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AN INVESTIGATION INTO THE PROFILE OF GHANAIAN
HIGH SCHOOL MATHEMATICS TEACHERS’ KNOWLEDGE
FOR TEACHING ALGEBRA AND ITS RELATIONSHIP WITH

STUDENT PERFORMANCE

presented by

ERIC MAGNUS WILMOT

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5/08 K'[ProllAcc8Pres/ClRC/DaIeDue‘indd

AN INVESTIGATION INTO THE PROFILE OF GHANAIAN HIGH SCHOOL
MATHEMATICS TEACHERS’ KNOWLEDGE FOR TEACHING ALGEBRA AND
ITS RELATIONSHIP WITH STUDENT PERFORMANCE
By

Eric Magnus Wilmot

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Teacher Education

2008

ABSTRACT
AN INVESTIGATION INTO THE PROFILE OF GHANAIAN HIGH SCHOOL
MATHEMATICS TEACHERS’ KNOWLEDGE FOR TEACHING ALGEBRA AND
ITS RELATIONSHIP WITH STUDENT PERFORMANCE
By

Eric Magnus Wilmot

Available literature on teaching and student performance is replete with
evidence that the teacher is one of the most important factors that influence
student performance, especially in the developing world. In spite of this, there is
widespread disagreement among researchers about which aspect of teachers’
subject matter knowledge best relates to student performance. Several studies
that have attempted to establish this link have relied on proxy measures Of
teacher knowledge (e.g., the number of university courses taken). In addition,
various conceptualizations of teacher knowledge have presented it as a domain
neutral domain. Consequently, there is the need for re-conceptualization of
teacher knowledge in ways that is both domain specific and lends itself to some
form of direct measurement instead of by proxy. One of the ground-breaking
works in this direction is currently being done by researchers in the Knowledge of
Algebra for Teaching (KAT) project at Michigan State University.

Through analyses of research literature and videos of teaching,
researchers in the KAT project have hypothesized that there are three types Of
knowledge for teaching high school algebra: knowledge of school algebra,

advanced knowledge Of mathematics, and teaching knowledge. In addition, the

KAT project is currently developing and validating instruments to measure their
conceptualization. This study used the KAT project’s conceptualization as the
framework. Rather than rely on proxy measures of teacher knowledge, the study
also adapted the instruments developed by the KAT project.

This study investigated whether the KAT conceptualizations could be
corroborated in Ghana. It also examined the differences in the knowledge for
teaching algebra among in-service and prospective high school mathematics
teachers in Ghana. In addition, this study investigated how Ghanaian high school
teachers’ knowledge for teaching algebra is related to the performance of their
students.

In all, 1565 high school elective mathematics seniors from eight public
schools in Ghana, as well as 38 mathematics teachers from these schools
participated in the study. In addition, 301 university seniors comprising 132
mathematics, 44 statistics and 125 mathematics education majors participated in
the study. Factor Analysis of data from this study did not corroborate the three
hypothesized types of knowledge. In addition, Analysis of Variance (ANOVA)
conducted on the data revealed that in-service teachers performed best on both
instruments used in the study. Finally, Linear Regression performed revealed
that student performance is positively related to teachers’ advance knowledge.
However, the relationship was found not to be significant. In the light Of the

findings, recommendations for further research have been made.

Copyright by
Eric Magnus Wilmot
2008

DEDICATION
This dissertation is dedicated to my loving spouse, Vida, and my wonderful
children, Dennis, Vanessa, and Shirley-Anne for their sacrifice, support and
prayers. It is also dedicated to my late mother whose sweat and inspiration laid
the foundation for every aspect of my life. Mother, I believe it is not a goodbye

but a goodnight because we shall meet in the morning.

ACKNOWLEDGEMENTS

I would like to thank family members and relatives whose support has
made this happen. My loving wife and children provided a serene atmosphere
that made it possible for me to do most of my readings and writings at home
while enjoying their company. I cannot thank Vida, my loving wife enough for
extensively proof-reading my writings throughout the program. I am grateful too
for the support and advice Of my elder brother Sam and his wife Krista.

This work would not have been possible without the financial support of a
number of people and institutions. Though not a direct financier, I am grateful to
the National Science Foundation for funding the Knowing Mathematics for
Teaching Algebra (KAT) Project (NSF REC-0337595), at the Division Of Science
and Mathematics Education, Michigan State University. The KAT project
provided the needed framework for this dissertation. Thus, this material is based
upon work supported, in part, by the National Science Foundation, award number
REC-0337595. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author and do not necessarily reflect
the views Of the National Science Foundation. I am grateful to the Principal
Investigators on this project for allowing me to use their instruments and other
project materials for this dissertation at such an early stage of the project.

I am also grateful to the University of Cape Coast, Ghana, for sponsoring
aspects of my fieldwork and especially to the Department of Science and

Mathematics Education for providing free accommodations during the fieldwork.

vi

I am particularly thankful to the Graduate School and the Teacher Education
Department at Michigan State University, as well as Professors Jack Schwille
and Joan Ferrini-Mundy for the financial support for my data collection data.

I am especially indebted to Prof. Jack Schwille for working with me as my
advisor throughout the time it took me to complete this doctoral course Of study.
He was very supportive of my career goals and he worked actively to provide me
with the professional guidance to pursue those goals. I am grateful for his time
and efforts.

My dissertation directors, Professors Joan Ferrini-Mundy and Sharon
Senk, and members of my Dissertation Committee, Professors Jack Schwille,
Mark Reckase and Natasha Speer, have generously given their time and
expertise to improve this dissertation. I am grateful for the extensive personal
and professional guidance I received from each member Of my Dissertation
Committee.

My special appreciation and thanks also goes to the many friends and
colleagues who assisted, advised, and supported my research and writing efforts
over the years. In particular, I express my gratitude to J. Y. Appiah for the
invaluable support he gave me during my fieldwork. I am thankful to J. Y. for
visiting schools with me and helping supervise students and teachers as they
completed forms for this dissertation. I appreciate, as well, the invaluable family
resources he placed at my disposal, including a car that helped me get so much
done in a Short period. I am also grateful to my friends, Hilda, lrini and Violeta,

fellow graduate assistants on the KAT project, who spent considerable time and

vii

energy to score responses to the open-ended items and give me the needed
second Opinions to shape my data. In addition, I express my gratitude to Hilda
and another graduate student, Xin, for the introducing me to SPSS. The
introductory sessions with them were the key to my ability to run my analyses on
time. I also sincerely thank Dr. Steve Sharra, a close friend of mine, who read the
very first draft and gave me feedback on my writing. In a special way, I express
my gratefulness to Dr. and Mrs. Addai whose friendship and generosity
supported my family throughout our stay in Lansing.

The journey to my doctoral degree started with the encouragement and
support of two special friends, Dr. Shirley Miske and Anne Schneller. I cannot
end without expressing my sincerest gratitude and thanks for their
encouragement and support beginning with my application into the doctoral
program and through to the end. But for Shirley and Anne I would have pursued
this program in a different university. Therefore, I owe the two Of them the unique
experience and learning I have enjoyed here at Michigan State University. I am
particularly grateful to Shirley for regularly coming down from Minnesota to visit
and make sure all was well with my family. While in these doctoral studies,
opportunities Shirley gave me to work with her on some of her projects were
valuable, both financially and for my professional experience. Thanks for being

there for my family and me and for helping to get us this far.

viii

TABLE OF CONTENTS

 

LIST OF TABLES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

xii

LIST OF FIGURES xiii
CHAPTER 1: INTRODUCTION AND RATIONALE 1
1.1 Rationale for the Focus of this Study 1
1.2 Background of Work lnforrning this Study 5
1.3 The Context Of Education in Ghana 12
1.4 School Mathematics in Ghana 15
1.5 Background of SSS Mathematics Teachers in Ghana 20
1.6 Purpose of the Study 25
1.7 Research Questions 27
1.8 Significance Of the Study 28
1.9 Organization Of Chapters 30
CHAPTER 2: REVIEW OF RELEVANT LITERATURE 32
2.1 The Issue Of Teacher Knowledge 32
2.1.1 Teachers’ Knowledge and their Teaching Practice 36
2.1.2 Teachers’ Knowledge and Student Performance 41
2.2 Conceptual Framework 46
2.2.1 Knowledge of School Algebra 46
2.2.2 Advanced Knowledge of Mathematics 47
2.2.3 Teaching Knowledge 50
2.3 Relationship between the Three Types of Knowledge 51
2.4 The KAT Project’s Item Development Matrix 52
2.4.1 The Y-Axis: Algebra Content 53
2.4.1.1 Expressions, Equations, and Inequalities 54
2.4.1.2 Functions and their Properties: Linear and Non-linear. 54
2.4.2 The Z-Axis: Domains of Mathematical Knowledge 55
2.4.2.1 Core Concepts and Procedures 55
2.4.2.2 Representation 55
2.4.2.3 Applications 56
2.4.2.4 Reasoning and Proof 56
2.5 Importance of this Conceptual Framework to this Study 57
CHAPTER 3: METHODS 59
3.1 Research Design 59
3.2 Target Populations 60
3.3 Accessible Population 63
3.4 Selection of Subjects 64
3.4.1 Selection of the University Students for the Study 65
3.4.2 Selection of In-service Teachers for the Study 67

ix

3.4.3 Selection Of Senior Secondary School Students for the Study ---- 72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.5 Instrumentation 76
3.6 Procedure 83
3.7 Scoring Of Content Items 87
3.8 Mode Of Data Analysis 89
3.8.1 Analysis of Data Related to the First Research Question—-—----—- 90
3.8.2 Analysis Of Data Related to the Second Research Question ------ 90
3.8.3 Analysis of Data Related to the Third Research Question------- 92
3.8.4 Data not Used in the Analyses 92
CHAPTER 4: ANALYSIS AND RESULTS 94
4.1 Comparing Data from This Study and the KAT Validation Study -------94
4.1.1 Item Difficulty Levels and Point—biserial Correlations 94
4.1.2. Reliability for the Ghana Study and the KAT Study 100
4.2 Research Question One 102
4.2.1 Factor Analysis on Form 1 105
4.2.1.1 Factor Analysis of Form 1 with Three Factors 105
4.2.1.2 Factor Analysis of Form 1 with Two Factors 114
4.2.1.3 Factors Analysis of Form 1 with Eight Factors 117
4.2.2 Factor Analysis on Form 2 120
4.2.2.1 Factors Analysis of Form 2 with Three Factors 124
4.2.2.2 Factor Analysis of Form 2 with Two Factors 126
4.2.2.3 Factor Analysis of Form 2 with Eight Factors 127
4.2.3 Conclusions Related to Research Question One 128
4.3 Research Question Two 130
4.3.1 Preliminary T-Test on Form 1 and Form 2 131
4.3.2 ANOVA on Total Scores of Forms 1 & 2 135
4.3.3 Conclusions Related to Research Question Two 139
4.4 Research Question Three 141
4.4.1 Linear Regression 142
4.4.2 Conclusions Related to Research Question Three 147

 

CHAPTER 5: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS --149

 

 

 

5.1 Summary of the Study 149
5.2 Conclusions from the Study 156
5.2.1 Limitations of the Study 156
5.2.2 How the Instrument Performed on the Sample from Ghana ----- 158
5.2.2.1 Differences in Curricular Emphasis 160

 

5.2.2.2 Possible Affordances of Handheld Technological devices - 162
5.2.2.3 Differences in the Level Of Familiarity to Types of Questions ---- 165
5.2.3 How Data from this Study CorTOborated the KAT Framework mm 167

 

 

5.2.3.1 Critique of KAT Framework and Instruments 171
5.2.3.2 Effect of Curriculum on Conceptualization Of Teacher Knowledge 175
5.2.4 Differences Between Prospective and In-service Teachers ----—- 178
5.2.5 Relationship Between Teacher and Student Perfonnances------ 180

5.3 Recommendations for Further Study 185

 

 

APPENDICES 191

 

 

 

 

 

 

 

Appendix I Content of School Algebra Used by the KAT Project --------- 191
Appendix II Algebra in the Core Mathematics Syllabus in Ghana ------------ 193
Appendix III Algebra in the Elective Mathematics Syllabus in Ghana ------- 194
Appendix IV Tasks of Teaching Defined by the KAT Project 196
Appendix V Invitation/Permission Letters 197
Appendix VI Consent Form 204
Appendix V|l High School Students’ Assent Form 207
Appendix VlllParental Notification for High School Students 209
Appendix IX Public Released Items of the KAT Project 212
Appendix X Sample Opportunity to Learn Questions 215

REFERENCES 216

 

xi

LIST OF TABLES

 

 

 

 

 

 

 

Table 1.5.1 Background of SSS Mathematics Teachers in Ghana 21
Table 3.2.1 Rationale for Inclusion of University Students and Teachers ----- - 62
Table 3.4.1 Number of Participating University Students by Major Area ----- 65
Table 3.4.2 Number of Participating In-service Teachers by School ---------68
Table 3.4.3 Courses Taken by ln-service Teachers in the Sample -- -------- - 70
Table 3.4.4 Teaching Experience of Participating ln-service Teachers -------- - 70
Table 3.4.5 Teaching Certificates of ln—service Teachers in the Sample -----71
Table 3.4.6 Participating High School Students by School and Class ————— --75
Table 3.4.7 Classes with Teachers Who Participated in the Study ---------- 76
Table 3.5.1 Categorization of Items on Forms 1 and Form 2 80
Table 3.7.1 Agreements in Scoring Open-ended Items on Forms 1 & 2 ----- 88
Table 3.7.2 Agreements in Scoring the SSS students’ Open-ended Items ----89
Table 4.1.1 T-Test of the Difference in Difficulty Of the Two Forms-- -------- -96
Table 4.1.2 Difficulty Level and Rank of Items on Forms 1 and 2 97
Table 4.1.3 Point-biserial Coefficients Of Items on Forms 1 and 2 99
Table 4.1.4 T-Test of the Difference in the Point-biserial Coefficients ----- 100
Table 4.1.5 Reliability for Form 1 and Form 2 101
Table 4.2.1 Total Variance Explained by Each of the Factors Of Form 1---- 106
Table 4.2.2 Communalities of Form 1 109
Table 4.2.3 Item Loadings on the Three Retained Factors Of Form 1 nnnnn 112
Table 4.2.4 Item Loadings on the Two Retained Factors of Form 1 -------- 115
Table 4.2.5 Item Loadings on the Eight Retained Factors of Form 1------ 118
Table 4.2.6 Form 1 Item Loadings and Categorization on the Eight Factors- 119
Table 4.2.7 Total Variance Explained by Each of the Factors of Form 2—--- 121
Table 4.2.8 Communalities of Form 2 123
Table 4.2.9 Item Loadings on the Three Retained Factors of Form 2------124
Table 4.2.10 Item Loading on the Two Retained Factors of Form 2--------126
Table 4.2.11 Item Loading on the Two Retained Factors of Form 2-—-------- 128
Table 4.3.1 Descriptive Statistics Of Form 1 and Form 2 by Item 132

Table 4.3.2
Table 4.3.3
Table 4.3.4
Table 4.3.5
Table 4.3.6
Table 4.3.7
Table 4.3.8
Table 4.4.1
Table 4.4.2
Table 4.4.3
Table 4.4.4

 

Descriptive Statistics of Forms1 and 2 by Knowledge Type —--- 133

 

 

 

 

 

 

 

Group Statistics of Form 1 and Form 2 133
Independent Samples T-Test 134
Descriptive Statistics Of Total Scores by Major Area 135
Test of Homogeneity of Variance in the Total Scores --------- 137
ANOVA of the Mean Differences in Total Score 137
Multiple Comparisons of Differences in Total Scores ---------- 139
Mean Class Scores and Teachers’ Total Scores 143
Linear Regression Model Summary 143
Coefficients of the Linear Regression Model 145

ANOVA Test of Significance of the Predictor in the Model ------ 146

xii

Figure 2.2.1
Figure 2.3.1
Figure 2.4.1

Figure 4.2.1
Figure 4.2.2
Figure 4.3.1
Figure 4.4.1

LIST OF FIGURES

 

 

 

Conceptual Representation of "Advanced Knowledge" ---------49
Conceptual Representation of the Three Types of Knowledge --- 52
The KAT Project’s Item Development Matrix 53
Graph of Factor Loadings for Items on Form 1 108
Graph Of Factor Loadings for Items on Form 2 122
Mean Total Score on Form 1 & 2 by Major Area 136

 

 

Scatter Plot with the Regression Line 146

xiii

CHAPTER 1: INTRODUCTION AND RATIONALE

This chapter opens with a discussion of the rationale for focusing this
study on high school teachers’ knowledge for teaching algebra and its
relationship to student performance. Thereafter, because the setting of the study
is in Ghana, a country different from the United States Of America (US),
background discussions about school mathematics, as well as about high school
mathematics teachers (both in-service and prospective) in Ghana have been
presented in turn. After this introductory material, the purpose Of the study and
the Significance of the study are presented. The chapter ends with the research
questions that guided the study, followed by an overview of how the entire

dissertation is organized.

1.1 Rationale for the Focus of this Study

The momentum for the “Standards Reform” in mathematics education in
the US began in the early 19805 as educators started responding to a “back-to-
basics” cry, a reaction to the “new mathematics” of the 19603 and 1970s (Van de
Walle, 2007). The origin and impetus of this reform can be traced to the low level
of students’ performance and stratification of mathematics instruction in US.
schools (see the National Commission on Excellence in Education’s “A Nation at
Risk” report of 1983; Clune 1998).

According to Usiskin (1995), however, advocates of the Standards Reform
did not only aim at improving the quality of teaching and Ieaming in schools, they
began challenging the notion that only some could learn algebra. The standards

reformers argued that because of the foundation algebra provides for other

mathematics courses and its application to other fields, every student in the US
has to be given the opportunity to study algebra (Usiskin, 1995). Senk and
Thompson (2003) have argued that such “algebra for all” calls were consistent
with the call for increased mathematics requirement for all US. schools,
highlighted in the “A Nation at Risk” report. The argument has been that, without
the opportunity to study school algebra, it would be almost impossible to raise the
mathematics performance of many high school students in the US. In addition,
without studying algebra, some students would be denied access to certain
careers.

A look at the National Council of Teachers of Mathematics’ (NCTM)
Standards (NCTM, 2000) indicate that the NCTM with regards to “algebra for all”
seems to favor teaching for algebraic thinking in all of school mathematics from
the early primary grades. Though specific courses in algebra are not completely
discouraged, the emphasis on algebra thinking in all of school mathematics is
quite different from specific course(s) in a specific grade that have traditionally
been emphasized prior to the standards reform era. This position Of the NCTM is
seen in its emphasis on the need to help all students develop reasoning and
problem-solving skills built upon exploration, modeling, describing/conjecturing,
explaining and generalizing, which are basic to the formation Of algebraic
reasoning (NCTM, 2000). It is this shift from focusing only on single course(s) in
algebra to content strands that are strengthened each year by weaving with other
domains of school mathematics that seem to be emphasized in the “algebra for

all” call.

AS Roberts (1991) appropriately puts it, “the student who closes the door
on school algebra (and so on all mathematics) closes the door on much more”
(Roberts, 1991, p. 15a cited in Chazan, 1996, p.456). Moses (1995) and Moses
and Cobb (2001) have aptly referred to algebra as “the new Civil right” and have
argued that because algebra opens the door to productive careers it serves as a
precious resource for providing equity to all students. In other words, in the US it
is largely accepted by mathematicians and mathematics educators that the
egalitarian vision that schools Should “provide each student with an opportunity
for social, political and economic equality” (Cusick, 1983, p. 1) is achievable if all
students are allowed to study algebra in school.

Strength has been provided to this “algebra for all” notion by studies such
as that of Gamoran and Hannigan (2000). Gamoran and Hannigan (2000)
studied the impact of high school algebra on 12,500 students who differed in their
mathematics skills prior to entering high school and found that the students
improved in their mathematics achievement after taking algebra. In the Gamoran
and Hannigan (2000) study, though the regression analysis showed differential
benefit across ability groups with students who had high prior achievement
benefiting more, the results showed that algebra is useful to all students. Now in
the US, almost every high school student studies algebra before graduating
(Dossey & Usiskin, 2000).

According to the Nation’s Report Card: Mathematics 2007, the report of
the 2007 National Assessment Of Education Progress (NAEP) results, after about

twelve years Of various states in the US responding to the call for “algebra for all”

there is evidence from the 2007 results of the NAEP that this is yielding positive
effects in the US. The mathematics report of the 2007 NAEP revealed that
among fourth- and eighth—graders of all ethnic, racial and income groups
generally improved in performance over earlier years. For instance, among
fourth-graders Black students improved by 35 points over their 1990 scores while
the gains White and Hispanic students were 28 and 27 points respectively. Even
among American Indian/Alaska Native students who showed no significant
improvements over their 2005 average score, in 2007 there was a 4 percent
increase in the number that performed at or above proficient level over the 2005
level. The NAEP 2007 report also showed improved performance of White, Black
and Hispanic eighth-grade students as well as students across all income levels.
Compared with the 1990 scores White, Black, and Hispanic students improved
by 21, 23, and 19 points respectively while Asian/Pacific Islander students
improved by 22 points. The only group of eighth-graders who showed no
Significant gain in scores over previous years was the American Indian/Alaska
Native students.

In addition, the 2007 NAEP results showed that the achievement gap
between students of color and their white peers as well as between students
from low-income families and others is narrowing. For example, among eighth-
graders, the Black-White achievement gap narrowed by eight points while the
Hispanic-White gap narrowed by four points from the 1996 outcomes. Within the
same period, the 2007 NAEP results for fourth-graders showed that though

improved achievement by Black students narrowed the White-Black gap by eight

points over the 1996 outcomes, the reduction was significantly lower than the
2005 gap. Similarly, reduction in the White-Hispanic gap in 2007 was not
significantly different from the 2005 or 1990 outcomes. Thus, by opening the door
of algebra to all students not only through specific course(s) in algebra but also
integrating algebraic reasoning in all domains of school mathematics, many more
doors (i.e., improved mathematics achievement for all and the closure of the
achievement gap among different racial groups Of students) are being Opened to
all.

After attaining this noble ideal of “algebra for all”, the next step is to,
perhaps, begin thinking about issues of how to ascertain the type of knowledge
teachers need or have for teaching high school algebra. Whereas it is true to say
that teachers’ subject matter knowledge is not the only factor that influences
instructional practice, it can also be said to be one of the major factors that
influenced teachers’ decisions. As Even (1989) puts it, “a teacher who has a solid
mathematical knowledge,. . ., is more capable of helping his/her students achieve
meaningful Ieaming” (p. 4). Such considerations provide the rationale for
focusing this study on high school teachers’ knowledge for teaching algebra and

its relationship to student performance.

1.2 Background of Work Informing this Study

The question of what teachers need to know about their subjects to help
them promote powerful and flexible knowledge and understanding in students
has been difficult to answer. In addition, this difficulty has further been

complicated by the different conceptualizations so far given of teachers’

knowledge (Kennedy 1991 ). For instance, Cochran and Jones (1998) identified

the following four components of it:

1) Content knowledge: facts, concepts and procedures within a
discipline

2) Substantive knowledge: knowledge of explanatory frameworks in a
field

3) Syntactic knowledge: knowledge of ways in which new knowledge
is brought to the field, as well as;

4) Beliefs about the subject matter.

The argument by Thompson (1984) that teachers’ beliefs, views and preferences

about mathematics are some of the factors that play a significant role in shaping

their instructional behavior has also implied that the type of mathematical

knowledge teachers draw upon in teaching could be influenced by their beliefs,

views and preferences about the subject. In addition, Shulman (1986b) also

introduced the following seven types of knowledge as the knowledge base for

teaching:

1) Content knowledge

2) General pedagogical knowledge (e.g. classroom management)

3) Curriculum knowledge (school “stuff,” “tools of the trade,” materials,
tests, etc.)

4) Pedagogical content knowledge (interaction between content and

Pedagogy)

5) Knowledge of learners and their characteristics

6) Knowledge of educational contexts (workings of the classroom,
school financing, etc.)

7) Knowledge of educational ends, purposes, and values
At the centerpiece of Shulman’s (1986b) conceptualization is the idea Of
“pedagogical content knowledge” (PCK) because it distinguishes the teacher as
professional from content experts, and those who simply understand kids.
Throwing more light on his conceptualization of PCK and its importance so far as
the knowledge base for teaching is concerned, Shulman (1987) argued that,

The key to distinguishing the knowledge base of teaching lies at the

intersection of content and pedagogy, in the capacity of a teacher to

transform the content knowledge he or She possesses into forms that are
pedagogically powerful and yet adaptive to the variations in ability and

background presented by the students. (p. 15)

Thus, Shulman argues that good teaching is contextualized and interwoven with
good understanding of an academic discipline. In addition, to ensure teacher
competence, there is the need to focus attention on developing the knowledge
base of teaching which supports such a complex view of teaching.

Ma (1999) also introduced the idea of profound understanding of
fundamental mathematics (PUFM). According to Ma, teaching and Ieaming which
demonstrates PUFM is “connected, has multiple perspectives, includes the basic
ideas of mathematics, and has a longitudinal coherence” (Ma, 1999, p.122). This
means that teachers with PUFM are able to explain the why of the mathematics

they teach and come up with multiple and flexible ways to teach. In this way,

Ma’s conceptualization of PUFM can be said to be related to Shulman’s PCK in
the sense that they both suggest that a teacher who demonstrates any of the two
types of knowledge, he/she should have a deep, broad, and thorough
understanding of mathematics and the ability to teach it in flexible ways. Earlier,
Leinhardt and Smith (1985) had proposed two types of teacher knowledge.
These are “lesson structure knowledge” (LSK) and “subject matter knowledge”
(SMK). Lesson structure knowledge comprised smooth planning and organizing
of lessons and providing clear explanations. Leinhardt and Smith’s (1985)
conceptualized subject matter knowledge to consist of concepts, algorithmic
operations, connections among different algorithms and knowledge of the types
of errors students make.

Recently, in their work on elementary school mathematics teachers’
knowledge, Deborah Ball and her colleagues have also contributed to the idea of
the type of knowledge needed by teachers by introducing the idea of
mathematical knowledge for teaching (Ball & Bass, 2000; Hill, Ball & Schilling,
2004; Hill, Rowan & Ball, 2005). Relying on existing theories about teacher
knowledge, Ball and her colleagues have developed and administered survey-
based questions based on teaching mathematics at the elementary school level.
Through factor analyses of their data, they have proposed, among other things,
the idea of “specialized knowledge of content". As they put it,

In addition to a general factor, specific factors representing knowledge of

content in number and operations, knowledge of students. . . [there is

also] a specialized knowledge of content (SKC) made up of several items:

representing numbers and Operations, analyzing unusual procedures or

algorithms and providing explanations for rules. (Hill, Ball & Schilling,

2004, pp 27-28)
Not only have there been such conceptualizations of the knowledge base for
teaching, a number of instruments have been developed to measure teachers’
knowledge of the content of school mathematics, as well as issues related to
pedagogy. In the US, for instance, to be certified to teach mathematics, various
states require pre-service teachers to pass a mathematics test. An example of
this is PRAXIS, a teachers’ licensing examination developed by Educational
Testing Service (ETS), which is currently used by over 30 states. Such efforts are
meant to ensure that mathematics teachers have a good knowledge of the
mathematics students are required to learn in school. In spite of this, the quality
of achievement of K-12 students in mathematics continues to be of national
concern. Consequently, the RAND Mathematics Study Panel (2003) made a
number of recommendations for improving teachers’ mathematical knowledge for
teaching. These include the need for further Clarification of the knowledge
demands Of teaching mathematics, and a deeper understanding of ways to
provide opportunities for prospective and practicing teachers to acquire this kind
of knowledge. In addition, the RAND Mathematics Study Panel (2003)
recommended the development Of instruments for assessing the mathematical
knowledge for teaching across grade levels and mathematical domains. The
RAND panel also singled out algebra as an important area of focus in all these

efforts.

As already explained, the development Of items to measure teachers'
knowledge for teaching mathematics at the elementary school level by Ball and
her colleagues can be said to be in line with the issues raised by the RAND
Mathematics Study Panel. At the secondary school level, and also in line with
RAND panel recommendations, Ferrini-Mundy and her colleagues on the
Knowledge of Algebra for Teaching (KAT) project have begun conceptualizing
and framing questions about mathematical knowledge for teaching algebra (see
for instance, Ferrini-Mundy, Burrill, Floden, & Sandow, 2003; Ferrini-Mundy,
Senk, & McCrory, 2005). In their conceptualization, The KAT project has
hypothesized that mathematical knowledge for teaching algebra consists of three
types of knowledge. These are, 1) knowledge of school algebra (referred to later
in this dissertation as “school knowledge” as used by the KAT project team), 2)
advanced knowledge (i.e., knowledge of the content of other mathematics
domains different from algebra) and 3) teaching knowledge. These three
categories are explained in the conceptual framework section of chapter two of
this dissertation. In addition to their conceptualization, the KAT project is
developing items and designing instruments to measure knowledge for teaching
algebra at the secondary school level.

Through several pilots of the instmments they are developing, the KAT
project is using empirical data from pre-service and in-service teachers in
different settings across the country to validate their framework for knowledge for
teaching algebra. In addition, KAT is studying the status and variation of

knowledge for teaching algebra among pre-service and in-service teachers

10

drawn from across the US. The importance of the KAT work can be seen in the
fact that, after two decades of various conceptualizations Of the knowledge base
for teaching, the KAT project is a specific attempt at focusing on conceptualizing
the knowledge required for teaching in one specific domain of mathematics at the
secondary school level (i.e., algebra). Another thing that makes the KAT project
unique is the work it is doing towards developing tools for assessing
mathematical knowledge for teaching algebra. The KAT conceptualization has
the potential of becoming a good framework for other researchers who may be
interested in conceptualizing knowledge demands for teaching other domains of
mathematics especially at the secondary school level. In addition, the KAT
instrument could serve as a potential tool for accessing and improving knowledge
of pre-service teachers or for professional development of in-service high school
algebra teachers. The KAT studies, which include investigating the status and
variations Of knowledge for teaching algebra among pre-service and in-service
teachers, also have the potential Of providing empirical data on what needs to be
done nationally to improve teachers’ knowledge.

This dissertation study is based largely on the work being done by the
KAT project, but conducted in a different context and setting. It used the
instruments that are being developed by the KAT project to examine issues of
knowledge for teaching algebra among prospective and in-service high school
mathematics teachers in Ghana. This dissertation study also extended the KAT
work by focusing on the relationship between high school teachers’ knowledge

and the performance Of the students in the Ghanaian context. As discussed

11

under the section titled “significance of the study”, examining these issues in the
Ghanaian context is important for a number of reasons. For instance, since
Ghana is a country with a Slightly different system Of secondary education and
teacher certification programs than what exists in most US. states, investigating
issues of teacher knowledge in Ghana, especially using KAT instrument and
framework, is a good first step to examining how the KAT conceptualizations
could be corroborated not only internationally but also across systems whose
Dre-service teacher education is different from the US. Such a corroboration or
otherwise Of the KAT conceptualization is useful not only for the team of
researchers in the KAT project at Michigan State University, but also the entire
mathematics education research community worldwide.

In the next three sections, the contexts of education in Ghana, school
mathematics and the background of senior secondary school (high school)

mathematics teachers in Ghana are discussed.

1.3 The Context of Education in Ghana

At the time of this dissertation study, the structure of formal education in
Ghana comprised six years of primary school, three years of junior secondary
school (JSS), and three years of senior secondary school (SSS) prior to entry
into various forms Of tertiary institutions. This structure has been in place since
the 1987 Educational Reform in Ghana. Basic school education (i.e., primary and
junior secondary school education) was tuition-free and compulsory to all
children Of school-going age in Ghana. The content of the curriculum of all public

primary, junior secondary and senior secondary schools in Ghana is centrally

12

controlled by the Ministry Of Education (MOE) and the Ghana Education Service’
(GES). This means that for each of the school subjects, every student in Ghana’s
public schools studies the same content (except in situations where students are
permitted to select different electives at the SSS level). In addition, the GES
controls the school calendar and mandates timetabling, re-Opening and closing
dates for all public schools up to the SSS level. All public primary schools, JSS
and SSS in Ghana run a trimester system each school year.

Students enter class one (i.e., the first grade) at age six and the medium
of instruction is English2 except when the subject Of study is Ghanaian Language
and Culture. School subjects Offered at the primary school include Mathematics,
English language, Ghanaian language, Integrated Science, Agriculture and
Environmental Studies, Vocational Skills, Life Skills, French (where a teacher is
available), and Religious and Moral Education (RME).

At the end of the JSS, students write a national examination, the Basic
Education Certificate Examination, which is also used for selection into senior
secondary schools. At the time of the dissertation study, the following subjects
were offered at the JSS level: Mathematics, English language, Ghanaian
language, General Science, Agricultural Science, Social Studies, Environmental

Studies, Vocational Skills, Life Skills, French (where a teacher is available),

 

I In Ghana, the Ministry of Education is responsible for enacting policies while the Ghana
Education Service is the body that implements educational polices.

2 Though Ghana is a multi-lingual and multi-ethnic country with over forty mutually unintelligible
indigenous languages spoken within her borders, English, the language of its colonizers has
historically become the official language of the country. In 2002, the early exit school language
policy was abolished for an all-English policy.

13

Music and Art3, and Religious and Moral Education (RME). Grades Obtained in
various subjects at the Basic Education Certificate Examinations (BECE) are the
aggregate of their scores at the BECE (70%) and students’ continuous
assessment scores throughout their primary and JSS education (30%).

After JSS, students may enter SSS into specific programs that prepare
them for university or other specialized tertiary institutions (e.g. the nurses’
training college, post-secondary teachers’ training colleges, etc). The programs
offered include General Arts, General Science, Business, Technical, and
Vocational programs. Not all senior secondary schools in the country Offer all
programs. Consequently, every student is given a special SSS entrance form
that lists every senior secondary school in the country and the programs
available in those schools. Counselors at the JSS collaborate with students and
their parents to help students to select three schools and programs that best suit
the student’s ability. Once a student’s grades qualifies him/her into the SSS,
assignment to specific schools and programs is made randomly using a special
computer software at the headquarters Of the Ghana Education Service based on
the quality of grades Obtained at the BECE in specific subjects considered
fundamental in the program chosen by the student.

At the senior secondary school level, students usually study a combination
of three (in some cases, four) elective subjects and a number of core subjects.
For instance a student in the General Science program could select Elective
Mathematics, Physics, Chemistry and Biology as his/her electives. A General
Arts student, on the other hand may select any three or four from subjects such

‘

3 Music and Art ware not examined at the BECE

14

as Economics, Geography, Elective Mathematics, and Literature in English,
French for his/her electives. In addition to the elective subjects, there are core
subjects, which every student studies at the SSS level. These include Core
Mathematics, Core English language, Ghanaian language and Culture,
Integrated Science, Social Studies, Life Skills, Religious and Moral Education

(RME).

1.4 School Mathematics in Ghana

In US. junior high and high schools, separate courses in algebra (e.g.,
Algebra I, Algebra II etc.) could be offered to students. However, in Ghana only
one integrated mathematics course is offered at the JSS (the equivalent of
seventh to ninth grade) to all students. This mathematics course is a national
curriculum, and is therefore, offered to all students in the public school system for
the entire three years of the JSS education. The Teaching Syllabus for Junior
Secondary Schools (Ministry of Education, 2001) lists the major content areas
covered on page (iv) as

. Number

. investigations with Numbers
0 Shape and Space “Geometry”
. Estimation and Measurement
. Introduction to the Set Theory
. Algebra

0 Collecting and Handling Data

15

In addition to these content areas, problem solving which does not appear as a
topic in itself is emphasized throughout the syllabus. Furthermore, these topics
are not covered in succession. They have been broken down into smaller content
pieces, called units (and sub-units) and have been sequenced in a spiral manner.
The various units are arranged in such a manner that the topics taught in the
early grades are not covered in complete detail but are returned to repeatedly
throughout the years and developed further, with increasing detail and depth, as
students progress through the grade levels. In The Process of Education, Bruner
(1960) made a case for this type of sequencing when he said, “A curriculum as it
develops should revisit the basic ideas repeatedly, building upon them until the
student has grasped the full formal apparatus that goes with them” (p. 13).

At the time Of this study, there were two types Of mathematics programs
offered in Ghana at the Senior Secondary School (SSS) level (the equivalent of
grades 10 to 12 in the US). These were Core Mathematics and Elective
Mathematics. As already discussed, in the public school system, every SSS
student took Core Mathematics for the entire three years of SSS education.
Elective Mathematics, on the other hand, was selected by students who require
further mathematics content preparation beyond the core mathematics coverage.
For instance, Elective Mathematics was an automatic elective course for
students in the Science and Technical programs. Other students in the General
Arts and Business programs could also select Elective Mathematics. Like the
mathematics course at the JSS level, both of the mathematics courses in SSS

were, at the time of this dissertation study, also integrated mathematics programs

16

with their content sequenced in a spiral manner. In addition, like all other school
subjects, the Ghana Education Service centrally controlled the syllabi for both
Core Mathematics and Elective Mathematics. Being a national curriculum, the
content of each Of these mathematics courses is also the same for all public
schools in Ghana.
According to the Ministry of Education (2003a), the major content areas to

be covered in Core Mathematics comprise the following:

. Number and Numeration

. Algebra

. Mensuration4

. Plane Geometry

. Trigonometry

0 Statistics and Probability

o Vectors and Transformation in a Plane

0 Investigations and Problem-solving

0 Use of Calculators and Computers
“Investigations and problem solving together with the use Of calculators are not
topics by themselves in the syllabus but nearly all topics include activities
involving them” (Ministry of Education, 2003a, p. iii). Core Mathematics is offered

for ten class periods of 40 minutes a week for the two terms of the first year.

 

‘ Mensuration, in the syllabus, comprises finding surface area and volume of solids such as
cuboids, prisms cylinders, rings, pipes, cones, spheres, pyramids and calculation of distances
between two points on the same latitude or longitude

17

Thereafter, Core Mathematics is supposed to be offered six periods a week for
the third term of the first year and the remaining two years of SSS.

In contrast, the syllabus for Elective Mathematics contains the following

content:

. Algebra

0 Logic

0 Coordinate Geometry

o Trigonometry

. Calculus

0 Linear Transformation

. Vectors

. Mechanics

. Statistics

0 Probability
The time allotted to Elective Mathematics in the SSS timetable is a minimum of
seven class periods of 40 minutes each per week (Ministry Of Education, 2003b).
Appendix II contains an extraction Of the content of algebra in both of these
mathematics courses at the SSS level in Ghana at time Of the study.

In spite of the two distinct approaches to offering algebra to all students in
both Ghana and the US, there continues to be national outcry over the
performance of students on algebra in national and international assessments.
For instance, in Ghana, students’ progression from the SSS level to the

university and other tertiary levels of the education system is through a national

18

examination, the Senior Secondary School Certificate Examinations (SSSCE).
This examination has been in place in Ghana since the National Educational
Reform of 1987. Following similar educational reforms in Anglophone West
Africa, beginning May 2006, this SSSCE was changed into the West African
School Certificate Examinations (WASCE). All high school leavers in Anglophone
West Africa now take the WASCE for selection into universities and other tertiary
institutions. The sad situation is that, since 1993, several reports by the Chief
Examiners Of the SSSCE have highlighted students’ poor handling Of some Of the
problems on algebra. For instance, in 2004, the Elective Mathematics Paper 25 of
the SSSCE had the following as one Of the questions,

Express 3x2 -6x+10 in the form of a(x-b)2 + c where a, b and c

are integers. Hence state the minimum value of 3x2 — 6x +10 and
the value of x for which it occurs (VVAEC, 2004).

The chief examiners’ report that year acknowledged that most of the candidates
attempted the question. However, the reported highlighted the fact that students
performed poorly on it because many of them either could not complete the
square or resorted to calculus to find the minimum value, a method that was not
accepted.

Due to the foundational nature of algebra, there is the need to reverse this
trend of students’ poor performance on algebra in Ghana. Though in Ghana no
studies have been conducted to examine the reasons for this poor performance,

worldwide, a number of studies on student performance in mathematics have

 

5 At the SSSCE, candidates write two papers at separate times during the examination period.
The first paper consists of multiple choice questions while the second paper has open ended
questions.

19

revealed that one of the factors that can improve student achievement in school
mathematics is teachers’ knowledge (see for example, Harbison & Hanushek,
1992; Mullens, Mumane 8. Willett, 1996; Hill, Rowan & Ball, 2005). Informed
decisions about the type of improvement needed in the knowledge base Of
teachers could be made if data about the nature of teachers’ knowledge and
which aspects of it best relate to student performance were available. In the case
of Ghana, the integrated nature of each of the two mathematics curricula means
that a good knowledge Of the entire content of school mathematics in some
integrated way is essential for teacher effectiveness. However, to study teacher
knowledge relative to the coverage of entire mathematics syllabus (either core or
elective) is too broad to cover in a single study. Therefore, given the foundational
nature of algebra and the issue of poor performance in this area this study is
limited to algebra.

It is in the light of this that this study was designed to investigate the
profile Of knowledge that Ghanaian senior secondary school mathematics
teachers and prospective teachers have for teaching the content of algebra in the

SSS syllabus, and how this knowledge is related to student performance.

1.5 Background of SSS Mathematics Teachers In Ghana

Across the spectrum of literature of teacher competence and student
performance are studies that project the value added by teacher education.
These studies Show that graduates of teacher education programs feel better
prepared for their job and can positively affect their students’ achievement than

those who enter teaching Without adequate teacher education background (see

20

Darling-Hammond, 2003; Laczko-Kerr & Berliner, 2003; Kennedy, Ahn 8. Choi,
2006). These issues of pedagogical preparation, subject matter preparation and
experience are especially important in the case of Ghana due to the manner in
which teachers are recruited for the senior secondary school level. The
population of senior secondary school mathematics teachers in Ghana can be
categorized into four main groups based on their courses of study at the

university as shown in the table below.

Table 1.5.1 Background of SSS Mathematics Teachers in Ghana6

 

 

Type of Bachelors Explanation Of Academic Background

Degree

B.Sc. (Math), Diploma in Graduated before 1991 from the University of

Education Cape Coast (UCC)

B.Ed. (Math) Graduated from UCC’S Faculty of Education
from 1991 or from University of Education,
Winneba

B.Sc. (Math) or related field Graduated in a non-education faculty/college

and Post Graduate from any university of Ghana but after entering

Certificate/Diploma the teaching field returned to pursue either a
certificate or diploma in education at UCC

B.Sc. (Math) or related field Graduated in a non-education faculty/college
from any university of Ghana and has been
teaching thereafter. But has not yet returned to

pursue either a certificate or diploma in
education at UCC

As shown in Table 1 above, first, there are those who graduated from the
University of Cape Coast (UCC) before 1991. This group pursued four-year

bachelor degrees with a major or minor emphasis in mathematics. In addition,

 

6 Percentage of teachers of Ghana’s teaching force in each category was not available at the time
of the studies.

21

they pursued a three-year university diploma in education alongside their degree
programs.

From 1991, the arrangement whereby a student at UCC could pursue both
a bachelor degree in mathematics and a diploma in education together was
discontinued. Students who gained admission into UCC from 1991 and were
interested in degrees in education or the teaching profession were admitted into
the faculty of education to pursue a bachelor degree in education with emphasis
in mathematics (i.e., B.Ed. (Math)). The University of Education, Winneba
(UEW), also in Ghana, now Offers a similar B. Ed. (Math) degree with two tracks:
secondary education and basic education. Graduates from these B.Ed. (Math)
programs constitute the second group of secondary mathematics teachers in
Table 1.5.1.

The third group comprises teachers who graduated from any of the
country’s universities with bachelor degrees in mathematics or in a related
discipline. Every year, Ghana loses a number of its teachers to developed
countries like the UK, US etc., as well as to other African countries. To fill the
vacuum left because of this exodus, the National Service Secretariat in Ghana
posts graduates of related content areas to teach mathematics in the senior
secondary schools. Many of these graduates eventually remain as mathematics
teachers without any pre-service teacher education background. The University
of Cape Coast (UCC) offers a summer sandwich diploma program in
mathematics education (until 1992, this was a certificate program) for this third

group of teachers over a minimum of two summers. Between the two summers,

22

participants of this sandwich program work with mentor teachers in the schools
where they teach while faculty from the university pay occasional visits to
oversee their progress.

The fourth group, like those in the third group graduated with Bachelor of
Science degrees in mathematics or a related field without any emphasis in
mathematics education. These may have entered the teaching field by either
being posted through the National Service Secretariat or having applied to teach
on their own. However, since entering the teaching field, this group of teachers
may not have been able to take advantage of UCC’s summer sandwich program
in education. They could therefore be teaching without going through any
college-level teacher preparation program.

Distinguishing between these groups is necessary because they may
have gone through different types of coursework in college. For instance,
teachers in Ghana who pursued bachelor degrees in mathematics take such
courses as abstract algebra, linear algebra, real analysis, complex analysis,
advanced calculus and ordinary differential equations prior to graduating. In
contrast, students in the B. Ed (Math) program are prepared in mathematics
content courses, mathematics education as well as foundations of education
courses. At the University of Cape Coast in Ghana, for instance, mathematics
education students take a minimum of 30 credits of mathematics content courses
with their counterparts in the mathematics departments. These comprise six
credits of university-based mathematics courses per semester from level 100 to

300 and the first semester of the 400 level. In addition, B.Ed. (Math) students

23

take other mathematics content courses targeted at the senior school curriculum,
such as algebra and trigonometry, advanced algebra, analytic geometry and
calculus, statistics and probability, vectors and mechanics at the Department of
Science and Mathematics Education. In addition, B.Ed. (Math) students take a
number of mathematics education courses, such as The Nature of Mathematics,
Psychological Basis of Mathematics Instruction, Curriculum Studies in
Mathematic Education, and Assessment in Mathematics Education. Other
mathematics education courses are, Methods of Teaching Secondary
Mathematics, Problem-solving in Mathematics, Introduction to Research Methods
in Mathematics Education, and a small research project that culminates into the
writing of what is referred to as a “long essay”. Mathematics education students
also take foundations of education courses such as History of Education in
Ghana, Social Foundations of Education, Psychology of Human Development,
Psychology of Human Learning, Guidance and Counseling, Educational
Measurement and Evaluation, Educational Administration, and Sociology of
Education in Ghana. In addition, during the third year of the undergraduate
teacher education program, mathematics education students undertake
microteaching sessions, under the guidance of experienced professors.
Thereafter, these pre-service teachers are attached to schools for one semester
of supervised internships (referred to as Off-Campus Teaching Practice), with the
help Of experienced mentors. During the internship, pre-service teachers get the

opportunity to observe lessons and take lead roles in teaching under the

24

mentorship of collaborating teachers in their subject areas while faculty from the
university occasionally visit to assess their performance and progress.

Thus, it is obvious that mathematics teachers at the secondary school
level in Ghana have varied backgrounds and experience. Such variation in
background preparation could mean that the four different group of teachers
discussed above could have different profiles of knowledge. With such variation,
the question that arises is the extent to which each aspect Of teachers’
knowledge reflects their program of study at the university, their experience or a
combination of the two.

In the light of the possibility Of this difference in background and, therefore,
knowledge base, this study was set up to investigate the knowledge for teaching
algebra Of each category of teachers. The same reason provides the rationale for
extending the study beyond the profile Of teachers’ knowledge to exploring how

teachers’ knowledge relates to student performance.

1.6 Purpose of the Study

This dissertation study had two main objectives. AS already explained, the
study is based on the KAT project’s three conceptualizations of knowledge for
teaching mathematics: school knowledge, advanced knowledge and teaching
knowledge. However, due to the varied paths to teaching of Ghanaian senior
secondary mathematics teachers, it is hypothesized that depending on their prior
university coursework and experience the four different types of teachers
identified in the previous section would have different levels Of each of the three

knowledge types proposed by the KAT project. Consequently, one aim of this

25

study was to answer the question, “How do the different types of teachers (both

prospective and in-service) who participated in the study differ in the profile of

knowledge they have?” To answer this, three different populations were

identified. These were:

1)

2)

3)

ln-service senior secondary school mathematics teachers; a working
hypothesis in this study was that this population could comprise teachers
in each Of the four groups discussed in the previous section;

Pre-service secondary school mathematics teachers in colleges Of
education in Ghana. It was hypothesized that this population’s knowledge
profile would define the profile of the in-service teachers with B.Ed. (Math)
degrees prior to entering the teaching profession; and,

Undergraduate final-year students in mathematics or related subjects in
Ghana’s universities who, through the national service scheme, had the
potential of being posted to senior secondary schools to teach after
graduation. It was hypothesized that this population’s knowledge profile
would define the profile of the in-service teachers with non-education

degrees prior to entering the teaching profession.

Since it is expected that their background could lead teachers and prospective

teachers to answer the questions on the instrument used for this study differently,

the study could explore whether there are any differences in their knowledge

base and how, if any, their knowledge differed. In addition, the in-service

teachers in the aforementioned four categories (see Table 1.5.1) would be

compared.

26

Second, of the in-service teachers selected, the study aimed at examining

the performance of their students and finding out how, if any, students’

performance and their teachers’ knowledge are related. In particular, the study

aimed at investigating the extent to which, if any, the different types of teachers’

knowledge for teaching algebra, hypothesized in the theoretical framework, relate

to their students’ performance.

1.7

Research Questions

The study was guided by the following research questions,

1)

2)

3)

To what extent does Ghanaian pre—service and in-service secondary
mathematics teachers’ knowledge for teaching algebra corroborate the
three categories Of knowledge hypothesized in the KAT framework:
knowledge of school algebra (or school knowledge), advanced knowledge
and teaching knowledge?

How does the knowledge for teaching algebra differ among the different
categories of secondary school mathematics teachers and potential
teachers in Ghana? AS already explained, the categories of teachers
aimed at here are in-service teachers, pre-service teachers in colleges of
education in Ghana, and undergraduate final-year students in
mathematics or related subjects. In addition, among the in-service
teachers, the aforementioned four categories (see Table 1.5.1) would be
compared.

What is the relationship between the performance of in-service secondary

school mathematics teachers in Ghana and the students of their classes?

27

1.8 Significance of the Study

As already discussed, this study is an attempt to examine issues of in-
service and pre-service teacher knowledge for teaching algebra at the high
school level. These are the issues of concern to the KAT project at Michigan
State University and the entire mathematics education research community
worldwide. This study is an attempt to use the KAT instruments to measure high
school mathematics teachers’ knowledge in a setting outside the US.

Research of this type, which studies issues of teacher knowledge, has
never been done in Ghana. Therefore, as far as Ghana is concerned, this is a
groundbreaking study. It is hoped that the findings of this study will contribute to
a better understanding of the knowledge Ghanaian teachers and potential
teachers have for teaching algebra. The findings could also contribute to
understanding how teachers’ knowledge relates to student Ieaming specifically in
Ghana. Such an understanding could be helpful in future curriculum planning in
mathematics at the SSS level, as well as pre-service teacher education. It could
also be useful in discussions about professional development of the Ghana’s
senior secondary school mathematics teachers.

Also, in one sense, this study was an attempt to give an international
dimension to the work that has been started by the KAT project. Ghana was a
good international site for three main reasons. First, unlike the US Where
standards-based curricula are sometimes used, in Ghana, schools are still using

syllabi that directly break down objectives for the content of school mathematics

28

at each grade level. Second, unlike most US. schools7, where separate
mathematics courses in algebra, geometry, calculus etc. are offered, in Ghana,
the core and elective mathematics courses offered to all high school students are
integrated and the content sequenced in a spiral manner. Third, the syllabi in
Ghana, unlike the US, are centrally controlled by the Ghana Education Service.
These differences in curricular arrangement between most US. schools and
schools in countries that have arrangements similar to those in Ghana could lead
to questions of generalizability Of findings from the KAT project across different
curricula and settings. Therefore, extending aspects Of the KAT study in Ghana,
as was done in this study, is essential in investigating whether KAT’s framework
will work in countries with different curricular arrangement such as Ghana.

In addition, the extension made in this study Of relating teachers’
knowledge to student performance has the potential to contribute to the
discussion of which type Of knowledge relates to improved instruction and
student learning. For researchers interested in finding ways of improving teacher
competence, findings from this study about which of the three hypothesized
knowledge types (i.e., knowledge of school algebra, advanced knowledge and
teaching knowledge) best relates student performance, as well as how the three
combined relate to student performance may be important. Replications of a

study like this could lead to decisions about what type of knowledge to

 

7 According to Reys, Dingman, Nevels & Teuscher (2007), six states in the U. S. (Florida,
Georgia, Indiana, New York, North Carolina and Tennessee) provide standards organized for
integrated high school mathematics courses. There are also states, such as Minnesota, with
individual schools (e.g., Wayzata High School in Plymouth, MN, which is personally known to me)
providing integrated mathematics courses using the Core Plus Mathematics Project (CPMP).

29

emphasize in pre-service teacher education, as well as professional development

programs.

1.9 Organization of Chapters

This section presents the organization of the chapters of this dissertation
in the order they are presented.

The first chapter provides the rationale for this study. In addition, because
the setting of the study is in Ghana, a country different from the United States of
America (US), background information about education in Ghana, the nature of
high school mathematics curriculum, as well an overview of the background of
high school mathematics teachers in Ghana is presented. The research
questions that guided the study are then presented. The chapter ends with the
purpose of the study and its significance.

The second chapter provides a review of literature relevant to the study.
The review is organized around issues that have influenced research on
teachers’ knowledge and teaching. It has been extended to cover studies
involving teachers’ knowledge and their practice, as well as teacher knowledge
and student performance. The theoretical framework used in the study is
presented after these reviews.

In Chapter 3, the research designs used in this study are presented. An
account of the procedures used in the study has also been given, as well as a
description Of the research instruments. The chapter ends with brief discussion of

how data for this study were analyzed.

30

Chapter 4 presents results and analysis of the study. This Chapter has
been organized in line with the research questions. In this way, analyses of data
related to a specific research question are conducted and the results presented
before moving on to another research question. To help make sense of the
statistical analyses, conclusions drawn are presented at the end Of each set of
analyses and results. These conclusions have been limited to issues related to
each of the research questions.

Chapter 5 summarizes the study and discusses conclusions drawn from
the analyses and results. The chapter ends with recommendations for further

research in the area.

31

CHAPTER 2: REVIEW OF RELEVANT LITERATURE

In this chapter, literature about teacher knowledge is reviewed. The review
draws out some of the issues that have influenced research on teachers’
knowledge and teaching. It has also been extended to cover studies involving
teachers’ knowledge and their practice, as well as teacher knowledge and
student performance. Further more, the research studies reviewed in this chapter
are mostly those that are grounded in the cognitive perspective (as opposed to,
for example, the sociO-cultural perspective). This reliance on studies framed by
cognitive perspectives is a result of the fact that the domains of knowledge
hypothesized in the theoretical framework, which guided this dissertation study,
emphasize the cognitive dimensions (i.e., on what is in teachers’ heads and
which they apply to their work) of teachers knowledge for teaching algebra. At
the time of this study, the knowledge domains hypothesized in the theoretical
framework, were in their early stages of conceptualization. As a result, the
possibility or othenrvise of the social dimensions Of teachers’ knowledge had not
yet been the focus Of study; hence the reliance on studies that emphasize the

‘ cognitive domains of teacher knowledge in the review of literature for this study.

2.1 The Issue of Teacher Knowledge

For many years, researchers have debated the issue of which school
factors influence student achievement. According to Duthilleul & Allen (2005),
this debate was started in the US when the report entitled Equality of Educational

Opportunity by Coleman et al., (1966) “concluded that family background

32

characteristics and community level variables accounted for more variance in
student achievement than school resource variables like . . . teacher
characteristics” (p.3). Coleman and his colleagues analyzed data from about
600,000 students and 60,000 teachers in more than 4,000 schools and
concluded that only about 10 percent of the variance in student achievement
could be explained by school factors. In the intense debate that ensued
thereafter, viewpoints emerged that questioned whether schools matter in
student Ieaming. Such negative findings and views about schools and teachers
in particular could be What underlined early attempts at conceptualizing the
knowledge base for teaching that was spearheaded by Shulman (1986b, and
1987). As Strom (1991) puts it, “at one level, concern about the knowledge base
focuses on improving the respect and status accorded teaching, thereby making
it a more rewarding career” (p. 1). The idea was that for teaching to be respected
as a profession that influences Ieaming outcomes, a case needed to be made
that it involved a wise application of a specialized body Of knowledge. It is to this
end that Shulman (1986b) introduced the idea of “pedagogical content
knowledge” as a type Of knowledge, which comprises such things as how
students understand, and how to use resources effectively to present ideas in
ways that make them more accessible to different types of students.

Substantial research has been conducted on the issue of the knowledge
base for teaching since the time Shulman (1986b, 1987) put fonlvard his
framework. Some of these have led to further conceptualizations. A good

example of this is the study by Liping Ma. In Knowing and Teaching Elementary

33

Mathematics: Teachers’ Understanding of Fundamental Mathematics in China
and the United States, Ma (1999) presents the results and analyzes of interviews
with 23 elementary schoolteachers from the US. and 72 from China. Through
the analyses of the interview data, Ma (1999) introduces a different
conceptualization Of the knowledge base for teaching mathematics, which she
calls, "profound understanding of fundamental mathematics” (PUFM). Ma's
(1999) PUFM involves a type Of knowledge more than content knowledge; it also
has to do with communicating the subject matter Of school mathematics to
students. In this regard, with the exception of the fact that Shulman’s (1986)
pedagogical content knowledge (PCK) is a generalized form of knowledge
(possibly not restricted to a particular subject matter like mathematics), Ma’s
(1999) conceptualization can be said to show some resemblance to Shulman’s
(1986b) PCK. At least both seem to involve a complex combination of content
knowledge and pedagogical knowledge.

Not only did the 1966 Equality of Educational Opportunity report by
Coleman and his colleagues influence the need for providing a framework aimed
at professionalizing teaching, it also fueled studies into the effects of instruction
on student learning. In the Handbook of Research on Teaching, Brophy and
Good (1986) reviewed several studies conducted in the 1970s, and drew the
following conclusion, ”The myth that teachers do not make a difference in student
Ieaming has been refuted” (p. 370). In fact, currently there is wide-ranging
agreement, from numerous studies that the availability and effectiveness of

teachers contribute in no small ways to student learning outcomes (see for

34

example, Jordan, Mendro, and Weerasinghe, 1997; Sanders and Rivers, 1996;
Wright, Horn, and Sanders, 1997). The Wright, Horn and Sanders (1997) study
for instance is one of the latest studies to reveal that the most important factor
affecting student learning is the teacher. After analyzing the achievement scores
Of more than 100,000 students, Wright and his colleagues concluded, “Effective
teachers appear to be effective with students of all achievement levels,
regardless of the level of heterogeneity in their Classrooms. If the teacher is
ineffective, students under the teacher's tutelage Will show inadequate progress
academically despite how similar or different they are regarding their academic
achievement (Wright et al., 1997, p. 63).

In the face of this renewed confidence in the teacher as one of the most
important factors that influence student performance, one issue that remains is
the question Of which aspects of the teacher’s knowledge best relate to student
performance. This issue is important to research into meaningful ways of school
improvement; an importance further enhanced by current not so good
performance in mathematics of US and Ghanaian high school students’ on
international comparative studies (e.g. TIMMS, 2003). For instance, though
eighth graders in the US performed above average in the 2003 Trends in
lntemational Mathematics and Science Study (TIMMS, 2003), they were
outperformed by ten of the twelve OECD-member countries. In the case of
Ghana participating for the first time in the TIMMS assessment, Ghanaian eighth
graders did not only perform below average, they were the second from last

among the 44 participating countries. Studying which aspects of teacher

35

knowledge best relates to student performance warrants new conceptualization
Of the knowledge needed to teach various content of school mathematics. Such a
new conceptualization would need to lend itself to being assessed both
qualitatively and quantitatively and on a large scale. The current study sought to
do exactly this. The study relied on the conceptualization of teachers’ knowledge
for teaching algebra provided by the Knowledge of Algebra for Teaching (KAT)
project and investigated whether data obtained by adapting the KAT instrument
on Ghanaian teachers would corroborate the KAT conceptualization. In addition,
the study sought to examine how the different types of teachers’ knowledge

hypothesized by the KAT project relates to student performance.

2. 1 . 1 Teachers’ Knowledge and their Teaching Practice

Lampert (1990) and Ball (1993) used their own classrooms to examine the
relationship between possession of subject matter knowledge In mathematics
and the teaching of mathematics. Their work confirms the argument by Shulman
and Quinlan (1996) that, “excellent teachers transform their own content
knowledge into pedagogical representations that connect with their prior
knowledge and dispositions of the learner” (p. 409). While it is important not to
diminish the significance of subject matter competency of teachers, it is also
important to consider how much of it helps teachers to transform their knowledge
in ways described by Shulman and Quinlan (1996). In addition, it is also
important to consider the role of experience in facilitating this transformation.

Research on teaching is replete with attempts at examining how teachers

transform their knowledge into their teaching practice (Peterson & Clark, 1978;

36

Leinhardt, & Greeno, 1986; Shulman, 1986, 1987; Wilson, Shulman & Richert,
1987; Leinhardt, 1988). For example, Leinhardt & Greeno (1986) described a
skilled teacher as one who has “a complex structure composed of interrelated
sets of organized actions [called schemata. . .] which are applied flexibly with little
cognitive effort in circumstances that arise in the classroom”(p.75). This
perspective of a skilled teacher emphasizes the strong belief in knowledge
transformation into teacher actions. However, there has been wide variation on
what aspects of teacher knowledge to focus on in research.

For instance, available literature Shows that initial research into how
teachers translate their knowledge into their teaching practice was in the form of
process-product research, which started in the US from the 1920 onwards (see
reviews Of these early types of studies by Brophy & Good (1986), Gage (1978)
and Doyle (1977)). The rationale behind the design of this early research into
teaching was the presumption that a direct connection could be established
between teacher actions in the classroom and student achievement.
Consequently, in studying teaching, process-product researchers coded teacher
actions and related them to student behaviors (outcomes) that were measured.
Coding teacher actions was an indirect attempt at breaking down which aspects
Of teachers’ knowledge are transformed into their teaching practice.

In the mid 19703, this early approach into studying teaching received four
main kinds of criticisms, including its over-reliance on correlational methods and
the types of teacher actions that were coded. A complete review of all these

critiques have been summarized and evaluated by Gage and Needels (1989).

37

Their review grouped the criticisms of process-product research on teaching into
four main categories. The first Of these categories are those criticisms that have
attacked the conceptualization of process-product research, including the
conception of causality implied in this research paradigm. The second category
Of criticisms is those that focused on the methods used by process-product
researchers (i.e., pre-determined coding categories, the need for experimental
methods, the role Of mediation and the quality of the outcome measures). Third
are critiques concerned with the predictive power Of process-product research.
The fourth and last of Gage and Needels’ (1989) categories are criticisms that
have focused on the use of meta-analysis and the conversion of findings into
rules for teaching by process-product researchers.

Based on these critiques, several researchers proposed a modification in
the process-product design (see for example, Berliner, 1979; Peterson & Swing,
1982). For instance, according to Berliner (1979), he and his colleagues in the
Beginning Teacher Evaluation Study (BTES) study introduced a variable, which
they called Academic Learning Time (ALT) in their modification of the process-
product research design. The BTES program insisted that this variable does not
only serve as the link between teacher behavior and student achievement but is
also an important operationally behavioral indicator Of student learning. ALT
therefore became the research variable of interest to the BTES program. One
aspect of ATL was what Berliner and his colleagues referred to as “engaged
time”; the actual time students spend on tasks provided by the teacher in learning

a particular content. Their argument was that if the time is spent on materials that

38

are too difficult for the student, he/she could not acquire any extra concepts,
skills and Operations that are needed for effective performance at that grade
level. On the other hand, engaging students over long periods on many easy
tasks will also not improve academic Ieaming. Unfortunately, in keeping the focus
on how teachers transform their knowledge into their teaching practice, the BTES
program failed to show what type of knowledge teachers use in judging the
difficulty level Of the tasks they give to their students, especially in heterogeneous
situations where students come with varied competencies. Their ALT construct
also failed to indicate how teachers are able to decide when to move to new
materials.

Later, other researchers brought the mental life Of the teacher to the
center of teaching research (see for example, Peterson and Clark, 1978;
Putnam, 1987). Focusing on the mental life Of teachers suggests that
researchers in this area hoped that the type Of knowledge teachers transform into
their practice could be seen in the thought process of teachers before, during
after teaching. The rationale for this line of research is that experienced teachers’
knowledge about teaching is organized into packages Of question and
explanations that make it possible for them to enhance student Ieaming and
overcome student misconceptions about subject matter they teach (Putnam,
1987; Shulman, 1987). Putnam (1987) refers to these packages as “curriculum
scripts” and argues it is these curriculum scripts, which shape teachers’ agenda
for teaching and not students’ prior knowledge (or intuitive knowledge of

students). The “curriculum scripts” that experience teachers possess enable

39

them to adopt flexible and interactive approaches to teaching and enhance their
efficiency. To researchers with this perspective, focusing on the mental life of the
teacher before, during and after teaching is one way of accessing what aspects
Of their knowledge they transform into teaching.

The work by Shulman and his colleagues threw the brightest light on how
teacher knowledge could influence teaching (see Shulman, 1986, 1987; Wilson,
Shulman & Richert, 1987). Shulman’s (1986) conceptualizations of “content
knowledge” and “pedagogical content knowledge” and the distinction between
them brought the attention Of researchers in several content domains to issues
involving the type of knowledge teachers need about content for teaching,
different from what an ordinary adult may have (see for example, Ball, 1988;
Wilson & Winneburg, 1988; Grossman, 1990).

All the aforementioned research paradigms produced mostly qualitative
information about teachers’ knowledge and its influence on their teaching
practice. At the time Of this study, work by Ferrini-Mundy and her colleagues on
the KAT project appears to be groundbreaking work aimed at developing
measures of knowledge Of mathematics for teaching algebra at the high school
level. Past approaches reviewed above either ended in conceptualization of the
knowledge for teaching or with measures of teacher knowledge and actions that
influence their teaching practice. Even when attempts were made to link to
student performance, as was done in the BTES study, the construct

conceptualized (e.g., the ATL construct) was mostly limited to teacher actions.

40

These later constructs, ATL, for example, failed to show which type of knowledge
teachers use to judge the difficulty level of the tasks they give to their students.

The approach by the KAT project is an improvement over past
approaches in a number of ways. For instance, after the initial
conceptualizations, researchers in the KAT program are designing items and
reliable and valid instniments to measure knowledge hypothesized in their
framework in large scale settings for teaching algebra among pre-service and in-
service secondary school mathematics teachers. These measures being

developed by the KAT project have been adapted in this study.

2. 1.2 Teachers’ Knowledge and Student Performance

Traditionally, education researchers have felt that somehow there has to
be a relationship between teacher knowledge and student achievement. In
particular, many have felt that teachers’ subject matter knowledge needed to be
related to student achievement. Even those who feel that a teacher’s actions in
the classroom have to be related to his/her subject matter knowledge also feel
that quality pedagogical representations have to be related to improved student
achievement. The argument by Shulman and Quinlan (1996) that, “excellent
teachers transform their own content knowledge into pedagogical
representations that connect with their prior knowledge and dispositions Of the
learner” (p. 409) summarizes this suspicion that teachers’ content knowledge
would lead to quality teaching practices and, perhaps, indirectly to student
achievement. It is therefore, not surprising the volume of critique directed at

process-product researchers when this was not made an explicit focus of their

41

study (see reviews by Brophy & Good, 1986; Gage, 1978; Doyle, 1977).
Consequently, at the time when the Coleman et al., (1966) report sparked a
wave of research aimed at correlating factors such as students’ family
background and sociO-economic status, etc (Hanushek, 1981; Greenwald,
Hedges & Laine, 1996), some researchers still focused on the relationship
between teacher knowledge and student performance. Proxy measures of
teacher knowledge such as performance on certification examinations or other
forms of examinations were used as variables to examine relationships with
student achievement (see for example, Hanushek 1972; Boardman, Davis &
Sanday, 1977; Strauss & Sawyer, 1986; Ferguson 1991; Harbison & Hanushek,
1992; Tatto, Neilsen, Cummings, Kularatna & Dharmadasa, 1993; Mullens,
Mumane & Willett, 1996; Rowan, Chiang, & Miller, 1997). A number of studies
have shown that secondary school students’ achievement in mathematics is
related to their mathematics teachers’ knowledge. In the developing world,
teachers’ subject matter knowledge was found to be a better predictor of student
achievement than other home-based factors by Harbison and Hanushek (1992)
in Brazil, as well as Mullens et al. (1996) in Belize.

Though this was a good development, and several studies found a
positive correlation between teacher knowledge and student achievement in
mathematics, there were a number of disagreements among researchers as to
what aspects of teachers’ knowledge are related to student achievement and
how much Of it. ROWan, Chiang, & Miller (1997), for example, revealed that

teachers' knowledge of subject matter has a direct effect on students'

42

achievement in mathematics. This study also found that that the size Of the effect
of teachers’ subject matter knowledge depends on the average levels of ability of
students in a school. Others have found, however, that students taught by
teachers with advanced degrees in mathematics performed poorer compared
with students taught by teachers without advanced degrees in mathematics (see
Monk, 1994; Rowan, Correnti & Miller, 2002). For instance, Monk’s (1994) study
revealed that the taking of advanced mathematics courses beyond five by
teachers produces virtually no returns in terms of the impact on the achievement
Of their students. This finding pointed to the possible existence of the economists’
law of diminishing returns on the number Of advanced mathematics courses
taken by teachers and the impact on their students’ achievement.

In addition, there were questions about how much of the relationship
between teacher knowledge and student achievement was due to experience.
For instance, the study by Rowan et al. (2002) pointed more, at the elementary
school level, to teachers’ years Of experience than subject matter competency as
perhaps the most consistent predictor Of students’ achievement.

In addition to these, there are also studies that seem to project the value
added by teacher education to teacher competence. It is well documented that
even with strong subject matter background, teachers without subject matter
pedagogical preparation could not engage their students in deep thinking about
the subject matter Of mathematics but only teach the way they were taught
(Darling-Hammond, 1999 & 1991; Fergusson & Womack, 1993; Grossman,

1989; Shulman, 1987). For instance, in a three-year study, Ferguson and

43

Womack (1993) found that course work in teacher education better predicted
teacher effectiveness than measures of content expertise alone. This study found
that education courses were better predictors of teaching success (i.e., student
achievement) than teachers’ grade point average (GPA) prior to entering teacher
education programs. Monk (1994) found similar results about science education
coursework and student achievement. Review of studies by Darling-Hammond
(1999), which show that students of certified teachers score higher on
standardized mathematics test than those of uncertified teachers, lend support to
the value of teacher education in teaching success. Earlier, Darling-Hammond
(1991) had argued in favor of the efficacy of subject-specific methods courses for
those preparing to teach. She cited several studies to support her conclusion that
teachers admitted to the profession through quick-entry alternative routes had
difficulty with pedagogical content knowledge and curriculum development (see
also Darling-Hammond, 2003; Lacsz-Kerr & Berliner, 2003 and Kennedy, Ahn &
Choi, 2006). The paper by Kennedy, Ahn 8. Choi, (2006) in particular provides a
synthesis of research on the relationship between teachers' educational
background and the mathematics achievement of their students. Their synthesis
concluded that “additional teacher courses in content, content pedagogy, and
pedagogy, all benefit students” (p. 36). Consequently, Kennedy et al. (2006)
recommended, that instead Of debating the relative merits of the various
domains, an intensive study in both content and teacher education domains

could be one way of improving the curriculum of pre-service teachers.

Unlike the situation in the US, not many studies correlating the value
added by teacher education or teachers subject matter knowledge to student
achievement have been examined in Africa. The few studies that have examined
similar issues in Africa have found that the relationship depends on the SES
status of the school. For instance, datasets collected by the Southern African
Consortium for Monitoring Educational Quality (SACMEQ) between 2000 and
2002 have led Duthilleul and Allen (2005) to examine the relative contribution of
teacher’s education, subject matter competency and pedagogical practices to
sixth graders’ mathematics achievement in Namibia. Duthilleul and Allen (2005)
found that the relative contributions of each of these factors depend on the socio-
economic status of the school. According to Duthilleul and Allen (2005), in low
SES schools, effective teachers had a high level of subject matter competency,
while in high SES schools teacher training continued to be associated with
effective teaching. These findings support the notion that effective teachers
should not only have a sufficient level of subject matter competency but should
also receive adequate teacher training in order to develop effective pedagogical
practices and contribute to student achievement.

The lack of agreement among researchers as to which aspect Of teacher
knowledge and how much of it, as well as how much experience contributes to
influencing student achievement point to the need for better measures of teacher
knowledge. Rather than rely on proxy measures, such measures need to be
related to conceptualization of teachers’ knowledge for teaching mathematics. It

is In the light of this that the conceptualization of knowledge for teaching algebra

45

at the high school level by the KAT project has been used as a framework for this
study. In the next section, this conceptual framework is discussed, as well as

how the conceptualizations fit into the present study.

2.2 Conceptual Framework

The Knowledge Of Algebra for Teaching (KAT) project currently in
progress at Michigan State University is working on conceptualization and
validation of the type of knowledge used by teachers in teaching algebra.
Through analyses Of research literature, recommendations by professional
organizations and videos of teaching, researchers in this project have
hypothesized that the knowledge used by teachers in teaching school algebra
consists of three types. These are “knowledge of school algebra” (referred to in
short as “school knowledge”), “advanced knowledge of mathematics” (also
referred to as “advanced knowledge”), and “teaching knowledge”. These three
types of knowledge, discussed below, constitute the theoretical frame of algebra

knowledge for teaching that guided this dissertation study.

2. 2.1 Knowledge of School Algebra

The KAT project defines “Knowledge of School Algebra” (or simply
“School Knowledge”) as the knowledge of mathematics in the intended
curriculum of middle school and high school. This is the content of school algebra
that teachers are expected to help students discover or learn in their algebra
classes. In the US, the big ideas of this type of knowledge are described in

documents such as the National Council of Teachers of Mathematics (NCTM)’S

46

Principles and Standards for School Mathematics (NCTM, 2000) while the
specific grade-level algebra content is described in the various states’ standards,
textbooks and other instructional materials used in the schools. In their work,
researchers in the KAT project delimited this type of knowledge by reviewing
content standards Of ten different states in the US (Appendix I lists content areas
generated by the KAT project from this review). At the SSS level in Ghana, the
content of this type of knowledge is included in both the Core and Elective
Mathematics Syllabuses. The content of this domain of mathematics, school
algebra, at the high school level in Ghana has been presented in Appendix II
(from Core Mathematics) and Appendix III (for Elective Mathematics), after a
review of the Core and Elective Mathematics syllabi from Ghana. This type Of
knowledge is considered important by the KAT project because unless teachers
understand the grade-level algebra content they are to teach, they would find it
difficult to influence student learning. Since students are expected to Ieam their
school algebra, it sounds reasonable to hypothesize that for teachers to influence
students Ieaming, they (teachers) need to understand the content Of school

algebra themselves.

2. 2.2 Advanced Knowledge of Mathematics

According to the KAT project, Advanced Knowledge of Mathematics (or
simply “Advanced Knowledge”) “includes other mathematical knowledge, in
particular college level mathematics, which gives a teacher perspective on the
trajectory and growth of mathematical ideas beyond school algebra” (Ferrini-

Mundy, Senk and McCrory, 2005, p.1).

47

The KAT project lists areas like calculus, linear algebra, number theory,
abstract algebra, complex numbers and mathematical modeling as some Of
these general areas (see Ferrini-Mundy et al., 2005). In addition, in the
conceptualization of advanced knowledge, members of the KAT project
acknowledge that “knowing alternate definitions, extensions and generalizations
of familiar theorems, and a Wide variety of applications Of high school
mathematics are also characteristics of an advanced perspective of
mathematics” (Ferrini-Mundy et al., 2005, p. 1). Thus, it can be argued that
having an advanced perspective of mathematics affords teachers with a deep or
profound understanding of school algebra. Figure l below is a diagram that
exemplifies how the possession of an advanced perspective of mathematics
could make it possible for a teacher to make connections across topics while
unpacking the complexity of a mathematics content to make that content more

comprehensible.

48

SUCCESSORS
(bridging, big ideas)

 

Complex Analysis

Representations of solutions
(numerical, graphical)

Importance of domains and
ranges of functions

Complex numbers

 
 
   

WHAT IS THE SOLUTION?

ITEM "KERNEL" 2
X < 0

Mathematical thinking: "It depends on . .

   
  

Operations: extending number
systems to "close" operations

   
    

Number domains: natural
numbers, intergers,

/
PREDECESSORS fractions, real numbers

(unpacklng)

Figure 2. 2.1 Conceptual Representation of “Advanced Knowledge”

Figure 2.2.1 above shows an example of a specific content of school
algebra, the content indicated by the item kernel, “what is the solution of x2 < O?”
In class, the KAT project hypothesizes that a teacher's task, when this item is the
focus, is unpacking content preceding the content of focus. In addition, teachers
would also need to engage in bridging the big ideas that come after the item Of

interest. As shown in the figure, predecessor content that need to be unpacked

49

by the teacher includes content such as sets of numbers, operations on numbers
and engage in getting students to mathematically think broadly about conditions
on which such solutions would depend. Also shown in the figure are examples of
related content that go beyond the content of school algebra as complex
numbers, domain and range Of functions, and various forms of representations of
solutions. As already indicated, the KAT project considers “advanced knowledge”
important because possessing it affords teachers with a deep or profound
understanding of school algebra. In addition, it is hoped that teachers who
possess this type of knowledge would have a good knowledge of the trajectory of
the content of school mathematics. This knowledge in turn could help teachers to
engage in bridging (making connections across topics), trimming (removing
complexity while retaining integrity and decompressing (unpacking complexity to
make content more comprehensible) of the content of school algebra to students;

processes that could be vital to effective teaching.

2. 2.3 Teaching Knowledge

The third category of knowledge in the KAT framework is the teaching
knowledge. In the KAT framework, this knowledge is described as “knowledge
specific to teaching algebra that may not be taught in advanced mathematics
courses. It includes such things as what makes a particular concept difficult to
learn and what misconceptions lead to specific mathematical errors. It also
includes mathematics needed to identify mathematical goals, Within and across
lessons, to Choose among algebraic tasks or texts, to select what to emphasize

with curricular trajectories in mind and to enact other tasks of teaching” (Ferrini-

50‘

Mundy, McCrory, Senk & Marcus, 2005, p.2). Thus, this is the type Of knowledge
that teachers have and which they use in the teaching the subject matter of
school algebra (see Appendix III for the tasks of teaching from the KAT
framework). This point is made by the KAT researchers when they say that, “the
knowledge referred to here may fall into the category of pedagogical content
knowledge or it may be pure mathematical content applied to teaching” (Ferrini-
Mundy et al., 2005, p.1). In addition, since this type of knowledge may not be
taught in advanced mathematics courses, it may not necessarily be available to
mathematicians. Consequently, this is the knowledge that could differentiate an

engineer or a mathematician from an algebra teacher.

2.3 Relationship between the Three Types of Knowledge

The KAT project conceptualizes that their hypothesized three types Of
algebra knowledge for teaching, School Knowledge, Advanced Knowledge and
Teaching Knowledge are not hierarchical in nature. Neither do they exist in a
continuum with well-definable boundaries. Rather, their boundaries are blurry in
the sense that they are interwoven in many ways. A schematic diagram of this
conceptualization is presented in Figure 2.3.1 below. In chapter 4, data from this

dissertation study has been used to confirm this conceptualization.

51

   
   

Advanced Mathematical Knowledge

  
    

Knowledge of
School Algebra

  

Teaching Knowledge

Figure 2.3.1 Conceptual Representation of the Three Types Of Knowledge.

2.4 The KAT Project’s Item Development Matrix

As already discussed, one aim Of the KAT project is to develop items and
design reliable and valid instruments to measure knowledge in large-scale
settings for teaching algebra among pre-service and in-service secondary school
mathematics teachers. To achieve this, two other constructs were conceptualized
by the KAT project in addition to the construct Of the “algebra knowledge for
teaching” (which comprised the three types of knowledge hypothesized). The
other two construct were defined as “Algebra Content” and “Domains of
Mathematical Knowledge”. The three constructs where presented in an item
development matrix that is cuboid in nature (or in three-dimensional space) as

shown in Figure 2.4.1 below.

52

 

///
///
////

 

 

 

 

 

 

 

 

 

B. Functions 8.
their Properties: / /
LInear 8' Nonlinear D. Reasoning 8. Proof
ALGEBRA CONTENT /C A IicationsDOMAINS OF
A Ex ' 9" MATHEMATICAL
presslons,
Equations 8. J/ B. Representations KNOWLEDGE
Inequalities
A. Core Concepts 8. Procedures
A. Knowledge C. Teaching

ofSchool B-Advanced Knowledge
Algebra Knowledge

ALGEBRA KNOWLEDGE FOR TEACHING

Figure 2.4.1 The KAT Project’s Item Development Matrix
As shown in Figure 2.4.1, the three types of algebra knowledge for teaching are
pictured on the x-axis, the “algebra content” is pictured on the y-axis and
“domains Of mathematical knowledge” are pictured on the z-axis.
In the sub-sections that follow, the content of the constructs of “algebra

content” and “domains of mathematical knowledge" are briefly explained.

2.4.1 The Y-Axis: Algebra Content

Though this dimension could have many categories, the KAT project
limited this to two areas or topics that are considered central mathematically,
included across K-12 curricula for school algebra (traditional as well as reform)

and which have been the focus Of research on student learning in algebra.

53

2.4.1.1 Expressions, Equations, and Inequalities

Researchers in the KAT project decided to formulate items on
expressions, equations and inequalities for a number of reasons. First, review of
the content of school algebra in various states indicated that at the heart of high
school algebra is the algebra of polynomial and rational expressions, equations,
and inequalities. It was therefore considered that the ability Of students to work
with these concepts is integral to their success in algebra and all higher-level
courses. Second, research has shown that students could have difficulties with
concepts involved in working with expression, equations and inequalities. For
instance, available literature is replete with students’ difficulties in understanding
differences between expressions and equations as well as the many uses of the
equal Sign (see, for example, Wagner 8. Kieran, 1989; Kieran, 1993; Bednarz,
Kieran 8. Lee, 1996). Third, according to Nathan and Koedinger (2000), whereas
students performed better on the word problems than on the symbolic equations,
teachers have consistently rated solving symbolic equations easier than solution
of similar word problems. Consequently, in designing valid and reliable
instruments for assessing teachers’ knowledge, researchers in the KAT project
have felt the need to formulate items that could be categorized as involving

expressions, equations and inequalities.

2.4.1.2 Functions and their Properties: Linear and Non-linear.

Function is the second topic of this study. The KAT project defines a
function broadly as “as a relationship between two sets of Objects, usually

numbers where every number in the first set is related to exactly one number in

54

the second set. The first set (domain) and second set (range) may be infinite or
finite and consist of real or complex numbers” (Ferrini-Mundy et al., 2005, p. 3).
The need to include this topic in the content domains is based in part of it being
considered a central mathematical idea. Another rationale for focusing on this
topic follows from the literature on the possible misconceptions of functions to
pre—service teachers (Even, 1993). In addition, like expressions, equations and

inequalities, concepts of functions are fundamental for higher-level mathematics.
2.4.2 The Z-Axis: Domains of Mathematical Knowledge

2.4.2.1 Core Concepts and Procedures

According to the conceptualizations by researchers in the KAT project,
core concepts and procedures consists of knowledge of the concepts, definitions,
axioms, theorems, and algorithms, as well as mathematical language, notation,
and conventions. This is the type of knowledge Shulman (1986) calls declarative

or substantive knowledge.

2.4.2.2 Representation

The term representation in the KAT framework is taken in this general
sense as Kaput (1985) talks as “involving some kind of relationship between
symbol and referent...” (P. 383). Thus, the KAT framework defines
representation as the various forms used to describe or picture mathematical
concepts and procedures. It includes number lines, tables, graphs, area models,

and matrices.

55

2.4.2.3 Applications

“By applications, we mean the representation of a real world situation by a
mathematical one” (Ferrini-Mundy, McCrory, Senk & Marcus, 2005 p.5). Because
of the application of algebra to many real situations, it is important that teachers
be able to link the algebraic concepts they teach with real situations; hence the

focus on applications.

2.4.2.4 Reasoning and Proof

According to the KAT project, “reasoning and proof includes knowledge of
the specialized vocabulary Of reasoning, the ability to find examples and
counterexamples of statements, and the ability to use analogies or geometric
arguments to justify statements, and the ability to use various proof techniques
within an axiomatic system to make convincing arguments. [It also includes]
the ability to judge the reasonableness Of conjecture” (Ferrini-Mundy, McCrory,
Senk 8. Marcus, 2005 p. 5). This conceptualization is consistent with the
components of reasoning and proof emphasized by NCTM (2000) Principles and
Standards for School Mathematics:

0 Make and investigate mathematical conjectures;
. Develop and evaluate mathematical arguments and proofs;
0 Select and use various types Of reasoning and methods of proof
(NCTM, 2000, p. 342)
This domain has been focused by the KAT project because of the need for
teachers to also be able to help students understand mathematical justifications,

and to be able to construct additional justifications that make sense to them.

56

2.5 Importance of this Conceptual Framework to this Study

The focus Of this study is in part influenced by conceptualizations of
content knowledge, curriculum knowledge, pedagogical knowledge, and
pedagogical content knowledge put forward Shulman and his colleagues
(Shulman, 1986b; Wilson, Shulman 8. Richert, 1987). My perspective on the
conceptualizations of content knowledge, curriculum and pedagogical knowledge
is that effective mathematics teachers, especially at the high school level, use
them in ways that blend these three types of knowledge into a somewhat new
form of knowledge. My personal intellectual work has gotten me to think of the
teacher knowledge in terms of connected or overlapping packages of knowledge
(see Ma, 1999) or curriculum scripts, to use the words Of Putnam (1987). In my
perspective, these connected packages of knowledge are a blend knowledge of
content of the subject matter they teach, knowledge of other content in the school
curriculum and their relationship, as well as why particular representations could
be problematic or easy to some students. To this end, the concept Of
pedagogical content knowledge, which Shulman defines to include
representations of specific content together with why the Ieaming of that content
is easy or difficult for students, have resonated well with my personal
perspective. Unfortunately, until now, earlier researchers who have relied on
Shulman’s conceptualizations have only concerned themselves with teacher
knowledge qualitatively.

The KAT conceptualization has illuminated the perspectives I had and

contributed to my learning about teaching in a number of ways. First, their

57

emphasis on both school knowledge and advanced knowledge confirm for me
the fact that teachers need to know not only the content they are teaching, but
also content Of other areas Of their subject (and sometimes beyond their subject)
that are connected to it. This perspective is inherent in the KAT project’s
construct of “advanced knowledge” covered content beyond the content of school
algebra that afford teachers with a deep or profound understanding of school
algebra. Second, the argument by Ferrini-Mundy and her colleagues that the
boundary between their three conceptualizations is blurry connects well with my
perspective of connected or overlapping nature of components of teachers’
knowledge. Third, attempts by the KAT project to develop measures of the profile
of knowledge for teaching algebra using specific school algebra curriculum and
ideas about the tasks involved in teaching is a departure from the qualitative
measures of earlier researchers. Finally, this conceptualization by KAT has
implications for my thinking about the types of knowledge teachers need for
teaching other content of high school mathematics curriculum.

It is my hope that this conceptual framework will not only help me to focus
on which of these three types of knowledge hypothesized by KAT is best related
to student performance, but also how they are transformed into Ghanaian high

school mathematics teachers’ teaching.

58

CHAPTER 3:.METHODS

This chapter opens with a discussion of research designs used in this
study. This discussion is linked to the research questions that guided the study.
This is followed by description Of the target and accessible populations. After this,
the selection of participants for this study is discussed. In addition, a description
of the instruments used in collecting data for the study is presented. Thereafter, a
detailed account Of the procedures used in data collection is presented. The

Chapter ends with a look ahead into how data for this study were analyzed.

3.1 Research Design

This study adopted two types of designs based on the nature of the
research questions used in the study. For instance, the first and second research
question involves four different populations based on their college-level
coursework and teaching experience. These were 1) pre-service mathematics
education students in the final year of their college preparation program, 2) final
year mathematics major students (considered possible prospective teachers), 3)
final year statistics major students (also considered possible prospective
teachers) and 4) in-service teachers from the participating senior secondary
schools. For these different populations, the quality of their school knowledge,
advanced knowledge and teaching knowledge was what was assessed using the
adapted KAT instruments. The three hypothesized knowledge types were
considered as three discrete factors of knowledge for each person. Therefore, a

factorial design was used for the first and second research questions.

59

The third research question was aimed at investigating the relationship
between in-service teachers and their students who participated in the study.
Thus, this research question emphasizes causal analysis as the main research
goal and multivariate linear regression as the main statistical tool. Thus, the
design used with respective to the investigation into the issues raised in the third
research question was measuring the relation between two variables and was

therefore conducted using a correlational design.

3.2 Target Populations

The target population from which participants of the study were selected
comprised three main populations. These were, 1) university seniors pursuing
mathematics education, mathematics and other mathematics related programs in
the country’s universities, 2) in-service mathematics teachers from senior
secondary schools (i.e., high schools) in Ghana, and their 3) high school seniors
taking elective mathematics.

The decision to include each of these subgroups was based on the aims
of the study and hypotheses made about the type of knowledge each subgroup
would have by the time Of the study. For instance, one aim of the study was to
investigate the profile of knowledge (i.e., the nature of and possible differences in
the level or quality Of knowledge) of prospective and in-service Of senior
secondary school mathematics teachers. Based on this objective, it became
necessary to focus on both in-service and prospective teachers. Consequently,
there was the need to target teachers who were at the time Of the study teaching

mathematics (either Core Mathematics or Elective Mathematics) in the senior

6O

secondary schools. Since some of these in-service mathematics teachers would
have been teaching for a long time, it was hypothesized that some of them would
have forgotten aspects Of the content of their university coursework. In addition, it
was hypothesized that, depending on the number of years of teaching
experience, some in-service teachers will have a high level of teaching
knowledge while for others the level could be low.

In addition, it was also necessary to focus on university students who
could be teaching soon after their programs in order to find out what their profile
of knowledge looked like before they entered the teaching field. In Ghana, this
group of pre-service teachers could be said to comprise not only mathematics
education students, but also students majoring in mathematics or a related area.
This is because, to fill vacancies left in the teaching field following the mass
exodus of Ghanaian teachers, the National Service Secretariat, every year, posts
both mathematics education majors and mathematics (or related area) majors to
the senior secondary schools to teach mathematics. Therefore, pre-service
teachers were thought of as comprising final year students of mathematic
education, mathematics and related discipline. It was hoped that by the time of
the study, final year students in the selected universities would be close to
completing their respective college coursework and would consequently have
acquired the advanced knowledge, which most teachers at the senior secondary
schOols covered while at college. Based on this, it was hypothesized that the
mathematics majors in this group will have high knowledge Of school algebra,

high advanced knowledge of mathematics but low teaching knowledge.

61

Similarly, it was hypothesized that the mathematics education majors in this
group will also have high knowledge Of school algebra, mixture of high and low
advanced knowledge of mathematics but a mixture of low and high teaching
knowledge because Of their teacher certification program. I

A summary of the hypothesized level of knowledge of the target
population of in-service teachers and the final year university students is

presented in Table 3.2.1 below.

Table 3.2.1 Rationale for Inclusion of University Students and Teachers

 

 

School Advanced Teaching
Target Population Knowledge Knowledge Knowledge
Final year mathematics
majors High High Low
Final year students majoring
in areas related to math High Mixture Low
Final year mathematics
education majors High Mixture Mixture
ln-service high school
mathematics teachers High Mixture High

 

Another aim Of the study was to examine the relationship between in-
service teachers’ knowledge for teaching algebra and the performance of their
students in algebra. Prior to the conduct Of fieldwork for this study, the content of
algebra in both the Core and Elective Mathematics syllabuses was reviewed and
extracted (see Appendix II). It was the initial review that provided the rationale for
deciding that Elective Mathematics students in SSS 3 in Ghana were the right
group Of students who would have had the best opportunity to cover what the

student instrument was testing.

62

3.3 Accessible Population

Because Of financial and other practical constraints, not everyone in the
target populations described in Section 3.2 above could be contacted. A decision
had to be made to narrow the selection to a smaller number of institutions. For
instance, at the time of this study, there were about twenty universities in Ghana.
Of these, six were public institutions and the remaining fifteen were private
institutions- mostly established by various religious missions. However, only
three of the public universities had mathematics departments that offered
degrees in mathematics. These were the University Of Ghana in Accra, Kwame
Nkrumah University of Science and Technology in Kumasi, and University of
Cape Coast in Cape Coast. In addition, only two public universities, University of
Cape Coast and University Of Education, Winneba, Offered degree programs in
mathematics education. Thus, in all, four Of the country’s universities hosted
departments from which final year undergraduate students in mathematics,
mathematics education or a related field could be selected. The accessible
population of university students (earlier referred to as pre-service senior
secondary school mathematics teachers) therefore consisted of students from
the relevant departments in these four universities. Overall, it was estimated that
a minimum of 100 students could be recruited in each of the five aforementioned
departments to participate in the study.

The location of the aforementioned four universities had implications for
the senior secondary schools to be selected for this study. To reduce cost and

facilitate fieldwork, a decision was taken to limit the selection of senior secondary

63

schools to schools located within and around cities in which the aforementioned
universities were located. Thus, the accessible population Of senior secondary
school mathematics teachers and their students were limited those in schools
located at Accra, Kumasi, Cape Coast, and Takoradi. Takoradi was added to the
list Of towns from which to select senior secondary schools because of its
proximity and easy assess from Cape Coast, as well as, the concentration of
schools there. Overall, it was estimated that about 200 in-service teachers and

2000 students could be accessed from the senior secondary schools.

3.4 Selection of Subjects

The original plan was to conduct the study between May 2006 and July
2006. As is the practice in Ghana, selection of subjects began with the seeking of
permission of the heads of departments or institutions. Since the researcher was
located in the US, the beginning of this process of securing such permission was
done with telephone contacts with heads of mathematics and mathematics
education departments in the aforementioned universities, as well as,
headmasters/headmistresses of the target senior secondary schools. During
these initial telephone conversations, the purpose of the study was explained to
them and a time for face-to-face meetings with them was set. The plan was to
get approval from the heads of departments and institutions for the study to be
conducted in their institutions and to allow access to the students and teachers
who would eventually decide to agree to participate or opt out of the study.

These initial telephone discussions were therefore aimed at facilitating

planned visits to each of the research sites originally in May 2006. Unfortunately,

64

approval for the commencement of the study from Michigan State University’s
Social Science, Behavioral Science and Education Institutional Review Board
(SIRB) was delayed due to issues that needed to be Clarified about the initial
application sent to the Review Board. By the time approval was Obtained, senior
secondary school teachers in Ghana were on strike over working conditions.
Consequently, fieldwork for the study was done in November and December

2006.

3.4.1 Selection of the University Students for the Study

In all, 301 students from three departments in three of the target universities
agreed to participate in the study. These students were from the Departments of
Mathematics at University of Ghana and Kwame Nkrumah University of Science
and Technology, and the Department of Mathematics and Statistics at University
of Cape Coast. Also included were students from the Department of Science and
Mathematics Education at University of Cape Coast. Table 3.4.1 shows the
distribution Of the participating university students by their major area of
specialization instead of their universities. This style of presenting this aspect of

data collected has been adopted to maintain confidentiality of participants

Table 3.4.1 Number of Participating University Students by Major Area

 

 

Major Area of Specialization N
Mathematics 132
Statistics 44
Mathematics Education 125

 

65

Selection of participants from these universities started with meetings with
heads of departments who had agreed during earlier phone discussions to permit
the study to be conducted in their institutions. These meetings took place before
the commencement Of the project and gave the heads of departments the
opportunity to further discuss the project and ask questions. At the initial meeting
in each university, the head of department was given two copies Of the consent
forms. One copy was for his/her records and the other copy was signed by
him/her and collected back. The meeting also provided opportunities for a
preliminary meeting with the students who participated in the study.

At the meeting with the students, they were informed about the project and
the voluntary nature of students’ participation was emphasized. In this way, they
were given the opportunity to decide, without any form of coercion, to participate
or not to participate in the study after any questions participants had were
addressed. The meeting also discussed how the privacy of participants was
going to be protected, as well as the risks and benefits to be derived in
participating in the study. In addition, steps taken in the study to protect or
minimize any risks involved in participating in the study were discussed.
Participants were also provided with the researcher’s contact information, as well
as those of relevant personnel at MSU who they could contact should they have
any further questions.

Those who agreed to participate in the project were then given the written
consent forms on the day of administration of the instruments. There were no

criteria to exclude any student in the mathematics and mathematics education

66

departments of any of the participating universities. Instead, any student in these
departments who agreed to participate was given the chance to do so and none

was paid to participate.

3.4.2 Selection Of In-service Teachers for the Study

A similar approach was taken in the senior secondary schools to select in-
senrice teachers who participated in the study. Through earlier telephone calls to
heads of institutions about the study, eight headmasters/headmistresses agreed
to allow the study to be conducted in their schools. These consisted of five
schools in Cape Coast and three in Takoradi. The Cape Coast schools
comprised Mfantsipim School, Wesley Girls’ High School, St. Augustine’s
Secondary School, Adisadel College and Holy Child School. Ghana Secondary
Technical School, Fijai secondary School and Archbishop Porter Girls”
Secondary School were the schools in Takoradi that participated in the study.

On arrival in Ghana, initial visits were made to these eight schools to meet
with headmasters or headmistresses in charge to further discuss the purpose of
the study and arrange to meet with the mathematics teachers. At the meeting
with teachers, the purpose Of the study was explained to them. In addition, steps
taken to protect their privacy were discussed, as well as, the risks and benefits
involved with their participation. Teachers were also given the opportunity to ask
questions and the written consent was shown to them. As was done in the
universities, teachers who agreed to be participants were also provided with
contact information of the researcher, as well as, those of relevant personnel

here at MSU who they could contact should they have any further questions. At

67

the end Of the meeting, a day was agreed upon for administration of the
instrument. Participating teachers only signed consent forms on the day of
administration of the instrument.

With in-service teachers too, there were no plans to exclude any
mathematics teacher in any Of the participating senior secondary schools from
the study. Participation was purely voluntary and the teachers’ instrument was
administered to all teachers who agreed to participate in the study. Unfortunately,
only 38 teachers from the eight schools participated in the study. Many of them
complained that their strike action had caused them to miss too many Class
hours and they needed to make up for the lost hours. As was the case of the
university students, none of the in-service teachers was paid to participate in the
study.

Table 3.4.2 below shows the breakdown of in-service teachers by school
who participated in the study. To ensure confidentiality, participating schools

have been identified by codes not traceable to the schools.

Table 3.4.2 Number of Participating In-service Teachers by School

 

 

School Number of Number of Number Of Classes
Code Classes Participating Teachers whose Teachers
participated
A 3 6 3
B 2 4 0
C 8 13 2
D 8 1 0
E 7 2 0
G 6 6 2
H 7 6 3
J 3 0 0
Total 44 38 10

 

68

As Table 3.4.2 indicates, with the exception of school J, where no in-
service teacher agreed to participate in the study, there was at least one teacher
from each of the participating school in the study. In addition, each of the
participating senior secondary schools had multiple classes per grade level.
Unfortunately, not all the teachers who, at the time of the study, were assigned to
teach the specific participating Classes agreed to participate in the study. Only
four out of the eight participating schools had classes whose assigned teachers
participated in the study. These were schools with codes A, C, G and H. School
A was the only school in which teachers of all the participating classes also
participated in the study. In School C, teachers from only two out of the eight
participating classes took part in the study. In Schools G and H, the number of
participating teachers who were assigned to participating student Classes was
two out of 6 and three out Of seven respectively. In the remaining four schools,
with codes B, D, E and J, none of the teachers who participated in the study was
assigned to any Of the participating student classes.

In the sections that follow, the demographic data of the in-service teachers
who participated in the study are discussed. Table 3.4.3, below, presents the
college level mathematics and mathematics education courses taken by the in-
service teachers who participated in the study.

A cursory look at Table 3.4.3 shows that though, a greater percentage of
teachers whose classes were tested than the entire 38 teachers in the sample
said they had taken each of the courses, the trend of courses by the two groups

was similar.

69

Table 3.4.3 Courses Taken by In-service Teachers In the Sample

 

 

Courses Taken Percent of Percent of Sample Whose
Sample, N = 38 Classes were Tested, N = 10
Calculus 73.7 90
Linear Algebra 71 .1 90
Abstract Algebra 63.2 70
Advanced Geometry / Topology 47.4 50
Real/Complex Analysis 13.2 20
Differential Equations 68.4 80
Number Theory/Discrete Math 36.8 40
Methods of Teaching Math 53.3 70
Psychology of Learning Math 53.3 70
Assessment Of Math Education 47.4 50

 

In addition, of the mathematics and mathematics education courses taken, only
two (i.e., Calculus and Linear Algebra) had been taken by more than 70 percent
of the 38 participating teachers. Among the 10 teachers whose classes were
tested in the study, the number of mathematics courses that had been taken by
70 percent or more increased to six, including two mathematics education
courses. Table 3.4.4 below presents the teaching experience Of the participating

in-service teachers.

Table 3.4.4 Teaching Experience of Participating In-service Teachers

 

 

 

Number Core Mathematics Elective Mathematics
of Years Percent of Percent of Percent Of Percent of
Teaching Sample, N=38 Sample whose Sample, N=38 Sample whose
Classes were Classes were
Tested, N=1O Tested, N=10
0 7.9 20 10.5 —
1-2 15.8 - 15.8 10
3-6 28.9 30 21.1 30
7-10 15.8 20 18.4 30
10+ 18.4 10 21 .1 10
Missing
Cases 13.2 20 13.2 20

 

7O

From Table 3.4.4 it can be seen that only 7.6% and 10.5% respectively of
the participating Core Mathematics and Elective Mathematics teachers indicated
that they were in their first year of teaching at the time of the study. Of the
sample whose classes were tested, 20% of the Core Mathematics teachers (i.e.,
2 out of 10) were in their first year Of teaching. None of the Elective Mathematics
teachers whose classes were tested was in the first year of their teaching
profession.

Also, about 63 and 60 percent respectively of the teachers in the sample
and those whose classes were tested had taught Core Mathematics for a
minimum Of 3 years. For Elective Mathematics, these percentages changed
slightly to 60 and 70 respectively.

The next table, Table 3.4.5, presents the nature of teaching certification

held by the participating in-service teachers.

Table 3.4.5 Teaching Certificates of In-service Teachers in the Sample

 

 

Type of Certification Percent of Percent Of Sample
Sample, Whose Classes were
N = 38 Tested, N = 10

Bachelor of Education (Math) 50.0 40

Diploma in Education 10.5 20

Post Graduate Diploma/Certificate in 7.9 10
Education

Other 1 0.5 10

Missing Cases 21.1 20

 

A quick look at Table 3.4.5 shows that the largest proportion of in-service
teachers in the sample, as well as those whose classes were tested had

Bachelor of Education (Math) certificates. The proportions were 50 and 40

71

percent respectively. As already explained in section 1.5, these were teachers
who undertook degree programs in mathematics education and graduated not
earlier than 1991. At the time of the study, only two universities in Ghana, the
University Of Cape Coast and University of Education, Winneba offered such
degree programs.

Only 10.5 % of the teachers in the sample and 20% of those whose
classes were tested indicated that they had Diploma in Education certifications.
These were teachers who graduated from the University of Cape Coast not later
than 1991. They pursued four-year Bachelor of Science degrees in mathematics
or a related field and while on that program spent the second to the fourth year
pursuing the education diploma alongside. This arrangement was mandatory for
every student at the University of Cape Coast those years and UCC was the only
university that offered such programs. Finally, the 7.9% and 10% respectively of
the sample and those whose classes were tested had obtained Post Graduate
Diploma/Certificate in Education. At the time of the study, this program was run
on sandwich basis only at the University of Cape Coast during the university’s
regular summer break. It was Opened to teachers who enter the teaching field

with bachelor’s degrees in mathematics or a related field other than education.

3.4.3 Selection of Senior Secondary School Students for the Study

The process of selecting the participating senior secondary school seniors
from the aforementioned eight senior secondary schools, however, followed a
slightly different approach. This was because in the senior secondary schools the

majority of the students were between age seventeen and eighteen.

72

Consequently, the consent process needed to be different since parental consent
was also needed for the students.

The senior secondary schools who agreed to participate in the study were
boarding schools. This system made it possible for parents across the country to
send their children or wards to those schools. Consequently, it was difficult to get
access to each parent for his or her consent at the time Of the study. Permission
was therefore sought from the Institutional Review Board at MSU to use parental
notification in place of the required parental consent. This was done by mailing
written parental notification of the study to the parents of all final year Elective
Mathematics students through the heads of the participating institutions. This
was done about two months prior to the conduct of the study to make it possible
for parents to discuss participation in the study with their wards or children ahead
Of the study. In Spite Of this parental notification, a meeting was conducted with
the students during the initial visit to each school. At this meeting, the purpose of
the study was explained to the students and questions they had were addressed.
In addition, steps taken to protect their privacy were discussed, as well as the
risks and benefits involved with their participation. Students who agreed to be
participants were also provided with contact information of the researcher, as
well as those Of relevant personnel at MSU who they could contact should they
have any further questions. Students who agreed to participate were given the
opportunity to provide a signed assent of their agreement to participate in the

study on the day Of administration Of the instrument.

73

In addition, there were no criteria to exclude any final year elective
mathematics student in any of the participating senior secondary schools. The
student instrument was administered to all students who agreed to participate in
the study. In all, 1565 students from the eight senior secondary schools
participated in the study.

The break down of the participating senior secondary school students in
the eight schools by class is shown in Table 3.4.6 below. In Table 3.4.6, the
participating senior secondary schools are identified by codes consistent with
those used earlier in Table 3.4.2 above for the in-service teachers.

As shown in Table 3.4.6, above each of the participating schools had
multiple Classes per grade level. In all, the participating students came from 42
classes across eight schools. However, as explained in the previous section,
teachers assigned to only ten out of these forty-two classes participated in the

study.

74

Table 3.4.6 Participating High School Students by School and Class

 

 

School Class Number Of students Subtotal

A1 48

A A2 45 134
A3 41
B1 33

B B2 42 75
C1 19
CZ 26
C3 08

C C4 28
CS 25 207
C6 42
C7 29
C8 30
D1 23
D2 37
D3 24
D4 25

D D5 32 218
D6 42
D7 35
E1 31
E2 52
E3 42

E E4 35
E5 75 322
E6 87
G1 46
62 42
G3 45

G G4 17 230
GS 35
G6 45
H1 39
H2 52

H H3 54
H4 13 280
H5 24
H6 49
H7 49
J1 41

J J2 17 99
J3 40

Total 1565 1 565

 

75

The breakdown of students in each of the Classes whose teachers took part Of

the study is shown in Table 3.4.7 below.

Table 3.4.7 Classes with Teachers Who Participated in the Study

 

School Code Class Code Number of Students Sub-total

 

A A1 48
A2 45 134
A3 41
C C1 19 45
CZ 26
G G1 46 88
G2 42
H H1 39 140
H2 52
H7 49
Total 407 407

 

3.5 Instrumentation

The study involved administering mathematics assessment instruments
adapted from the Knowledge Of Algebra for Teaching (KAT) project at Michigan
State University (MSU) to participants of the study. The adaptations involved
Changing the contexts and wording of questions in the KAT instrument to reflect
Ghanaian contexts. For example an item that originally read,
“At a storewide sale, shirts cost $8 each and pants cost $12 each. If S is the
number of shirts and P is the number of pants bought, which of the following
is a meaning for the expression 8S + 12P?”

was adapted into,
“At a storewide sale, shirts cost ¢80000 each and a pair of trousers cost
¢120000 each. If S is the number of shirts and P is the number of trousers

bought, which of the following is a meaning for the expression 80000S +
120000P?"

76

In this way, not only was the US currency changed into the Ghanaian currency,
the prices of the items were also changed to reflect market values in Ghana at
the time Of the study. In addition, variations in names commonly used for the
commodities used in the item were also Changed to reflect the right contexts in
Ghana. For example, in the adaptation made in the item used above, “pants” was
changed into “trousers” as is commonly called in Ghana.

Similar changes were made in the survey questions to reflect Ghanaian
contexts. For example, one Of the original survey questions in the KAT
instrument asked teachers to select algebra courses they have taught in the past
as shown below,

Which of the following algebra courses have you taught in the last five

years? Check all that apply.

El Pre-algebra

Cl Remedial algebra

II] First year algebra

El Second year algebra

El Advanced algebra

a Algebra in an integrated program
El

El

Other (please specify)
l have never taught any kind of algebra courses

 

This question was changed to focus on the type Of mathematics (core or elective)
and the grade level in the following way,

Which Of the following algebra courses have you taught in the last five
years? Check all that apply.

El Core Mathematics in SSS 1

El Core Mathematics in SSS 2

Cl Core Mathematics in SSS 3

Cl Elective Mathematics in SSS 1

Cl Elective Mathematics in SSS 2

Cl Elective Mathematics in SSS 3

El Other (please specify)

 

77

The process of adapting the items to suit Ghanaian contexts occurred at
two levels. First, the researcher used his experience of the Ghanaian school and
examination system to change the wording and contexts Of the original KAT
instrument appropriately. Next, the resultant versions of the instrument were
given to two mathematics education professors in Ghana to look at and further
modify appropriately, before they were administered. In all, only two content
items on Form 1 were in contexts that were not applicable in Ghana. These two
items were the only ones whose contexts were modified into Ghanaian contexts.
One of the items was the example given in the preceding paragraph, in which the
word “pants” was changed into “trousers” and the prices changed from dollars
into the then Ghanaian currency. The other item only had the following phrase in
it, “In a first year algebra class” modified into “In a first year elective mathematics
class”. In this way, the mathematics needed to answer the adapted questions, as
well as their levels of difficulty were kept the same as they were in the original
KAT instrument. The other aspects of that item were maintained. None of the
items on Form was based on contexts that were not applicable to Ghanaian
contexts. As a result, no item on Form 2 was modified. Thus, the instruments
administered in this study were comparable with the instrument administered in
the US by the KAT project.

Two main types of assessment instruments were administered. One of
these was administered to the participating university students and in-service

teachers. There were two main versions of this instrument; Form 1 and Form 2.

78

The other instrument was administered to final year senior secondary school
students taking elective mathematics. Each instrument consisted of two sections:
- a first section of survey questions and a second section of content items to be
answered by participants. None of the instruments required the use of any
identifiers like names, gender, school, identity numbers or anything that could be
traced to participants.

The first section of the instrument administered to the participating
university students and in-service mathematics teachers contained ten survey
questions about participants’ prior university coursework, their major and minor
areas of specialization and their years of experience as teachers. The second
section of each version of the instrument contained seventeen multiple-choice
and three open-ended content items. These items were based on the algebra in
the senior secondary school syllabus (i.e., school knowledge items), related
advanced mathematics items, and items based on the tasks of teaching (i.e.,
teaching knowledge items).

Table 3.5.1 below presents the complete categorizations and distribution
of the items among the three hypothesized types of knowledge in the framework
that guided the study. In the categorizations of items used, the first number
represented the type of knowledge in the domain of algebra knowledge for
teaching (see the horizontal or x-axis of Figure 2.5.1 ). Among these first
numbers, X = 1 represents “schools knowledge”, X = 2 represents” advanced
knowledge”, while X = 3 represents “teaching knowledge”. The second and third

numbers represents the domains on y-axis (algebra domain) and the z-axis

79

(domains of mathematical knowledge) respectively. Y = 1 represents content in
“expressions, equations and inequalities” and Y = 2 is an indication that the
content is item based on “functions. For the third number in the categorization, Z
= 1, 2, 3, and 4 respectively indicates that the item was categorized to be based
on core concepts and procedures, representations, applications, and reasoning
and proof (see Fig 3 for these domains). For instance, the first item on Form 1
was categorization (1 2 3). This means that it was a school knowledge item

based on function and their applications.

Table 3.5.1 Categorization of Items on Forms 1 and Form 2

 

 

 

 

Fonn‘l Fonni!
Item # Categorization Item # Categorization
1 123 1 112
2 321 2 321
3 111 3 113
4 211 4 312
5 321 5 212
6 122 6 121
7 312 7 221
8 213 8 322
9 221 9 113
10 314 10 311
11 313 11 321
12 224 12 122
13* 224 13* 224
14* 121 14* 121
15* 324 15* 324
16* 211 16* 211
17 112 17 211
18* 311 18* 311
19 121 19 124
20 224 20 214

 

* Items in both forms

80

As shown in Table 3.5.1 above, on Form 1, six of the 20 content items
were classified as assessing “school knowledge”. The remaining 14 items were
split exactly between “advanced knowledge” and “teaching knowledge”. On Form
2, there were seven “school knowledge” items, six “advanced knowledge” and
seven “teaching knowledge” items. In addition, between the two versions of this
instrument were five common items (four multiple choice and one open-ended).
Each Of these versions of the instrument was completed by participants in no
more than 60 minutes.

The first section of the instrument administered to the participating senior
secondary students was a four-item questionnaire about participants’
parents/guardians’ educational background, and the extra resources (both
human and material) available to them for Ieaming their school mathematics. The
second section of this instrument consisted of 12 Of the 13 school knowledge
items on the teachers’/university students’ instrument. These comprised ten
multiple-choice and two Open-ended content items based on the algebra in the
secondary school curriculum. The secondary students’ instrument was also
completed in no more than 60 minutes.

Samples of the content items on these instruments are shown in the
“Public Released Items of the KAT Project” in Appendix IX. None of the actual
instruments (i.e., Form 1, Form 2, or the high School students’ Form) could be
displayed in the appendix. The KAT project that owns the copyright was still in
progress at the time of the writing of this study and so their items had not been

made public.

81

Furthermore, it could not be guaranteed that by the time of the study each
participating senior secondary school would have covered all the content being
examined on the student instrument. As a result, an Opportunity to Learn Form
(see Appendix X for a sample) was also created to give teachers of the
participating classes the chance to indicate the extent to which students in their
classes have had the Opportunity to cover the mathematics required to answer
each of the items on the student instrument. Using the KAT project public
released items, an example of the opportunity to Ieam questions that followed a
content item is as follows;

3. A stgd7e5nt solved the equation, 3(n - 7) = 4 — n and obtained the solution

n = . .

What might the student have done wrong?

Your students’ opportunity to learn the mathematics in the question
above:

Please, indicate whether or not your students had the Opportunity to learn the
mathematics needed to answer the question above.

A. Yes, I taught it because it is part of the required curriculum

8 Yes, I taught it, even though it is not part of the required curriculum
C. No, it is not part of the required curriculum so I did not teach it

D

No, even though it was part of the required curriculum, I did not/have not
taughtfi.

The actual Opportunity to Learn form used in the instruments was based

on the instrument adapted for study. However, because the KAT project was still

82

in progress at the time of the writing of this dissertation and the actual items had

not been made public the form could not be displayed.

3.6 Procedure

Between April 24 and May 5, 2006, telephone contacts were made with
the universities that participated in the study as well as with heads of senior
secondary schools in Accra, Kumasi, Cape Coast and Takoradi. During these
discussions, the rationale Of the study was discussed and verbal approval
Obtained to use the institutions as sites for the study. However, it was not until
August 17, 2006, after approval for the study was obtained from Institutional
Review Board (IRB) at MSU, that official invitation letters and parental notification
for the senior secondary schools were mailed. The invitation letters were
addressed to the heads of departments in the universities that had agreed to
participate, the headmasters and headmistresses Of the eight senior secondary
schools as well as the heads of the mathematics departments in these secondary
schools. Unfortunately, no follow-up for actual fieldwork could be done because
teachers in senior secondary schools in Ghana were on strike by the time the
IRB approval was Obtained. However, continued telephone calls were made to
keep options for the study Open until the strike action was over on November 8,
2006 following a court order on October 31, 2006. The period between October
31, 2006 and November 10, 2006 was used to complete flight arrangements (i.e.,
renegotiating for ticket and transit visa issuance) for the study. In addition,
between October 31, 2006 and November 10, 2006, final phone calls were made

to heads of the participating departments and secondary schools. This provided

83

opportunities for a timeline to be agreed upon for initial meetings and data
collection.

Following these developments, the study was conducted towards the end
of the fall semester of the 2006/2007 academic year. Fieldwork for this study
began on November 13, 2006. The first two weeks, November 13 to 24, were
spent visiting with heads of the participating departments in the three universities
and the eight participating senior secondary schools. As already discussed,
these initial meetings were aimed at further discussing the purpose of the study
and agreeing on dates and times for administration Of the instruments. At these
meetings, each head Of department or institution was given two copies Of the
consent forms. One copy of this form was signed by the head and collected while
the other was meant for their records. In addition, each headmaster or
headmistress of the participating senior secondary school was given the chance
to complete the headmasters/headmistress’ questionnaire during these initial
meetings. The meeting also provided opportunities for preliminary meetings with
the teachers and students who participated in the study. In addition, on
November 13, 2006, the adapted instruments for the study were given to two
Ghanaian experts (i.e., mathematics education professors) for review. By Friday,
November 24, 2006, this review was completed and meetings had been held with
each of the experts to discuss their points Of view. It also made it possible for
their suggestions to be incorporated during the weekend and for the instruments

to be printed.

84

As already discussed, this study involved administering the instrument
adapted from the KAT project to final year mathematics and mathematics
education students in three universities, as well as teachers and final year
elective mathematics students in eight senior secondary schools in Ghana.

Administration Of the instruments commenced from the senior secondary
schools. Data collection was done in the 3 participating schools in Takoradi on
November 27 and 28 2006. The five schools in Cape Coast were done from
November 29 to December 1, 2006. In each school, the student instrument was
administered after the normal school hours so as not to disrupt classes. This was
also possible because of the boarding facilities available in the schools. In
Ghana, senior secondary schools usually have big halls where the entire school
usually meets for morning devotion, entertainment and other whole school
programs. These same rooms are usually converted into examination centers for
the conduct Of national examinations for the final year students each year. It was
originally envisaged that all the students who agreed to participate in the study
could be brought together in each of these rooms in the various schools so the
instrument could be administered in one room. Unfortunately, it became clear
during the initial meetings with the headmasters and headmistresses that there
was insufficient time to get tables and chairs arranged in these rooms and make
them usable as was envisaged. Consequently, participating students in each
school were brought together in their own classrooms where they were given 60
minutes to complete the instruments. To make this possible, provision was made

to travel with six graduate students from the University of Cape Coast (UCC) to

85

help supervise the students. In addition, one lecturer from UCC, who is a friend,
also traveled with us to help with supervision of students during the
administration of the instruments. In the same way, the in-service teachers in
each Of the schools who agreed to participate were brought together to complete
the instruments at a sitting lasting no more than 60 minutes. The Class teachers
completed the OTL forms during the time when the sessions with students were
in progress.

Administration Of the instruments in the universities was done in a slightly
different manner. The universities were writing their end-Of-semester
examinations in December. These examinations notwithstanding, the heads of
the participating departments had agreed with the participating students to
incorporate the administration of the instruments with the examinations.
Consequently, opportunity was provided during the examination weeks, from
December 4 to December 8, 2006 for students who agreed to participate to come
together to complete the instruments. This made it possible for participants in
each department of the three universities to complete the instrument at a sitting
lasting no more than 60 minutes. During the administration of the instruments,
available forms, Form 1 and Form 2, were distributed in such a way that as
students in a particular row completed Form 1, those in the next adjacent row
completed Form 2. This was done for all the university students except with the
statistics majors who were all given Form 1 because of their relatively small
number. At the University of Ghana, the instrument was completed on December

4, 2006 and at Kwame Nkrumah University of Science and Technology on

86

December 6, 2006. At the University of Cape Coast it was completed on
December 7, and 8 respectively by mathematics (and statistics) majors and their

mathematics education counterparts.

3.7 Scoring of Content Items

For the content items on both the high school students’ form and the
teachers’ forms, responses to the Open-ended items were scored on a four-point
scale. The exact scoring rubrics used could not be presented in the appendix
because the KAT project, whose instruments was used in the study was still in
progress at the time of the writing of this dissertation and the actual items had not
been made public. The following, however describe the general spirit of the
rubrics used for scoring the open-ended items:

Score of 4: All steps of the solution have carefully been laid. There does

not have tO be a reason for each step but each step follows
reasonably from the one before. The solution can be shown as

a model solution to any audience.

Score of 3: All steps Of the solution have carefully been laid but there are
minor errors.

Score of 2: There is an evidence Of a chain of reasoning but some major
conceptual mistake was made or there is an evidence of a
chain of reasoning but the solution is not complete.

Score of 1: There is at least one correct statement.

Score Of 0: Something mathematical is said but is not valuable for the
question.

Score 777: Nothing mathematical is said (e.g. “no clue , I don’t Know”)

Score 999: Blank

87

For each question, specific explanations have been given to make the
scoring less difficult. For instance, the difference between what constitutes a
minor error and a major conceptual error is made for each question. In addition,
the KAT project had anchor papers for each score that could be used to resolve
issues of disagreement.

To ensure reliable scoring, three doctoral students who have worked as
graduate assistants on the KAT project, and were familiar with the rubrics used
provided a second scoring independent Of the scoring of the researcher. Papers
on which there were variations in scoring were then identified and meetings
arranged to discuss the differences and settle on agreed scores. Second opinion
on the score for teachers’ Form 1 and Form 2 were given by two different
graduate assistants. Tables 3.7.1, below, summarizes the agreements among
the scores. As shown in Table 3.7.1, below the agreement in scoring the two
teachers’ forms was more that 96% for each of the items. Similar high
percentages in agreement were obtained in the scoring of the senior secondary

school students’ form.

Table 3.7.1 Agreements in Scoring Open-ended Items on Forms 1 8. 2

 

 

 

 

Outcome of Form 1 Form 2

Scoring Item 18 Item 19 Item 20 Item 18 Item 19 Item 20
Number 150 150 150 189 189 189
Scored

Number of 150 147 145 186 188 185
Agreements

Percentage 100 98 96.67 98.41 99.47 97.88
Agreement

 

88

As shown in Table 3.7.2 below, the percentage agreement between the
scores given to the two Open-ended items on the student form by two scorers

was between 98% and slightly over 90%.

Table 3.7.2 Agreements in Scoring the SSS students’ Open-ended Items

 

 

Outcome of Scoring Item 11 Item 12
Number scored 1565 1565
Number of agreements 1533 1540
Percentage agreement 99.04 98.85

 

For the purposes of analyses, the four-point scale of the open-ended
items was reduced to a two-point scale by dividing each open-ended score by
two. This was done to avoid putting tOO much weight on the open-ended items
since responses to the multiple-choice items were scored as zero (0) for wrong

responses and one (1) for right responses.

3.8 Mode of Data Analysis

Data for this study is both qualitative and quantitative in nature. The
qualitative data comes from participants’ responses to the demographic survey
questions while the quantitative data comes from participants’ responses to
content items on the instruments used. The responses to the demographic
survey were intended to be used to analyze the background of participants. In
the case of the in-service teachers, the plan was to use these data to distribute
them into the clusters shown in Table 1.5.1 using factor analysis performed on
the data. Unfortunately, since very few in-service teachers participated in the

study, this initial factor analyses was not performed.

89

Since the purpose of the analysis Of data collected in the study is to help

answer the research question, data analysis was done by research question.

3. 8.1 Analysis of Data Related to the First Research Question

The first research question that guided this study was,

“To what extent does Ghanaian pre-service and in-service secondary

mathematics teachers’ knowledge for teaching algebra corroborate the

three categories of knowledge hypothesized in the KAT framework?”
To answer this question data from the university students and in-service teachers
were used. Factor analysis was performed on each item in the instrument using
the full set of data across both groups. The factor loading for each item was then
analyzed to determine whether conclusions could be based on three factors. The
decision to look for three factors is because the theoretical framework used for
this study had Lug conceptualizations of teachers’ knowledge: school
knowledge, advanced knowledge and teaching knowledge. In addition, the
nature Of the loading of the items was also examined to confirm the extent to
which items originally categorized as assessing the same type of knowledge
loaded-together on separate factors. This was done to determine whether the
three types Of knowledge hypothesized by the KAT project came out distinctively
and whether a differentiation could be made among the three different types of

knowledge.

3. 8.2 Analysis of Data Related to the Second Research Question

The second research question was,

90

“How does the knowledge for teaching algebra differ among the different

categories Of secondary school mathematics teachers and potential

teachers in Ghana?”
Data for this research question came from the scores obtained by in-service
teachers and university students on each Of Forms 1 and Form 2. The university
students who participated in the study comprised students majoring in three
different areas; mathematics, mathematic education and statistics. Originally, two
types of analyses were envisaged to answer this research question. The first
involved investigating the difference in knowledge among the different categories
of in-service teachers discussed in Section 1.5 (see Table 1.5.1) as well as how
differences in their teaching experience affect their knowledge. The second
involved comparing the four broad groups of teachers (in-service teachers, final
year university students majoring in mathematics, mathematics education and
statistics). However, because of the small number of in-service teachers who
participated in the study the first type of analysis could not be performed. As a
result, Analysis Of Variance (ANOVA) was performed on the sub-total scores on
the school knowledge items, advanced knowledge items and teaching knowledge
items by in-service teachers and the various groups of prospective teachers (i.e.,
final year university students majoring in mathematics, mathematics education
and statistics) on Forms 1 and 2 together. This was useful in establishing how
the three types of knowledge (school knowledge, advanced knowledge and
teaching knowledge) differed among the different categories Of secondary

mathematics teachers and potential teachers who participated in the study.

91

The outcomes from the analyses have been presented in Chapter 4.

3. 8.3 Analysis of Data Related to the Third Research Question

The third question that guided this study was,
‘What is the relationship between the performance of in-service secondary
school mathematics teachers in Ghana and the students of their classes?”
Data from the participating senior secondary school students and their
corresponding in-service teachers were be used to analyze this question. Linear-
regression was the main analyses done to answer this research question. The
rational for this study is provided in the Section 4.4 (the rationale for using linear

regression is discussed in this section).

3.8.4 Data not Used in the Analyses

Because of the small number of participants, especially the in-service
teachers (38 in all and 10 for the high school classes used) and the statistics
majors (44 in all), the demographic data of the university students and in-service
teachers could not be used as effectively as planned in the analyses. Those data
could, for instance, have been useful in dividing the in-service teachers into
groups depending on the number of years they have taught, the mathematics
courses and grade levels they have taught, and their teachers’ certification. It
could also have been used to group the entire data into the number of algebra
courses taken. Analysis of Variance could have been performed on the

subgroups that could have been generated to answer aspects of the research

92

question. However, the number of participants in these subgroups would have
been too small for any meaningful analysis.

Another data type that was not used was data generated using the
Opportunity to Learn (OTL) forms. In all, 19 teachers completed these OTL
forms. Only six of these were for the ten classes with teachers who participated
in the study. Using this data therefore would have further reduced the number of
classes used for the linear regression in Section 4.4.3 (i.e., Chapter 4, Section

3.2). A decision was therefore taken not to use these data.

93

CHAPTER 4: ANALYSIS AND RESULTS

In this chapter, the analyses of data and the results are presented. As
already discussed in chapter three, the present study adapted the instruments
generated form the KAT project for use in Ghana. The adaptation maintained the
mathematics content of the items but only changed item contexts from the US
contexts to Ghanaian contexts. This chapter has, therefore, been organized to
Open with a discussion of some of the item statistics, difficulty levels Of the items,
the point-biserial coefficients, and instrument reliability, generated from this
study. However, instead of discussing these item statistics in isolation, a
comparison is made with data collected from the KAT project as at the time of the
study. After that, the analyses relevant to the research questions that guided this
study have been presented. These analyses have been linked to the research
questions in succession. In this way, beginning with the first research question
that guided the study, data, analyses and results related to a specific research

question are presented before attention is directed to the next research question.

4.1 Comparing Data from This Study and the KAT Validation Study

4. 1. 1 Item Difficulty Levels and Point-biserial Correlations

In this section, the present study has been referred to as the Ghana study.
The difficulty levels presented in this section are the proportion of correct
responses obtained from participants in the two studies (Ghana and KAT)
respectively. As already explained in Chapter 3, multiple-choice items (items 1 to

17) on each Of the two forms were given a score of 1 or 0 for right or wrong

94

responses respectively. The Open-ended items (items 18 to 20) were scored on a
4-point scale (i.e., scores from 0 to 4). The score Obtained were later rescaled to
a 2-point scale (i.e., scores from 0 to 2). On each Of the two forms, the difficulty
levels and point-biserial coefficients presented were obtained by further rescaling
the open-ended scores to 0 to 1.

The item difficulty levels presented in this section are the proportions of
responses to the item that were correct. In this way, a low proportion for an item
is indicative of the item being more difficult than another item with a higher
proportion.

The point-biserial correlation coefficients presented in this section are the
item-total correlations between performance Of participants on the individual
items and their performance on the entire instrument administered in the study.
In other words, the point biserial coefficients reflect how well items were
“discriminating” between low- and high-performing participants. Ideally, high
point-biserial coefficient Of an item means that participants providing correct
responses to that item are participants who obtained high total scores and low
coefficients would mean that there is no correlation between score on an item
and score on the test.

As discussed in Chapter 3, 449 participants in the KAT project and 150
from Ghana completed Form 1 while Form 2 was completed by 392 and 189
participants respectively in the KAT and Ghana studies. Table 4.1.1 presents
results of a paired-sample t-test performed to compare the difficulty level of items

on Forms 1 and 2 to the two samples.

95

Table 4.1.1 T-Test of the Difference In Difficulty of the Two Forms

 

Mean Difficulty Variance Pearson
Form Ghana KAT Ghana KAT Correlation df t-stat p-values

1 .4086 .5509 .0778 .0325 .7897 19 -3.62 .0018“
2 .4022 .5450 .0681 .0316 .4867 19 -2.71 .0138“

** Significant difference in difficulty level

From Table 4.1.1 each of the two forms was significantly more difficult to the
sample from Ghana than the KAT sample in the US. The means and variances of
the two samples on Forms 1 and 2 indicate that the items were more difficult to
the sample who participated in this study (i.e., the sample from Ghana) than their
US counterparts. In addition, the correlation coefficients computed for the two
forms (see Table 4.1.1) reveals that the difficulty levels of the items on Form 1 to
the Ghanaian sample were more strongly correlated to the levels on the US
sample than the items on Form 2.

TO examine the difficulty of each of the individual items, Table 4.1.28 was
used. Table 4.1.2 presents the difficulty levels of items on Form 1 from the
Ghana study and from the KAT project. A cursory look at this table reveals that,
in general, majority of the items on Form 1 were more difficult for participants in
the Ghana study than participants in the KAT study. Only six out of twenty items
appeared slightly less difficult to the Ghana participants than the KAT
participants. These were items 1, 2, 3, 11, 15 and 17. On Form 1, item 3, with
difficulty .889 for KAT participants and .933 for the Ghana participants was the

least difficult to participants in both studies.

 

8 Values of difficulty levels, point-biserial coefficients and Cronbach’s alpha of the KAT project
were taken from the KAT project’s draft summer 2007 technical report of their validation study.

96

Table 4.1.2 Difficulty Level and Rank of Items on Forms 1 and 2

 

 

 

 

Item Ghana Study KAT Study
Form ID Difficulty Rank Difficulty Rank
1 .880 3 .771 2
2 .653 5 .760 3
3 .933 1 .889 1
4 .147 17 .314 20
5 .347 10 .394 14
6 .247 14 .510 1 1
7 .487 6 .731 6
8 .060 20 .356 18
9 .340 1 1 .379 15
1 10 .427 8 .751 4
1 1 .753 4 .708 7
12 .200 16 .368 12
13 .130 18 .560 9
14 .201 15 .584 8
15 .457 7 .339 19
16 .298 12 .546 10
17 .913 2 .748 5
18** .270 13 .482 12
19*“ .362 9 .451 13
20** .067 19 .377 16
1 .989 1 .684 5
2 .979 2 .620 8
3 .328 9 .625 7
4 .296 12 .635 6
5 .169 18 .235 20
6 .640 5 .758 3
7 .222 15 .490 14
8 .275 13 .513 13
9 .730 3 .847 1
2 10 .682 4 .745 4
1 1 .217 16 .783 2
12 .164 19 .306 18
13 .130 20 .560 10
14 .201 17 .584 9
15 .457 6 .339 16
16 .298 1 1 .546 1 1
17 .381 8 .281 19
18** .270 14 .482 15
19** .314 7 .529 12
20** .302 10 .317 17

 

** Open-ended items

97

In addition, four of the top five least difficult items to participants in the
KAT study (i.e., items 3, 17, 1, and 2), were also in the top five least difficult
items in this study. However, while item 4 (with difficulty .314) was the most
difficult item for participants in the KAT study, the most difficult item for the
Ghana study was item 8 (with difficulty .060). Also, four Of the five most difficult
items to participants in the KAT study (i.e., items 20, 12, 8, and 4), were also
among the five most difficult items in the Ghana study.

Also, Table 4.1.2 reveals that, the items on Form 2 were more difficult to
the Ghanaian sample than the US On Form 2, as'with Form 1, the Ghana
participants performed better than the KAT participants in the only six out of
twenty items. These were items 1, 2, 6, 10, 15 and 17. This is further discussed
in Chapter 5. In addition, item 9, with difficulty .847, and item 1, with difficulty level
of .989 were the least difficult items for KAT participants and Ghana participants
respectively. In addition, as with Form 1, four of the top five least difficult items to
participants in the KAT study (i.e., items 1, 6, 9, and 10), were also in the top five
least difficult items in this study. However, while item 5 (with difficulty .235) was
the most difficult item for participants in the KAT study, the most difficult item for
the Ghana study was item 13 (with difficulty .130). Also, unlike the trend in Form
1, only two of the five most difficult items to participants in the KAT study (i.e.,
items 5 and 12), were also among the five most difficult items in the Ghana

study. These are further discussed in chapter 5.

98

Another statistic of importance is the point-biserial coefficients. Table 4.1.3
presents the point-biserial coefficients of items on the two forms from the Ghana

study and from the KAT project.

Table 4.1.3 Point-biserial Coefficients of Items on Forms 1 and 2

 

 

 

Item Form 1 Form 2
ID Ghana KAT Ghana KAT
1 .083 .315 -.024 .421
2 .044 .397 .062 .330
3 .154 .341 .221 .389
4 .145 .234 -.062 .366
5 .149 .312 .244 .439
6 .013 .376 .139 .415
7 .183 .435 .385 .426
8 .173 .468 .206 .288
9 -.003 .313 .151 .195
10 .174 .378 .313 .229
11 .284 .285 .216 .363
12 -.002 .310 .410 .439
13 .155 .546 .242 .510
14 .070 .328 .278 .298
15 -.002 .254 .1 1 1 .342
16 .272 .596 -.131 .565
17 .286 .534 .129 .445
18** .426 .610 .442 .577
19** .392 .684 .384 .628
20** .285 .61 1 .499 .603

 

** Open-ended items

A cursory look at Table 4.1.3 reveals that, compared with data from the
KAT study, the point-biserial coefficients obtained in this study were smaller for
the items on both forms. To determine whether the average discriminations of the
two samples on the two forms were significantly different or not, a paired-

samples t-test was conducted.

99

The output of this t-test is presented in Table 4.1.4. Table 4.1.4 reveals
that there is a significant difference in the point-biserial coefficients Of the two

samples on each Of Forms 1 and 2.

Table 4.1.4 T-Test of the Difference in the Point-biserial Coefficients

 

Mean of Variance of Pearson
Form Discriminations Discriminations Correlation df t-stat p-values
Ghana KAT Ghana KAT
1 .1641 .4164 .0165 .0184 .7406 19 -11.62 .000**

2 .2108 .4134 .0288 .0143 .2683 19 -5.05 .000**
** Significant difference in difficulty level
From Table 4.1.4 the coefficients obtained from the sample in Ghana are
statistically smaller than those obtained from the US. sample on both forms. It
was therefore concluded that the items on the two instruments could, worked

significantly better in discriminating between high and low performing candidates

in the KAT project’s sample than their counterparts in Ghana sample.

4.1.2. Reliability for the Ghana Study and the KAT Study

The following table, Table 4.1.5, presents a comparison of the internal
inconsistencies of the data from this study and the KAT Validation study
conducted at the time of this study as calculated using Cronbach's Alpha. In
Table 4.1.5, the internal consistencies of the total scores on each of Form 1 and

Form 2 are presented.

100

Table 4.1.5 Reliability for Form 1 and Form 29

 

 

 

Form Cronbach's Alpha Mean Score Std. Deviation
Ghana KAT Ghana KAT Ghana KAT
1 .521 .837 0.4086 0.5509 0.2789 0.1802
2 .643 .842 0.4022 0.5440 0.2610 0.1777

 

Since Forms 1 and 2 are instruments not only with the same number of
items but also designed to measure the same construct, Cronbach's alpha can
be interpreted as the correlation between scores obtained on the two forms. In
this way, instrument reliability as measured by Cronbach's alpha can be seen as
a measure or form of the inter-item correlation or an estimate of the proportion of
variance that is consistent in the set scores on each Of Forms 1 and 2. As a
result, a high reliability would mean that the items on the instrument are
measuring the same underlying construct well. Sometimes Cronbach’s alpha is
reduced because of a small standard deviation. As a result, in comparing the
Cronbach’s alpha obtained in this study (i.e., using the sample from Ghana) to
value obtained from the KAT project’s sample, it is necessary to also look at the
standard deviations from the two samples. From Table 4.1.5, the Cronbach’s
alpha obtained on each of the forms from this study was smaller than the
corresponding values obtained from KAT’s sample at the time of this study.
Compared with the KAT project, the small reliabilities values Obtained in this
project is worthy of note because of the higher standard deviations obtained from

this study on each of the two forms.

 

9 Scores for the open-ended items range from 0 to 2 while those for the multiple choice items
range from 0 to 1.

101

According to Nunnally and Bernstein (1994), though a reliability coefficient
of .70 is sufficient in exploratory research, a value of .80 or higher should be
required in basic research. Applying this cut-Off criterion, it is clear that the
coefficients Obtained in this study were too small (see Table 4.1.5 above). In
other words, whereas the instruments performed quite well in terms of reliability
in measuring the underlying construct of knowledge for teaching algebra among
the US participants (refer to coefficients Obtained from the KAT data in Table
4.1.5 above), in this study conducted in Ghana, they did not perform well in
measuring this construct.

Another interpretation of the reliability coefficient is to consider the nature
of the construct being measured by the instrument. In general, if the construct
being measured (in this case, knowledge for teaching high school algebra) is
multi-dimensional in nature, reliabilities measured by Cronbach's alpha could be
low. From this perspective, the low reliability coefficients Obtained in the present
study could be an indication that the instrument could be measuring a multi-
dimensional type of knowledge. When this happens, a factor analysis could
reveal the items which load strongest on which dimensions or factors. This
provides one Of the bases for the factor analysis presented in the next section to

answer research question one.

4.2 Research Question One

The first research question that guided this study was,

102

“To what extent does Ghanaian pre-service and in-service secondary

mathematics teachers’ knowledge for teaching algebra corroborate the

three categories Of knowledge hypothesized in the KAT framework?”
To answer this question, data from the university students and in-service
teachers were used. Altogether, 150 and 189 participants respectively completed
Form 1 and Form 2 (refer to Table 4.3.1 for descriptive statistics on each form).
Exploratory factor analysis was performed on the data for each instrument
separately (i.e., Form 1 and Form 2). Factor analysis was chosen because it
helps, among other things, to examine the number of variables, called factors,
which could be used to either completely or to a large extent explain the variation
in scores in the data collected. In the conceptual framework, three types of
knowledge had been hypothesized. However, in this study, no prior assumption
was made about the truth or othenrvise of this hypothesis. In other words, no
specific decision was made earlier in this study about the exact number and
nature of the underlying factor structure (i.e., of the type of knowledge measured
by the instrument). Consequently, Exploratory Factor Analysis was used. The
idea was to allow as many factors as items on each of the instruments to be
extracted so that a decision could be made, based on the factor loadings, as to
the number Of factors that could be retained to explain the pattern of relationship
among scores in the data. It helped to answer the question of whether there is
enough evidence to conclude that three factors could be distinguished, a number
corresponding to the types of knowledge hypothesized in the theoretical

framework. In addition, exploratory factor analysis helped to determine whether

103

the factors that emerged could be described, using the three types of knowledge
hypothesized in the framework. The extraction method used was principal
component analysis and the rotation method used was Oblimin with Kaiser
Normalization. Oblimin rotation was used because of its ability to allow the
factors extracted to be correlated.

The number of variables or factors needed to explain the variation in the
data could have been modeled by using structural equation modeling (SEM) and
factor analysis could have been incorporated in SEM to confirm these variables.
However, SEM could not be used in this analysis because the 339 scores
collected from the university students and in-service teachers in this study were
far less than the minimum required for a very good SEM‘0 analysis. Therefore
Factor Analysis", as a stand-alone test, was the best for the small sample size
(150 subjects or participants for Form 1 and 189 for Form 2) obtained in this
aspect of the study. The analysis was initially done to retain three factors and
later with two and eight factors. Three reasons accounted for this. First, as the
next sections reveal, the strength Of factor loadings in the initial analysis (using
three factors) was not effective in allowing distinct labeling of the factors. Second,
a scree-plot was used to confirm the three factors. However, the use Of the
scree-test is usually criticized for being subjective as different researchers could

focus on different points on the curve as the points where the steepness

 

'° To use SEM, Pedhazur (1997) has argued the subject to variable ratio must be at least 30:1.
Comfrey & Lee (1992) have also argued that to use SEM, “the adequacy of sample size might be
evaluated very roughly on the following scale: 50 - very poor; 100 — poor; 200 - fair; 300 — good;
500 — very good; 1000 or more - excellent” (p. 217).

” According to Gorsuch (1983) and Hatcher (1994), in Exploratory Factor Analysis, a subject to
item ratio of at least 5:1 is recommended while Nunnally (1978) argues for a ratio of 10:1.

104

smoothens. As a result, two such contentious points were focused on for
analysis. As will be seen in the next section, the scree-plot appeared to have its
elbow between 2 and 3 factors, suggesting that two factors needed to be
retained. Consequently, the analysis was done with 2 factors as well. Third, the
Kaiser-criterion applied in the test also pointed to the retention of 8 factors. The
idea was to determine which of the three dimensions, 3, 2, and 8, gave the best
interpretation of the factors.

To do this interpretation, loadings of absolute value above 0.30 were
considered strong enough to be indicative of the nature of the factor. Therefore,
item loadings on the various factors were compared and the ones with loadings
0.30 and above used. Also, since cross loading (i.e., loading of 0.30 or above on
more than one factor) is indicative that an item cannot be uniquely assigned to
any of the factors, such items were removed and not used to determine the
nature of that factor (see Guadagnoli and Velicer, 1988). Finally, factors with
fewer than three items were considered unstable and not labeled (see Costello
and Osborne, 2005). In the next two sub-sections, results of the factor analysis

are presented and discussed beginning with Form 1.

4. 2.1 Factor Analysis on Form 1

4.2.1.1 Factor Analysis of Form 1 with Three Factors

The first step in the factor analysis was the examination Of the number of
factors needed to explain the variation in scores on the various items on Form 1.
The table below, Table 4.2.1 shows how items loaded on various factors and the

variance explained by all possible factor loadings when Factor Analysis was

105

done to retain three factors. This table, Table 4.2.1, shows results of the number
of possible factors that could be extracted from the data to explain the variation
among the scores and their corresponding eigenvalues. The eigenvalues give an
indication of the strength level Of each Of the extracted factors. Consequently, the
eigenvalues could be used to decide on the required number Of factors needed to

represent the relationships in the data.

Table 4.2.1 Total Variance Explained by Each of the Factors Of Form 112

 

Initial Eigenvalues

 

 

Factor % of
Total Variance Cum. %

1 2.526 12.628 12.628
2 1.649 8.244 20.872
3 1.468 7.342 28.214
4 1 .368 6.838 35.052
5 1.304 6.520 41.573
6 1.196 5.978 47.551
7 1.137 5.686 53.236
8 1.056 5.281 58.517
9 .992 4.962 63.479
10 .940 4.700 68.179
11 .882 4.412 72.591
12 .785 3.924 76.516
13 .705 3.523 80.039
14 .699 3.495 83.534
15 .638 3.192 86.726
16 .610 3.048 89.774
17 .596 2.981 92.755
1 8 .498 2.492 95.248
19 .482 2.409 97.656
20 .469 2.344 100.000

 

 

'2 Extraction Method: Principal Component Analysis.

106

In Table 4.2.1, a low eigenvalue for a given factor implies that factor’s
contribution to the explanation of variances in the variables is small and may be
ignored. Consequently, in this analysis, the Kaiser-criterion (also referred to as
the K-1 rule) of retaining only the factors with eigenvalues greater than 1.0 was
initially used. Based on the initial eigenvalues, it is clear that by the K-1 rule,
eight main factors could be retained. Together these eight explain about 58.5%
Of the variance. However, since the theoretical framework guiding this study
hypothesizes three main knowledge types, the eight factors revealed by the
Kaiser criterion did not make much sense in this analysis. Consequently, the
scree-test plot was used for further Check. Essentially, scree-test plots are
graphs of the factors (as shown in Table 4.2.1 above) on the horizontal axis
against the corresponding eigenvalues on the vertical axis. On this graph, as the
number Of factors increases (i.e., as one moves from left to right along the
horizontal axis), the eigenvalues decrease (refer to this also from Table 4.2.1).
However, the change in slope of the graph resulting from these variations is
usually not constant but decreases as the number Of factors increases.
Conventionally, the steepness of the slope of various sections of the graph is
examined and the x-coordinate of the point beyond which the variation in slope
begins to be uniform is chosen as the needed number of factors.

The graph below, Figure 4.2.1, shows the plot needed for the scree-test. It
can be seen from Fig. 4.2.1 that from between factor numbers 2 and 3 there is a
major variation in the slope of this graph. The variation in the steepness of the

graph reduces relatively more beyond factor 2 or 3. Hence, from the scree-test,

107

only the first two or three factors were retained for further analysis. Thus, from
the factor analysis, one inference was that Mg or glee factors were needed. In
other words, whereas the factor analysis extracted 20 factors for examination
(because of the 20 content items on the research instrument), it can be inferred
from the scree plot that the number of factors needed to explain the variation in
scores in the data is either two or three. This subjective interpretation of the

scree-plot was what provided the rationale for the analysis with not only 3

2.5 1

2.0 -

1.5“

Eigenvalue

1.0-

0.5 a

 

 

i '2 5:15 6 7 8 9 10111213141516 17131920
Factor Number

Figure 4. 2.1 Graph Of Factor Loadings for Items on Form 1

Focusing first on three factors, it can be seen from Table 4.2.2 that these three
factors together explain about 28.2% percent of the variation in scores (see

Table 4.2.1).

108

The next item from the SPSS output was Table 4.2.2, a table of
communalities shown below. This communalities table shows how much (the
percent) of the variance in the scores of the items has been accounted for or
explained by all the extracted factors. The “communality” column shows the
percent of variance in any given variable that is explained by the extracted
factors. In general, high values of variances in the “communality” column of the
table would mean that the items had a lot of characteristics in common and low

values would mean that the items have little in common with each other.

Table 4.2.2 Communalities of Form 113

 

Item ID Communality

 

1 .185
2 .273
3 .282
4 .113
5 .247
6 .398
7 .379
8 .165
9 .145
10 .310
11 .344
12 .276
13 .233
14 .125
15 .198
16 .419
17 .203
18 .504
19 .476
20 .365

 

 

‘3 Extraction Method: Principal Component Analysis. The communalities shown in this table is
from the output Of the Factor Analysis run to retain three factors.

109

In general, communalities need to be interpreted with respect to the
interpretability of the factors. As a result, an item or variable with communality as
high as 0.82, for instance, may tell much unless the factor on which that item is
loaded is interpretable. On the other hand, communality as low as 0.28 could be
meaningful if the item contributes to the interpretation of the factor on which it
loads. In other words, the most important thing to focus on is not the
communalities alone but the extent to which the items contribute in the
interpretation of the factors though frequently this contribution is greater when
communalities are high.

A cursory look at Table 4.2.2 reveals that in general, the variance in each
item that was accounted for by the extracted factors was very low. It can be seen
from Table 4.2.2 that twelve out of the 20 items had less than 30% of their
variance explained by the other factors extracted. It can be seen from Table 4.2.2
that nineteen out of the 20 items had less than 50% of their variance explained
by the other factors extracted. In addition six of the items had communalities
below 0.2 meaning less than 20% of their variances are explained by the
extracted factors. Thus, though test items generally have low reliabilities which
may result in low communalities, the low communalities in Table 4.2.2 are worth
noting because, as would be seen in the sections that follow, the pattern of item
loadings on the factors did not help to interpret the factors.

A rotation matrix was used to help interpret how the items were loaded on
the three retained factors. Generally, each loading on a table such as this

corresponds to the correlation between the item and the factor. Conventionally

110

for each item or variable the table is examined for the factor on which the
variable loads (i.e., where the correlation is greatest) by using the absolute
values of the loadings. As already explained, in an attempt to label the three
extracted factors, the criterion of considering loadings of absolute value greater
than 0.3 as illustrative of the potential nature of the factor was used. The idea
was to see whether a case could be made, using the pattern of the strength of
loading to uniquely label each of the three retained factors using one of the three
hypothesized knowledge types in the theoretical framework that guided the
study. Then by examining the types of items or variables loading strongly on a
particular factor, an attempt was made to define or label the factor. Table 4.2.3
below presents the how the items on Form 1 were loaded on the three factors
retained.

Table 4.2.3, reveals that four items (items 3, 16, 18, and 19) cross loaded
on more than one factor. Consequently, they were not assigned to any of the
three retained factors. Four other items (items 5, 9, 12, and15) uniquely loaded
strongly on Factor 1. Of these two each were classified by the members of the
KAT project as measuring advanced knowledge (items 9 and 12), and teaching
knowledge (items 5 and 15). Five items (items 1, 2, 6, 8, and 20) were uniquely
loaded on Factor 2. Two of these (items 1 and 6), according to the KAT
classification, were school knowledge items, another two (items 8 and 20) were
advanced knowledge items, and the remaining one (items 2) was a teaching

knowledge item.

111

Table 4.2.3 Item Loadings on the Three Retained Factors of Form 114

 

 

 

Factor
Item ID 1 2 3

1 -.109 -.419 .014
2 -.120 -.514 .051
3** .375 .007 .396
4 .148 -.043 .31 1
5 .440 -.035 .257
6 .015 .521 .284
7 -.235 -.060 .553
8 -.128 -.305 .272
9 -.352 -.158 -.016
10 .268 .209 .432
1 1 .091 -.080 .584
12 -.507 -.128 .069
13 -.163 -.092 .443
14 -.266 -.098 .209
15 .426 -.067 -.056
16** .363 -.448 .349
17 .017 -.217 .419
18** .345 -.604 .149
19** .444 -.481 .262
20 -.113 -.579 .216

 

** Items with cross-loadings

Six items (4, 7, 10, 11, 13, and 17) uniquely loaded on Factor 3. According to the
classification by the KAT project (refer to Table 3.5.1), two of these were
advanced knowledge (items 4, 13) items, three (items 7, 10, and 11) were
teaching knowledge items and the remaining one was a school knowledge item
(i.e., item17).

From the foregoing discussion, items measuring the same type of

knowledge, at least as classified by KAT as, did not load selectively on the

 

‘4 Extraction Method: Principal Component Analysis.
Rotation Method: Oblimin with Kaiser Normalization.

112

respective factors. It was, therefore, not possible to label any of the factors
exclusively using the three hypothesized knowledge.

The nature of the factor loadings was also analyzed using the other two
dimensions (Algebra Content and Domain of Mathematical Knowledge). The idea
was to test whether, instead of the types of knowledge these items were
categorized (according to KAT) to be measuring, a case could be made about
the nature of the item loadings using their attributes in the other two domains in
the KAT framework (see Figure 2.4.2). Of these two domains, only the Algebra
Content dimension showed some promise on one of those factors. On Factor 1,
all the four strongly loaded items (items 5, 9, 12 and 15) were originally classified
as items based on the content of “Functions” (see Figure 2.4.2). Factors 2 and 3
had items loading strongly on both categories of “Algebra Content” in the KAT
framework. For instance, of the six items loading strongly on Factor 3, five had
prior to this study been classified by members of the KAT project as based on the
content of “Expressions, Equations and Inequalities” (items 4, 7, 10, 11, and 17)
while the remaining one had been classified as based on “Functions” (item 13).
Also, of the five items that loaded strongly and uniquely on Factor 2, four (items
1, 2, 6, and 20) were based on “Functions”, while the remaining one, item 8, was
based on the content of “Expressions, Equations and Inequalities". Thus, since
items measuring only one of the two kinds of algebra content did not load
exclusively on any of the three factors, it was not possible to label the factors
using any of the two content of algebra categories as developed in the KAT

project.

113

Similarly, analyzing the nature of item loadings on the three retained
factors in terms of the third dimension of the items as defined in the KAT project,
the domain of mathematical knowledge, did not help to uniquely label any of the
factors. For instance, of the four strongly loading items of Factor 1, two (items 5
and 9) were classified by the KAT project as measuring core concepts and
procedures and the other two (items 12 and 15) as measuring reasoning and
proof. On Factors 2 and 3, items measuring each of the four domains of
mathematical knowledge were strongly loaded. On Factor 2, one item, item 2,
was based on core concepts and procedures, two items (items 1 and 8) were
based on applications, one item (item 6) on representation and another one (item
20) on reasoning and proof were strongly loaded. Finally, on Factor 3 one item
each was based on core concepts and procedures (item 4), and applications
(item 11), while two each were based on representations (items7 and 17) and
reasoning and proof (items 10 and 13).

Thus, none of the three dimensions of the KAT framework could be used
as the basis for labeling the three retained factors. Based on the possibility of
interpreting the elbow shown of the scree-plot in Figure 4.2.1 as corresponding to
two factors, factor analysis was done with two dimensions. In the next sub-

section, this analysis is presented.

4.2.1.2 Factor Analysis of Form 1 with Two Factors

The total variance explained using two dimensions (i.e., when the factor

analysis was run to retain two factors are presented in Table 4.2.1.

114

A cursory look at this table shows that only about 20.9% of the total
variance is explained by the two retained factors. Also as in the earlier analysis,
applying the Kaiser-criterion suggests that eight factors could be retained
(explaining 58.5% of the variance).

The factor loadings were also examined using the two factors. Table 4.2.4
below how the items on Form loaded on two factors. As revealed by Table 4.2.4,
using a minimum strength of 0.3 as illustrative of the potential nature of the
factors, the factors could not be labeled using any of the three types of

knowledge hypothesized in the KAT project.

Table 4.2.4 Item Loadings on the Two Retained Factors of Form 1

 

 

 

Factor
Item ID 1 2

1 .392 -.1 01
2 .486 -.088
3 .095 .51 7
4 .098 .330
5 .132 .423
6** -.475 .326
7 .077 .352
8 .304 .127
9 .095 -.204
10 -.1 1 5 .527
1 1 .153 .530
12 .050 -.201
13 .109 .288
14 .073 .039
1 5 .128 .144
16** .532 .405
17 .257 .335
18 .659 .203
19** .568 .364
20 .568 .046

 

** Items with cross-loadings.

115

As can be seen from Table 4.2.4, there were cross loadings of items 6, 16,
and 19. They were, as a result, not uniquely assigned to any of the factors. For
the items that strongly loaded on only one of the two factors, five (items 1, 2, 8,
18, and 20) loaded on Factor 1. Of these, two each had been categorized in the
KAT project as measuring advanced knowledge (items 8 and 20) and teaching
knowledge (items 2 and 18), while the remaining one, item 1, was categorized in
the KAT project as a school knowledge item. On Factor 2, seven items (items 3,
4, 5, 7, 10, 11, and 17) loaded strongly. These were distributed as follows using
the KAT categories: two school knowledge (items 3 and 17), one advanced
knowledge (item 4) and four teaching knowledge items (5, 7, 10, 11). Thus, on
each of the two factors, items earlier categorized to be measuring all the three
hypothesized knowledge were loaded. In other words, the nature of the loadings
on the two retained factors did not permit a differentiation in the nature of the
factors on the basis of the three hypothesized knowledge categories.

The items loading were also analyzed with respect to the classification of
the items in terms of the algebra content (i.e., “Expressions, Equations and
Inequalities” or “Functions”) they were measuring. In this regard, it was observed
that the items that loaded strongly on Factor 1 comprised two “Expressions,
Equations and Inequalities” items (items 8 and 18) and three items (1, 2, and 20)
based on “Functions”. Then on Factor 2 there were six “Expressions, Equations
and Inequalities” items (items 3, 4, 7, 10, 11, and 17) and one item, item 5, on

“Functions”. Thus, both types of content loaded together on each of the two

116

retained factors. Consequently, the factors could not be defined in terms of the
two algebra content areas on which the items were based in the KAT project.

The third dimension of the KAT framework, the domain of mathematical
knowledge, was also used to analyze the nature of item loadings on the two
factors. The items did not load on the factors in any unique manner even on the
basis of this third domain. For instance, of the five items that strongly loaded on
Factor 1, two each (items 2 and 18 and items1 and 8), were respectively
classified as measuring core concepts and procedures, and reasoning and proof.
The remaining one, item 20, measured application. On Factor 2, there were
three items on core concepts and procedures (items 3, 4, and 5), two on
representation (items 7 and 17) and one each on application (item 11) and
reasoning and proof (item 10). As a result of this non-unique nature of the factor
loadings, it was not possible to distinctively label the two factors in terms of the
domain of mathematical knowledge as hypothesized in KAT.

Thus, attempts at using all the three dimensions of the KAT framework to
label each of the two retained factors were not successful. Consequently, the
next level of analysis was done using eight dimensions. As has already been
explained the basis of using eight factors was the outcome of applying the

Kaiser-criterion. This analysis is discussed in the next subsection.

4.2.1.3 Factors Analysis of Form 1 with Eight Factors

Table 4.2.1, below, presents, the variance explained when the analysis
was run with eight factors. A cursory look at this table shows that when eight

factors are the focus, about 58.5% of the total variance is explained.

117

Next, the factor loadings were examined using eight factors. As in the
earlier analyses, 3 minimum strength of 0.3 was used as illustrative of the
potential nature of the factors. Also, consistent with the earlier analyses, in an
attempt to interpret the factors, items with cross loadings could not be uniquely
assigned to any of the factors. Table 4.2.5 below presents how the items on

Form 1 loaded on the eight retained factors.

Table 4.2.5 Item Loadings on the Eight Retained Factors of Form 1

 

 

 

Factors
Item ID
1 2 3 4 5 6 7 8

1** .182 -.126 -.120 .578 .073 .351 .066 .049
2 .282 -.040 -.100 .054 -.117 .100 .706 .168
3 .165 .164 .242 -.061 .044 .170 .059 -.724
4 .086 .283 .145 .596 .069 -.287 .171 -.268
5** .388 .418 -.103 -.248 -.233 -.072 -.077 -.065
6** -.250 .650 .019 -.025 .049 -.035 -.353 .145
7 .097 .095 .757 -.119 -.063 .100 -.010 .071
8 .103 .080 .190 .068 -.816 .063 .056 -.037
9 -.133 -.173 .143 .602 -.254 .014 -.266 .210
10 .050 .61 1 .040 -.076 -.051 .006 .069 -.263
1 1 .247 .602 .169 .084 .089 .190 .278 .000
12 -.028 .000 .190 -.066 .130 .162 .189 .702
13 .026 .015 .650 .262 -.074 -.132 .102 -.166
14 -.106 .119 .000 .004 .020 .788 .010 -.007
15** .378 -.012 -.168 .050 -.014 .023 -.613 -.020
16 .719 .096 .189 -.059 .002 .008 .105 .008
17** .392 .105 .384 .127 .462 .316 -.056 .141
18*“ .622 -.011 -.136 .332 -.180 .070 .218 -.164
19 .695 .016 .063 .071 -.056 .069 .042 -.188
20** .319 -.096 .051 .053 -.487 .540 .172 -.025

 

** Items with cross loadings

To simplify the discussion with these eight factors, the strongly loaded

items on each of the factors have been extracted and presented with their KAT

118

categorization (in parenthesis) in Table 4.2.6 below (without items with cross
loadings).

As shown in Table 4.2.6, three factors, Factors 5, 6, and 7 had only one
item each loading strongly on them. In addition, four factors, Factor 1, 3, 4, and 8
had only two items each strongly loading on each of them respectively. Again,
one or two item loadings were considered too small to be indicative of the nature

of these factors.

Table 4.2.6 Form 1 Item Loadings and Categorization on the Eight Factors

 

Number
Factors of Items Item ID and CategorizatioMin parenthesis)

 

1 2 16(2,1,1) 19(1,2,1)

2 3 5(3, 2,1) 10(3,1,4) 11 (3,1,3)
3 2 7(3, 1,2) 13 (2, 2, 4)

4 2 4(2,1,1) 9(2,2,1)

5 1 8(3,2,1)

6 1 14(1,2,1)

7 1 2(3,2,1)

8 2 3(1,1,1) 12(2,2,4)

 

As a result, seven of the eight retained factors (i.e., Factors 1, 3, 4, 5, 6, 7,
and 8) were considered too unstable to be correctly interpreted in this study. The
only factor on which the minimum of three items loaded was Factor 2. These
three items were items 5, 10, and 11. All three items were teaching knowledge
items. They did not have any other attribute in common. Thus, only one of the
eight retained factors could be labeled, if necessary in terms of one of the types

of knowledge hypothesized in the study.

119

Thus, none of the three factor analyses helped to label all the factors
retained on Form 1. This result is further discussed in chapter five. Similar

analyses were done on Form 2. The next section presents these analyses.

4. 2.2 Factor Analysis on Form 2

As was done with Form 1, exploratory factor analysis using the principal
component analysis method of extraction was conducted on participants’ score
on each of the items on Form 2. The aim was to examine whether variations in
participants’ score on the 20 items on the research instrument (i.e., Form 2)
correspond to multiple underlying factors or to three factors (e.g. the three types
of knowledge for teaching algebra; school knowledge, advanced knowledge and
teaching knowledge) as hypothesized in the theoretical framework that guided
this study.

As was done with Form 1, factor analysis was performed with three, two
and eight components. The procedure started with an examination of the number
of factors needed to explain the pattern of relationships among these items on
Form 2. Table 4.2.7 below shows the possible number of factors initially
extracted by the factor analysis. The eigenvalues obtained were the same for
each of the three analyses. In addition, values of the “Extraction sum of squares”
obtained for three and two components were respectively contained within the
first three and the first two rows of the eight component output. As a result, the
output with eight factors is what has been presented in Table 4.2.7.

From Table 4.2.7, three retained factors account for about 31.3% of the

variation in scores while two and eight factors account for about 23.7% and

120

60.3% of the variance respectively. Also, as Table 4.2.7 indicates, using the
common rule of thumb of the Kaiser-criterion for dropping all factors with
eigenvalues less than 1.0 from the analysis, 8 factors needed to be retained.
However, since the Kaiser-criterion always has the tendency to overestimate the
number of true factors needed to explain a given data set (see (Lance, Butts, and

Michels, 2006), the scree-test was used for confirmation.

Table 4.2.7 Total Variance Explained by Each of the Factors of Form 215

 

Initial Eigenvalues

 

 

Factor % of
Total Variance Cum. %
1 3.186 15.929 15.929
2 1.560 7.798 23.727
3 1.513 7.565 31.292
4 1.363 6.815 38.106
5 1.227 6.136 44.342
6 1.154 5.768 50.010
7 1.061 5.306 55.316
8 1.001 5.003 60.319
9 .940 4.698 65.017
10 .903 4.515 69.532
1 1 .853 4.363 73.795
12 .785 3.924 77.719
13 .759 3.794 81.513
14 .673 3.365 84.878
15 .642 3.211 88.089
16 .550 2.750 90.840
17 .539 2.693 93.533
18 .466 2.330 95.863
19 .442 2.209 98.072
20 .386 1 .928 100.000

 

 

'5 Extraction Method: Principal Component Analysis.

121

Figure 4.2.2 below shows the scree plot of the extracted factors. It can be
inferred from Figure 4.2.2 that the variation in the slope of the graph greatly
reduces beyond factor number 2 or 3. As a result of the possible of settling on
the factor number of 2 or 3 as the point beyond which a gentle variation in slope
occurs, the analysis was done with both factors 2 and 3 to see which better

explained the pattern of relationships among these items.

Eigenvalue
"l

1-

 

L

l l 1 l l l I l l l l l I l l l l l I
1234567891011121314151617181920
Factor Number

Figure 4.2.2 Graph of Factor Loadings for Items on Form 2

The next table from the SPSS output, Table 4.2.8, shows how much of the

variation in the score of each item was explained by the extracted factors. As

122

discussed earlier, values in the “communality” column show the percent of

variance in any given variable that is explained by the other extracted factors.

Table 4.2.8 Communalities of Form 216

 

Item ID Communality

 

1 .293
2 .039
3 .330
4 .110
5 .352
6 .226
7 .358
8 .146
9 .076
10 .228
11 .264
12 .459
13 .292
14 .380
15 .522
16 .428
17 .319
18 .598
19 .407
20 .431

 

A cursory look at Table 4.2.8 reveals that in general, the variance in each
item that was accounted for by the extracted factors was very low. It can be seen
from Table 4.2.8 that eighteen out of the 20 items on Form 2 had less than 50%
of their variance explained by the other factors extracted. In general,
communalities need to be interpreted with respect to the interpretability of the
factors. As a result, an item or variable with communality as high as 0.82, for

instance, may tell much unless the factor on which that item is loaded is

 

'6 Extraction Method: Principal Component Analysis.

123

interpretable. Whether items have high or low communalities is important only
when the nature of factor loadings helps to interpret the factors (see section 4.2.3
for further discussion of these communality values). In the next section the

analysis done with three components is discussed.

4.2.2.1 Factors Analysis of Form 2 with Three Factors

Table 4.2.9 below presents how the items loaded on three retained

factors.

Table 4.2.9 Item Loadings on the Three Retained Factors of Form 2"

 

 

 

Factor
Item ID 1 2 3

1 .058 .445 .266
2 .094 .184 -.037
3** .450 .370 .077
4 -.304 .062 -.044
5 .582 .106 .018
6 .005 .048 -.471
7 .585 .035 -.204
8 .379 .055 -.102
9 .1 19 .255 -.089
10 .190 .091 -.461
1 1 .473 .062 -.270
12** .‘630 .121 -.342
13“ .084 .382 -.413
14 .608 .179 -.102
15 -.039 .705 -.130
16 .033 -.644 .045
17 .069 -.200 -.508
18** .506 .212 -.646
19** -.018 .316 -.563
20 .280 .037 -.628

 

** Items with cross loading

 

17 Extraction Method: Principal Component Analysis.
Rotation Method: Oblimin with Kaiser Normalization.

124

From Table 4.2.9, it can be seen that five items (items 3, 12, 13, 18, and 19)
cross loaded on more than one factors. They were therefore not assigned to any
of the retained factors and not used in interpreting any of the factors. In terms of
the strength of their loadings, six items (items 4, 5, 7, 8, 11, and 14) loaded on
Factor 1, three items (items 1, 15, and 16) loaded on Factor 2 and four items
(items 6, 10, 17, and 20) loaded on Factor 3. The KAT categorizations of the
items that loaded on each of the three factors are shown in parenthesis besides
the item ID below (refer to the explanations accompanying Table 3.5.1 for
explanations of the KAT categorizations);

Factor 1- items 4(312), 5 (212), 7(221), 8(322), 11(321), and 14(121)

Factor 2- items 1(112), 15(324), and 16(211)

Factor 3- items 6(121), 10(311), 17(211), and 20(214)

As can be seen from these listings, items originally categorized as
measuring each of the three types of knowledge hypothesized in the KAT
framework loaded on each of the three factors. Similarly, some of the items
originally categorized to be based on each of the two types of algebra content
loaded on each of the factors. In addition, none of the factors was uniquely
loaded with items based on one aspect of the KAT domain of mathematical
knowledge. As a result, none of the three factors could be labeled using any of
the three dimensions from the KAT framework. The next section presents results

of the factor analysis performed using two components.

125

4.2.2.2 Factor Analysis of Form 2 with Two Factors

The scree-plot obtained for this analysis was the same as the one
obtained in the previous section (see Figure 4.2.2). Consequently, no new scree—
plot is presented in this section. To interpret the two factors retained the factor
loadings were essential. Table 4.2.10 below presents these factor loadings when

two components or factors were retained.

Table 4.2.10 Item Loading on the Two Retained Factors of Form 218

 

 

 

Factor
hentfl) 1 2
1 n137 .322
2 .077 .171
3 .274 .262
4 «217 .112
5 .431 .018
6 .254 .196
7 .560 .022
8 .343 .034
9 .118 .251
10 .387 .209
11 .506 .083
12 .661 .141
13 .254 .479
14 .510 .121
15 a022 .705
16 .057 n619
17 .345 a032
1 8** .721 .340
19 .262 .478
20 .552 .200

 

** Item with cross loadings
As will be observed from Table 4.2.10, only one item, item 18, loaded

strongly on both factors. For that reason, item 18 was the only strongly loaded

 

'8 Extraction Method: Principal Component Analysis.
Rotation Method: Oblimin with Kaiser Normalization.

126

item that could be assigned to any of the two retained factors and used in the
interpretation of the factors. After taking out item 18, nine items (items 5, 7, 8, 10,
11, 12, 14, 17, and 20) loaded strongly on Factor 1 and four items (items 13, 15,
16, and 19) on Factor 2.

Analysis of the item attributes indicated that the items on each of the two
retained factors did not load uniquely on the basis of any of the three dimensions
(domain of algebra knowledge, algebra content domain, or domain of
mathematical knowledge) of the items (refer to Table 3.5.1 for their
categorization). The two factors could therefore not be labeled in terms of
aspects of any of the three dimensions. In the next section, the analysis using

eight components is presented.

4.2.2.3 Factor Analysis of Form 2 with Eight Factors

Table 4.2.11 below presents the strengths of the factor loadings of each of
the items on the eight retained factors. A cursory look at Table 4.2.11 shows that
nine items loaded on more than one of the factors. These were items 4, 5, 6, 7,
10, 11, 13, 17, and 18. They were, therefore, excluded in the attempt at labeling
the factors.

After taking out these nine cross loading items, items that strongly loaded
on each of the factors were items 19 and 20 on Factor 1, items 15 and 16 on
Factor 2, items 3 and 7 on Factor 3 and items 12 and 14 on Factor 4. The rest
were item 8 on Factor 5, item 2 on Factor 6, item 9 on Factor 7 and item 1 on

Factor 8.

127

Table 4.2.11 Item Loading on the Two Retained Factors of Form 219

 

 

 

Item Factor
ID 1 2 3 4 5 6 7 8

1 -.038 .212 .144 .100 .062 -.149 .042 -.741
2 -.021 .206 -.075 -.038 .121 .739 -.040 .029
3 .081 .182 .723 -.279 .090 -.074 -.032 .050
4** -.146 .216 .140 .258 .113 -.389 .021 .546
5** .184 -.218 .540 -.238 .111 -.060 .379 -.172
6** .380 -.023 -.028 .200 .533 .121 .066 .209
7** .230 -.21 1 .356 -.225 .280 .344 .523 .133
8 -.013 -.192 .184 -.120 .721 .207 .081 -.151
9 .006 .189 -.072 -.040 -.008 -.082 .804 -.035
10** .132 .106 -.192 -.41 1 .475 -.289 .285 .169
11** .522 -.178 .200 -.156 -.010 .390 .313 -.151
12 .279 .043 .147 -.772 .093 .167 .204 .098
13*“ .282 .321 .049 -.090 .566 -.068 -.053 .042
14 .124 .025 .269 -.692 .115 .054 .056 -.158
15 .090 .768 .114 -.035 -.011 .100 .208 .101

16 -.052 -.644 -.002 .099 -.077 -.025 .042 .273
17** .213 -.022 -.491 -.481 .052 -.134 .183 .273
18** .693 .041 .129 -.417 .329 .156 .197 .049
19 .701 .231 -.004 .096 .227 -.148 -.066 -.069
20 .738 -.033 -.014 -.281 .043 -.01 1 .097 .099

 

** Items with cross loadings
Thus, none of the eight retained factors had at least three items uniquely
loading onto it. Consequently, it was that there were too few items on each factor

to use for any reliable labeling of the factors.

4. 2.3 Conclusions Related to Research Question One

To answer the first research question, factor analysis was initially done
using three and two components based on the possibilities of selecting two or
three as elbows of the scree-plot. A third analysis was also run using eight

components based on interpretation of the eigenvalues obtained by applying the

 

‘9 Extraction Method: Principal Component Analysis.
Rotation Method: Oblimin with Kaiser Normalization.

128

Kaiser-criterion. However, on both forms, factor analysis using eight factors was
not pursued because after taking out the cross-loading items, the factors had
insufficient items (less than three) left on them to make any meaningful
interpretation.

In general, factor analysis run on the data from the two forms revealed
that the items had small communality values. On Form 1 and Form 2, only one
and two items respectively had at least 50 percent of its variance explained by
the extracted factors. Though test items generally have low reliabilities which
may result in low communalities, the low communalities obtained in the factor
analysis in this study are worth noting especially because the pattern of item
loadings on the factors did not help to interpret the factors.

Also, when the analysis was done with three factors, the three factors
could explain only about 28% and 31% respectively of the variation in scores on
Form 1 and Form 2. When the analysis was done with two factors, the proportion
of the variation in scores that could be explained dropped to about 21% and 24%
respectively.

In addition, the nature of the factor loadings, after the cross loading items
were removed for any of the analysis run using 2 and 3 factors did not permit
interpretation of the retained factors. The analyses revealed that items originally
categorized as measuring any of the three types of knowledge hypothesized in
KAT (i.e., school knowledge, advanced knowledge, and teaching knowledge) did
not uniquely load any of the retained factors. In other words, items meant to

assess different types of knowledge loaded together. Even when the item

129

loadings were analyzed in terms of the other two dimensions of the KAT Item
Development Matrix, (i.e., “algebra content” or “domain of mathematical
knowledge”), the nature of the loadings did not provide evidence for labeling the

factors. These results are further discussed in the last chapter of this dissertation.

4.3 Research Question Two

The second research question was,

“How does the profile of knowledge for teaching algebra differ among the

different categories of secondary school mathematics teachers and

potential teachers in Ghana?”
As already explained in chapter three, data for this research question came from
the scores obtained by in-service teachers and university students on Forms 1
and Form 2. The university students who participated in the study comprised
students majoring in three different areas; mathematics, mathematics education
and statistics. Thus, together with the in-service teachers there were four groups
whose sub-scores on items classified as assessing the three types of knowledge
specified in the KAT project needed to be compared. Consequently, it was
initially decided that Analysis of Variance (ANOVA) performed on the sub-total
scores on the school knowledge items, advanced knowledge items and teaching
knowledge items (as well as the total scores) by the various categories of in-
service and prospective teachers on Forms 1 and 2 together was the best
analysis needed to answer this research question. However because the planned
dimensional structure of the instruments was not supported in the factor analysis,

ANOVA could not be performed on the sub-total scores on the three types of

130

knowledge hypothesized by KAT. The analysis was therefore limited to only the
total scores of participants on Forms 1 and 2.

Before merging the data on Forms 1 and 2, it was necessary to test to see
whether there was any significant difference between the performances of
participants who completed Form 1 and Form 2. The two groups were assumed
to be independent samples. Consequently, Independent Samples t-test was
performed on their total mean scores on Form 1 and From 2. This preliminary t-
test was essential because the results had implications for whether there was
any justification to proceed with ANOVA on the combined forms or on the
individual forms separately. For instance, if the t-test test had revealed significant
differences between the two groups’ performances, then there could have been
no basis to consider them as equivalent groups and thereby run ANOVA on the
combined forms. The preliminary t-test revealed no significant differences in
performance on Form 1 and Form 2. This was what provided the basis for
proceeding with ANOVA on the combined Form 1 and Form 2 scores as
explained in the preceding paragraph. The results of the preliminary t-test are

presented in the next section.

4. 3.1 Preliminary T-Test on Form 1 and Form 2

Table 4.3.1 below shows descriptive statistics of the performance of these
participants on each of the items on the two forms. As shown in Table 4.3.1,
altogether, 150 and 189 participants respectively completed Form 1 and Form 2.

A cursory look at the table reveals that in general, performance of

participants on both forms was quite low. For instance on Form 1, participants

131

were able to score an average of more than 60 percent of the maximum
allowable points (see mean scores for items 1, 2, 3, 11, and 17) only on five

items (out of the twenty items) .

Table 4.3.1 Descriptive Statistics of Form 1 and Form 2 by Item20

 

 

 

Form 1 (N = 150) Form 2 @l = 189)
Item ID Mean Std. Deviation Mean Std. Deviation

1 .880 .326 .989 .103
2 .653 .477 .979 .144
3 .933 .250 .328 .471
4 .147 .355 .296 .458
5 .347 .477 .169 .376
6 .247 .432 .640 .481
7 .487 .501 .222 .417
8 .060 .238 .275 .448
9 .340 .475 .730 .445
10 .427 .496 .682 .467
11 .753 .432 .217 .413
12 .200 .401 .164 .371
13 .147 .355 .116 .322
14 .173 .380 .222 .417
15 .493 .502 .429 .496
16 .540 .500 .106 .308
17 .913 .282 .381 .487
18 .510 .554 .563 .540
19 .723 .462 .627 .644
20 .133 .404 .603 .571

 

Participants scored above 50% of the maximum score on only six of the
items on Form-- items 1, 2, 3, 11, 16, and 17 (items 18 and 19 were open-ended
items so their respective .510 and .723 mean scores were out of the scaled down
score of 2 points and are therefore less than 50% of the maximum score for each

of those items). On Form 2, the performance appears even slightly lower. As in

 

2° For the multiple choice items (items 1-17) participants either scored 1 for a correct response or
0 for a wrong response. The open-ended items (items 18 to 20) were originally scored on a 0 to 4
scale and then scaled down to 0 to 2 for this analysis.

132

Form 1, though, participants were able to score an average of more than 50
percent of the maximum allowable points on only five items (items 1, 2, 6, 9 and
10). Again, items 18, 19, and 20 on Form 2 were open ended items with
maximum score of 2 points each. Thus the .563, .627, and .603 respectively
shown as the mean scores were less than 50% of the allowable maximum score
in each case.

Table 4.3.2 below presents the performance of participants on items that
were originally categorized in the KAT project as measuring each of the three

types of knowledge.

Table 4.3.2 Descriptive Statistics of Forms1 and 2 by Knowledge Type

 

 

 

Form 1 (N = 150) Form 2 (N = 189)

Type of Knowledge Mean Std. Deviation Mean Std. Deviation
School Knowledge 3.870 .965 3.701 1.394
Advanced Knowledge 1.567 1.187 1.598 1.232
Teaching Knowledge 3.670 1.503 3.442 1.372

 

From Table 4.3.2 performance of participants was highest on both forms
on items originally categorized to be measuring school knowledge. On both
forms, their worst performance was on the subtest of advanced knowledge items.

The total mean performance on the two forms is summarized in Table

4.3.3 below.

Table 4.3.3 Group Statistics of Form 1 and Form 2

 

 

Std. Std. Error
Form used N Mean Deviation Mean
1 150 9.107 2.670 .218
2 189 8.741 3.135 .228

 

133

From Table 4.3.3 participants who completed Form 1 performed slightly
better than those who completed Form 2. To test whether the difference in
performance on the two forms was statistically significant or not, independent
samples t-test was run on the total mean scores at 5 percent level of

significance. Table 4.3.4 presents the results of this test.

Table 4.3.4 Independent Samples T-Testz1

 

Levene's Test
for Equality of
Variances t-test for Equality of Means

. Sig. (2- Mean Std.
F 3'9' t df tailed) Diff. Error

Equal variances
assumed 2.758 .098 1.139 337 .256 .366 .321

Equal variances 1.160 335.291 .247 .366 .315

not assumed
From Table 4.3.4, it is clear that the group variances and the mean scores on
Form 1 and Form 2 were not significantly different. Thus, the two groups that
completed Form 1 and Form 2 could be considered equivalent group. As already
explained the subtotal scores on the school knowledge, advanced knowledge
and teaching knowledge items of participants could not be compared because
the dimensional structure of the instruments was not supported in the factor
analysis. As a result, in an attempt to answer the second research question,
participants’ scores on the two forms were combined and ANOVA was performed

on their total score. These analyses and results are shown below.

 

2‘ The t-test performed was two-tailed at 5% level of significance

134

4.3.2 ANOVA on Total Scores of Forms 1 & 2

Table 4.3.5 shows the descriptive statistics obtained from the data

analysis of scores from the Form 1 and Form 2 combined.

Table 4.3.5 Descriptive Statistics of Total Scores by Major Area

 

95% Confidence
Interval for Mean
Std. Std. Lower Upper

 

Major N Mean Dev. Error Bound Bound Min Max
Math 132 9.140 2.711 .236 8.670 9.607 3.00 17.50

Math Ed 125 7.676 2.030 .181 7.317 8.035 2.50 12.50
Statistics 44 7.977 2.277 .343 7.284 8.670 .50 1 1.00
ln-service 38 13.184 2.822 .458 12.257 14.212 7.50 19.50
Overall 339 8.903 2.940 .160 8.589 9.217 .50 19.50

In Table 4.3.5 above, minimum and maximum scores have been calculated as
mixed numbers (numbers with whole number and fractional components). For
example, the minimum score of the in-service teachers has been given as 7.50.
Though none of the items had a possible score of 0.5, the fractions were the
result of the scaling down of the open-ended items’ score from a maximum of 4
(used during the scoring) to 2 (for the analyses). In this way, a participant who
scores only 7 multiple choice items and receives a score of 1 on only one open-
ended item will have his/her total score reduced from 8 to 7.50 after halving the
open—ended score during the analyses of data. Similar, minimum and maximum
scores with fractional components are presented in Table 4.3.9 for the same
reason.

A cursory look at Table 4.3.5 suggests that the mean the total scores on

the two combined forms may be different for each of these fours groups of in-

135

service and prospective teachers. As shown in Table 4.3.5 above, in this study,
the group that performed best on the total score of the combined forms was the
in-service teachers. On the other hand, the group that scored least was the
mathematics education majors. These relative mean scores of participants

majoring in the different domains is presented graphically in Figure 4.3.1 below.

 

 

 

 

 

‘E

I
0 i
U- 12.00
08
‘- l
E
u.
g: 8.00 4
O
2
O
0
m
g 4.00 .
'—
h
0 i
C i
I“ l
0
E I

 

 

 

 

 

 

 

 

 

I I I
Math Math Ed Statistics In-service

Major Area

Figure 4.3. 1 Mean Total Score on Form 1 & 2 by Major Area

One assumption implicit in Analysis of Variance is that the groups under
consideration have approximately equal variances. That is, even if the group
means are different, the dispersion of each group’s scores about the group mean

will be approximately equal (i.e., there will be no significant difference in the

136

variances of the various group’s score). In this study, Levene’s test was used to
test this assumption. In general, if in the output table of Levene's Test, the value
under the column marked "Sig." is less than .05 then an inference is made that
the group variances are significantly different at the .05 level (i.e., the group
variances are not approximately equal). On the other hand, if the value under
“Sig.” is greater than .05, then the group variances are not significantly different
(i.e., the group variances are approximately equal). The Table 4.3.6 below

presents the results of the Levene’s test on the combined Form 1 and Form 2.

Table 4.3.6 Test of Homogeneity of Variance in the Total Scores

 

Levene
Statistic df1 df2 Sig.
3.343 3 335 .019

 

From the outcome of the homogeneity test, it was observed that the value under
Sig. is less than .05. Thus, the group variances are significantly different. This
means that the group variances are not approximately equal.

To examine whether the group means of the total scores on the two forms
combined were significantly different or not, the ANOVA table was used. The

results of this test are presented in Table 4.3.7 below.

Table 4.3.7 ANOVA of the Mean Differences in Total Score

 

 

Sum of Mean

Squares df Square F Sig.
game” 929.815 3 309.938 52.150 .000
roups
W'th'" 1990.973 335 5.943
Groups
Total 2920.788 338

 

137

Table 4.3.7 above shows that there is a large F-ratio (i.e., 52.150) for the
Between-Groups variance. In addition, there is an associated F probability (.000),
smaller than the 0.05 significance level. Thus, there seems to be adequate
evidence that there is a difference overall among the total mean scores for at
least one pair of these four groups.

All ANOVA tables, such as Table 4.3.7 above, are able to reveal the
possibility of at least one pair of groups differing significantly on their mean
scores or the difference between their means. ANOVA tables do not, however,
reveal the actual pairs of groups whose means differ significantly. To investigate
which particular pairs of groups have significant between mean differences,
Tukey’s HSD Tests was used at the 5% level of significance. Tukey’s HSD Test
is useful because it presents results of multiple comparisons among all the
possible pairings of the groups of focus, which in this case are final year
university students majoring in mathematics, mathematics education and
statistics majors, as well as in-service high school mathematics teachers in
Ghana. The result of this multiple comparison is presented in Table 4.3.8 below.

Table 4.3.8 shows that at the 5% level of significance, the in-service
teachers had significantly higher scores on the combined forms than any group
of university students. In addition, the mathematics majors had higher score than
the statistics and mathematics education majors. However, there was no
significant difference between the total mean score of the statistics and

mathematics majors on the combined forms.

138

Table 4.3.8 Multiple Comparisons of Differences in Total Scores

 

 

 

95% Confidence
(I) Major (J) Major Mean Interval
Area Area Difference Std. Upper Lower
(I'J) Error 319- Bound Bound
Math Math Ed 1.464* .304 .000 .679 2.250
Statistics 1 .163* .424 .033 .067 2.259
ln—service -4.044* .449 .000 -5.202 -2.885
Math Ed Math -1.464* .304 .000 -2.250 -.679
Statistics -.301 .427 .895 -1 .405 .802
ln-service -5.508* .452 .000 -6.674 -4.442
Statistics Math -1.163* .424 .033 -2.259 -.067
Math Ed .301 .427 .895 -.802 1.405
In-service -5.207* .540 .000 -6.601 -3.813
ln-service Math 4044* .449 .000 2.885 5.203 ‘
Math Ed 5508* .452 .000 4.442 6.674
Statistics 5207* .540 .000 3.813 6.601

 

* The mean difference is significant at the .05 level.

4.3.3 Conclusions Related to Research Question Two

A number of conclusions can be drawn from the foregoing analysis and
results. First, a look at the raw total mean score of participants who completed
Form 1 and Form 2 (see Table 4.3.2) shows that the mean score on Form
1(9.107) was slightly higher that the mean score on Form 2 (8.741). The
observed difference suggests that perhaps the items on Form 2 were of a slightly
higher difficulty level than those on Form 1 (except that data from this study does
not provide evidence that the difficulty level is significant).

Second, though different participants completed Form 1 and Form 2, the t-
test conducted revealed that overall the performance of those who completed
Form 1 was not significantly different statistically from the performance of Form 2
participants. In each of the participating universities, students majoring in the

same area who agreed to take part in the study completed the forms at one

139

sitting. At this sitting, the number of available Forms 1 and 2 were randomly
distributed to participants. A similar approach was used to distribute the forms to
the in-service high school mathematics teachers who agreed to participate in the
study. The result of no significant difference in performance between those who
completed Form 1 and Form 2 was therefore not surprising following the random
manner of distribution of the two forms.

Third, results of the Analysis of Variance (ANOVA) discussed in the
preceding section reveal that in Ghana, the sub-group of in-service teachers
performed best on the two forms. The analysis of variance performed on the data
revealed that in-service high school teachers in Ghana performed significantly
better than each sub-group of prospective mathematics teachers majoring in
mathematics, mathematics education and statistics from the country’s
universities. It was therefore concluded that knowledge for teaching algebra of
in-service high school mathematics teachers is significantly different from that of
prospective teachers.

Among university students in Ghana, ANOVA revealed differences in the
performances among students in the different majors. In general the
mathematics majors performed significantly better than their counterparts
majoring in statistics and mathematics education. Between the statistics and
mathematics education students, the statistics majors did slightly better than the
mathematics education students. However, this difference was not significance at
the .05 level. These results are further discussed in the last chapter of this

dissertation.

140

4.4 Research Question Three

The third question that guided this study was,

‘What is the relationship between the performance of in-service secondary

school mathematics teachers in Ghana and the students of their classes?”
As already discussed in chapter three, data from the participating senior
secondary school students and their corresponding in-service teachers were to
be used to analyze this question. Essentially, this research question asks for the
relationship between student performance (on the high school students’ form)
and the performance of their teachers on the school knowledge, advanced
knowledge, and teaching knowledge items (of teachers’ Form 1 and Form 2).
Analysis of the variables that could be extracted from this research question
reveals one dependent variable (students score) and three predictor (or
independent) variables. Structural Equation Modeling (SEM) and multiple
regression, because they can each be used to investigate the relationship
between a dependent variable and a number of independent variables, were
thought of as the possible analytical methods to use to analyze data for this
research question. Prior to commencing with fieldwork for this study, SEM was
the preferred method of analysis of data for this third research question because
of its inherent advantages over linear regression. For instance, SEM makes room
for measurement errors and estimates them (Byrne, 2001, pp. 3-4) while in
multiple regression it is assumed that the variables are devoid of any
measurement error. However, SEM could not be used for this analysis because

the number of in-service teachers who participated in the study was too small.

141

Multiple regression could also not be performed using the three types of
knowledge hypothesized by KAT because the factor analysis used to test the first
research question did not support the dimensional structure in the KAT
framework. Consequently, the teachers’ scores were combined and linear
regression was performed between the mean total teachers’ score and the mean
class scores of the students in their classes.

As already discussed in Chapter 3, of the 42 senior secondary school
classes who participated in the study (refer to Tables 3.4.7 and 3.4.8), teachers
of only ten of them also participated. Consequently, the linear regression was
performed using only the ten participating classes whose teachers also
participated in the study. These were the classes with codes A1, A2, A3, CI, C2,

G1, G2, H1, H2 and H7. This analysis is presented in the next section.

4.4. 1 Linear Regression

A Linear regression model involving the mean class score for the ten
classes and the mean scores of their teachers’ total score was estimated and
tested to answer the third research question. Consequently, for the ten classes of
interest, the dependent variable to be used in the model was the mean score for
of all ten classes of focus. The independent variable was the mean total score of
teachers of the ten classes. The model tested was

Y = ,60 + fl1X+ e, where,
Y = the dependent variable representing the mean class score,
X = the independent variable representing the mean of the total knowledge score

for teachers of the classes used

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o = the intercept (mean class score when all independent variables are equal 0).
,61 = the slope for predictor X1 , where the slope is the change in mean class

score for one unit increase in mean teacher total score, holding all other
independent variables constant.

e = the error for the class score (e’s are assumed to be independently and

identically normal distributed with a mean of 0 and variance, 02)
Table 4.4.1 shows the mean scores for students and teachers of these ten

classes used to perform the linear regression.

Table 4.4.1 Mean Class Scores and Teachers’ Total Scores

 

 

Class Mean Teachers’ Mean Class
ID Total Score Score
A1 13.50 3.65
A2 12.00 4.40
A3 7.50 3.23
C1 16.00 5.68
CZ 19.00 4.39
G1 16.00 5.00
G2 10.00 4.39
H1 14.50 3.50
H2 13.50 2.66
H7 13.00 3.42
Mean 13.50 4.03
Std Dev 3.24 0.91

 

Table 4.4.2 shows the correlation coefficients for the linear model.

Table 4.4.2 Linear Regression Model Summary22

 

Multiple R R Square Adjusted R Square Std. Error
.428 .183 .081 .868

 

22 Predictors: (Constant), Mean teachers’ total score
Dependent Variable: Mean class score

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The “R-square" value in the model is the coefficient of determination. It is
indicative of how much of the variability in the dependent variable (mean class
score) is explained by the model. Table 4.4.3 shows that 18.3% of the variability
in the mean score of the ten classes of focus (i.e., A1, A2, A3, C1, C2, G1, GZ,
H1, H2, and H7) could be explained by the model.

The “Multiple R” in Table 4.4.2 is the square root of R-square. It is the
correlation between the predictor variable and the dependent variable. Table
4.4.2 shows that in the linear regression model the correlation between the mean
teachers’ total score and the mean class score is .428.

In general, the “Adjusted R-square” value in the model is the modification
or correction made in the R-Square as the number of predictor variables is
increased. It takes into account the number of predictor variable and the sample
size, and is therefore adjusted based on the number of degrees of freedom. In
reality, what happens is that increasing the number of predictor variables only
increases the possibility that the estimated regression model may not be
significantly better than what could be obtained by chance. Therefore, when
there are many predictor variables present, it is the adjusted R—square values
that need to be used to report the percentage of the variability in the dependent
variable explained by the predictors present. As a result, it becomes more
relevant as a diagnostic tool when used in multiple regression. Since the linear
regression model does not have many predictor variables, attention was not

focused on the adjusted R- square value in Table 4.4.2.

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In linear regression models, there are always differences between the
observed values and the values predictable from the model. These differences
are called residuals. In general, the “Standard Error of the Estimate” in the model
summary is a measure of the standard deviation of the residuals. In the model
that was estimated from this study, the standard error of the estimate was .868.
Compared with the standard deviation of the dependent variable of 0.91 (see
Table 4.4.1), this standard error of estimate from the model is much smaller.

To estimate this linear regression model, values of coefficients of the
independent variables and the constant term were used. Table 4.4.3, presents

these coefficients.

Table 4.4.3 Coefficients of the Linear Regression Model

 

 

 

Unstandardized 95% Confidence
Variable Coefficients Interval
B Std. Error t p-value Lower Upper
Constant 2.419 1.236 1 .957 .086 -.431 5.269
Tchr Total .119 .089 1.338 .218 -.086 .325

 

Usually, the linear regression output (shown in Table 4.4.3) gives the
needed coefficients for the model (refer to the column marked “B” under the
“Unstandardized Coefficients”). Therefore, using Table 4.4.3, the linear
regression model was estimated as

Y = 2.419 -I- .119X+ e, where,
Y = the dependent variable representing the mean class score,
X = the independent variable representing the mean of the total knowledge score

for teachers of the classes used,

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2.419 = the intercept (mean class score when the independent variable is zero),
.119 = the slope for predictor X , where the slope is the change in mean class

score for one unit increase in mean teacher total score, holding all other

independent variables constant, and

e = the error for the class score (e’s are assumed to be independently and

identically normal distributed with a mean of 0 and variance 02 ).

Figure 4.4.1 presents the scatter plot with the linear regression line.

y = 0.119X + 2.419

5.00 R2 = 0.183
5.001 0
§ 4.00-
5 3.00- ,
2.00 -
1.00-

0.00 . . . a
0.0 5.0 10.0 15.0 20.0

 

 

= 0.183

Teachers

Figure 4.4.1 Scatter Plot with the Regression Line

Next the ANOVA table presented in Table 4.4.4 was used to ascertain

whether the effect of the predictor was significant.

Table 4.4.4 ANOVA Test of Significance of the Predictor in the Model

 

 

Sum of Mean

Squares df Square F Significance
Regression 1.349 1 1.349 1.791 .218
Residual 6.025 8 .753
Total 7.374 9

 

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The model was tested at 5% level of significance. From Table 4.4.4, it was clear
that predictor variable, mean teachers’ knowledge, was not a significant predictor

of the mean class score. This is further discussed in the next chapter.

4.4.2 Conclusions Related to Research Question Three

After testing for no violation in the assumptions inherent in linear
regression, the final model was estimated to be of the form,
Y = 2.419 + .119X-I- e, where,
Y = the dependent variable representing the mean class score,
X = the independent variable representing the mean of the total knowledge score
for teachers of the classes used,
2.419 = the intercept (mean class score when the independent variable is zero),
.119 = the slope for predictor X , where the slope is the change in mean class
score for one unit increase in mean teacher total score, holding all other
independent variables constant, and

e = the error for the class score (e’s are assumed to be independently and

identically normal distributed with a mean of 0 and variance 02 ).
Thus, in terms of answering the third research question it was concluded that a
linear relationship existed between the performance of in-service teachers who
participated in the study and the students of their classes. This relationship was
estimated by the equation above.
Though the coefficient of mean teachers’ total score in the model is
positive, the ANOVA test of significance shows that the effect is not statistically

significant. This is also shown by the p-value of the independent variable. In

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theory, the p-value of the independent variable is indicative of whether the
independent variable has statistically significant predictability. As shown in Table
4.4.3, the p-value (.218) shows that the mean teachers’ total knowledge has no
statistically significant predictability. These results are further discussed in the

last chapter of this dissertation.

148

CHAPTER 5: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

This chapter summarizes the study and discusses conclusions drawn from
the analyses and results. In the discussion of the conclusions, the limitations of
the study have been highlighted since they have implications for the extent to
which the conclusions of the study can be generalized. In addition, because
different statistical analyses were done with respect to each research question,
conclusions from this study have been organized in line with the research
questions. This style of organization was adopted because it helped avoid mixing
up conclusions from the different analyses and results. In addition, it helped to
avoid rushing into conclusions beyond the results of the analyses. The chapter

ends with recommendations for further research in the area.

5.1 Summary of the Study

This study was premised on the fact that the teacher is one of the most
important factors that influence student performance, especially in the developing
world (see for example, Mullens et al., 1996; Sanders and Rivers, 1996; Jordan,
Mendro, and Weerasinghe, 1997; Wright, Horn, and Sanders, 1997). The study
acknowledges that though literature on teaching and student performance is
replete with evidence that supports this premise, there is widespread
disagreement among researchers about the exact nature of the link between
teachers’ subject matter knowledge and student performance. The lack of

agreement between the findings of, for instance, Monk (1994) and Rowan, et al.

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(2002), on one side, and Harbison and Hanushek (1992) and Mullens et al.
(1996), on another side, exemplifies the disagreement.

In the face of the many different general conceptualizations of teacher
knowledge, this study also acknowledges the need for re-conceptualization of
teacher knowledge in ways that is not only domain specific, even within coverage
of school mathematics, but also lends itself to some form of direct measurement
instead of by proxy. This is because it is unclear whether the type of knowledge
teachers need for teaching calculus, for instance, would be the same as that
needed to teach a different domain like algebra. In addition, because of the many
different proxy measures that have so far been used by various researchers, the
study concedes the need for improved ways of measuring the different
components or types of teacher knowledge that could result from such re-
conceptualizations.

As discussed in the second chapter of this dissertation, these
considerations underIay the pivotal role of the work of the Knowledge of Algebra
for Teaching (KAT) project at Michigan State University to this study. As already
discussed, through analyses of research literature, recommendations by
professional organizations and videos of teaching, researchers in the KAT project
have hypothesized that the knowledge used by teachers in teaching school
algebra consists of three types: “knowledge of school algebra” (referred to in
short as “school knowledge”), “advanced knowledge of mathematics” (also

referred to as “advanced knowledge”), and “teaching knowledge”. In addition, the

150

KAT project is currently developing instruments to measure and validate their
conceptualization (refer to Ferrini-Mundy et. al., 2003 and 2005).

This study used the KAT conceptualization as the framework for studying
Ghanaian high school teachers’ knowledge for teaching algebra. Rather than rely
on proxy measures of teacher knowledge, the study also adapted the
instruments developed by the KAT project for its data collection in Ghana.

The aim of the study was three-fold. First, though the KAT project was at
its initial stages, at the time of this study, the present study was an attempt at
investigating whether the KAT conceptualizations could be corroborated in
settings outside the US such as Ghana. As explained in chapter one, Ghana
becomes a good international setting for this exploration because of the
differences in the type of mathematics offered its high schools, as well as the
background of the country’s high school mathematics teachers. This is important
because of the integrated mathematics curriculum in Ghana, as well as
differences in the background of teachers in Ghana’s high schools as against
what prevails in many US schools.

Second, the study aimed at investigating the differences, if any, in the
level or quality of the three hypothesized knowledge types in KAT by Ghanaian
high school teachers of different backgrounds. As explained in chapter one, the
importance of this aspect of the study could be seen in the differences in
background of prospective, as well as in-service high school mathematics
teachers. In Ghana, university students majoring in mathematics education,

mathematics and other related areas could be posted to teach high school

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mathematics by the National Service Secretariat and many of them could remain
in such capacity until they retire. In addition, not all the non-mathematics
education graduates could take advantage of summer sandwich programs run by
the University of Cape Coast to obtain a Diploma in Education, one of the
certificates considered as a teaching certificate in Ghana. As a result, the prior
university coursework of high school mathematics teachers in Ghana could vary
as could their professional development experience; hence the need to examine
differences in the profile of their knowledge for teaching algebra.

Third, this study also aimed at investigating how Ghanaian high school
mathematics teachers’ knowledge for teaching algebra is related to the
performance of their students. As can be seen from some of the literature
reviewed in the first two chapters of this dissertation, attempts at using proxy
measures of teacher knowledge to estimate this type of relationship have so far
produced mixed results. In the face of these mixed results, it was important to
find out the type of results the KAT project's domain specific conceptualization
and instruments could provide to the larger research community worldwide.

Participants of the study came from three main types of populations.
These were, 1) high school seniors studying elective mathematics in senior
secondary schools in Ghana, 2) in-service senior secondary school mathematics
teachers from the participating high schools, and 3) prospective senior secondary
school mathematics teachers comprising final year students with majors in
mathematics, statistics and mathematics education from three public universities

in Ghana. In all, 1565 high school seniors offering elective mathematics from

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eight public schools and thirty-eight mathematics teachers from the participating
high schools participated in the study. In addition, 301 university students
comprising 132 mathematics majors, 44 statistics majors and 125 mathematics
education majors participated in the study. Their participation was voluntary and
no one was paid to participate.

As already explained, the study involved administering assessment
instruments adapted from the KAT project at Michigan State University to
participants. The adaptation involved changing the contexts and wording of
questions to reflect Ghanaian contexts. Two adapted instruments, Form 1 and
Form 2, from the KAT project were administered to the prospective and in-
service teachers. Each instrument consisted of two sections. The first section of
each of these two instruments was a survey questionnaire that was focused on
finding out the college level courses participants have taken, the grade level and
mathematics courses that they have been teaching or have taught before (and
for how many years), and the type of teaching certification they have, if any. The
second section consisted of twenty content items based on the content of
algebra in the high school curriculum this content, related advanced mathematics
content, as well as items based on the tasks of teaching. In the case of Ghana,
making contextual adaptations to a few of the items to suit Ghanaian contexts
helped to make the instrument more appropriate because analysis of the senior
secondary school mathematics curriculum at the time of the study revealed that
the high school algebra content was consistent with the algebra covered in the

core and elective mathematics curricula. As a result, at each research site, every

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participant (in-service and prospective teachers) completed only one of the two
forms. At the universities, participants in a particular department or major
completed their forms at a sitting and in no more than 60 minutes. A similar
approach was adopted in each senior secondary school. In all, 150 participants
completed Form 1 and 189 completed Form 2.

The participating final year senior secondary school students, on the other
hand, completed one common instrument. This instrument was also an
adaptation of the KAT instruments. It also consisted of two sections, a survey
questionnaire and a set of assessment items. The survey questionnaire section
of the instrument contained questions about the educational background of
students’ parents/guardians, whether they have extra resources for Ieaming
mathematics apart from what they receive at school, and whether their parents
hire the services of other teachers to provide extra tutorials for them. The second
section of content items comprised items from the KAT instruments that were
based on the algebra content in the senior secondary school core and elective
mathematics curriculum. In each of the participating schools, students in the
elective mathematics classes in the third year (seniors) who agreed to participate
were brought together to complete the instrument at one sitting lasting no more
than 60 minutes.

To protect participants” privacy, no identifiers like participants’ names, their
gender, identity numbers or anything that could be traced to them was required
to complete any of the instruments. In addition, participants’ performance on the

instrument was not shared with their academic departments or institutions.

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Analyses and discussion of the results of the study were also devoid of the
names of the participating institutions. Where names were needed for the
purpose of analyses, pseudo names or codes not traceable to any institution or
individual have been used. Finally, analyses of data have not been done to
compare performance of participants from the different institutions. Instead, in the
universities students in a specific major were grouped together and the analysis
was done to compare them by their major area. For instance, students majoring
in mathematics in this study came from more than one university. Students
majoring in the same area from the different participating universities were
combined and no attempt was made to compare them as separate units from
their individual universities. In the same way, the participating high schools were
not compared but only used in the linear regression to build the linear regression
model.

Factor analyses was done on the performance of the prospective and in-
service teachers on each item of Form 1 and Form 2 to find out whether a case
could be made for the three types of knowledge: school knowledge, advanced
knowledge and teaching knowledge, hypothesized in the theory framework. The
aim was to test whether it was reasonable to talk about three possible knowledge
types (i.e., of three factor loadings and the manner of loadings on them) to
explain the variation of knowledge used in completing the instruments. To
investigate the differences, if any, in the level or quality of knowledge of by
Ghanaian high school teachers of different backgrounds, Analysis of Variance

(ANOVA) was performed using the mean of the total scores of participants on the

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items on Forms 1 and 2 combined. The results of an earlier t-test that confirmed
that participants did not perform significantly different on the two forms and this
provided the justification for combining both forms for the Analysis of Variance.
Finally, linear regression was used to estimate a model of the relationship
between the performance of senior secondary school students who participated
in the study and their assigned mathematics teachers’ knowledge. Due to the fact
that not every participating class’ teacher agreed to participate in the study, only
those classes whose teachers also participated were used for the linear

regression.

5.2 Conclusions from the Study

5.2.1 Limitations of the Study

Before proceeding with the conclusions from this study, it is worthwhile to
note that this study is one of the first attempts, especially in Ghana, at
investigating issues related to high school mathematics teachers’ knowledge and
its relationship with student performance. However, the study is fraught with a
number of limitations that could have implications on the degree to which the
study’s outcomes could be generalized.

First, due to limited financial resources for carrying out the study, it was
not possible to select the participating senior secondary schools (i.e., high
schools) randomly from all the high schools in Ghana. Instead, schools were
selected from towns like Cape Coast and Takoradi that are not too far from each
other and had large concentration of schools. At the time of the study, there were

over 500 public senior secondary schools in Ghana. Therefore the number, 8,

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that participated was less than two percent of the schools in Ghana. Traditionally,
the schools used are among the top class of high schools in terms of their
students’ performance in national examinations. As a result, they do not reflect
the full spectrum of high schools in Ghana.

Second, as was the case in the selection of high schools, there was no
random sampling of in-service and pre-service teachers for this study. ln-service
teachers from the participating high schools who voluntarily agreed to participate
were allowed to participate. In the same way, pre-service teachers majoring in
mathematics education from only one of the two universities where mathematics
education programs are offered participated. The mathematics and statistics
majors who participated also did so voluntarily and were not selected randomly.

Third, the strike action, at the beginning of the 2006/2007 academic year,
by members of the National Association of Graduate Teachers in Ghana limited
the number of days that could be spent on the fieldwork. It also limited the
number of high schools that could be used in the study. The same constraints
limited the number of university students and in-service teachers that participated
in the study. For instance, there are students pursuing degrees in engineering
and other related science and mathematics degrees who could enter the
teaching profession through initial National Service postings and other means but
few such students could be used in the study. Thus, the sample of university
students did not cover the full range of all such related mathematics majors. This

has implications for the generalizability of the results.

157

Fourth, the limited time available to the in-service teachers to complete the
school term, after their strike action, did not make it possible for many of them
who had originally agreed to participate to do so during the fieldwork. The small
number of participating in-service teachers made it impossible for certain type of
analysis to be performed. It also limited the number of cases used for the linear
regression. Consequently, caution need to be exercised in generalizing from the
model estimated in this study.

Fifth, as will be seen from the discussions of the results of the study, there
is a possibility that the KAT framework is either not enough to explain the data
collected or not all the items are measuring what they had been categorized as

measuring in the Ghanaian context.

5. 2.2 How the Instrument Performed on the Sample from Ghana

From the results of the study, a number of conclusions could be drawn
about how the performance of the participating in-service and pre-service
teachers in Ghana compares with their counterparts in the US who had, as at the
time of the study, completed the two instruments for the KAT project. First, in
terms of the difficulty levels of the items, some of the items that were most
difficult and least difficult items to the US participants were also the most difficult
and least difficult respectively to the Ghanaian participants. For instance, on
Form 1, four of the five least difficult items to participants in the KAT study (i.e.,
items 3, 17, 1, and 2), were also in the five least difficult items in this study.
Similarly, on Form 2, four of the five least difficult items to participants in the KAT

study (i.e., items 1, 6, 9, and 10), were also in the five least difficult items in this

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study. However, whereas on Form 1, four of the five most difficult items to
participants in the KAT study (i.e., items 4, 8, 12, and 20) were also among the
five most difficult items in the Ghana study, on Form 2, only two (items 5 and 12)
of the five most difficult items to the KAT participants were also among the five
most difficult items to the Ghana participants.

Second, in general, the KAT participants performed significantly better
than the Ghanaian participants in this study on both Forms 1 and 2. A majority of
the items, 14 out of twenty, on each of Form 1 and Form 2 (i.e., 12 out of the total
of 40 items on the two Forms) were more difficult for participants in Ghana than
their counterparts in the US who had as at the time of the study completed the
two forms for the KAT project. The only items that appeared slightly less difficult
to the Ghana participants than the KAT participants were items 1, 2, 3, 11, 15
and 17 on Form 1 and items 1, 2, 6, 10,15, and 17 on Form 2. Of these only item
17 on Form 2 was an advanced knowledge item. The rest were either school
knowledge or teaching knowledge items.

Three possible explanations could be given for the difference in
performance between the US participants and their Ghanaian counterparts.
These are; 1) differences in curricular emphasis, 2) possible affordances of
handheld technological devices which the US participants use and which their
Ghanaian participants did not have, and 3) familiarity to the nature of type of

questions on the KAT forms.

159

5. 2.2. 1 Differences in Curricular Emphasis

As already mentioned, one possible explanation of the differences in
performance of the US participants and their Ghanaian counterparts on items
such as those cited above would the differences in the curricula emphasis
between US schools and those in Ghana.

To illustrate this point, let us begin by taking a look at the issue of proof.
What constitutes a proof varies from course to course. For instance, in Ghana,
informal proofs are accepted in the core mathematics course at the high school
level. Therefore, a Ghanaian teacher who teaches only core mathematics may
take a good informal proof as a valid proof especially when the proof is presented
to him as a students’ work. This could possibly explain why only about 13% of
the Ghanaian participants answered item number 13 (one of the common items
in Forms 1 and 2) correctly versus 56% of their US counterparts. In this question
different proofs of a statement by three high school students and participants
were asked to determine which of the constituted valid proofs. Since this was a
multiuple-choice item, Ghanaian core mathematics teachers who conceptualize
the students being referred to as similar to those who could be in their classes
were likely to selected the option that included the informal proof as the correct
answer. In another item (i.e., item 19 of Form 2), a mathematical statement was
given and participants were asked to determine if it was true and justify their
answers. Compared with about 53% of the US. participants, only 31% of

Ghanaians answered this questions right because many of the Ghanaian

160

participants gave a number of correct examples as their justification and earned
only 1 out of the maximum 4 points.

Another area where differences in curricula emphasis could have caused
the difference in performance is the approaches projected in textbooks or the
mathematics books available to teachers in the two countries. Presently in the
US textbooks promote graphical approaches to dealing with functions more than

textbooks in Ghana. For instance, in their precalculus book, Stewart, Redlin and

Watson (2006) use graphs of standard functions such as f (x) = x2, g(x) =%

and so on, and the idea of transformations to lead students into drawing graphs

of functions such as p(x) = (x + 4)2 and h(x) = —:—E. On the other hand,
x

because of the fact that national examinations tend to emphasize analytical
methods more than graphical approaches in dealing with functions, Ghanaian
textbooks also tend to emphasize analytical methods more than such graphical
solutions. Consequently, Ghanaian students and teachers would most likely
graph the functions p(x) and h(x) by finding the intercepts, turning points,
asymptotes and the behavior of the curves at the critical points. Such an
approach, though effective, is not economical especially for “timed tests” such as
the KAT instruments. The point being made is that with such graphical emphasis
in most of the US. books, the KAT participants may have developed the ability to
sketch graphs and use them to answer questions faster than their Ghanaian
counterparts. That could explain why only about 20% of the Ghanaian

participants answered item 14 of Form 2 correctly while 58% of US. participants

161

got it right. In that question”, an equation involving two distinct expressions
under radical signs on each side of the equal sign was given and participants
were asked to determine how many solutions the equation has. Using the graphs
of the requisite standard functions and appropriate transformations, U. S.
participants can save more time answering this question than their Ghanaian
counterparts who could lose on other questions because of too much time spent

on this one.

5. 2. 2.2 Possible Affordances of Handheld Technological devices

To explain the graphical calculator affordance it will be good to take a look
at two multiple-choice questions on the KAT form that both groups of participants
answered. One of the questions gave participants a logarithmic function and
asked students to determine which of three other logarithmic functions have the
same graph as the original function. Only about 15% of the Ghanaian
participants answered this question correctly compared with about 31 % of the US
participants (refer to Table 4.1.2).

For the first of this question, from known properties of logarithmic
functions, it is easy to see that the function that one of three options was the
same as the given function. The question therefore reduces to making a decision
about the other two options. However, by simply graphing these other options on
the calculator together with original function one can decide between these two

faster than graphing them from first principles without the calculator. In this case,

 

23 The KAT project had not released the items on their instruments for public consumption.
Consequently, instead of quoting the actual questions, the two items being referred to are
described generally.

162

any of the US. participants who had access to graphing calculators and had the
opportunity to use them in completing the KAT forms were more likely to solve
problems such as this faster than their Ghanaian counterparts“.

On another item, of participants in Ghana who completed both Forms 1
and 2, only 28% answered this item correctly compared with about 58% of the
US participants (see Table 4.1.2). In the question, an equation involving functions
with the radical sign on both sides of the equal sign was given and participants
were asked to solve it. By drawing the function defined by the expression on the
left hand side and that on the right hand side of the equal sign on the same axes,
using the graphing calculator, it is considerably easier to answer this question
than to draw the same graphs without the calculator. Typical Ghanaian teachers
and students who do not use the graphing calculator are more likely to solve this
second problem analytically by squaring both sides and grouping like terms, a
process that could be more time consuming that the graphical solution. This way,
the Ghanaian participants could spend much more time on questions such as
these to the detriment of the other questions.

In a synthesis of peer-reviewed, published research on the impact of
graphing calculators on student performance, Burrill, Allison, Breaux, Kastberg,
Leatham, and Sanchez (2003), concluded that, “overall, the use of handheld
technology [in the form of graphing calculators] had a positive impact on student

performance” (p. 38). Such positive impacts, according to Burrill et al. (2003), are

 

2‘ At the time of the study, graphing calculators were not permitted in Ghanaian high schools. As
a result, not only were such devices not available for teachers to use, teachers did not develop
the competence to use them. Teachers in Ghana were expected to lead students to graph
functions from first principles.

163

possible when “calculator-friendly tasks [are used than when] parallel tasks that
removed the calculator advantage were presented” (p. 41). In completing the
instruments, the KAT project did not discourage the use of handheld technology
in completing the instruments. As a result, any of the US teachers in the KAT
sample who had their graphing calculators available and used them would
perform better on the calculator-friendly tasks such as the two items described
than the Ghanaian teachers who, not only, do not have access to them but are
also not exposed to using them even for instruction.

Even for the US teachers who may not have had graphing calculators
available when completing the KAT forms, it is possible that their sense about the
way the graphs might look could be more fully developed than their Ghanaian
counterparts because of their experience with graphing calculators and other
types of software than enhance graphing abilities. Ghanaian participants in this
study, on the other hand, had not been exposed to the use of graphing
calculators either in school or in their teaching practice”. The affordances of the
graphing calculator technology could contribute to the lower performance of the
Ghanaian sample. The point being made is that given two people of identical
knowledge, one in the US sample and the other in the Ghana sample, the US
participant who uses the graphing calculator could work faster and have a higher
chance of going through the items than the Ghana participants within the 60

minutes allowed. In addition, for some of the questions, drawing a quick graph on

 

25 At the time of this study, senior secondary schools in Ghana were not allowed to use graphing
calculators in their national examinations. As a result, teachers themselves do not learn to use it
for teaching and are not allowed to use them in their classes. It came as no surprise when during
the study none of the participating in-service or pre-service teachers in Ghana was seen using
any graphing calculator.

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the calculator could improve one's chances of getting it right. Under these
conditions, therefore, higher scores from the US participants should not come as

a surprise.

5. 2. 2.3 Differences in the Level of Familiarity to Types of Questions

A third possible explanation, as already mentioned, lies in the differences
in the level of familiarity with the types of questions on the instruments used in
the study. After subjecting the items on the KAT instruments to review by two
mathematics educators in Ghana, they both agreed that the content being
measured by the KAT instrument is covered in the mathematics curriculum of
Ghanaian high schools. However, the items developed here in the US are not
necessarily the type of items the sample in Ghana are used to in their curriculum.
Some of the Ghanaian in-service teachers made anecdotal remarks to this effect
(i.e., their unfamiliarity with some of the items on the instruments) during informal
discussions after the administration of the instruments in Ghana. Those remarks
confirmed for me similar remarks made by a section of mathematicians and
mathematics educators with whom I shared a table with during one of the MSU
item development workshops about an item I had formulated. One of the items I
had formulated for our group’s discussion did not see the light of the day
because, as I was told, “it was not the typical question teachers in the US were
exposed to”. Therefore, the possibility of teachers in the US being familiar with
some of the items could improve their chances of performing better than the

teachers in Ghana.

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Third, compared with data from the KAT study, on both forms, the point-
biserial coefficients obtained in this study were significantly smaller for the items
of both forms. It can therefore be concluded that the items on the instruments
could not discriminate as well between high and low performing candidates in
Ghana as they did among the US participants in KAT at the time of this study.
As has already been mentioned, there was no random selection of teachers for
this study. Therefore, the sample used cannot be said to be representative of
high school mathematics teachers in Ghana. In addition, since there was a small
variation in teaching experience and teaching certification among the in-service
teachers (see Tables 3.4.5 and 3.4.6), the small number of in-service teachers
used in this study (38 out of the total of 339) may not have been large enough to
bring much variation in knowledge of the teachers. The possible lack of enough
variation in knowledge of the small number of participating in-service teachers in
Ghana is important especially because of the centrally controlled national
curriculum used in the country. The teachers in Ghana, at the time of the study,
were all teaching the same topics and the same textbooks. In addition, Ghanaian
high school mathematics teachers collaborate among themselves through in-
school departmental meetings, as well as workshops organized at annual
regional and biennial conferences of the Mathematical Association of Ghana. All
these factors may have been responsible for reducing any differences in
knowledge that existed among these few participating teachers (i.e., 38 in-

service teachers from only eight schools of comparable performance) before their

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teaching career started. These may explain the inability of the KAT instrument to
discriminate much between low and high performers in Ghana.

Fourth, the reliability coefficients obtained in this study were too small
compared with coefficients obtained from the KAT data. This means that the
instruments performed quite well in measuring the underlying construct of
knowledge for teaching algebra in the US but not well in this study (see reliability
coefficients in the Summer 2007 draft technical report for the KAT validation
study). Another interpretation of the reliability coefficient is to consider the nature
of the construct being measured by the instrument. In general, if the construct
being measured (in this case, knowledge for teaching high school algebra) is
multi-dimensional in nature, reliabilities measured by Cronbach's alpha could be
low. From this perspective, the low reliability coefficients obtained in the present
study are an indication that the instrument could be measuring a multi-
dimensional type of knowledge. When this happens, a factor analysis could
reveal the items which load strongest on which dimensions or factors. This
provides one of the bases for the factor analysis performed to answer research

question one.

5. 2.3 How Data from this Study Corroborated the KAT Framework

Related to the first research question, factor analysis was initially done
using three components. Then based on the possibilities of selecting two or three
as elbows of the scree-plot, the analysis was repeated using two components. A
third analysis was also run using eight components based on interpretation of the

eigenvalues obtained by applying the Kaiser-criterion. The idea was to use the

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nature of the factor loadings to determine which of these analyses best explained
the retained factors and therefore, provided evidence of the extent to which the
data on teachers’ performance corroborated the three-dimensional
conceptualization of knowledge for teaching algebra in the theoretical framework.

However, on both forms, the eight-component factor analysis revealed
that only one of the retained factors had three items loading uniquely on it. The
rest had one or two items loading uniquely on them. Therefore, using the
Costello and Osborne (2005) recommendation not to interpret factors when less
than three items are uniquely loaded, the eight-component factor analysis was
not pursued much further because no meaningful interpretation could be drawn
with the resultant small unique factor loadings.

With the three-component factor analysis, the three factors could explain
only about 28% and 31% respectively of the variation in scores on Form 1 and
Form 2. When the analysis was done with two factors, the proportion of the
variation in scores that could be explained dropped to about 21% and 24%
respectively. In addition, there were several cross-loading of items even with the
two- and three-component factor analysis. For instance, on the three-component
factor analysis, four items on Form 1 and five items on Form 2 cross-loaded on
more than one factor. The two-component analysis also resulted in three items
on Form 1 and one item on Form 2 cross-loading on both factors.

In addition, on both the two- and three-component analyses, items
originally categorized as measuring any of the three types of knowledge (school

knowledge, advanced knowledge, and teaching knowledge) did not uniquely load

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on any of the retained factors. In other words, items meant to assess different
types of knowledge loaded together on the retained factors. Consequently, after
the cross-loading items were removed, the nature of the factor loadings did not
permit interpretation of the retained factors from both the two- and three-
component analyses. Even when the item loadings were analyzed in terms of the
other two dimensions of the KAT Item Development Matrix, (i.e., “algebra
content” and “domain of mathematical knowledge”), the nature of the loadings did
not provide evidence for labeling the factors.

These results imply that data from this study could not corroborate the
KAT framework. In other words, the profile of knowledge of the participating pre-
service and in-service high school mathematics teachers in Ghana did not
corroborate the three categories of knowledge, knowledge of school algebra,
advanced knowledge and teaching knowledge, hypothesized in the KAT
framework.

As already discussed in the preceding chapter, the low reliability figures
obtained for the instruments pointed to the possibility that the construct
measured by the instrument- knowledge used by teachers for teaching school
algebra- was multi-dimensional in nature. It was therefore expected that factor
analysis would result in item loadings that could help identify the components.
The inability of the items of the same nature, according to the KAT
categorizations, loading uniquely on any particular factor in all the analysis could

be explained in a number of possible ways.

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First, as has already been mentioned, there was no random selection of
teachers for this study. Therefore, the sample used could not be considered as
representative of the broad variations in knowledge of senior secondary school
mathematics teachers in Ghana. In addition, the small number of in-service
teachers used in this study (38 out of the total of 339) may not have been
sufficient to bring enough variations in knowledge not only among the in-service
teachers, but also between the in-service and pre-service teachers necessary to
be detected by the research instrument.

Second, related to the issue of sample size used for the factor analysis
are the low communalities resulting from this study, which together may have
influenced results of this study. Hogarty, Hines, Kromey, Ferron, and Mumford
(2005) have shown that, in factor analysis, sample size is less likely to influence
the quality of factor solutions when communalities are high than when there are
low communalities. In the case of this study, very low communalities were
obtained. On both forms no item had communality of 0.6 or above (see Tables
4.2.2 and 4.2.10. These low communalities, coupled with the relatively small
sample sizes of 150 and 189 for Forms 1 and 2 respectively (compared with 449
for Form 1 and 392 for Form 2 in the KAT project) may together have contributed
to the poor quality of factor solutions obtained in this study.

Third, there may be the need to take another look at the categorization of
the items. This is because, if the underlying construct being elicited by the item
does not fit the type of knowledge it has been categorized to be measuring, then

the items will load differently than expected. The fact that items categorized to be

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measuring one of the three types of knowledge (school knowledge, advanced
knowledge and teaching knowledge) are not loading uniquely on any of the
factors could, thus, point to possible errors in some of the original
categorizations. This possibility of wrong categorization of some of the items may
have accounted for some of the cross-loadings that resulted from the factor
analysis. The other possibility could be that the KAT framework is not sufficient in
explaining the type of knowledge needed to teach algebra at the high school
level (i.e., the hypothesis of the three distinct dimensions may not be accurate).
In the next two sub-sections two other possible explanations to how data
from this study failed to support the dimensional structure hypothesized in the
KAT framework. The first of these provides a critique of the KAT framework and
uses the critique to offer a possible explanation about why the KAT framework
was not supported by data from this study. After this, an alternative explanation is
given based on how it is possible for a different curriculum context, such as

Ghana’s could lead to different conceptualization.

5. 2.3. 1 Critique of KA T Framework and Instruments

As discussed in chapter two, the KAT project’s conceptualization of
teachers’ knowledge is important in the sense that it attempts to focus on looking
at the knowledge required for teaching in one specific domain of mathematics at
the secondary school level (i.e., algebra). In addition, the work by researchers in
the KAT project towards re-conceptualization of the knowledge used by teachers
in teaching algebra at the high school level and the development of tools for

assessing this type of knowledge has the potential of being valuable to the

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mathematics education community worldwide. However, in trying to work with the
KAT framework and their instruments in Ghana, some shortcomings or potential
issues have come to light, which suggests the possibility that the framework is
not sufficient in explaining the type of knowledge needed to teach algebra at the
high school level.

For instance, in the KAT framework knowledge of mathematics content
that precedes algebra (e.g., the content of arithmetic), so far as the trajectory of
content of school mathematics, has not been separated from the mathematics
content that comes after school algebra. Both types of content, content preceding
and content beyond high school algebra, seem to have been put together in what
the project calls advanced knowledge (refer to the KAT project’s conceptual
representation of advanced knowledge in Figure 2.2.1). It is generally true that, to
be effective, teachers would need to engage in unpacking of content preceding
the content of focus, as well as in bridging of big ideas that come after the
content of interest. Unfortunately, the KAT framework does not cater separately
for knowledge of school mathematics preceding algebra (e.g., arithmetic and
number sense) and the knowledge that comes after algebra. Findings from the
present study, for instance, points to teachers’ advanced knowledge as the form
of knowledge that is most likely to positively relate to students’ performance (see
section 5.2.5). Unfortunately, even in the light of such finding, it is impossible to
discuss which aspect of the KAT project’s advanced knowledge is more helpful to
teachers. This issue is important because even if the KAT framework, in its

present form, becomes corroborated through large scale research, it would still

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be difficult to be explicit as to whether the advanced knowledge teachers need to
teach algebra well is the aspect of knowledge of school mathematics that
precedes algebra or the knowledge that comes after algebra or whether they
both contribute significantly. Separating these two types of knowledge in the
framework will, therefore, help to better understand the aspect of advanced
knowledge teachers need to teach algebra well.

In addition, though the KAT framework acknowledges that the boundary
between the three types of knowledge emphasized in the KAT framework (i.e.,
knowledge of school algebra, advanced knowledge and teaching knowledge) is
blurry, the discussion of the three types of knowledge in the framework is silent
on any conceptualization on the type of knowledge that results from the complex
interaction of the three types of knowledge in flexible ways by teachers. In
practice, teachers use curriculum scripts (Putnam, 1987) or complex structure
composed of interrelated sets of organized actions, which Leinhhardt and
Greeno (1986) have termed schemata. This notion of how teachers transform
their knowledge into pedagogical representations that connect with their prior
knowledge and dispositions of the Ieamer could be seen as a multifaceted
combination of content knowledge (either school or advanced knowledge or both)
and teaching knowledge consistent with Ma’s (1999) conceptualization of PUFM.
Unfortunately, the KAT framework is silent on the type of knowledge produced
from the interaction or combination of two or more of their hypothesized three
types of knowledge. If the framework had taken this into account, the cross-

Ioadings, as well as the loading of items of different categorizations on a common

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factor in this study could have been explained. This, in turn, could have improved
the analyses of the knowledge of teachers in this study’s sample using the KAT
framework. In the face of this, elaboration in the KAT framework is
recommended to incorporate the type of knowledge formed by interactions
among the present three hypothesized knowledge.

Further more, even in its present form, the KAT item development matrix
has 24 different cells. However, the items on the KAT instruments do not span all
the 24 categories. In addition, for the statistical procedures such as factor
analysis, performed to retain three to eight factors, the twenty items on each of
the KAT instruments seems to be too small. Gorsuch (1983) prefers six variables
per factor but suggests a minimum of four variables per factor except in
situations where the factors have been exceptionally defined in previous
research. In addition, the possibility of cross-loadings in factor analysis that could
reduce the number of items loading on each of the factors even when only three
factors are retained. Consequently, on the basis of the theoretical framework, it
seems that twenty items per instrument such as the KAT instruments may affect
the analysis. Further more, a close look at Table 3.5.1 reveals that though the
KAT item development matrix has 24 cells on each of Form 1 and Form 2, the
items fitted only 15 cells. In other words, assuming that each cell has a unique
attribute, the 20 items are showing only 15 of such attributes. This number, 15,
further affects the analysis (e.g., the mode of factor loadings) more than if there
were 20 uniquely categorized items. The implications of all these is that since,

the development of items by the KAT project was in part based on their

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framework, any downside of the framework could mean that the instruments do
not have the right types of items or the full range of items necessary for good
factor solutions. It is important, for future work using the KAT framework, even as
it exists now, to not only consider increasing the number of items but also
ensuring that items that fit each of the cells are incorporated. Due to the
possibility of cross-loadings in Factor Analysis, as was observed in this study,
increasing the number of items on the forms could ensure sufficient numbers of
unique loading items on the factors retained and improve the chances of being
able to label the factors. In addition, ensuring that each of the 24 cells in the item
development matrix has an item on the instrument would help produce a form
with a wider variation of items and possibly improve the quality of factor loadings

as well as the item communalities.

5. 2.3.2 Effect of Curriculum on Conceptualization of Teacher Knowledge

In Ghana Algebra courses are therefore not offered as separate courses
from other aspects of mathematics at the high school level. Instead, both the
core mathematics and elective mathematics offered at the senior secondary
school level are forms of integrated mathematics (see discussion of this in
section four of chapter one). Elective mathematics teachers, for example, teach
all the branches of mathematics to students in an integrated manner (refer to
section 1.4 for major content areas covered in both core and elective
mathematics syllabi). The curriculum for both Core and Elective Mathematics is a
form of spiral curriculum. To use the words of Bruner (1960), the high school

mathematics curriculum “as it develops [from the tenth grade is designed] to

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revisit the basic ideas repeatedly, building upon them [until the twelfth gradel”
(p.13). By this spiral arrangement of topics, elective mathematics teachers, for
example teach mathematics content that covers not only algebra but also
geometry, trigonometry, vectors, mechanics, statistics and probability.

As a result, items on topics such as the calculus in the elective
mathematics syllabus that would be categorized as measuring KAT’s advanced
knowledge could measure school knowledge in the Ghanaian context. A good
example of such an item is the one shown below”, which appeared on one of the
elective mathematics papers of the 2004 senior secondary school certificate

examinations in Ghana.

a) The gradient function of y = px2 + qx + r is y = 8x + 4. The

function has a minimum value of 1. Find the values of p, q, and
r.

b) Find the value of c for which y = 3x + c is perpendicular to the

tangent of y = x2 —1

Such a question, if used on one of the KAT instruments would be
categorized as advanced knowledge. However, in Ghana, it is meant for twelfth
graders and would therefore be categorized as a school knowledge item in the
Ghanaian context.

This issue is important because in Ghanaian high schools a particular
mathematics teacher is expected to teach the entire content of any one of the
two mathematics courses assigned to him or her. Unless a teacher transfer from

a school most schools, especially those who participated in this study will keep

 

26 This was question number 12 of the Elective Mathematics Paper 2 of the SSSCE for twelve
graders in Ghana in 2004

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one teacher with a class from the ninth through the twelfth grade; an
arrangement that makes teachers informally accountable for the success of their
students at the SSSCE. As a result of this arrangement, teachers are given the
opportunity to teach across the entire mathematics syllabus are supported to do
so through collaborations with each other at the departmental, professional
development programs sponsored occasionally by the Ghana Education Service,
and workshops provided by the Mathematical Association of Ghana (MAG) at the
district, regional and national levels.

Consequently, to Ghanaian teachers, many of the advanced knowledge
items may appear to them as would KAT's school knowledge items. This may
cause items categorized as school knowledge by KAT to load together with
advanced knowledge items. And this could possibly explain why the KAT
framework did not work well Ghana.

The foregoing implies that in order to conceptualize teachers’ knowledge
for teaching algebra for a country such as Ghana in which the curriculum and
experience of teachers is different from their counterparts in the US, it is
necessary to take into account such contextual differences. This will ensure that
findings from studies using such conceptualizations would be directly applicable
in the local (i.e., the country’s) context by curriculum developers, teacher
educators and other stakeholders of education. It will also contribute to a more
open discussion among mathematics educators and researchers worldwide
about the need to be cautious about the best way to go to scale worldwide with

ideas, conceptualizations, theories etc. developed in different contexts.

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5. 2.4 Differences Between Prospective and In-service Teachers

Related to the second research question, a number of conclusions can be
drawn. First, though different participants completed Form 1 and Form 2, initial t-
test conducted revealed that, overall, the performance of those who completed
Form 1 was not significantly different from the performance of Form 2
participants, at the .05 level of significance. The implication of this is that should
there be the need to consider using only one of these forms, especially in
situations where large sample sizes could not be guaranteed, anyone one of
them would work just fine.

Second, results of the analysis of variance (ANOVA) discussed in the
preceding section reveal that in Ghana, the sub-group of in-service teachers
performed best on the two forms. The analysis of variance performed on the data
revealed that in-service high school teachers in Ghana performed significantly
better than each sub-group of prospective mathematics teachers majoring in
mathematics, mathematics education and statistics from the country’s
universities. It was therefore concluded that knowledge for teaching algebra of
in-service high school mathematics teachers is significantly different from that of
prospective teachers (i.e., the three categories of university seniors majoring in
mathematics, mathematics education and statistics).

This finding from the study is consistent with the argument by Sherin
(2002) that in the course of teaching new curriculum, especially reform oriented
curricula, teachers adapt their knowledge and in the process develop new

content and pedagogical content knowledge in order to cope with the demands of

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the new curriculum. In Ghana, in-service teachers are required to teach an
integrated mathematics curriculum. While in the university, these teachers took
mathematics courses on different aspects of mathematics not in an integrated
manner. To be successful in teaching the integrated mathematics curriculum at
the high school level, in-service teachers in Ghana have to adapt their
knowledge. They are helped in most cases by new textbooks developed and
revised since the 1987 educational reforms embarked on by the country and the
school-level collaborations that exist among teachers in high schools in the
country. Since the 1987 reform new curriculum materials, including textbooks
have periodically been introduced to help teachers adapt their knowledge and
cope with the demands of the new materials and changes in content. In addition,
the Mathematical Association of Ghana organizes annual conferences at the
regional and district level, as well as biennial conferences at the national level. At
these conferences teachers and mathematics educators share their research and
best practices and provision is made for professional development on areas
where teachers have earlier indicated they need help with. With these changes
and activities constantly going on, teachers could be engaging in various forms of
adaptation that may have improved their content and teaching knowledge
beyond the level at which they were when they were in college. It is therefore not
surprising that in-service teachers outperformed all categories of university
seniors in this study.

Among university students in Ghana, ANOVA revealed differences in the

performances among students in the different majors. In general the

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mathematics majors performed significantly better than their counterparts
majoring in statistics and mathematics education. It is unclear whether these
differences existed before the different categories of students entered the
university or whether the differences are the result of the type of courses they
have taken at the university level. It is hoped, as recommended in Section 5.3
that future research will take this into account in the design. Between the
statistics and mathematics education students, the statistics majors did slightly
better than the mathematics education students. However, this difference was

not significance at the .05 level.

5. 2.5 Relationship Between Teacher and Student Performances

In this study, a linear model was estimated for the relationship between
Ghanaian high school mathematics teachers’ knowledge and the performance of

their teachers. This relationship was estimated by the equation below.

Y = 2.419 + .119X-I- e, where,

Y = the dependent variable representing the mean class score,

X = the independent variable representing the mean of the total teachers’
knowledge (i.e., the mean score of teachers whose classes participated in
the study),

2.419 = the intercept (i.e., the mean class score when the independent variable

and other confounding variables are zero,

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.119 = the slope for Predictor X, where the slope is the mean change in mean
class score for one unit increase in the mean teachers’ total score, holding
all other independent variables constant, and

e = the error term
Thus, in terms of answering the third research question it was concluded

that a linear relationship existed between the performance of in-service teachers

who participated in the study and the students of their classes. This relationship
was estimated by the equation above.

The positive coefficient of X, the mean teachers’ total score, in the
regression model implies that, granted that all possible confounding variables
have been included in the model, a unit increase in mean total score of the
participating in-service teachers would correspond to an increase in mean class
score of their students (by a factor of .119). As was discussed in Chapter 4, this
coefficient of .119 is not statistically significance as would be expected in theory.
Studies in the developing worId have established that teachers’ subject matter
knowledge is a better predictor of student achievement than other home-based
factors (see for instance, Harbison and Hanushek, 1992; Mullens et al., 1996).
However, the findings from this study suggests that in terms of teaching high
school algebra in Ghana, for the schools that participated in the study, teachers
knowledge does not significantly affect students’ performance. Two possible
explanations could be given to explain the non-significant coefficient of teacher

knowledge obtained in this study.

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The first possible explanation has to do with the small number of teachers
who participated in the study as well as the type of students these schools
attract. Apart from the fact the number of sample points (i.e., the ten teachers
and their classes) used for the regression analysis, few schools that participated
in this study were not selected randomly. In addition, at the time of the study, the
participating high schools were among the best performing and most resourced
schools in Ghana. In Ghana, high schools are schools of choice and students
across the country compete at the Basic Education Certificate Examinations for
admission in the traditionally good performing high schools. This makes it
possible for the best performing high schools to attract the cream of students
across the country each year. As a result, there is the possibility that other
factors such as student ability and school resources, not provided in the model,
could be latent in the model. Such hidden factors may have acted as
confounding variables that have introduced systematic errors in the model. It is
the possibility of such systematic errors that may have affected the degree of
significance of the coefficient of the predictor variable (teacher knowledge).
Hence, caution needs to be exercised in interpreting the coefficients or making
generalizations with this model over schools that are not as high performing as
the ones used in the study. As recommended in the next section, making room
for random sampling of schools and increasing the sample size in future studies
could lead to the building of a better or more stable model.

The second possible explanation is to consider this finding in the light of

literature on the influence of teacher knowledge on student performance in

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algebra (see for instance, Begle, 1972; Copeland and Doyle, 1973, and
Eisenberg, 1977). Begle (1972), for instance, found no significant correlation
between teacher knowledge and student performance in algebra. Begle (1972)
explained his finding by arguing that perhaps there is a lower bound of teacher
knowledge below which there is a relationship between teacher knowledge and
student performance. And that this lower bound could be so low that in algebra
there may be no need to worry about the influence of teacher knowledge on
student performance. One of criticisms leveled against the Begle (1972) study
was that he used teachers who were participants of the National Science
Foundation Institute and could therefore be considered highly motivated
teachers. However, the fact that the study by Eisenberg (1977) that took care of
this weakness also produced findings similar to Begle (1972) means that Begle’s
interpretation could not be easily discarded.

Applying Begle’s (1972) interpretation on the traditionally high performing
nature of students from the schools used in this study, it can be argued that
perhaps when very high performing high school students are the focus of the
study, such students could be so motivated that teachers’ knowledge may have
to be very high for its relationship with student performance to be significant. In
other words, for high performing students such as those used in this study,
instead of a lower bound below which teacher knowledge can significantly
influence student performance, it may be necessary to think of a form of
threshold level of teacher knowledge, above which an increase in teacher

knowledge will cause corresponding significant improvement in student

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performance. This is because such high performing and highly motivated
students may be able to understand enough of school algebra on their own with
the availability of curriculum materials (for example, their mathematics
textbooks). This last point is particularly significant because literature on student
performance in developing countries is replete with the fact that school
resources, such as textbooks, are some of the biggest predictors of student
performance in the developing world, after family inputs have been taken care of
(see Hanushek, 1997).

The positive constant term, 2.419, implies that controlling the effect of
teachers’ knowledge the mean class score of students in the participating school
would be positive. In the participating high schools in Ghana parents, through the
Parent Teacher Associations (PTAs), pay for teachers to provide extra tuition for
the students beyond what is stipulated on the academic time tables by the Ghana
Education Service. This improves the performance of the students generally and
especially on the national examinations used for selection into the country’s
universities. As a result of this, children learn a great deal of the content of their
school subjects, including the algebra in the mathematics curriculum outside of
class (and sometimes independent of their teachers). This, together with the
originally strong academic background of the students prior to their admission
into these high schools could explain the high positive constant term in the

model.

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5.3 Recommendations for Further Study

The conclusions drawn from this study have far-reaching implications for
further studies into teacher knowledge and student performance especially in
Ghana. The following options are recommended for further studies:

First, universities in Ghana have other mathematics-related programs from
which students have exited into the field of teaching either through initial national
service postings or sometimes due to the limited job opportunities available in the
country. For instance, currently there are high school mathematics teachers who
majored in various engineering programs or even in economics (with
mathematics as a minor area of emphasis). This study did not involve students
from the full spectrum of all the possible programs. The results of the study
therefore apply to the selected major areas of mathematics, statistics and
mathematics education and it is not known whether university students majoring
in these other areas would compare differently with in-service teachers or with
any of the other groups who took part in this study. Because research in the
developing world has found teachers’ subject matter knowledge as a better
predictor of student performance than students’ home-based factors (see
Harbison and Hanushek, 1992; Mullens et. al., 1996), further research is needed
to include all these groups. This is important because knowing how all the
possible groups compare would help to provide a framework for investigating
what experiences in their programs of study could be responsible for the trend
observed. This, in turn would be useful for professional development of teachers

and for improving pre-service mathematics teacher education in Ghana.

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Second, the limitations imposed on this study due to the small number,
especially of the participating in-service teachers call for the need to get more in-
service teachers involved in further studies such as this. Extending fieldwork to
cover a period of at least one year is one recommended option. This extension
could offset the effects of any unexpected disruption in the school calendar as
happened when teachers went on strike at the time the study was to take off.
Also, spending this extended time in fieldwork could improve the chances of
involving a larger sample of in-service teachers in any future studies. Another
option involves seeking enough funding to pay participants to get more of them
involved. Taking such steps to increase the participation of more in-service
teachers could help investigate how the number of years of teaching experience
could improve or expand teachers’ knowledge.

Third, in theory, it would be expected that the interaction of school
knowledge, advanced knowledge and teaching knoWIedge should produce
significant effects on mean class score. Unfortunately, because of the limited
number of cases that were available for use in estimating the regression model in
this study structural equation modeling could not be employed in estimating a
regression model that included the interaction terms. In addition, the findings of
the factor analysis could not permit multiple regression to be performed as an
alternative model. Consequently, linear regression with a single predictor (the
mean of the teachers’ total scores) was used in the analysis. Further research
involving a larger number of schools and teachers Is needed to increase the

number of cases and study the effects of such interaction terms.

186

In addition, future studies may need to make modification in the KAT
framework to possibly split KAT’s advanced knowledge into knowledge of
mathematics content that precedes school algebra and knowledge that comes
after school algebra. That way, items based, for instance, on concepts in
arithmetic could load on a factor different from those items on content such as
calculus. This modification is necessary because in its present form, if KAT’s
advanced knowledge is found to be significantly positively related to student
performance it will still be difficult to know what aspect of advanced knowledge is
being referred to.

It is further recommended that such re-conceptualization be done to
include the types of knowledge formed as a result of the interaction of the
individual knowledge types since in practice teachers do not only use KAT’s
hypothesized knowledge types in isolation but also rely on a complex interaction
of any two or more of them. Further more, if factor analyses in such future
studies are able to interpret the extracted factors using the KAT framework or a
modified framework that takes into account other types of teacher knowledge
(e.g., knowledge generated from the interaction of the main hypothesized
knowledge types), such studies would contribute to the discussion of which
aspect of teacher knowledge is best related to student achievement. In this
regard, adapting an instrument such as the KAT instruments or modifying them in
the light of new conceptualizations is helpful in moving away from the use of
proxy measures to more direct measures of teacher knowledge. In this way, the

model that is eventually estimated will help answer the question of which aspect

187

of teacher knowledge is the best predictor of student performance at least in
Ghana. Such a contribution is also necessary for curriculum policy maker and
implementers who are responsible for the content of teacher education
curriculum in Ghana. It is also useful for deciding the focus of in-service program
meant aimed at improving high school mathematics teacher knowledge for
improved student performance.

The case for further research using a larger number of schools and
teachers is strengthened by the fact that the coefficient of regression in the
model was found not to be significant. As already explained, taking this step in
future research could help build a more stable model or confirm a model of the
form developed in this study. Findings of such studies have the potential to
contribute to the discussion on the relationship between teacher knowledge and
student performance especially in developing countries such as Ghana. As
already discussed, so far there is widespread disagreement among researchers
about the exact relationship between teacher knowledge and student
achievement. Though part of the disagreement could be due to the proxy
measures so far used to represent teacher knowledge, even among those who
argue for positive effect of teacher knowledge, it is unclear which aspect of
teacher knowledge shows what effect. Further research that uses a larger
number of schools and teachers will not only help build a more stable model, it
can also improve the generalizability of the results across all senior secondary

mathematics teachers in Ghana.

188

In addition, Shulman and Quinlan (1996) have argued that is teachers’
ability to transform their content knowledge into pedagogical actions that connect
to their students’ dispositions could make them effective in their. In this regard, it
is recommended that future studies should include classroom observations and
interviews of teachers about how they prepare for their work and why they
responded in specific ways in class to student difficulties and questions. Including
these observations and interviews in future studies could help contribute to the
discussion of how teachers transform their subject matter knowledge into
effective pedagogical practices.

One aspect of the KAT instruments adapted in this study is that the school
knowledge items were developed with US school curriculum in mind. Therefore,
extracting the school knowledge items and administering the resulting student
instrument to high school students without taking into consideration what
students have had the opportunity to cover in their curriculum could result in false
conclusions. This issue is important especially depending on the grade level of
the students and the time of the school year the study is conducted. For this
reason, initial decisions regarding this study involved administering opportunity-
to-leam forms so students’ performance could be assessed only on the items
they have had the opportunity to learn. Unfortunately, this could not be done, as
several teachers did not agree to complete it. It is therefore recommended that
further studies build in ways of including such opportunity-to-learn measures.

Finally, this study has found that among university seniors, there are

marked differences in the level of the three types of algebra knowledge for

189

teaching. It is not clear whether the differences in knowledge found in this study
were the result of the differences in their university coursework or the differences
in performance prior to entering the different programs at the university level. A
longitudinal study that helps students’ entry knowledge levels to be determined
and the changes in them as they progress in their programs is needed to assess
the effect of their experiences in the growth in their knowledge base. Findings
from such a study could be useful in decisions about curriculum development,
especially for the mathematics education students who are actually being
prepared for the classroom, as well as for professional development of high

school mathematics teachers in Ghana.

190

APPENDICES

Appendix I Content of School Algebra Used by the KAT Project

 

Functions and their Properties: Linear and Nonlinear

 

A. Analyze or interpret linear relationships expressed in symbols, graphs, tables,
diagrams, or written desorptions.

 

B. Generalize a pattern appearing in a numerical sequence or table or graph
using words or symbols. Analyze or create patterns, sequences or functions
given a rule.

 

Understand relations and functions, and select, convert flexibly among, and
use various representations for them.

 

Express the function in general terms (either recursively or explicitly), given a
table, verbal description, or some terms of a sequence.

 

01.0.0

Recognize, describe, or extend numerical and geometric patterns using
tables, graphs, words, or symbols

 

.71

Identify or analyze distinguishing properties of linear and non-linear including:
quadratic, inverse (y = k/x) logarithmic, power, radical, polynomial or
exponential functions from tables, graphs, or equations.

 

G. Understand and perform functional transformations such as arithmetically
combining, composing, and inverting commonly used functions

 

H. Solve problems involving functional concepts such as composition, defining
the inverse function and performing arithmetic operations of functions.

 

l. Describe and analyze functions of one variable by investigating domain and
range, rate of change, intercepts, zeros, and local and global behavior.

 

J. Recognize and analyze the general forms of linear, quadratic, inverse, or
exponential functions (e.g., in y = ax + b, recognize the roles of a and Q). In
other words, analyze functions with parameters as to their behavior and how
they change as parameters change.

 

K. Identify or represent functional relationships in meaningful contexts including
proportional, and common nonlinear (e.g., compound interest, bacterial
growth) in tables, graphs, words, or symbols.

 

L. Express linear and exponential functions in recursive and explicit form given
a table or verbal description

 

 

M. Use an algebraic model of a situation to make inferences or predictions.

N. Given a real-world situation, use a linear, quadratic, inverse, logarithmic,
power, radical, polynomial or exponential function to fit the situation (e.g.,
half-life bacterial growth) and/or use the function to solve problems.

 

O. Approximate and interpret rates of change from symbolic, graphical and
numerical data.

 

 

P. Solve systems of linear or nonlinear systems of equations by graphical
methods or using functional concepts.

 

191

 

Appendix I continued

 

Equations, expressions, and inequalities

 

A. Write algebraic expressions, equations, or inequalities to represent a

sfiuafion.

 

B.

Perform basic operations, using appropriate tools, on linear, polynomial,
logarithmic, exponential and rational expressions (including grouping and
order of multiple operations involving basic operations, exponents, roots,
simplifying, and expanding).

 

Solve linear, rational, or quadratic exponential, polynomial, logarithmic
equafions.

 

Solve linear, rational, quadratic or polynomial inequalities.

 

Use, evaluate, and solve problems involving formulas. Formulas include both
common (e.g. relationship between a circle’s circumference and diameter (C
= pi d), distance and time under constant speed) and more advanced (e.g.,
the volumes and surface areas of three dimensional solids; or such formulas
as: A = P(1 + r)‘, A = Pe”).

 

F.

Analyze situations or solve problems using linear, quadratic equations, or
inequalities symbolically or graphically.

 

 

G. Solve analyze, represent or interpret systems of equations or inequalities

 

Copyright 2006, Knowing Mathematics for Teaching Algebra (KAT) Project, NSF RE C-0337595,
Division of Science and Mathematics Education, Michigan State University. Not for reproduction
or use without written consent of KA T.

192

 

Appendix II Algebra in the Core Mathematics Syllabus in Ghana

 

Functions and their Properties: Linear and Nonlinear

 

A.

Recognize relation between two sets and establish the relation between the
two sets by mapping, distinguish between various types of relations,

 

B.

Identify functions from relations, draw graphs of sets of points lying on a line

 

C.

Solve quadratic functions by factorization and graphical method, solve
simultaneous equations in two variables, one linear one quadratic by
graphical methods only, find solution of analogous equations from a given
equation, use graphs of quadratic function to solve problems, determine
maximum and minimum points.

 

Equations, expressions and inequalities

 

A.

Express statements in mathematical symbols, formulate algebraic
expressions from given situations, evaluate algebraic expression for a given
value, perform operations on simple algebraic fractions with monomial and
binomial denominators

 

Perform simple operations on algebraic expression, multiply two binomial
expressions, change the subject of a relation, factorize algebraic expression
of various forms

 

. Apply difference of two squares to solve problems

 

Find the solution set of linear equations in one variable, illustrate the solution
set of inequalities in one variable on the number line, illustrate graphically the
regions corresponding to inequalities on two variables

 

Translate word problems into mathematical sentences, find solution set of
simultaneous linear equations algebraically and graphically, graphical solution
of inequalities in two variables, translate practical non-linear programming
situations into inequalities and solve graphically

 

Solve quadratic equations by factorization and graphical method, find lines of
symmetry from quadratic graphs, solve simultaneous equations in two
variables, one linear one quadratic by graphical methods only, find solution of
analogous equations from a given equation, solve equations involving
exponents, use logarithms and antilogarithms to evaluate expressions

 

. Use, evaluates, and solves problems involving formulas. Formulas include

both common (e.g. relationship between a circle’s circumference and
diameter; finding area of a ring, surface area of a cone, a pyramid and other
regular solids; and volume of a pipe and other regular solids) and more
advanced (e.g., computing compound interests, depreciation).

 

Solve problems involving direct, indirect, partial variations and joint variation
including their application to real life situations

 

 

Solve and explain problems involving income tax, value added tax and
custom duties; calculate monthly amount payable to social security fund on a
given income

 

193

 

Appendix III Algebra in the Elective Mathematics Syllabus in Ghana

 

Functions and their Properties: Linear and Nonlinear

 

A. Mappings and Functions (recognize the difference between functions and

relations; determine inverse of one-to-one functions, composite functions)

 

B.

Polynomial functions (recognize linear, quadratic and other polynomial
functions; sketch graphs of quadratic functions; use graphs of quadratic
functions to solve problems; determine parallel and perpendicular lines;

 

Rational functions (recognize a rational function; carry out the four basic
operations on rational functions; resolve rational functions into partial fractions

 

Binomial theorem (write down the binomial expansion of expressions with
positive integral indices; application of binomial theorem)

 

. Logarithmic and exponential functions (recognize a logarithmic function, use

the laws of logarithms, draw graphs of logarithmic functions, reducing
relations involving exponents into linear forms using logarithms and draw and
use such linear graphs to predict/estimate the value of the dependent variable
given the value of the independent variable and vice versa)

 

. Linear transformation and matrices (use linear transformations to find image

and object points, find the inverse of a linear transformation, find the
composition of two linear transformations, recognize the identity
transformation, reflections in the x- and y—axes, and in the line y=x, rotation
about the origin and enlargement from a point with a scale factor as some
special linear transformations; state the matrix representing a linear
transformation, recognize some special function (e.g. identity matrix,
triangular matrices), perform algebraic operations on matrices, find the
determinant of a matrix, properties of determinants, and apply determinants
to find area of triangles and quadrilaterals, solve 2 and 3 simultaneous liner
equations, find inverse of a matrix)

 

Equations, expressions and inequalities

 

A. Use the method of completing the square to solve quadratic equations, apply

the concepts of surds to solve equations; use the sum and product of the
roots of quadratic to solve equations, factor and remainder theorems and their
application to cubic expressions and the solution of cubic equations, perform
algebraic operations on polynomial

 

Binary operations on numbers and sets(closure, commutative, associative,
and distributive properties; identity elements and inverse of an element)

 

Sequences and series (recognize a linear sequence including finite and
infinite sequences, find the expression for the general term of the sequence,
obtain the finite sum of a linear sequence; recognize an exponential
sequence, find the expression for the general term of the sequence, obtain
the finite sum of a exponential sequence, find the limit of the sum of an
exponenfialsequence)

 

 

Inequalities in 2 variables (draw graphs of linear inequalities in two variables,
graphical solution of two simultaneous linear inequalities in 2 variables,
analytical method to find solutions to simultaneous linear inequalities)

 

194

 

Appendix IV Tasks of Teaching Defined by the KAT Project

 

KAT Label

Full Explanation

 

Analyzing
students’
mathematical
work and
thinking

a Teacher's mental ideas about what students have in their
heads. Listening and interpreting students’ explanations.

. Determining the mathematical validity of a student strategy,
solution or conjecture.

0 Teachers’ models, representations, or understandings of
the ideas, mental representations, thought processes, or
mathematical reasoning that students may exhibit or
express in some way.

. Interpreting and responding to students’ questions.

0 Teachers’ attempts to understand comments, reasoning, or
processes that students have made.

0 Evaluating students’ work from a mathematical point of
view. This includes challenging students with the need for
using sound mathematical reasoning I argumentation and
what counts as a sound argument.

. Making sense of student thinking even when it is incorrect.

0 Teachers generating questions, problems, or other activities
that will lead students to a deeper understanding of
mathematics.

0 Deciding whether a surprising idea that a student provides
is worth capitalizing on and exploring.

0 Solving mathematical problems that were not anticipated.
Such problems may emerge in the class from students’
observations or questions; sometimes they are problems in
textbooks.

 

Designing,
modifying and
selecting
mathematical
tasks

0 Making the task accessible to a range of learners.

. Modifying a task to fit the needs of particular groups of
Ieamers while preserving the mathematical intentions of the
tasks.

0 Appraising and assessing the mathematical elements of the
task.

0 Teachers generating questions, problems, or other activities
that will lead to a deeper understanding of mathematics.

 

 

Establishing
and revising
mathematical
goals for
students

 

- Consider policy documents.

0 Deciding the central ideas in a given domain, what should
be emphasized.

o Deciding how to approach particular topics.

0 Teachers’ understanding and knowledge of how school
math curriculum develops both across the years and within
one year to build upon and further develop specific math
concepts and their relation to other math concepts.

 

195

 

 

Appendix IV continued

 

0 Evaluating the appropriateness of instructional materials eg.

 

 

 

Accessing technology applications, concrete materials etc for a given
and using math task and a given set of students.
tOOlS and 0 Determine how to integrate or coordinate ideas from across
resources for a set of resources.
teaching . Organizing the use of instructional materials so as to support
student Ieaming of math.
0 Teachers’ attempts to communicate or explain math
concepts, definitions, or ideas from school math to students.
Explaining 0 Teachers’ attempts with students to make connections
mathematical between math concepts.
ideas at?" o Presenting and encouraging multiple representations of
solvrng math concepts.
mathematical 0 Teachers' attempts to explain deep mathematical ideas in 3
problems mathematically accurate and appropriate manner so that
students may be able to understand. This goes beyond
simple definitions or formulas to the “Big ldeas” that may lie
behind them.
. Putting content in more elementary form that is still
intellectually respectable.
0 Teachers find themselves doing math problems publicly in
their classrooms.
- Deciding what a class will hold as mathematical
Building and assumptions, axioms, or starting points for justification and
supporting argument. Such decisions involve trimming.
mathematical 0 Helping students Ieam to interact with one another
community mathematically, respond appropriately to each others’ claims
and discourse and arguments.

 

o Collectively build a body of math knowledge for the
classroom.

. Teachers’ attempts to understand or discern students’
conceptions or misconceptions behind questions or
comments that do not immediately make good mathematical
sense.

0 Teachers’ attempts to locate students’ questions/comments
in mathematical space.

0 Designing instmction and instructional responses to
maximize student interest and motivation.

 

Copyright 2006, Knowing Mathematics for Teaching Algebra (KAT) Project, NSF REC-0337595,
Division of Science and Mathematics Education, Michigan State University. Not for reproduction
or use without written consent of KAT.

196

 

Appendix V Invitation/Permission Letters

Permission from heads of institutions/departmen ts

Eric M. Wilmot

118 Erickson Hall

Teacher Education Department
Michigan State University

East Lansing, MI 48824

Tel: Home (in US): 517-394-0561
Cell (While in Ghana): 0246-226-244
Email: wilmoter@msu.edu

July 12, 2006
Dear Sir/Madam,

This letter is to seek permission and your approval for the conduct of a research
project in your institution or department. The study is being conducted using final
year students at the mathematics and mathematics education departments of the
University of Ghana, Kwame Nkrumah University of Science and Technology,
University of Cape Coast and the University of Education. Also participating are
mathematics teachers in senior secondary schools in Accra, Kumasi, Takoradi
and Cape Coast, all in Ghana and the final year elective mathematics students in
the participating schools. This research project is being conducted for a PhD
dissertation by a Ghanaian doctoral student at the Michigan State University in
the US. It is being conducted between October and December 2006. The
purpose of the study is to investigate the profile of knowledge for teaching
algebra of potential Ghanaian senior secondary school (SSS) mathematics
teachers, as well as, that of in-service SSS mathematics teachers in the selected
senior secondary schools in Ghana.

The study aims at investigating how previous university coursework in
mathematics and/or mathematics education, as well as previous teaching
experiences, if any, of participants is related to the knowledge used in teaching
algebra in the SSS core and the elective mathematics syllabus. In addition, the
study seeks to explore the possible relationship between teachers’ knowledge for
teaching algebra and the achievement of their students on the algebra in the
SSS mathematics curriculum.

To achieve the objectives of the study, a survey and an assessment instrument
will be administered to participants who agree to participate in the study. This

197

instrument has been adapted from the Knowing Mathematics for Teaching
Algebra project currently underway at the Michigan State University in the US. It
has been designed to take not more than one hour to complete in a single
session. The aspect of the instrument for you consists of items based on the
algebra in the SSS mathematics syllabus; some advanced knowledge items, as
well as, items related to the task of teaching. In fact you have the right to accept
or not to accept your institution’s participation in the study.

All the data collected from the study will be confidential, to the maximum extent
allowable bylaw. It will be kept in a safe place, under lock and key. To ensure
confidentiality of the students and teachers who agree to participate in the
research, no identifiers like your names, or any personal data that could be
traced to participants will be required of them in completing the instrument. All
the names of the university or institution in which you are currently enrolled or are
teaching will be changed in the dissertation write-up and any subsequent
publications. Finally, because the study is not part of your usual course
assessment, your responses will not be shared with any lecturer in your
academic department or institution; neither will your performance be compared
with those in the other universities or institutions for the purpose of making any
claims about the different universities or schools participating in the study. Only
the researcher will have access to the data generated from the study.

All the data collected from the study will be confidential, to the maximum extent
allowable by law. It will be kept in a safe place, under lock and key. To assure
confidentiality of the students’ who agree to participate in the research, no
identifiers like their names, or any personal data will be required of them in
completing the instrument. All the names of the university in which they are
currently enrolled will be changed in the dissertation write-up and any
subsequent publications. Finally, because the study is not part of your students’
or teachers’ usual assessment, their responses will not be shared with anyone in
your academic department or institution. Neither will their performance be
compared with those in the other institutions. Only the researcher and the
professors supervising this dissertation study will have access to the data
generated from the study.

There are no known risks associated with participation in this study. Teachers
and students who agree to participate in the study may benefit from working
through and reflecting on some of the questions related to mathematics teaching
and Ieaming. For teachers, this reflection will alert them to possible areas they
need to research further in order to improve their teaching. For the students,
such a reflection is likely to expose areas they need further reading on.

It is hoped that the finding of this study will contribute to a better understanding of
the knowledge Ghanaian teachers and potential teachers have for teaching
algebra at the SSS level and how it relates to improved instruction and student
learning in Ghana. Such an understanding could be helpful in future curriculum

198

planning in mathematics at the SSS level, as well as, pre-service teacher
education and professional development of the nation’s senior secondary school
mathematics teachers.

You have the opportunity to ask questions and express concerns to me, the
researcher, about your students' and/or teachers’ participation in the research, as
an ongoing process. You will be provided with a statement of any significant new
findings developed during the course of the research that may relate to your
students' and/or teachers’ willingness to continue participating. You will be given
a copy of your signed consent form.

There might be occasions when you might need to consult me, my dissertation
directors, or the Director of Human Research Protections at the Michigan State
University. Such occasions might be when you are concerned about the
confidential nature of your child/ward’s participation in the research or that of
his/her institution.

If you have any questions about the research or research related inquiries, you
may contact me directly or by phone: (517)394-0561, or e-mail:
wilmoter@msu.edu or regular mail: 118 Erickson Hall, East Lansing, MI 48910.

Alternatively, if you have questions about the research or research related
inquiries you may contact — anonymously, if you wish, my dissertation directors
whose names, contact addresses, telephone numbers and e-mail addresses are
shown below:

Prof. Joan Ferrini-Mundy Prof. Sharon L. Senk

Assoc. Dean & Dir, Sci & lVlath Ed D320 Wells Hall

211 N Kedzie Hall Mathematics Department
Michigan State University Michigan State University
East Lansing MI 48824-1031 East Lansing MI 48824-1027
Phone: (517) 432-1490 Phone: (517) 353-4691
Email: jferrini@msu.edu Email: senk@msu.edu

If you have any questions or concerns regarding your rights as a study
participant, or are dissatisfied at any time with any aspect of the study, you may
contact - anonymously, if you wish - Dr. Peter Vasilenko, Director of human
research protections by phone: (517)-355-2180, fax: (517)-432-4503, email:
irb@msu.edu, or regular mail: 202 Olds Hall, Michigan State University, East
Lansing, MI 48824

In signing below, you grant permission for students’ written work to be included in
the research. The participation of students of your department in the study is

199

completely voluntary. You reserve the right to refuse your institution’s
participation in this project. If you give consent now, you still reserve the right to
discontinue your institution’s participation in the study at any time, without giving
reasons.

Please, feel free to contact me directly if you have any concerns or questions.

Sincerely,

Eric M Wilmot
Teacher Education Department
Michigan State University

Please put your signature in the spaces provided below to indicate your
acceptance to permit the researcher to conduct the study in your
department/institution.

 

 

Your signature below indicates your voluntary agreement to allow your
institution/department to participate in this study:

Name & Signature: Date:

 

 

200

Invitation of university students and in-service teachers

Eric M. Wilmot

118 Erickson Hall

Teacher Education Department
Michigan State University

East Lansing, MI 48824

Tel: Home (in US): 517-394-0561
Cell (While in Ghana): 0246-226-244
Email: wilmoter@msu.edu

July 12, 2006

Dear Sir/Madam,

This letter is to invite you to participate in a research project, which is being
conducted using final year students at the mathematics and mathematics
education departments of the University of Ghana, Kwame Nkrumah University
of Science and Technology, University of Cape Coast and the University of
Education. Also participating are mathematics teachers in senior secondary
schools in Accra, Kumasi, Takoradi and Cape Coast, all in Ghana and the final
year elective mathematics students in the participating schools. The project is
being conducted for a PhD dissertation and is being conducted between May,
2006 and June 2006.

The purpose of the study is to investigate the profile of knowledge for teaching
algebra of potential Ghanaian senior secondary school (SSS) mathematics
teachers, as well as, that of in-service SSS mathematics teachers in the selected
senior secondary schools in Ghana. The study aims at investigating how
previous university coursework in mathematics and/or mathematics education, as
well as previous teaching experiences, if any, of participants is related to the
knowledge used in teaching algebra in the SSS core and the elective
mathematics syllabus. In addition, the study seeks to explore the possible
relation ship between teachers’ knowledge for teaching algebra and the
achievement of their students on the algebra in the SSS mathematics curriculum.
To achieve the objectives of the study, therefore, a survey and an assessment
instrument will be administered to participants who voluntarily agree to participate
in the study. This instrument has been designed to take not more than one hour
to complete in a single session. The aspect of the instrument for you will consist
of items based on the algebra in the SSS mathematics syllabus, as well as, some
advanced knowledge items and items related to the task of teaching.

It is hoped that the findings of this study will contribute to a better understanding
of the knowledge Ghanaian teachers and potential teachers have for teaching

201

algebra at the SSS level and how it relates to improved instruction and student
learning in Ghana. Such an understanding could be helpful in future curriculum
planning in mathematics at the SSS level, as well as, pre-service teacher
education and professional development of the nation’s senior secondary school
mathematics teachers.

Eligibility and criteria
You are being invited to participate in this study because you meet any one of
the following criteria:

You are a final year student of good academic standing in the mathematics
department or in mathematics education of your university and have the potential
of being able to teach mathematics at the SSS level should you be posted to one
of the senior secondary schools for your national service.

You have in the past taught or are now teaching Core or Elective Mathematics in
one of the senior secondary schools in Accra, Kumasi, Takoradi or Cape Coast,
all in Ghana.

Due to the fact that your participation in the study is voluntary, you may decline to
participate in the research or withdraw from the research anytime you wish,
without providing reasons for doing so. In the event of this happening, there will
be no penalty for your withdrawal.

How your privacy will be protected

All the data collected from the study will be confidential, to the maximum extent
allowable by law. It will be kept in a safe place, under lock and key. To assure
confidentiality of your participation in the research, no identifiers like your name,
the name of your institution or any personal data traceable to you will be required
of you in completing the instrument. In addition, all names, including that of the
university or school you are currently enrolled in, or the school in which you have
taught or are currently teaching will be changed in the dissertation write-up and
any subsequent publications. Finally, because the study will be conducted under
the auspices of your school or academic department, permission will be sought
from the head of your school or department. However, your performance will not
be shared with your academic department or institution. Only the researcher will
have access to the data generated from the study.

Risks and benefits

There are no known risks associated with participation in this study. Teachers
and students who agree to voluntarily participate in the study may benefit from
working through and reflecting on some of the questions related to mathematics
teaching and learning. For teachers, this reflection will alert them to possible
areas they need to research further in order to improve their teaching. For the
students, such a reflection is likely to expose areas they need further reading on.
The privacy and confidentiality measures being taken by the researcher are

202

aimed at protecting you from possible consequences as a result of your
responses.

You will have the opportunity to ask questions and express concerns to me, the
researcher, about your participation in the research, as an ongoing process. You
will be provided with a statement of any significant new findings developed during
the course of the research that may relate to your willingness to continue
participating. You will be given a consent form to sign to indicate your voluntary
agreement to participate in the study. And you will be given a copy of your signed
consent form for your records.

There might be occasions when you might need to consult me, my dissertation
directors, or the Director of Human Research Protections at the Michigan State
University. Such occasions might be when you are concerned about the
confidential nature of your child/ward’s participation in the research or that of
his/her institution.

If you have any questions about the research or research related inquiries, you
may contact me directly or by phone: (517)394-0561, or e-mail:
wilmoter@msu.edu or regular mail: 118 Erickson Hall, East Lansing, MI 48910.

Alternatively, if you have questions about the research or research related
inquiries you may contact - anonymously, if you wish, my dissertation directors
whose names, contact addresses, telephone numbers and e-mail addresses are
shown below:

Prof. Joan Ferrini-Mundy Prof. Sharon L. Senk

Assoc. Dean & Dir, Sci & Math Ed D320 Wells Hall

211 N Kedzie Hall Mathematics Department
Michigan State University Michigan State University
East Lansing MI 48824-1031 East Lansing MI 48824-1027
Phone: (517)432-1490 Phone: (517) 353-4691
Email: iferrini@msu.edu Email: senk@msu.edu

 

If you have any questions or concerns regarding your rights as a study
participant, or are dissatisfied at any time with any aspect of the study, you may
contact — anonymously, if you wish — Dr. Peter Vasilenko, Director of human
research protections by phone: (517)—355-2180, fax: (517)-432-4503, email:
irb@msu.edu, or regular mail: 202 Olds Hall, Michigan State University, East
Lansing, MI 48824

Yours sincerely,

Eric Wilmot

203

Appendix Vl Consent Form

Consent Form for University Students and ln-Service Teachers

This research project involves final year students at the mathematics and
mathematics education departments of the University of Ghana, Kwame
Nkrumah University of Science and Technology, University of Cape Coast and
the University of Education. Also participating are mathematics teachers in senior
secondary schools in Accra, Kumasi, Takoradi and Cape Coast, all in Ghana and
the final year elective mathematics students in the participating schools. The
study is being conducted between October and December 2006.

The purpose of the study is to investigate the profile of knowledge for teaching
algebra of potential Ghanaian senior secondary school (SSS) mathematics
teachers, as well as, that of in-service SSS mathematics teachers in the selected
senior secondary schools in Ghana. The study aims at investigating how
previous university coursework in mathematics and/or mathematics education, as
well as previous teaching experiences, if any, of participants is related to the
knowledge used in teaching algebra in the SSS core and the elective
mathematics syllabus. In addition, the study seeks to explore the possible
relationship between teachers’ knowledge for teaching algebra and the
achievement of their students on the algebra in the SSS mathematics curriculum.
To achieve the objectives of the study, the following survey and assessment
instrument have been designed to be administered to participants who agree to
participate in the study. This instrument has been adapted from the Knowing
Mathematics for Teaching Algebra project currently underway at the Michigan
State University in the US. It has been designed to take not more than one hour
to complete in a single session. The aspect of the instrument for you consists of
items based on the algebra in the SSS mathematics syllabus, as well as, some
advanced knowledge items and items related to the task of teaching.

All the data collected from the study will be confidential, to the maximum extent
allowable by law. It will be kept in a safe place, under lock and key. To assure
confidentiality of your participation in the research, no identifiers like your name,
the name of your institution or any personal data traceable to you will be required
of you in completing the instrument. In addition, all names, including that of the
university or school you are currently enrolled in, or the school in which you have
taught or are currently teaching will be changed in the dissertation write-up and
any subsequent publications. Finally, because the study will be conducted under
the auspices of your school or academic department, permission will be sought
from the head of your school or department. However, your performance will not
be shared with your academic department or institution. Only the researcher and
the professors supervising this dissertation study will have access to the data
generated from the study.

204

There are no known risks associated with participation in this study. Teachers
and students who agree to participate in the study may benefit from working
through and reflecting on some of the questions related to mathematics teaching
and Ieaming. For teachers, this reflection will alert them to possible areas they
need to research further in order to improve their teaching. For the students,
such a reflection is likely to expose areas they need further reading on. The
privacy and confidentiality measures being taken by the researcher are aimed at
protecting you from possible consequences as a result of your responses.

It is hoped that the findings of this study will contribute to a better understanding
of the knowledge Ghanaian teachers and potential teachers have for teaching
algebra at the SSS level and how it relates to improved instruction and student
Ieaming in Ghana. Such an understanding could be helpful in future curriculum
planning in mathematics at the SSS level, as well as, pre-service teacher
education and professional development of the nation’s senior secondary school
mathematics teachers.

The research project is being conducted for a PhD dissertation by a Ghanaian
doctoral student at the Michigan State University in the US. Due to the fact that
your participation in the study is voluntary, you may decline to participate in the
research or withdraw from the research anytime you wish, without providing
reasons for doing so. In the event of this happening, there will be no penalty for
your withdrawal. If you voluntarily agree to participate in the research, , you will
have the opportunity to ask questions and express concerns to me, the
researcher, about your participation in the research, in the course of the study, as
an ongoing process. You will be provided with a statement of any significant new
findings developed during the course of the research that may relate to your
willingness to continue participating. You will be given a copy of your signed
consent form.

There might be occasions when you might need to consult me, my dissertation
directors, or the Director of Human Research Protections at the Michigan State
University. Such occasions might be when you are concerned about the
confidential nature of your child/ward’s participation in the research or that of
his/her institution.

If you have any questions about the research or research related inquiries you
may contact me directly or by phone: (517)394-0561, or e-mail:
wilmoter@msu.edu or regular mail: 118 Erickson Hall, East Lansing, MI 48910.

Alternatively, if you have questions about the research or research related
inquiries you may contact - anonymously, if you wish, my dissertation directors
whose names, contact addresses, telephone numbers and e-mail addresses are
shown below:

205

Prof. Joan Ferrini-Mundy Prof. Sharon L. Senk
Assoc. Dean & Dir, Sci & Math Ed D320 Wells Hall

211 N Kedzie Hall Mathematics Department
Michigan State University Michigan State University
East Lansing MI 48824-1031 East Lansing MI 48824-1027
Phone: (517) 432-1490 Phone: (517) 353-4691
Email: jferrini@msu.edu Email: senk@msu.edu

If you have any questions or concerns regarding your rights as a study
participant, or are dissatisfied at any time with any aspect of the study, you may
contact — anonymously, if you wish — Dr. Peter Vasilenko, Director of Human
Research Protections by phone: (517)-355-2180, fax: (517)-432-4503, email:
irb@msu.edu, or regular mail: 202 Olds Hall, Michigan State University, East
Lansing, MI 48824

Please put your signature in the spaces provided below to indicate your
acceptance to participate in the research. The absence of a signature indicates
your decision not to participate in the study.

 

 

Your signature below indicates your voluntary agreement to participate in
this study:

Name & Signature: Date:

 

 

206

Appendix VII High School Students’ Assent Form

Secondary School Students’ Assent Form

The purpose of this study is to investigate the profile of knowledge for teaching
algebra of potential Ghanaian senior secondary school (SSS) mathematics
teachers, as well as that of in-service SSS mathematics teachers in the selected
senior secondary schools in Ghana. The study aims at investigating how
previous university coursework in mathematics and/or mathematics education, as
well as previous teaching experiences, if any, of participants is related to the
knowledge used in teaching algebra in the SSS core and the elective
mathematics syllabus. In addition, the study seeks to explore the possible
relationship between teachers’ knowledge for teaching algebra and the
achievement of their students on the algebra in the SSS mathematics curriculum.

The following survey and assessment instrument have been designed to take not
more than one hour to complete in a single session. The content items are based
on the algebra in the SSS mathematics syllabus. It is not compulsory that you
participate. In fact, you have the right to accept or not to accept to participate in
the study.

All the data collected from the study will be confidential, to the maximum extent
allowable by law. It will be kept in a safe place, under lock and key. To ensure
your confidentiality, no identifiers like your name, or any personal data is needed
to complete the instrument. All the names of the institution in which you are
currently enrolled will be changed in the dissertation write-up and any
subsequent publications. The study is not part of your usual school assessment.
As a result, your responses and performance will not be shared with anyone in
your institution, including your teachers and headmaster/headmistress. Neither
will your performance be compared with students in other institutions. Only the
researcher and the professors supervising this dissertation study will have
access to the data generated from the study.

There are no known risks associated with your participation. There will be no
monetary or other types of compensation for your participation. Since the items
are all based on your mathematics curriculum, you may benefit from working and
reflecting on the questions. Such a reflection is likely to expose areas you may
need further reading on, as well as, improve your confidence on areas you have
mastered in your curriculum.

It is hoped that your participation in this study will contribute to a better
understanding of the knowledge Ghanaian teachers and potential teachers have
for teaching algebra at the SSS level and how it relates to improved instruction
and student learning in Ghana. Such an understanding could be helpful in future
curriculum planning in mathematics at the SSS level, as well as, pre-service

207

teacher education and professional development of the nation’s senior secondary
school mathematics teachers.

This research project is being conducted for a PhD dissertation by a Ghanaian
doctoral student at the Michigan State University in the US. It is being conducted
between October and December 2006. Your parents have been notified about
your participation in this study. In spite of that parental notification, your personal
agreement to participate is still essential. Due to the fact that your participation in
the study is voluntary, you have the right to decline to participate in the research
or may withdraw from the research anytime you wish, without providing reasons
for doing so. In the event of this happening, there will be no penalty for your
withdrawal.

If you voluntarily agree to participate in the research, please put your signature in
the space provided below to indicate your agreement to participate in the study.

 

 

Yours signature below indicates your voluntary agreement to participate in
this study.

Name & Signature: Date:

 

 

208

Appendix VIII Parental Notification for High School Students

Eric M. Wilmot

118 Erickson Hall

Teacher Education Department
Michigan State University

East Lansing, MI 48824

Tel: Home (in US): 517-394-0561

Cell (While in Ghana): 0246-226-244
Email: wilmoter@msu.edu
July 12, 2006

Dear Sir/Madam,

This letter is to inform you of the conduct of a research project in your
child/ward’s senior secondary school. The study is being conducted using final
year students at the mathematics and mathematics education departments of the
University of Ghana, Kwame Nkrumah University of Science and Technology,
University of Cape Coast and the University of Education. Also participating are
mathematics teachers in senior secondary schools in Accra, Kumasi, Takoradi
and Cape Coast, all in Ghana and the final year elective mathematics students in
the participating schools.

The purpose of the study is to investigate the profile of knowledge for teaching
algebra of potential Ghanaian senior secondary school (SSS) mathematics
teachers, as well as, that of in-service SSS mathematics teachers in the selected
senior secondary schools in Ghana. The study aims at investigating how
previous university coursework in mathematics and/or mathematics education, as
well as previous teaching experiences, if any, of participants is related to the
knowledge used in teaching algebra in the SSS core and the elective
mathematics syllabus. In addition, the study seeks to explore the possible
relationship between teachers’ knowledge for teaching algebra and the
achievement of their students on the algebra in the SSS mathematics curriculum.

To achieve the objectives of the study, therefore, a survey and an assessment
instrument will be administered to participants who agree to participate in the
study. This instrument has been adapted from the Knowing Mathematics for
Teaching Algebra project currently undenrvay at the Michigan State University in
the US. It has been designed to take not more than one hour to complete in a
single session. The aspect of the instrument for your child or ward consists of
items based on the algebra in the SSS mathematics syllabus. In addition, there is
a second part of the study that will involve observing and recording lessons of

209

selected number of teachers. It is not compulsory that you allow your child to
participate. In fact you have the right to accept or not to accept your child’s
participation in the study.

Apart from this notification, your child will also be given the opportunity to decide
to participate in the study or not. Your child’s participation is not compulsory. The
study is not part of his/her usual assessment in school so his/her decision not to

participate will not cause her to lose anything at school.

All the data collected from the study will be confidential, to the maximum extent
allowable by law. It will be kept in a safe place, under lock and key. To ensure
confidentiality of the students and teachers who agree to participate in the
research, no identifiers like their names, or any personal data will be required of
them in completing the instrument. All the names of the institution in which they
are currently enrolled or are teaching will be changed in the dissertation write-up
and any subsequent publications. Finally, because the study is not part of your
child’s usual course assessment, his/her responses will not be shared with
anyone in their institution. Neither will his/her performance be compared with
those in the other institutions. Only the researcher and the professors supervising
this dissertation study will have access to the data generated from the study.

There are no known risks associated with your child/ward’s participation in this
study. There will be no monetary or other types of compensation for your
participation. Your child/ward may benefit from working through and reflecting on
the questions which are all related to mathematics (3) he is learning. Such a
reflection is likely to expose areas (s)he may need further reading on as well as
improve his/her confidence on the mathematics (s)he is currently Ieaming. The
privacy and confidentiality measures being taken by the researcher are aimed at
protecting your child/ward from possible consequences as a result of his/her
responses.

It is hoped that the findings of this study will contribute to a better understanding
of the knowledge Ghanaian teachers and potential teachers have for teaching
algebra at the SSS level and how it relates to improved instruction and student
learning in Ghana. Such an understanding could be helpful in future curriculum
planning in mathematics at the SSS level, as well as, pre-service teacher
education and professional development of the nation’s senior secondary school
mathematics teachers.

This research project is being conducted for a PhD dissertation by a Ghanaian
doctoral student at the Michigan State University in the US. It is being conducted
between October and December 2006. Due to the fact that your child’s
participation in the study is voluntary, you may decline to agree that helshe
participates in the research or ask him/her to withdraw from the research anytime
you wish, without providing reasons for doing so. In the event of this happening,
there will be no penalty for his/her withdrawal.

210

During the course of the project, you have the opportunity to ask questions and
express concerns to me, the researcher, about your child/ward’s participation in
the research, as an ongoing process. You will be provided with a statement of
any significant new findings developed during the course of the research that
may relate to your willingness to allow your child/ward to continue participating.

There might be occasions when you might need to consult me, my dissertation
directors, or the Director of Human Research Protections at the Michigan State
University. Such occasions might be when you are concerned about the
confidential nature of your child/ward’s participation in the research or that of
his/her institution.

If you have any questions about the research or research related inquiries you
may contact me directly or by phone: (517)394-0561, or e-mail:
wilmoter@msu.edu or regular mail: 118 Erickson Hall, East Lansing, MI 48910.

Alternatively, if you have questions about the research or research related
inquiries you may contact — anonymously, if you wish, my dissertation directors
whose names, contact addresses, telephone numbers and e-mail addresses are
shown below:

Prof. Joan Ferrini-Mundy Prof. Sharon L. Senk

Assoc. Dean & Dir, Sci & Math Ed D320 Wells Hall

211 N Kedzie Hall Mathematics Department
Michigan State University Michigan State University
East Lansing MI 48824-1031 East Lansing MI 48824-1027
Phone: (517) 432-1490 Phone: (517) 353-4691
Email: jferrini@msu.edu Email: senk@msu.edu

If you have any questions or concerns regarding your child’s rights as a study
participant, or are dissatisfied at any time with any aspect of the study, you may
contact - anonymously, if you wish — Dr. Peter Vasilenko, Director of human
research protections by phone: (517)-355-2180, fax: (517)432-4503, email:
irb@msu.edu, or regular mail: 202 Olds Hall, Michigan State University, East
Lansing, MI 48824

Sincerely,

Eric M Wilmot
Michigan State University

211

Appendix IX Public Released Items of the KAT Project

1. Which of the following situations can be modeled using an exponential

function?

i. The height h of a ball tseconds after it is thrown into the air.
ii. The population P of a community after 1 years with an increase of n
people annually.

iii. The value Vof a car after t years if it depreciates d% per year.

A. i only

B. ii only

C. iii only

D. i and ii only

E. ii and iii only

2. For which of the following sets S is the following statement true?

For all a and b in S, if ab = 0, then either a = 0 or b = 0.

i. the set of real numbers
i. the set of complex numbers

iii. the set of integers mod 6

212

iv. the set of integers mod 5

v. the set of 2x2 matrices with real number entries

i only
i and ii only
i, ii and iv only

i, ii, iii and iv only

91.0.01”?

i, ii, iii, iv, and v

3. A student solved the equation, 3(n - 7) = 4 — n and obtained the solution
n=215

What might the student have done wrong?

Copyright 2006, Knowing Mathematics for Teaching Algebra (KA 7') Project, NSF RE C-0337595,
Division of Science and Mathematics Education, Michigan State University. Not for reproduction
or use without written consent of KA T.

213

 

4. Hot tubs and swimming pools are
sometimes surrounded by borders of tiles.
The drawing at the right shows a square
hot tub with sides of length s feet. This tub
is surrounded by a border of 1 foot by 1 foot 3
square tiles.

 

 

 

 

 

 

 

How many 1-foot square tiles will be needed for the border of this pool?

a. Paul wrote the following expression:
25 + 2(s+2)
Explain how Paul might have come up with his expression.

b. Bill found the following expression:
(5 + 2)2 — s2
Explain how Bill might have found his expression.

0. How would you convince the students in your class that the two
expressions above are equivalent?

Copyright 2006, Knowing Mathematics for Teaching Algebra (KAT) Project, NSF REC-0337595,
Division of Science and Mathematics Education, Michigan State University. Not for reproduction
or use without written consent of KAT.

214

Appendix X Sample Opportunity to Learn Questions

1. Which of the following situations can be modeled using an exponential
function?
i. The height h of a ball tseconds after it is thrown into the air.
ii. The population P of a community after t years with an increase of n
people annually.

iii. The value Vof a car after t years if it depreciates d% per year.

A. ionly
B. ii only
C. iii only
D. iand ii only
E. ii and iii only

Your students’ opportunity to learn the mathematics in the question
above:

Please, indicate whether or not your students had the opportunity to Ieam the
mathematics needed to answer the question above.

A. Yes, I taught it because it is part of the required curriculum

B Yes, I taught it, even though it is not part of the required curriculum
C. No, it is not part of the required curriculum so I did not teach it

D

No, even though it was part of the required curriculum, I did not/have not
taughtfi.

Copyright 2006, Knowing Mathematics for Teaching Algebra (KAT) Project, NSF RE C-0337595,
Division of Science and Mathematics Education, Michigan State University. Not for reproduction
or use without written consent of KAT.

215

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