency of Teachers in Each Pattern

 

 

Textbook Lesson taught Frequency

Problems Problems Teacher questions (N= 1 69)
High High High 1 17
High High Low 8
Low High High 19
Low High Low 13
Low Low High 6
Low Low Low 6

 

Among teachers who had textbook problems that required higher level, eight teachers

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reported lowering the cognitive demand of problems in teaching by using lower level

teacher questions. Among teachers who increased the cognitive demand of textbook

problems to a higher level, thirteen teachers reported lowering the increased cognitive

level of problems used in teaching by using lower level teacher questions.

The tracking from textbook problems through teaching problems to teacher

questions used in teaching suggests two important aspects to consider. First,

comparatively, the simple relationship between textbook problems to problems used in

teaching in terms of the cognitive demand suggests that whether textbooks provide

higher level problems or lower level problems may be one indicator that describes what

problems teachers use in their teaching in terms of the cognitive demand and therefore

what learning opportunities students might be provided to students.

However, the complicated relationship between the cognitive demand of

problems and teacher questions used in teaching in the problem-question enactment

phase suggests that we need to explore learning opportunities for students by looking not

only at the problems written in a textbook or the problems set up by teachers, but also at

the ways in which the problems used by teachers are actually carried out by the questions

teachers pose for students to work on. In addition to that, the teachers are influenced by

not only the cognitive demand of textbook problems and questions but also teachers’

236

individual characteristics (e.g., teaching goals) as their learning opportunities (e.g., PD

experiences).

This study expanded upon previous research in that it attempted to explore

teachers’ transformation patterns from a different angle, in order to understand what

learning opportunities for students teachers provide with their textbooks. This study also

attempted to consider this topic based on what teachers report on the use of their

textbook and on how they actually use them in practice. This study suggests the

importance of the cognitive demand of problems and teacher questions presented in

textbooks and argues that if teachers have textbooks that provide higher level problems

and teacher questions, they are likely to more frequently provide problems and questions

that require students to think mathematically, and to reason out solutions and explain and

justify their answers with various representations, than teachers who have textbooks that

provide lower level problems and teacher questions. Students in this class are thereby

more likely to be provided learning opportunities for knowing and doing mathematics

that foster their mathematical thinking, reasoning and problem solving. Conversely, if

teachers have textbooks that provide lower level problems and teacher questions, they

are less likely to use problems and teacher questions that require higher level cognitive

demand, and thereby provide students with learning opportunities that involve following

237

predetermined steps to compute correct answers and chiefly manipulating numbers

without connections to underlying ideas or meaning.

However, the previous statements are not meant to suggest that textbooks that

require a higher level of student thinking are a panacea. I pointed out earlier that various

factors are associated with teachers’ decisions--in particular, teachers’ use of questions,

factors that decide the ultimate fate of the cognitive demand of problems students work

on (Henningsen & Stein, 1997; Stein et al., 1996). Next, I discuss what factors influence

these various transformation patterns, by synthesizing the findings fiom the survey and

the case study.

Influential Factors Associated with Various Transformation Patterns

Using the survey of in-service teachers (n=169 teachers), this study first

examined the factors that influence the transformation patterns in three phases and then,

using the case study data (8 teachers), I examined that what teachers actually perceived

constrained or supported their use of textbooks.

Several factors were commonly identified that distinguish the H-H pattern from

the L—L or from the L-H pattern in three phases of transformations as well as in the case

study. For example, the type of textbook (standards-based vs. conventional cuniculum)

was the most critical factor that distinguished teachers in the H-H pattern from teachers

1238

in the L-L pattern (see Odds in Table 5.11 and Table 5.14 of Chapter 5). Teacher teaching

goals (use of understanding-oriented student objectives) and teacher fidelity to textbooks

were also found to be important factors that characterized teachers in the H-H pattern

compared to teachers in other patterns. In addition, teacher view on textbook was a

commonly identified factor that distinguished L-H pattern from the L-L pattern. The type

of textbook (conventional curriculum) was found to be an insignificant factor in

distinguishing teachers in the L-H pattern from teachers in the L-L pattern. Using the H-

H and L-L patterns as a reference, 1 next summarize which factors characterize each

individual transformation pattern

Factors Distinguishing H—H Pattern from L-L Pattern

Four factors were identified as distinguishing teachers in the H-H pattern from

teachers in the L-L pattern: standards-based curriculum, teacher fidelity to the standards-

based curriculum, understanding-oriented teaching goals, and teacher knowledge.

Influences of standards-based curriculum The type of textbook was the most

critical factor that distinguished teachers who used higher levels of problems and teacher

questions in their teaching from teachers who used lower levels of problems and teacher

questions in their teaching. When teachers used standards-based textbooks, they were

more likely to use higher cognitive demand of problems and teacher questions in their

239

teaching, thereby providing the potential for high levels of student thinking. Conversely,

when teachers used conventional textbooks, they were more likely to be using lower

cognitive demand of problems and teacher questions in their teaching. This conclusion

was drawn not only from the survey but also the case study and is consistent with

findings in previous studies (e.g., Remillard, 2004) as well as current textbook adoption

policies. I argue that the teachers can “develop a vision of reform instruction” through

the use of the standards-based textbooks (Drake & Sherin, 2002; Lloyd & Behm, 2002;

Remillard & Bryans, 2003).

Influence of teacher fidelity to textbook Teacher fidelity to textbook was also an

important factor that characterized teachers providing higher levels of problems and

teacher questions compared to teachers providing lower levels of problems and teacher

questions in their teaching. Taken together with the type of textbook, when teachers

closely use standards-based curriculum, they are more likely to be categorized as being

in the H-H pattern and are more likely to provide learning opportunities for students that

know and do mathematics by fostering mathematical thinking, reasoning, and problem

solving (Star & Hoffman, 2005). Conversely, teachers in the L-L pattern who provided

lower level problems and teacher questions in teaching were more likely to use

conventional textbooks and less likely to report that they followed their textbooks.

240

The analysis of case studies confirmed this finding. For example, Brad,

categorized in the H-H pattern said that “as long as the lesson matched with the grade

level content expectations, I closely follow[ed] the textbook.” On the contrary, Randy,

categorized in the L-L pattern, said that he usually skipped some activities and problems

presented in a textbook because of “assessment.”

The influence of teacher fidelity to textbooks on teachers in the H-H pattern

extends previous studies in two important ways. First, consistent with previous studies

(e. g., Freeman & Porter, 1989), this finding questions the long-held conviction that

teachers who closely follow the textbook do not serve their students. This study argues

that the teachers who deviated most from the standards-based textbook were likely to

place an emphasis on drill and practice of computational skills, while those who followed

the text more closely were likely to emphasize applications and conceptual

understanding.

Second, although teachers’ fidelity to their textbook is often criticized by

reformers or educators (e.g., Apple & J ungck, 1990), the findings of this study suggest

that teachers’ fidelity to their textbooks itself should not be considered the indicator to

gauge teachers’ practice, but should be considered in conjunction with the type of

textbook teachers use and how they see their textbook. In addition to statistical results

241

from the survey which showed that the type of textbook and teacher fidelity to text were
crucial ingredients in deciding teachers’ textbook transformation patterns, case studies of
the H-H pattern also confirmed the influence of these factors. Moreover, I argue that
these factors should be considered with teachers’ individual characteristics such as
teachers’ notions of how students should learn (Barr, 1988; Stodolsky, 1989; Schmidt, et
aL,1987)

Influence of well-aligned teaching goals with the goals of standards-based
curriculum Teachers’ teaching goals were also found to be a significant factor in
characterizing teachers who provided higher level problems and teacher questions
compared to teachers who provided lower level problems and teacher questions (Ball &
Feiman-Nemser, 1988; Freeman, & Porter, 1989). Taken together with type of textbook
and fidelity to textbook, teaching goals constituted a critical factor that led teachers to
sustain high level cognitive demand in their teaching. First of all, all teachers in the case
study who were identified as using higher level. problems and teacher questions said that
their notion of how students learn mathematics matched the notion of teaching and
learning embedded in standards-based curricula, which are designed to help students
develop conceptual understanding, problem-solving, and mathematical reasoning

(Remillard, 2005).

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Moreover, comparison between teachers who maintained the cognitive demand of

problems and those who didn’t suggests that teachers’ teaching goals for student learning

was a critical ingredient in the ultimate fate of high level problems when teachers use

standards-based curriculum. For example, although Brad and Karen used the same

standards-based curriculum, Brad maintained the cognitive demand of the textbook

problems by using higher level teacher questions, whereas Karen decreased the level by

using lower level teacher questions. The reason behind this different use of the textbook

is the difference in their teaching goals for student learning. While Brad wanted students

to develop the underlying ideas between equivalent fractions, generate the rule and apply

the rule to find equivalent fractions, Karen wanted students to follow the rule to find

equivalent fractions. Different teaching goals led them to use the same textbook in

different ways, which in turn provided different notions of knowing mathematics and

doing mathematics to students. In keeping with previous studies, this study argues that

the extent to which teachers’ ideas about how mathematics is learned matches the

teaching and learning philosophies of standards-based curricula contributes significantly

to their use of curriculum (Lloyd, 1999; Remillard, 1999; Sherin & Drake, 2004;

Stodolsky, 1989).

Influence of teacher knowledge Teacher knowledge, in particular pedagogical

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content knowledge, was also found to be significant in characterizing teachers who

provided higher level cognitive demand problems and teacher questions, compared to

teachers who used low cognitive demand problems and teacher questions in their

teaching. In the survey, teachers categorized in the H-H pattern rated their knowledge

higher than teachers in the L—L pattern. This finding is consistent with previous studies,

in particular those that emphasize the importance of teacher knowledge (e. g., Shulrnan,

1986).

Standards—based curricula are much more challenging for teachers to use because

they include many more high-level tasks than teachers typically encounter in the

conventional curricula to which they are accustomed (Stein & Kim, 2007). Specifically,

because with the standards-based curriculum teachers are asked to teach in ways that are

unfamiliar to them and they did not experience as students, it is plausible that teachers

with better understanding of mathematics and how students learn mathematics maintain

higher level problems in their teaching (Wilson, 1999; Remillard, 2000). Based on this

study, I suggest that if teachers have deeper understanding of content and students’ ways

of thinking, they will be more likely to provide higher level problems and teacher

questions, thus helping students learn mathematics through mathematical thinking,

conjectures, explanations, and justification as opposed to the way teachers in the L-L

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pattern would approach teaching.

Factors Distinguishing H-L Pattern from H-H Pattern

The significant factor that distinguished teachers in the H-L pattern from teachers

in the H-H pattern was teachers’ notions of how students best learn, notions that were

embedded in their teaching goals.

Influence of competing conceptions of learning and teaching mathematics to

those of standards-based curriculum Only one factor, teachers’ teaching goals, was

found to be significant in distinguishing teachers in the H-L pattern from teachers in the

H-H pattern. The degree of emphasis on understanding-oriented student objectives

differs between teachers in the H-H pattern and teachers in the H-L pattern. The findings

from the survey revealed that teachers in the H-H pattern were more likely to report

heavier emphasis on understanding-oriented student objectives than teachers in the H-L

pattern. The case study supported this conclusion. For example, although Brad and Karen

used the same standards-based curriculum emphasizing mathematical thinking and

understanding, Karen changed this understanding-oriented goal into a procedural

fluency-oriented goal, such as following the rules and getting the correct answer. This

difference in her teaching goals for student learning led her to lower the cognitive

demand of problems. This trend set her apart from teachers in the H-H pattern, for

245

example, Brad. Indeed, her teaching goals came from her notion of how students learn

mathematics best. She believed that practice makes perfect. This difference in belief

about student learning provides a possible explanation for why teachers in the H-L

pattern emphasized understanding less than procedural fluency when using teacher

questions in their teaching.

Factors distinguishing L-H pattern from H—H pattern

Even though the L-H patterns only appeared in the survey, there is a possibility

that teachers begin with low cognitive demand problems but end up with high level

cognitive demand of student thinking by using high level teacher questions in actual

classrooms (consider the video clips of Deborah Ball teaching third graders). Several

factors significantly distinguished the teachers categorized as an L-H pattern from

teachers in other patterns. For example, in the survey, these teachers reported using

conventional textbooks and less reliance on their textbook than teachers in the H-H

pattern.

Influence of teacher perception of student achievement Interestingly, teachers in

L-H pattern perceived their students’ mathematics achievement to be higher than teachers

in other patterns. This study revealed that teachers in this pattern indicated their students’

mathematics achievement level as mostly high more often than teachers in other patterns.

246

This factor provides a plausible reason why teachers in this pattern increased the lower

cognitive demand of textbook problems into higher levels by using questions that

required students to reason out solutions and explain and justify their answers. In contrast

to teachers in the L-H pattern, the case study revealed that teachers in the L-L pattern,

like Randy, attributed their emphasis on low cognitive demand of tasks and low cognitive

demand of problems to their students’ diversity in terms of mathematics achievement.

This finding is consistent with previous studies (e. g., Remillard, 2005; Saracho, 2003),

implying that teachers’ use of textbooks was also shaped by students, since teachers do

not implement their plans in a vacuum, but in a classroom with students. I argue that

teachers who perceive their students as mostly high-achieving are more likely to enact

lower level textbook problems in ways that increase the cognitive level by using

questions that require students’ higher mathematical thinking.

Factors Distinguishing L-H Pattern from L-L Pattern

Three factors significantly distinguished teachers in the L-H pattern fi'om teachers

in the L—L pattem—teacher views on conventional textbook, understanding-oriented

teaching goals, and observation of experts’ teaching during professional development

programs.

247

Influence of teacher view on conventional textbooks Both teachers in the L-H

pattern and teachers in the L—L pattern were found to be using the same type of textbook

(conventional textbooks). Teacher views on conventional textbook were the critical

factor in distinguishing teachers in the L—H pattern from those in the L-L pattern. The

findings fiom the survey revealed that teachers in the L-L pattern are more likely to agree

that they were pleased with their textbook than teachers in the L-H pattern; these teachers

moreover are more likely to use closely the conventional textbook than teachers in the L-

H pattern.

Influence of different teaching goals from the goals of con ventional textbooks

Teachers’ teaching goals were also an important factor that distinguished teachers in the

L-H pattern from teachers in the L—L pattern. Teachers in the L-H pattern were more

likely to report heavier emphasis on understanding-oriented teaching goals than teachers

in the L-L pattern. In conjunction with dissatisfaction with conventional textbooks, this

difference in emphasis on student learning provides a possible explanation for why

teachers in the L-H pattern increase the lower level cognitive demand of the conventional

textbooks to higher level in their teaching. Because teachers were dissatisfied with their

textbooks and believed their students were more capable, they were more likely to

modify the level of cognitive demand.

248

Influence of lesson observation experience in professional development

opportunities Teachers’ lesson observation experience during PD was also found to be

significant in distinguishing teachers in the L-H pattern from teachers in the L-L pattern.

Analyses of the survey revealed that teachers in the L-H pattern were more likely to

report their frequent opportunities to observe expert teachers and to be observed teaching

during their professional development programs than teachers in the L-L pattern.

Consistent with previous studies (Carey & Frechtling, 1997; Darling-Hammond, 1997;

Lieberman, 1997), teachers in this study who had more opportunities to observe expert

teachers and to be observed teaching in their own classrooms and had more opportunities

to obtain feedback were more likely to increase the lower level problems and questions

in ways that allow students learn mathematics through mathematical thinking, conjecture,

explanations, and justification than teachers in the L-L pattern.

Other Important Findings to Consider

I have summarized thus far findings from the survey and the case studies, which

associated directly with the text-teacher relationship (i.e., whether and how teachers

change the cognitive demand of textbook problems and questions in their teaching and

what factors influence their transformations). In addition, there are several other aspects

that we need to consider in order to understand better the relationship between texts,

249

teachers and teaching. First, consistent with findings from previous studies, this study

found that textbooks did not dictate what to teach. Rather, it was the Michigan grade-

level content expectations (or assessment derived from the GLCEs) that determined what

was to be taught. Therefore, the first thing for teachers to do when planning lessons is to

determine whether their textbooks provide mathematics content “they should cover.”

This suggests that we also need to consider the text-teacher relationship in relation to

content framework.

Second, textbooks were the primary sources for teachers to make plans and create

their lessons. As long as the textbooks provided the lessons teachers are required by the

GLCEs to cover, some teachers closely followed the textbook whereas other teachers

selectively used problems and activities presented in textbook lessons. However, when

textbooks did not provide the content they needed to cover, teachers tended to refer to

available sources such as the Internet, other textbooks that contain the contents, or to

create their lessons based on their own experience or their knowledge.

Third, this study did not consider “time” as an influential factor that distinguished

teachers in one pattern from teachers in other patterns. However, teachers in the case

study referred most frequently to time pressures as influencing their use of textbooks (or

teaching). For example, in keeping with previous studies (e. g., Wilson, 1990), teachers in

250

the L-L pattern mentioned that they could only work on the basic skills—rules and

procedures. Teachers also commonly mentioned that they adapted their textbooks

according to student needs.

Discussion

This study looked at two dimensions of teachers’ textbook use in terms of

cognitive demands (i.e., problems and teacher questions) and provided interesting

findings. For example, this study suggested that when teachers used problems from their

textbooks, they at least maintained the cognitive demand of textbook problems. However,

this study also suggested that the enactment of mathematical problems was a multi-

dimensional practice. As mathematical problems are enacted in the mathematics class,

they become intertwined with teachers’ goals, intentions, and questions, as well as

interactions among teachers and students, and are thereby transformed into different

cognitive levels. Although teachers used the higher level of textbook in their teaching,

they decrease its cognitive demand by using a lower level of teacher questions. These

findings suggest the importance of the ways in which the problems are carried out by the

questions teachers use for students to work on as well as the importance of examining

these two dimensions (problems and teacher questions) in understanding the text-

teaching relationship. Some researchers examined only the enactment of problems to

251

understand teaching practice in terms of the cognitive demand (e.g., Stein & Smith,

2000) whereas other researchers explored only the use of teacher questions in teaching

(e.g., Nicol, 2006). By investigating these two dimensions together, this study expanded

research in the areas of teachers’ use of textbook in mathematics education.

In addition, this study identified influential factors on elementary teachers’

textbook use that need to be considered in order to change teaching practices in ways that

promote students’ understanding and sense-making. This study illustrated the factors that

influence teachers who decrease the higher level text into lower level cognitive demand:

teachers’ conflicting conception of learning and teaching mathematics and teachers’

frequent use of procedural-oriented teaching goals. For example, in keeping with

previous studies (e. g., Remillard, 2005), this study revealed that teachers who decreased

the higher level textbook problems and questions have a competing conception of

learning and teaching mathematics with those of standards-based curriculum designed to

help students develop conceptual understanding, problem-solving, and mathematical

reasoning. These teachers put more emphasis on procedural aspects of the mathematics

(i.e. how) rather than conceptual aspects (i.e., why). Although these teachers use the

higher level problems presented in standards-based cuniculum, they decrease the

cognitive demand of the higher level problems by using teacher questions that emphasize

252

the procedure of “how to get an answer”.

In a similar way, this study also identified the factors that influence teachers who
increase the lower cognitive demands of the text into higher level: teachers’
dissatisfaction with conventional textbooks, hi gh-achieving students, and experts’ lesson
observations in professional development. For example, this study revealed that teachers
who increased the lower level textbook problems and questions have a competing
conception of learning and teaching mathematics with that of conventional textbooks so
that these teachers put more emphasis on conceptual of the mathematics (i.e. why) rather
than procedural aspects (i.e., how). In particular, these teachers perceived their students
as mostly hi gh-achieving and had more opportunities to observe expert teachers and to
be observed teaching in their own classroom in professional development than teachers
who maintained or decreased the lower cognitive demand in teaching.

Among three-level factors—individual-level, contextual-level, and teachers’
opportunity-to-learn, this study revealed that contextual-level factor, type of textbook
(standards-based curriculum) is the most significant factor that plays an important role in
deciding the cognitive demand of problems and questions in teaching. However, this
study also revealed that the enactment of mathematical problems is a multi-dimensional

practice intertwined with contextual-level factors (type of textbook and student

253

mathematics achievement), individual-level factors (teachers’ goals, intentions, and their

knowledge), and teachers’ opportunity-to-learn factors (observation of experts’ teaching

in PD). This understanding is essential when curriculum materials are considered the

primary strategy for improving practice. These findings will help reformers, school

administrators, and teacher educators to consider in what ways they can support teachers

when teachers attempt to revise their practices according to reform ideas. Therefore, this

study will contribute to current reform efforts that change teaching practices in ways that

promote students’ understanding.

Furthermore, this study developed the fi'amework that describes teachers’

modifications in four components of a lesson plan--student objectives, classroom

activities, mathematical problems, and teacher questions of a lesson, which is associated

with maintenance or change in the level of cognitive demand. This framework was

developed based on the revised Bloom’s Taxonomy (2001) and Stein and Smith (2000)

and was utilized in this study to minimize methodological limitations of the survey and

to provide valid findings. In particular, this framework helped me not only to analyze

teachers’ lesson plan modification but also to examine analytically teachers’ actual

teaching in terms of cognitive demand. Many perspectives on the text-teacher

relationships tend to place the teacher and the text at odds with one another, each vying

254

for the same authority over curriculum-decision (of. Russell, 1994). This perspective is

likely to be emphasized in teacher education program (Ball & Feiman-Nemser, 1988).

Instead, preservice teachers should be encouraged to develop an interactive relationship

with textbooks and curriculum guides rather than antagonistic relationship. Teacher

education programs should provide aspiring teachers with opportunities to critically

analyze, to examine the mathematical and pedagogical assumptions embedded in

curriculum materials, and to revise their lesson in ways that increase student

understandings. In conjunction with such efforts, this framework can be a useful tool for

helping pre-service and in-service teachers be aware of the cognitive demand of their

lesson plans and analyze the cognitive demand of their textbooks. In addition, this

framework can be a validation tool to minimize methodological limitations of future

research that will use a survey in examining teachers’ use of textbooks in terms of the

cognitive demand.

However, despite these theoretical and practical contributions of this study, there

are several caveats that should be carefully considered: First, in contrast with previous

studies (e. g., Stigler & Hiebert, 2006), a large number of teachers in this study were

categorized in the H—H pattern. In addition, none of teachers were categorized as the H-L

pattern, most frequently identified by other researchers (e.g., Cohen, 1990; Cohen,

255

McLaughlin, & Talbert, 1993; Darling-Hammond, & McLaughlin, 1995; Kennedy, 2006;

Porter & Brophy, 1988; Stigler & Hiebert, 2006), in describing the text-teaching

relationship. Why do the findings in this study differ from those of other researchers?

One possibility is that the use of the code of 3 as the cut off score is too

generous to classify an individual teacher’s average frequency rating of high-level

problems presented in textbooks and those used by teachers in teaching into either higher

cognitive demand level or lower cognitive demand level. What would happen if a code of

4 was used? We might have seen the H-L pattern and a smaller number of teachers in H-

H pattern.

However, the cut off score chosen in this study was based on previous studies

(e.g., Sanders, 1966). The accuracy of this cut off score was tested in two ways: (1)

examining the type of textbook as an influential factor distinguishing H-H and H-L

patterns from L-L and L-H pattern; (2) examining group membership among

transformation patterns. First, this study used the type of textbook (standards-based

textbooks and conventional textbooks) as an indicator that distinguishes teachers in the

H-H pattern and the H-L pattern from teachers in L-L pattern and L-H pattern. If the cut

off score is accurate, the type of textbook should be an indicator of different

transformation patterns. The findings from multinomial logistic regressions revealed that

256

teachers in the H-H and H-L patterns were more likely to use the standards-based

curriculum whereas teachers in the L-L and L-H patterns were more likely to use the

conventional textbooks.

In addition, multinomial logistic regressions allowed me to examine group

membership among transformation patterns exhibited. In particular, the classification

results from multinomial logistic regression reveal whether transformation patterns

exhibited in each phase are correctly classified in the analysis. The overall rate of correct

classification is 78.5% for the analysis of group membership among transformation

patterns in phase 1; 73.9% for the analysis in transformation phase 2; and 81.6% for the

analysis in phase 3. Although these overall rates of correct classification do not directly

examine whether the use of the code of 3 is accurate, they provide me a rationale for

using a code of 3 as the cut off score.

In conjunction with the cut off score, another possible explanation for the absence of

the H-L pattern results concerns the methodological limitation of the survey as a research

tool. In the survey much of the data are self-reported by teachers. One limitation of self-

report for this study is the potential for teachers to provide biased responses to the survey

because they feel that they should (for a variety of reasons) respond to the questions in

“acceptable” or “socially desirable” ways; or teachers might unknowingly provide

257

misleading responses to the survey questions. Researchers reported that teachers sometimes

truly believe they are embracing pedagogical reforems although their teaching does not, in

fact, match the vision of the reformers (Cohen, 1990; Eisenhart, Borko, Underhill, Brown,

Jones, & Agard, 1996).

Further, convenience samples of this study which do not represent the entire

population of teachers might lead to different results from previous studies. Participants were

recruited among teachers who were taking Master’s courses at a large suburban Midwestern

University or those who were registered for professional development programs. Given their

participation, these teachers might be very familiar with reform ideas (e. g., NCTM standards)

or might be influenced by these programs and might thereby show different transformation

patterns from those identified by other researchers (e. g., Cohen, 1990; Stein & Smith, 1998;

Stigler & Hiebert, 2004).

There is an alternative way of examining transformation patterns in the survey

that may avoid teachers' overestimation of their teaching. This study used high-level

cognitive demand problems and questions (which often cause teachers’ embracing of

theoretically fashionable terms). If one compared teachers on their frequency of using the

low-demand options in questions 18 and 19, one might have different results. Due to the

small number of survey items measuring the low-demand options, in particular, relatively

258

low reliability, high-level cognitive demand problems and questions were selected and

used. Future research should explore transformation patterns using both hi gh-and low-

level problems and questions in a survey in order to capture a more accurate portrait of

what teachers do in their classrooms and how they trasnform their textbook in their

classroom. This limitation calls for future research that avoids each of the above potential

criticisms (e.g., to address the recruitment issue, teachers could be recruited fi'om a more

representative pool, perhaps a national survey).

The second caveat is that the case studies focus on only one specific topic

(fractions units) and the findings from the case studies are short-term observation. After

analyzing the survey, in order to examine how teachers actually use their textbooks in the

classroom, I observed eight teachers at least once (in most cases, twice) when they were

teaching fraction units. This limited observation might not capture an accurate portrait of

what these teachers do in their classrooms and how they intract with the students using

their textbook in their classroompractice. Or a limited observation might produces a

pattern that may or may not hold true for that teacher over time.

The third and final caveat is that there are inconsistencies regarding

transformation patterns between some findings from the survey and the case studies.

Among the eight teachers whose lessons were observed for the case study, two teachers

259

showed inconsistencies between their reports on the survey and their actual teaching.
This implies that there might be more cases that show inconsistencies between reports on
the survey and actual teaching. Yet, there are six cases in this study confirming what the
survey results suggest about transformation patterns as well as influential factors. In
addition, combining the results fiom both the survey and the case study helped me infer a
possible reason for the discrepancies between teachers’ responses on the survey and their
actual teaching. In light of these explanations, the findings of this study seem credible
and plausible and allow me to provide implications for policy makers, curriculum
developers, professional developers, and teacher educators.
Implications

Implication for Policy Makers

As many policy makers assume, teachers use their textbooks as primary resources.
A textbook can be a reasonable candidate for communicating and providing guidance for
change. In particular, this study showed that, in contrast to conventional textbooks,
standards-based curriculum has the potential to influence teaching practice in ways that
emphasize students’ mathematical thinking, reasoning, and sense-making.

There are various textbooks available on the market. In the selection of new

textbooks or new curriculum, policy makers and reformers need to first consider whether

260

the selected textbook can provide a vision for teachers to take on new roles in helping

children to construct knowledge in ways that they intend.

However, textbooks alone do not guarantee different practice. Policy makers and

reformers need to consider various factors that affect teachers’ use of textbooks, such as

teachers’ conception of learning and teaching mathematics and provide systemic support

and resources for teachers to use textbooks. All teachers need support. In particular,

teachers in the H-L pattern and the L-L pattern need to teach about mathematics as a field

of inquiry, not as a body of procedures. They need to learn to think about the goals of

learning mathematics as greater than the mastery of computational skills. Teachers

cannot make fundamental changes in their teaching without several kinds of support,

such as time and assistance in examining and evaluating their own assumptions about

how children learn mathematics and comparing their assumptions to those represented in

standards-based curriculum. This study revealed that the extent to which teachers’ ideas

about how mathematics is learned matches the teaching and learning philosophies of

standards-based curricula contributes significantly to their use of curriculum. Reformers

and policy makers must find ways to communicate about change in a way that makes

sense and respects where teachers are, while still helping them realize that they are being

asked to rethink what they do, and in a way that provides guidance for that change.

261

Moreover, policy makers need to help teachers have a clear understanding of

what their goals are. This study showed that the vision of learning and teaching

mathematics presented in the curriculum framework was seen by teachers as subtly

contradictory, due to the ambiguities of the term “mastery.” For example, since the term

“mastery” can be interpreted in multiple ways, Karen, one of the case participants,

perceived its meaning from a procedural standpoint. Her conception of “mastery” of

procedures hindered her from sustaining higher levels of student thinking in her class.

Schmidt et al. (1 98 7) pointed out that teachers are often confronted with the ambiguities

and weaknesses of curriculum policies in practice. Even in the situation that curriculum

frameworks are provided, teachers must still choose the content and methods of their

instruction. Karen’s case shows that such mismatch or miscommunication about the

intentions of the framework may function as a constraint for teachers’ use of higher level

problems and teacher questions from their textbooks. Policymakers and reformers may

need to clarify what they intend by “mastery” and find ways to communicate with

teachers.

Furthermore, policy makers should provide systemic support and resources for

teachers’ use of textbooks through professional development activities. Teachers need to

learn about the range of methods, including their respective strengths and weaknesses,

262.

because their lack of knowledge about alternative methods constrains their ability to

implement the textbooks adopted by district or states

Implications for Curriculum Developers

One hallmark of standards-based reform is its attempt to build on student thinking

in meaningful ways as opposed to teaching algorithms that often don’t connect with

students’ prior understandings. Research has documented the challenges that many

teachers face when they try to conduct lessons that take into account and productively

build on student responses (e. g., Ball, 2001; Brown & Campione, 1994; Chazen & Ball,

2001; Lampert, 2001; Leinhardt & Steele, 2005; Schoenfeld, 1998; Sherin, 2002). For

example, during whole-class discussions that typically follow exploratory group work,

teachers must make rapid assessments of students’ understandings, compare them with

the desired response, and then fashion a response that will simultaneously help move

both the responding student and the rest of the class towards a more sophisticated

understanding of the mathematics in question.

To help teachers make better use of their textbooks, curriculum developers need

to be aware of these teachers’ difficulties when using standards-based curriculum and

consider including more detailed information about how to use their textbooks. Ball &

Cohen (1996) have identified ways in which curriculum materials can be educative for

263

teachers, two of which I have identified as particularly relevant implications for

curriculum developers: (a) making visible developers’ rationales for including particular

tasks in terms of the mathematical understandings to be gained; and (b) helping teachers

learn how to anticipate what learners may think about or do in response to instructional

activities. In addition, curriculum developers might suggest a range of pedagogical

options that teachers could use to address the particular mathematical idea along with

sample student work or responses including the correct and incorrect student thinking.

Implications for Professional development

Teachers, like their students, are learners who need to be taught in innovative,

flexible ways. Teacher should have opportunities to learn and participate with their

textbooks in professional development. First, professional development activities should

provide teachers with opportunities to change their notions of learning and teaching

mathematics. As Brad’s case shows, matching a textbook with the teacher’s own

philosophy has important consequences. Professional development supporting textbook

adoption has focused on fidelity of implementation. If teachers can reconcile their own

views with those advocated by the developers, frequent adaptations by the teacher will

change the content being taught. A more flexible understanding of what a textbook is

might enrich the possibilities of professional development activities. By acknowledging

264

that teachers might not change radically their own views, middle ground solutions can be

a better strategy for promoting teacher change. Professional development can focus more

on understanding the particular needs of a community of teachers, rather than trying to

impose an implementation.

Second, professional development activities should help teachers become

consciously aware of the cognitive demand of activities, problems, and questions

presented in curriculum materials and examine and reshape textbook suggestions to fit

their classroom needs. Content in professional development activities should be also

specifically designed for improving teachers’ pedagogical content knowledge (e.g.,

Specific ways of teaching strategies/methods; specific ways of improving students’

conceptual understanding).

Furthermore, professional development activities should provide the

opportunities for teachers to work with other teachers in forming collaborative-inquiry

partnerships. Those facilitating teacher groups that involve examining students’

thinking and classroom discourse using textbooks have found that such activities can

enhance teachers’ mathematical knowledge (Russell et al., 1995; Smith & Featherstone,

1996) and their ideas about pedagogy (Hammerman, 1995). Collaborative analysis in

professional development activities would be identifying the mathematical roots of

26:5

students’ activities, assessing their understanding with respect to mathematical concepts
and exploring possible curricular actions. In those activities, teachers need to have
opportunities to critique other lessons and explore alternative possibilities with others.
Implications for Future Study

Limitations and findings from this study lead me to several implications for
future research. First, this study focused on the topic of fi'actions. Future studies might
consider other topics and examine how teachers use their textbooks in terms of cognitive
demands across different topics. It is possible that different topics require different
opportunities for student learning, which lead teachers to use different types of activities
and problems. For example, learning and teaching geometry involves more work for
students and teachers to use representations and concrete materials than learning and
teaching the operations for whole number and fi'actions. Comparisons of teachers’ use of
textbooks across different topics would shed light on the role that the specific topics play
in this process.

Second, due to the lack of participants categorized as the L-H pattern, this study
could not fully describe why and how these teachers modify their textbooks when using
problems and teacher questions fiom their textbooks. The analyses from the survey

revealed three factors that distinguish teachers in the L-H pattern from teachers in the L-

266

L pattem—( l) Participations of profession development programs that provide

opportunities to observe expert’s lesson, (2) teachers’ perception of students as high-

achieving, and (3) teachers’ frequent use of understanding-oriented teaching goals.

However, these findings do not tell how teachers in the L-H pattern modify the lower

level of textbooks into higher level. This finding calls for more future research that

examines how these factors influence teachers’ decision-making in the selection and use

of problems and questions in their teaching.

Third, one suggestion from the findings of this study is that the type of textbook

plays an important role in deciding student leaming opportunities. This study revealed

that when teachers use standards-based curriculum, they are likely to provide questions

leading to higher level of student thinking whereas when teachers use conventional

textbooks, they are less likely to provide higher level of student thinking. However, this

study does not examine relationship among student mathematics achievements, type of

textbook teacher use, and teachers’ use of textbooks. Findings of this study call for future

research that examines these relationships.

Fourth, this study focused on in-service teachers’ textbook use and influential

factors. Teachers in different stages require different need and support in order to help

them teach better. Future comparative studies of experienced teachers and novice

267

teachers (including student teachers) using standards-based curriculums and/or

conventional textbooks could contribute to research on teacher development and its role

in current reform efforts that change teaching practices in ways that promote students’

understanding.

Fifth and finally, this study developed a framework that illustrates teachers’

modifications in student objectives, classroom activities, mathematical problems and

teachers questions of a lesson, which is associated with maintenance or change in the

level of cognitive demand. This framework can be a useful tool for helping pre-service

and in-service teachers be aware of the cognitive demand of their lesson plans and

analyze the cognitive demand of their textbooks. This framework could be utilized in

teacher education programs designed to prepare teachers to be skilled in curriculum

development (Ball & F eiman-Nemser, 198 8) and in professional development programs.

Future research needs to examine the effectiveness of this framework in teacher

education programs and profession development programs.

268

Concluding Remark

This study shows the complexity of the relationship between texts, teacher and
teaching. If policy makers and reformers intend to change practice by using new
curriculum materials, they need to first choose a textbook that can provide a vision and
guide for teachers to take on new roles in helping children to construct knowledge in
ways that the change was intended. In addition, they should consider adding scaffolding
systems to help teachers make better use of their textbooks. Teachers are affected by
both contextual-level factors and individual-level factors (Johnston & Woodbury, 2003;
Knapp, 2002; McLauglin, 1991). Teachers need time and assistance in examining and
evaluating their own assumptions about how children learn mathematics and comparing
these assumptions to those represented in standards-based curriculum. Through
professional development activities, teachers should be provided with learning
opportunities for new methods and pedagogy. Working together with fellow teachers for
specific tasks that are closely related to their curriculum materials, teachers can be given
releaming opportunities for how to teach mathematics. Policy makers and reformers,
curriculum developers and professional developers should work together to provide the
systemic support described above and resources for teachers to make better use of their

textbooks.

269

APPENDICES

270

Appendix A

Survey

A. Background
1. Gender: Male Female

2. Please mark ALL teaching certifications that you hold.
Certified in K-8 grade education: If you have elementary certification, do you
have an endorsement for teaching 6-8 grade mathematics?_Yes, No.
Certified in 7-12 grade education. If yes, in what subject?
Other certification (Please specify):
None

 

 

 

3. How many years have you taught each of the following grade bands in schools? Please

include this year.

 

 

 

 

 

o K-2 Year(s)
o 3-5 Year(s)
o 6-8 Year(s)
0 9-12 Year(s)
0 Others (describe); Year(s)
"' Did you teach in 2005-06? Yes __ No __

4. Grade level(s) you taught in 2005-06. Please specify:

 

5. Type of School:_Public Private, non-religious_ Private, religious _Other

(Specify);
6. School Size (# of classes in your school): ; approximate number of students:

7. Number of students in your classroom:

 

271

8. Mathematics achievement level of students in your classroom in 2005-06. Check one.
__ Mostly high achieving
__ Mostly average achieving
Mostly low achieving
Students at a range of achievement level

9. Which of the following best describe your district’s policy regarding the use of the
school mathematics textbooks? Please mark (v) one.

a. All teachers in the building are expected to use the same mathematics textbook
series.
b. All teachers at the same grade level are expected to use the same
mathematics textbook; different series are allowed across grade level.
c. Each school can adopt its own textbook series; different schools may
adopt different series.
(1. The district has no policy, None of the above
e. The district has a policy but it is different from those listed above
(Please explain):

 

10. Do you generally use the textbook assigned to you by your district or school?
Yes No

l 1. Do you use any other textbooks? Yes No

1.2. List the textbook(s) students used in your class in 2005-06. Write it down in detail,

such as, the title of textbooks, year, and edition (e. g., Everyday Mathematics, Grade 3,
2001, 2nd edition). Mark (v) which one of these was your main resource.

 

Title & Publisher Grade Year of publication Edition

 

 

 

 

 

 

 

272

13. Think about the main textbook you used as a resource in planning lessons for
your mathematics classes. Indicate how much you agree or disagree with each of the
following statements.

l= Strongly disagree, 2=Disagree, 3=Neither agree nor disagree, 4=Agree, 5=Strongly agree

a. You are pleased with the textbook 1 1 2 3 4 5

b. The textbook adequately covers the topics on the state tests 1 2 3 4 5

c. The textbook series matches what you think is important in
mathematics
(I. The textbook reflects NCTM principles and standards for school
math.
e. The teacher’s manual of the textbook helps you learn’new ' '
methods to use in planning lessons

12345
12345

1 2 3 4 5

* Are you planning to teach the same grade you taught last year in the 2006-2007

academic year?

Yes, No.

* Would you be willing to be interviewed in a follow up study (Part II of this study)?

 

 

 

 

 

 

_Yes, _No.
If yes, please provide your contact information below.
Name
District
School
Email Address

 

 

B. Textbook Use on Rational Numbers

The questions in this section are about the main textbook you used as a resource when
you planed lessons on rational numbers in the 2005-06 academic year.

0 Did you teach anything about fiactions during the 2005-2006?
Yes No

If yes, go to next question.
If no, go to Question 20.

273

 

14. Think about the main TEXTBOOK YOU USED in planning and teaching fraction
lessons. Indicate how much you agree or disagree with each of the following

statements.

l= Strongly disagree, 2= Disagree, 3= Neither agree nor disagree, 4 =Agree, 5= Strongly agree

”a. You_used the textbook as a primary resource ' ‘fi ._ .,,1_ L321, ' 3 . 4 25.
b. Your planned in-class learning activities came direCtly froIII the 1 ' 2 . ‘4 5
textbook
c Your leed PrOblemS _came directly from the textbook i ‘ .1 V ' '2 , ’ 3 4' 5'
c. Most of the homework assignments were typically drawn from 1 2 3 1 4 5

the textbook.

d Most Of examples you planned’to’ use'to' explain content came
. directly from the teacher’s manual of the textbook. , . ,
e. Most of questions you planned to use came directly from the

teacher’s manual of the textbook.

15. Think about the fraction unit(s) you taught in 2005-06. Which of the following
SOURCES, if any, did you mainly depend on when planning lessons about the topic of
fractions? Indicate what percentage of the following sources you used when you planned
your lessons. (The sum of the percentage in 1.-7. below should be 100%).

 

Percentage

Sources you used

 

%

%
%
%

%

%

%

1. Textbook assigned to me by district or school

.....

 

2. Other textbooks (specify; H I i)
3. Other commercial prepared supplementary materials I I V I
4. Worksheets or other materials prepared by me or my colleagues

- 5. MEAP-test questions

6. Internet materials

" ‘ 7;”othere non-textbookltea'chingmate’rials (speCIfy; ' ) ' ‘

 

 

= 100%

Total

 

16. Think about STUDENT OBJECTIVES in your lesson plans for teaching fractions in
2005-06. How much emphasis did each of the following student objectives receive?
Please circle one in each row.

- 1= No Emphasis, 2= Minimal Emphasis, 3= Moderate Emphasis, 4= Heavy Emphasis

a Learn mathematical concepts . l 2 3 4 I
b. Learn mathematical algorithms/procedures 1 2 3 4
c Develop students’ computational skills I 1 2 V 3 j 4

274

 

(1. Learn to reason mathematically 1 2 3 4
j e Increase students’ interest mmathematlcs I, l I H j l 23 4
f. Learn to perform computations with speed and accuracy 1 2 3 4
I g. Learn to explain ideas mmathematrcs effectively. , ii 1 2 34
h. Learn how mathematics ideas connect with one another 7 ’ I 1 2 3 4
j. Learn how to solve problems H - " I H '* l i ’7 l 2 M3 4

17. When you planned lessons for fraction units, how often did you do the following?
1=Almost never, 2=Rarely (about once a week), 3=Sometimes (2—3 times a week), 4= Often (4
times a week), 5= (Almost everyday)

3. Use the questions in the textbook exactly as they are phrased l 2 3 4 , 5

b. Modify difficult questions in the textbook to make them easier for

students. , 1
c. Modify easy questions in the textbook into more challenging for (I l 1 t 2‘ 1'3 4 l 5
students ......................................
d. Omit questions or activities. 1 2 3 4 5

18. Think about the fraction PROBLEMS presented in your main textbook and those you
used in your planning and teaching. Indicate fi'equency of the following types of
problems in textbook and in your teaching. l= Never, 2= Rarely (once a unit), 3= Sometimes
(2-3 times a unit), 4= Often (4-5 times a unit), 5= Almost all lessons

 

In TEXTBOOK In TEACHING

     
   
   

2%“ 3: w. '; mfg”, ‘ “he . I
b. Problems that allow students to explain and justify 1 2 3 4 5 1 2 3 4 5
their ideas.
c. Problems involving communication with peers and 1 2 3 4 5 1 2 3 4 5
1 teachers.

g.’Problemsthatreqirue stutdens appl mathcncepts 1 2 3 4 5' 1 2 3 4 5
to real world problems.

 

275

19. Think about the QUESTIONS in the fraction unit(s) presented in your main textbook
and those you used in your planning and teaching of fi'actions. Indicate frequency of
the following types of questions in textbook and in your teaching.

l= Never, 2= Rarely (once a unit), 3= Sometimes (2-3 times a unit), 4= Ofien (4-5 times a unit), 5=

Almost all lessons

 

..: X . .
b. Require students to explain their reasoning when
giving an answer

d. Ask students to con51der alternative methods for
solutions

have read.

 

20. Suppose that you are planning to teach the concept of equivalent fractions to fourth
graders. The next page shows a daily lesson. Would you use this lesson plan exactly
as it is presented? Yes No

If no: Please describe in detail the modification(s) you would make including your

student objectives, classroom activities, problems, and questions you would use.

 

 

 

276

 

 

12-11 Daily LCSSOD Plan Objective: \Vrite equivalent fractions.

 

Introducing the Lesson

Manipulatives: fraction bars

Display 2 fraction bars, one showing % and the other 2.

 

Ask, 2What can you say about these two fraction bars? (They are equivalent fractions,

1

72’

Write on the chalkboard: 3:- = -(-9—2
What number do you multiply by 3 to get 9? (3) The next step is to multiply by 1
what number? (3)

119 _ _3_
3x3 9
Repeat the questioning for the fo(IIC3wing questions. ( )
1 1
— = — 2 — = — 4
7 l4( ) 4 16( )

Share with the students, We all know that g:— = 1, :— =1, ~2— =1, etc. You can find
2 3 4 5
many equivalent fractions by multiply any fraction by — ,-3—, 4 —, 5 —, etc. So

-1—— = fl— = 3,1: 111.: i , etc. The diagram at the top of page 350 helps students
2 2x3 6 2 2x4 8

see that when you double the number of pieces (multiply the numerator and denominator

by 2) you have equivalent fractions: 2:? Go over each equivalent fraction given for

g. Make sure students understand the procedure for multiplying by fractions that equal

1 1 1 2 2
1,forexample, —=—x1=—x—_—
2 2 2 2 4

The Class Exercise should be completed orally. Call on various students to respond,
noting those who may need further help.

 

Source: Burbank, MC et al. (1987). Houghton Mifllin Mathematics Teachers Edition, level 4,
Houghton Mifllin Canada Ltd..

277

 

 

 

EQUIVALENT FRACTIONS

You can write an equivalent fraction by multiplying the numerator and the denominator
by the same number.

 

1
You can write other equivalent fractions for2 -.

.9 09.9

 

 

l_1x3_ _ l_1x4___ 4 ___1_t___5_
2 2x3 6 2 2x4 8 2 2x5 10
A calculator is handy for finding equivalent fractions with large denominators.
5 ?
E ' 23?
Think: How many nines are in 261? 261 + 9 = 29
5 _ 5x29 _ 145
9 9x29 261
CLASS EXERCISES
Complete.
l_le.—l_) 2 1-12—Q 3 1-1“ _(1).
'2 2x24 '22x36 '22x4=(8)

By what number is the denominator multiplied? Complete the equivalent fractions.

lzfl 5 i=(_) 6 lz(_)_ 7 LL). 8 32$)

'36 '412 '510 '7 21 '416

PRACTICE

Complete.

24.) 10,14.) 11.3-9 ,ztzfi 132;)
8 24 5 20 4 24 7 21 8 16

350 LESSON 12-11

 

 

 

Source: Burbank, I.K. et al. (1987). Houghton Mzfllin Mathematics Teacher 19 Edition, level 4,
Houghton Mifilin Canada Ltd..

278

C. Factors Influencing Your Lessons

21. How would you rate your knowledge about each of following? Please circle only one
in each row.

l= Very poor, 2: Poor, 3= Adequately, 4 Good, 5= Excellent

la. Mathematics content knowledge on whole numbers 1 2 3 i I .4 5 i
A b. Mathematics content knowledge on rational numbers A 1 2 3 4 5

*c.Mathemat1cs content knowledge on geometry - l 2 3 I 4 5 -.
i (1. Mathematics content knowledge on measurement 1 2 3 4 5

i e. Mathematics content knowledge on data analysis and probability ’ l A 2 3 i '4 I 5 M
f. General classroom management 1 2 3 ' 4 I

.. g. Knewledge of how students learn (e.g., common

.. mistakes/confusion)
h. Specific ways of teaching Strategies/methods (e. g cooperative 1 2 3 4 5
groups) .. . _ . V { _
i. Specific ways of unproving student basic skills 1 2 3 4 5 ‘
j. Specific ways of improving student’ 3 conceptual understanding 1 2 3 4 5
' 1e Specific ways of improving students’ problem selving 1n ” ' l i I ' 7 i
1 2 3 4 5
,_mathematics -., 1 ,. _ .. 1 . ,_ _ _, _ g
1. Specific ways of using curriculum materials (e. g., new textbooks) 1 2 3 4 5

_ m Specific ways 0f modifying difficult questions in the textbook to
_ , make them easier for students. _

11. Specific ways of modifying easy questions in the textbook to make
them more difficult for students. 4

0. Specific ways of developing higher order thinking skills - l 2 3 4 l S '2

279

22. The followings are the statements of your school district and/or state curriculum
frameworks and test. Indicate your degree of agreement or disagreement by circling one
in each row.

1 = Strongly disagree, 2 = Disagree, 3 = Neither agree or disagree, 4 = Agree, 5 = Agree

b. District curriculum frameworks match my beliefs about what is important in

2 3 4 5
mathematics.
c. District curriculum frameworks reflect adequately NCTM reform ideas. _ V l 2 ‘ 3 _ 4 5
d. District teachers have an obligation to teach the content of and state-wide 2 3 4 5

tests.

e. The state-wide MEAP tests match my beliefs about what is important in math. 1 '2 3 4 .. 5

f. The state -wide MEAP tests reflect adequately NCTM reform ideas. 1 2 3 4 5

23. Different teachers have different philosophies. For each of the following pairs of
statements, check the box that best shows how closely your own beliefs are to each of the
statements in a given pair. The closer your beliefs to a particular statement, the closer the
box you check. Please circle only one for each set.

 

b. “The most important part of
instruction is the content of the
curriculum. That content is the
community’s judgment about what
children need to be able to know and

“The most important part of instruction
2 3 4 is that it encourages thinking among
students. Content is secondary.”

 

“It’s more pract1cal to give the whole

d. “It is a good idea to have many class the same assignment, one that has
activities going on in the class at the 2 3 4 clear directions, and one that can be
same time. Students learn by doing done in short intervals that match
different activities” students’ attention spans and the daily

7 _ 4 , ‘ , _ , _ 4 class schedule”

e. “Students learn mere When the: ' ' 7 L I '7 " ' “Students learn more when the teachers
teacher give background and directly 1 2 3 4 use active approaches like student ..
teach concepts? , . . .. . ,, .. .. . , ., . . , . discussion. presets, and presentatiQnS-.”

280

24. Approximately how many total hours have you spent on professional development
(PD) in the last 24 months? (Include attendance at professional meetings, workshops,
conferences, and formal courses for which you received college credit.)

Hours

 

* Please indicate how much time you have spent on PD activities on the following

content areas in the last 24 months?

1= Never, 2=Less than 6 hours, 3=6-15 hours, 4=16- 35 hours, _=5 35-65 hours
, 1‘

,‘a. Mathematics content knowledge on Whole numbers
. b. Mathematics content knowledge on rational numbers
, c. MathematiCS content knowledge on geometry

(1. Mathematics content knowledge on measurement

’ fie. Mathematics content knowledge on data analysis and probability

f. General classroom management

g. Knowledge of how students learn (if: g., common mistakes or
confusion)

h. Specific ways of teaching strategies/methods (e. g., grouping methods) .

ii. Specific ways of improving student basic skills 4 4
j. Specific ways of improving student’s conceptual understanding

' k. Specific ways of improving students’ problem solving in mathematics 1

1. Specific ways of using curriculum materials (e. g, new textbooks)
111. Specific ways of modifying questions in the textbook to make them i
_ more appropriate for the level of the students. , , ,

11. Specific ways of developing higher order thinking skills

—

N_N Nuisance) N NM.NN;~N

ft»

w.wwuwmwwwwwww

A hikh-h-brbq-h- isnarsanisss.

25. Indicate how often you did the following during the professional development.

Please circle only one per row.

1 = Never, 2= Rarely (Less than 20%), 3= Sometimes (21-59%), 4=Often (60-80%), 5= Almost

. (More than 80%) , _
, (a. Presentation
b. Listen to a lecture
c. Observe ademonstration of a lesson or unit
(I. Observing an expert teachers’ teaching
I e. Being observed teaching in classroom & obtaining feedback
f. Planning and designing classroom implementation
4. g Examining and discussmg student work i g
h. Leading a whole- gr-oup or small-group discussmn

Eli.Demonstrat1ng a lesson unit, or skill _

.1-

NNNNNNNNN

“THANK YOU so much for completing this questionnaire.”

281

w,uw_mwwwwmi

use}; base. 4:. A A,

u. m mulch-mu: Vi Lauri/1 01mm:

mmmmmvmmm'm

 

Appendix B

Interview Protocol

I. Post-observation interview
This is the copies you gave me before I observed your class.

1. Is there anything you changed while you were teaching this lesson?

If yes:

0 What changes were made?

0 What caused the changes? Why?

0 Did that change made things more or less complex?
If no: ask Q2

2. How did you use your textbook when planning this lesson? Looking at your
textbook lesson, there are things you seem to have kept the same and some things
you changed. Can you tell me how you think about this?

0 What did you add or delete from textbook? Why?

0 What made you decide to use them?
Probe: Did you think that modification changed the level of
difficulty/complexity? Why?

0 What were the particular activities/examples/problems you included and
emphasized in your planning? Why?

0 How did you teach and lead these activities/examples/problems? Please step me
through your activities and problems—before, during, and after.

0 What questions did you pose to have students get involved with these
activities?

0 What was your goal in this lesson? What did you expect your students to
achieve or accomplish in this lesson? Is your goal the same with the goals
provided in this textbook?

282

II. Planning for next lesson

This is the copies of planning for the next lesson.

. Could you tell me how you prepared this lesson with your textbook? Is there

anything you would like to do differently from the textbook? Why?
Please step me through your activities and problems—before, during, and after.
o Is there any particular activity, problems, or example in this lesson you
would like to include and emphasize? What are they? Why?
For this lesson, how would you teach and lead the classroom activity?
Probe: What kinds of questions would you use?
Can you illustrate questions you would like to use in each activity?

4. What would be student difficulties/errors when learning this lesson?

What was your goal in this lesson?

III. Factors

1.

What are the big idea(s) in this unit?

2. What is the hardiest concept to teach here? Why?
3.
4. How does the textbook help you to teach this content?

What is the hardiest concept for students to learn in this particular unit? Why?

Probe:

o How helpful is your textbook in helping you address what is hard about

this content?

0 In what ways is your textbook is not helpfiil?

o Are there parts of the book you find particularly helpful?
In general, what do you want your students to learn/accomplish by studying
mathematics?
I am also interested in the possible factors which influence your planning with
textbooks. Is there anything else you would like to say about some factors
influencing your use of textbooks?

283

REFERENCES

284

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