addressing math content knowledge and math anxiety in a
TRANSCRIPT
Addressing Math Content Knowledge and Math Anxiety in a Teacher Education Program
By
Pamela Brittain
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Department of Curriculum, Teaching and Learning Ontario Institute for Studies in Education
University of Toronto
© Copyright by Pamela Brittain 2021
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Addressing Math Content Knowledge and Math Anxiety in a Teacher Education Program
Pamela Brittain Doctor of Philosophy 2021
Department of Curriculum, Teaching and Learning Ontario Institute for Studies in Education
University of Toronto
Abstract
This study is an in-depth mixed methods study of a math content knowledge (MCK)
course from a large scale, urban university’s faculty of education teacher program. The focus of
this research is to describe the effect of this course on the math content knowledge and math
anxiety levels of elementary teacher candidates. The participants were the course creators, the
instructors and the teacher candidates. Quantitative data, in the form of pre- and post-course
diagnostic tests and anxiety scales (collected by the course creators), and qualitative interviews
were assessed to answer the research questions.
This research found that the course had a significant effect on both improving the math
content knowledge and decreasing the math anxiety levels of the teacher candidates enrolled in
the course. It also helped the teacher candidates to improve their self-efficacy and confidence
with mathematics. From an administrative perspective, the course seems to have met the
expectations that teacher candidates improve their scores on the diagnostic tests and the teacher
candidates interviewed for this study found a direct benefit from the course. In addition, there
were suggestions for improvement in the areas of instructor selection, breadth and depth of
topics covered in the course, and the creation of targeted course materials.
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Implications of this research indicate that this course could be applied to other faculties of
education, that it could be improved to more closely align with the newly implemented
provincial math proficiency testing, and that there are some suggestions for expanding the
effectiveness of the course. Further areas of research are also provided.
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Acknowledgements
I know I am going to miss someone, so let me start with a sincere thanks to everyone who
helped me on this incredible journey; if you were missed here, you were not missed in my heart!
My mentors: Kumar Murty, who gave me a chance to build something great and then
pushed me to pursue my own dreams. Gurpreet Sahmbi, who provided insights and guidance
through this whole process. Mary Reid and Jim Hewitt, my amazing committee, who also gave
me this amazing topic of study.
My friends: Andrea MacLeod and Carolyn Hoessler, my wonderful, amazing, supportive
book club, who helped me learn to be brave and limitless and have been with me through almost
this entire process. Jess, Chad, Melissa, Jeff, Claire, Josh, Kate and Shane, who helped keep me
sane with regular online trivia nights during a crazy, pandemic fueled final year.
My family: Peter and Lori, my incredible parents, who have always believed in me and
have spent the past four years telling everyone they meet about their Dr. Daughter! Herb and
Val, my wonderful in-laws, who have been just as proud of me as if I was their own. My puppies
(WJ and R) who spent hours on the couch with me while I struggled to write that perfect
sentence, and my kitties F and H (who sadly could not make it to the end, but was always there).
My supervisor: Doug McDougall, who went through every line, every edit, every indent,
every piece of formatting and every, single, comma; even working over Christmas break to
ensure I could defend and graduate by June. Thank you for everything, I am so proud and
honoured to be the 50th student you have helped to reach this point!
Finally, my love: Justin, whose love and support got me through every step of the way,
and I know this entire thing would not be possible without him. I owe him so very, very much!
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Table of Contents
Acknowledgements iv
Table of Contents v
Chapter One: Introduction 1
1.1 Introduction 1 1.2 Research Context 2 1.3 Purpose of the Study 3 1.4 Statement of the Problem 4 1.5 Significance of the Study 5 1.6 Background of the Researcher 6 1.7 Limitations of the Study 7 1.8 Plan of the Thesis 8
Chapter Two: Literature Review 11
2.1 Introduction 11 2.2 Math Anxiety 11
2.2.1 Teacher Beliefs and Influences 15 2.2.2 Gender Differences 16 2.2.3 Mathematics and Equity 18
2.3 Strategies to Address Math Anxiety 19 2.3.1 Self-Efficacy and Learning to Fail 21 2.3.2 Math Content Knowledge 22
2.3.3 Numeracy as a Foundation for Math 23 2.3.4 Teaching Methods 24
2.4 Elementary Teacher Candidate Requirements 25 2.4.1 Provincial Math Teacher Qualifying Test 29
2.4.2 Building and Evaluating Math Content Knowledge 30 2.4.3 Teaching Teacher Candidates Mathematics 32
2.5 Summary 34
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Chapter Three: Methodology 36
3.1 Introduction 36 3.2 Research Design 37
3.2.1 The Math Content Knowledge Course 38 3.2.1.1 Structure 41
3.3 Participants 43 3.4 Data Collection 43
3.4.1 Interviews 43 3.4.2 Surveys 44
3.5 Data Analysis 44 3.6 Ethical Considerations 45
Chapter Four: Findings 47
4.1 Introduction 47 4.2 The MCK Course 47
4.2.1 The Course Structure 49 4.3 The Course Creators 50
4.3.1 Henry 50 4.3.2 Lila 52
4.4 Course Expectations 54
4.5 Distribution of Data and Initial Findings 56 4.5.1 Diagnostic and Summative Test Results 56 4.5.2 Anxiety and Confidence 59
4.5.2.1 Correlation Between Math Content Knowledge and Math Anxiety 64 4.5.3 Distribution of Scores by Topic 66 4.5.4 Individual Question Analysis 68 4.5.5 Distribution of Teacher Candidates into Tiers 72 4.5.5.1 Tier 1 Analysis 75 4.5.5.2 Tier 2 Analysis 76 4.5.5.3 Tier 3 Analysis 78
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4.5.5.4 Tier 4 Analysis 80 4.5.5.5 Summative Tier Distribution 81 4.5.5.6 Individiual Question Analysis by Tier 84
4.6 Data Confidence and Course Effectiveness 85
4.7 The Instructors 87 4.7.1 Julia 88 4.7.2 Justin 88 4.7.3 Isabel 89 4.7.4 Gerdie 89 4.7.5 Rashida 90
4.7.7 Comparing Instructor Distributions and Scores 90 4.7.8 Instructor Perspective 91 4.7.8.1 Benefits 92 4.7.8.2 Challenges and Suggestions for Improvements 93 4.7.8.3 Changes in Teacher Candidate Performance and Evaluating Success 94 4.7.9 Advice to Future Instructors 95
4.8 The Teacher Candidates 96 4.8.1 Jack (Year One, PJ) 96 4.8.2 Lisa (Year One, PJ) 97 4.8.3 Zed (Year One, JI) 98 4.8.4 Kira (Year One, JI) 99 4.8.5 Juno (Year Two, PJ) 100 4.8.6 Maya (Year Two, PJ) 103 4.8.7 Francis (Year 2, JI) 103 4.8.8 Teacher Candidate Perspectives 105 4.8.8.1 Benefits 105 4.8.8.2 Challenges and Suggestions for Improvement 107 4.8.8.3 A Magic Wand for the Course 108
4.8.9 Teacher Candidate Expectations of Instructors 109 4.9 A Response from Lila and Henry 111 4.10 Seeing the Future 114
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Chapter Five: Discussion 115
5.1 Introduction 115
5.2 Research Questions 115 5.2.1 Research Question 1: What is the purpose of a mathematics content course in a teacher education program? 115 5.2.2 Research Question 2: What was the focus of the course? 117 5.2.3 Research Question 3: What are the perceived benefits and challenges of the MCK course? 118 5.2.3.1 Benefits 118 5.2.3.2 Challenges 119 5.2.4 Research Question 4: What effect did the course have on math content knowledge and math anxiety? 122 5.2.5 Research Question 5: How was success in the course evaluated and did the course meet expectations? 125
5.3 Major Findings 127 5.3.1 Effectiveness Across Tiers and Topics 128 5.3.2 Standardization and Instructor Backgrounds 129 5.3.3 Course Content Presentation 130 5.3.4 Effectiveness of Anxiety Scales 132 5.3.5 Targeted Course Content 133
5.4 Implications for Future Research 136 5.5 Conclusion 137 5.6 Researcher Reflection 140
References 141
Appendices 153
Appendix A: Letter of Consent – Course Creators 153
Appendix B: Letter of Consent - Instructors 155 Appendix C: Letter of Consent – Teacher Candidates 157 Appendix D: Additional Letter of Consent – Teacher Candidates 159
Appendix E: Interview Questions – Course Creators 160
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Appendix F: Interview Questions – Course Creator (Follow-up with Lila) 161 Appendix G: Interview Questions – Instructors 162 Appendix H: Interview Questions – Teacher Candidates 163
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List of Tables Table 1 57
Sample Size Distribution (Total Sample Size = 483) 57 Table 2 58
Diagnostic vs Summative Test Score (out of 36 points) 58 Table 3 60
Pre-Anxiety and Post-Anxiety Scales (out of 50) 60 Table 4 61
Diagnostic vs Summative Number of Blank Responses 61 Table 5 66
Diagnostic vs Summative Test Score Averages by Topic 66 Table 6 69
Individual Question Analysis 69 Table 7 70
Average improvements for teacher candidates who could improve 70 Table 8 72
Distribution of Teacher candidates in Each Tier 72 Table 9 82
Distribution of Teacher candidates in Each Tier at End of Course 82 Table 10 83
Diagnostic to Summative Changes in Student Tiers 83 Table 11 86
T-stat Tables 86
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Table 12 86
Summative and Anxiety Changes of > 10% 86
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List of Figures Figure 1 57
Sample Size Distribution 57 Figure 2 59
Diagnostic vs Summative Test Score Averages (out of 36) and Overall Change in Scores 59
Figure 3 60
Pre-Anxiety vs Post-Anxiety Scales (out of 50) and Overall Changes 60 Figure 4 62
Diagnostic vs Summative Number of Blank Responses and Changes 62 Figure 5 63
Regression Analysis of Blanks vs Anxiety 63 Figure 6 65
Regression Analysis of Math Content Scores vs Math Anxiety 65 Figure 7 67
Diagnostic vs Summative Test Score Changes by Topic (Percent Improvement) 67 Figure 8 67
Combined Topic Score Changes and Initial Scores by Topic (Including Proportional Ratios) 67
Figure 9 71
Difficult Level of Individual Questions by Proportion Correct (Diagnostic and Summative) 71
Figure 10 71
Overall Improvement by Teacher Candidates Who Could Improve 71
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Figure 11 73
Distribution of Teacher candidates in Each Tier 73
Figure 12 75
Tier 1 (>=75%) Changes (Content Score, Anxiety, and by Topic Area) - Years 1 & 2 (PJ & JI) 75
Figure 13 77
Tier 2 (<=50 % < 75%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI) 77 Figure 14 79
Tier 3 (<=25% < 49%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI) 79 Figure 15 80
Tier 4 (<=24%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI) 80 Figure 16 82
Summative Tier Distributions 82 Figure 17 83
Changes in Tiers 83 Figure 18 85
Tiers 1 – 4 Individual Question Analysis 85 Figure 19 91
Normalization of Instructor Content Score Changes (Expected – Actual) 91 Figure 20 91
Normalization of Instructor Anxiety Score Changes (Expected – Actual) 91
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Chapter One: Introduction
1.1 Introduction
Elementary teachers in Ontario are required to teach mathematics as a part of the
curriculum on a daily basis. Yet, many elementary teacher candidates self-report as having high
levels of math anxiety and struggle with math concepts at the elementary level (Finlayson, 2014;
Gresham, 2007; Novak & Tassell, 2017; Reid et al, 2018). Additionally, mathematics education
in Ontario has become increasingly focused upon in the curriculum (Education Quality and
Accountability Office (EQAO), 2019). This includes student testing in mathematics at the grades
3, 6 and 9 level to determine proficiency of students in the subject, and, recently, teacher
qualification testing in the subject as well. Based on the amount of media coverage and resources
dedicated to the subject of mathematics in our schools, this is clearly an area of intense and on-
going interest to parents, teachers and politicians alike. Yet, EQAO test results, which test
student proficiency in Ontario, and PISA test results, which compare Canadian students to other
countries, have shown a marked decline in student math grades and attitudes; leading this
researcher, and many others, to question the reasons and how they can be addressed.
One Canadian university is attempting to tackle this issue in a very direct way; through
the creation of a Math Content Knowledge (MCK) course aimed directly at providing elementary
teacher candidates with basic skills in elementary numeracy. In this study, I investigate how and
why this course was created, the challenges and benefits the course has faced, its intended and
perceived effects, and how successful the course has been over the first two years of
implementation. This chapter provides research context and questions, outlines the significance
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of the study and its limitations, and includes my personal background as a researcher.
1.2 Research Context
Math anxiety has been well studied: from its definitions, to its proposed causes, and
methods of addressing it. However, with all the research available on this topic, there is still no
consensus on the issue. Researchers have spent decades studying math anxiety and the impacts it
has on students, teachers, and the general population (Ashcraft & Krause, 2007; Chernoff &
Stone, 2014; Harper & Daane, 1998). Countless research papers have been written on the
proposed foundations of math anxiety and on strategies and methods to address it (Humayun,
2016; Meier, 2015; Ramirez et al., 2016). Significant research has also taken place as to the role
of math teachers in either lessening, or increasing, math anxiety’s influence and impact on
students (Beilock et al., 2010; Gunderson et al., 2012; Soni & Kumari, 2017). One significant
finding of this research deals with female teachers’ impact on their female students and, while
this thesis does not focus on gender differences at the pre-service teacher level, it is an important
point.
Even the field of mathematics itself has been studied and researched extensively. It has
had its place in education questioned (Muller, 2009), and its role, purpose, and instructional
methods continuously researched, scrutinized, and debated (Government of Ontario, 1994;
Ontario, 1965; Rankin, 2003). Governments have used math instruction and assessment as a
political platform (Alphonso, 2017; Ontario Ministry of Education, 2019), and universities have
questioned its place as a science, a philosophy, a language, or simply a tool for other subjects to
use (Muller, 2009). Mathematics has also had a long and complicated history in both our schools
and in our society in general, even dividing teachers themselves on the best ways to teach the
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subject (Bueckert, 2017). Regardless of the debates that have taken place as to math’s purpose,
and its method(s) of instruction, one point remains consistent: that the field of mathematics holds
a significant place in our school system and is a large (if somewhat contentious) part of the
Ontario elementary and secondary curriculum.
Since mathematics holds such a vital role in both our education system and the future
paths available for students’ post-graduation it is important to understand the role teachers play
in teaching mathematics and how their math anxiety, math content knowledge, and
approaches/attitudes towards math affect their students. This study attempts to answer some of
these questions by looking at how one Faculty of Education has addressed the issue of math
anxiety in elementary teacher candidates by creating a course specifically focused on building
math content knowledge in first-year Masters of Education elementary teacher candidates.
1.3 Purpose of the Study
This study seeks to understand and evaluate the methods, effectiveness, and underlying
principles of a math content knowledge (MCK) course. By interviewing various parties involved
in the course, it seeks to understand the reasons behind the creation of such a course, how it was
implemented by the instructors teaching it, and how it was received by the teacher candidates
taking it and its overall impacts. By studying such a course, this thesis is attempting to
understand the intended purpose of this course and to uncover the unintended impacts of such an
undertaking. Furthermore, it seeks to create a template of sorts for other faculties of education to
follow, along with suggestions for improvements to the MCK course that is part of this study.
This research will seek to provide insight and knowledge of this issue to both the
participants and creators of education programs and policies, with the intention of informing
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future programming and policy decisions. In addition, through the qualitative interviews, teacher
candidate participants will gain a deeper knowledge of their own history with math anxiety and
will have opportunities to form a greater understanding of these issues and reflect on their own
learnings from the MCK course; allowing them to recognize these patterns in their own history
and potentially have a (positive) impact on their future interactions with their students. Course
creators and instructors will also have the opportunity to evaluate not only the course but their
own involvement as well, allowing them to reflect and improve on their own teaching practices
and make overall improvements to the course itself.
1.4 Statement of the Problem
Numerous studies have shown that math anxiety is a real and measurable issue that is
experienced by a significant number of people in North America (Richardson & Suinn, 1972;
Young et al., 2012). Furthermore, studies have indicated that both pre- and in-service elementary
teachers tend to experience high levels of math anxiety and significant gaps in mathematics
content knowledge (Reid & Reid, 2017; Stoehr, 2017). Other studies have also shown that a
teacher’s attitudes towards mathematics can be passed onto their students (Beilock et al., 2010;
Gunderson et al., 2012), and that negative experiences by students in mathematics can cause
them to abandon further studies in mathematics and careers that require math and can have a
significant long-term effect on future wage earnings as well (Ahmed, 2018; Rose & Betts, 2004).
This research is a mixed methods study of a MCK course from a Canadian Faculty of
Education (FoE) that has taken a unique approach to addressing this issue; by teaching
elementary teacher candidates the basic math skills required to teach mathematics at the
elementary level. This required course for all first-year teacher candidates provides them with
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step-by-step instructions on how to do things such as long division, operations with fractions and
decimals, percents, and basic geometric principles; all with the underlying principle that, by
providing elementary teacher candidates with such skills, they are preparing them to more
effectively teach their students.
This research focuses on the following questions:
• What is the purpose of a mathematics content course in a teacher education program? • What was the focus of the course? • What are the perceived benefits and challenge of the MCK course? • What effect did the course have on math content knowledge and math anxiety? • How was success in the course evaluated and did the course meet expectations?
1.5 Significance of the Study
This research provides feedback to the creators of this course in order to improve its
outcomes and to provide other faculties of education with a template to follow in their own
institutions should they wish to implement a similar course or evaluate their own offerings in this
area. This study also provides insights into how MCK and Math Anxiety (MA) are closely
correlated and the effects that this course has on both math knowledge and confidence. By
profiling teacher candidates from various backgrounds and varying initial levels of math content
knowledge, this study seeks to provide a broad insight as to the effects such a course can have on
various types of teacher candidates and provides profiles on both the backgrounds of these
teacher candidates and their reactions towards the course.
This research also attempts to lay the groundwork for further research into math content
knowledge courses and their effectiveness. It opens the door for topics related to how most
teacher education programs in Canada approach math content knowledge in their pre-service
teachers, and how they compare to this course. This paper will also touch briefly on the newly
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instated Math Proficiency Test (MPT) in Ontario and its effects on the teacher candidates taking
the MCK course.
1.6 Background of the Researcher
While I am new to the field of research, I have been involved in mathematics education,
and education program and curriculum development for over 15 years. I am a certified teacher
with the Ontario College of Teachers at the Intermediate/Senior level with qualifications in
Chemistry and Mathematics. While I chose not to pursue classroom teaching after graduating, I
stayed involved in the field of education through program development in mathematics, science
and engineering. For nearly 10 years, I developed curriculum and programs for, at first, the
Faculty of Engineering at the University of Toronto, and then later developed an outreach
program for the Mathematics Department, also at the University of Toronto. During this time, I
obtained a Master of Mathematics for Teachers degree from the University of Waterloo and
further focused my education and experience on mathematics instruction to a wide range of
students.
The pre-university students I worked with and developed materials for ranged from
students struggling with basic math concepts to those preparing for national and international
math Olympiad competitions. I also worked closely with undergraduate and graduate-level
students in both mathematics and education to mentor and develop their mathematics instruction
skills; teaching them how to develop engaging, and accurate, mathematics curriculum and how
to engage and evaluate their students. On a personal level, I have been intrigued by mathematics
my entire life; although the subject has never come easily to me. Throughout my elementary and
high school career, I struggled with math anxiety and being told by teachers that ‘math wasn’t
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for girls’, but I persevered because I enjoyed the challenge and the subject. This has led me to
develop a passion and understanding for the subject that I do not think I would have otherwise
had.
The reason I have chosen this particular topic for my research thesis is because I fully
believe that the key to addressing math anxiety in students, and improving math skills and
attitudes towards the subject starting in elementary school, is to find ways to change the attitudes
of those who teach them. I also believe that there is more to math anxiety than simply
understanding the math concepts and that we need to address societal issues related to
mathematics and change underlying attitudes towards the subject. In short, I believe that we need
to find a way to position mathematics as a useful ‘language’ that, once learned, can unlock an
entirely new world of understanding.
1.7 Limitations of the Study
This study is an in-depth mixed methods study of a single MCK course. As such, it will
be limited by a number of factors. The first, and most significant, will be from the course
creators who have a vested interest in the course being seen in a positive light. They are proud of
the course they created and want to show it in the best way possible. They may also feel that they
have an obligation to their organization to show that the course was worth the time, effort, and
finances invested by their department. This may influence their responses to questions in the
interview and cause an unintentional skewing of recalled facts and information. This may also
occur in the instructors interviewed, where, even anonymous, they may wish to be seen as having
a positive impact on the course and may not wish to speak poorly of their employer. A further
limitation of the study is that only a limited number of teacher candidates were interviewed.
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While the teacher candidates interviewed do represent a broad cross-section of the course, and
represent multiple different backgrounds, ability levels, and overall engagement with the course,
they cannot capture all possible feedback of teacher candidates who have taken the course.
Additionally, the information gathered from the participants interviewed in this study will
be specific to those participants and may not be generalizable to all participants in the program,
or to all teacher candidates in general. However, this is offset by the inclusion of quantitative
data from 483 teacher candidates, collected from two years’ worth of participant results from the
MCK course. Furthermore, as this program is unique to a large urban university, it may not
capture the experiences of teacher candidates at other institutions and may not be generalizable
to their programs.
1.8 Plan of the Thesis
This thesis is divided into five chapters. The first chapter provides a justification for the
study and outlines its significance, limitations, and puts it in context of where the study lies in
the broader field of research. It also outlines my background, the connection to both the field of
research and my interest / experience in the topic being researched. This section also seeks to
provide an initial context for why the study is being done and what is hoped to be gained from
such an investigation.
Chapter two involves a literature review investigating the past and current research into
math content knowledge and math anxiety, with a specific focus on teacher beliefs and
influences. This chapter begins with an attempt to define math anxiety and then provides
information on ways to evaluate math anxiety. It then looks at strategies to address math anxiety,
along with how self-efficacy and building math content knowledge can be an effective strategy
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to reducing math anxiety. Finally, this chapter concludes by providing a section on the historical
context of the course, and of math education in general, and a comparison with other programs
and courses at other Canadian universities with education programs.
The third chapter provides an overview of the mixed methods methodology this study
follows, including a detailed breakdown of the research design, participant selection methods,
and overall framework this study follows. It also contains information on how the data was
collected and analyzed and how and why the study utilized mixed methods. Finally, issues
related to how the research was designed, how it connects to a larger theoretical framework, and
the ethical considerations of the study will be discussed.
Chapter four describes the data collected in this study. It includes an overview of the
MCK course, and profiles of faculty and teacher candidates involved. It provides an overview of
the structure of the course, how and why it was created, how it has undergone changes in its 2-
year pilot phase and details the experiences of the teacher candidates taking part in it. This
chapter contains stories and personal experiences from some of the teacher candidates who took
part in the class and also provides a more general overview of the teacher candidates who were
enrolled in the class by utilizing quantitative data obtained over the past two years. Both this
quantitative data and the qualitative information gained from the interviews is paired together to
tell a story of the impact of this course to both its teacher candidates along with those who have
developed and taught the course.
The final chapter involves a discussion of major themes encountered throughout this
study including highlights from interviews and feedback on where teacher candidates and
instructors saw the course as succeeding along with suggestions for improvement. Additionally,
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this section contains implications for future research and areas where this current study could be
expanded.
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Chapter Two: Literature Review
2.1 Introduction
Math anxiety has been studied for decades, from a number of perspectives, and in many
different contexts. Researchers have sought to understand its causes, its effects, its implications,
and to develop strategies to overcome it (Finlayson, 2014; Liu, 2017; Tok, 2013). There have
also been numerous studies on gender differences in math anxiety and performance (Devine et
al., 2012; Lindberg et al., 2010; Preckel et al., 2008), and how math anxiety affects different age
groups (Wigfield & Meece, 1988). Yet, while the breadth and depth of research on math anxiety
are substantial, many questions still remain.
This chapter gives a broad background on areas relevant to this study including: an
overview and definition of math anxiety, strategies utilized to address math anxiety,
requirements for becoming an elementary teacher in Ontario, and implications of teachers with
math anxiety for their students. A brief discussion on gender differences in mathematics and
math anxiety is included but will not be the focus of this study. Furthermore, this section
provides details on other education programs in Canada and how they seek to address issues of
gaps in math content knowledge of elementary teacher candidates.
2.2 Math Anxiety
The definition of math anxiety most widely employed today primarily derives from
(Richardson & Suinn, 1972) and their development of the 98-item scale known as the
Mathematics Anxiety Rating Scale (MARS). In it, they define math anxiety as:
Mathematics anxiety involves feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations. Mathematics anxiety may prevent a student from
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passing fundamental mathematics courses or prevent his pursuing advanced courses in mathematics or the sciences. (p. 551) Since the publishing of Richard and Suinn’s paper, other researchers and organizations
have built upon this definition. The National Council of Teachers of Mathematics (2013)
described math anxiety as “a real, biological condition and can be scientifically detected” (p.
405). This is complemented by researchers such as Young et al. (2012) who have been able to
further the field of research by studying the neurobiological effects of math anxiety showing,
through functional MRIs on children age 7 – 9, that math anxiety has a measurable impact on the
region of the brain responsible for negative emotions. Tobias and Weissbrod (1980) have also
included a definition of math anxiety to “describe the panic, helplessness, paralysis, and mental
disorganization that arises among some people when they are required to solve a mathematical
problem” (p. 65).
Yet, defining math anxiety is only one part of researching it. We must also turn our
attention to the effects of math anxiety on students, and adults, as it often leads to lower math
achievement, decreased participation in math topics and conversations, and even avoidance of
subjects and studies that involve math, including many choices made by teacher candidates
(Lake & Kelly, 2014). Stoehr (2017) states that “[m]athematics education researchers have
demonstrated that students’ experiences with mathematics anxiety can impede their participation
and achievement in mathematics” (p. 69). By limiting their study of mathematics, due to anxiety
of the subject, there is often a direct impact for students on future available career paths and
opportunities (Ashcraft & Krause, 2007), with more significant differences being seen for female
students (Huang et al., 2018).
Research into math anxiety and working memory has also helped to shed further light on
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the issues of math anxiety, performance and its effects (Ashcraft & Kirk, 2001). In their paper,
Ashcraft and Kirk (2001) showed that participants with math anxiety were dedicating a
significant portion of their working memory to their math anxiety and, even on non-conventional
mathematics problems, math anxious participants “show especially compromised performance
when the experimental task is itself demanding of working memory” (p. 235).
The definition of math anxiety, and its implication have evolved since the 1970’s, but the
underlying principles of the condition have remained consistent. Mathematics anxiety arises
amongst some people when they are required to solve a mathematical problem and it can
interfere with the manipulation of numbers and the solving of mathematical problems in a wide
variety of ordinary life and academic situations.
Gresham (2007), who studies math anxiety in pre-service teachers specifically, describes
mathematics anxiety as:
[…] a feeling of uncertainty and uneasiness when asked to do mathematics. It may manifest as an inability to perform well on tests, a feeling of physical illness, helplessness and panic, faintness, and mental disorganization (Bursal & Paznokas, 2006). It may be a dread of not being able to do well in mathematics or with numbers. Tobias (1978) gave the shortest definition of mathematics anxiety, describing it as the “I can’t” syndrome. She stated, “People almost experience sudden death with mathematics anxiety, as if a curtain has been drawn, like an impenetrable wall ahead, or seemingly standing on the edge of a cliff ready to fall off” (p. 45). It is a phenomenon where individuals suffer from the irrational fear of mathematics to the extent they become paralyzed in their thinking and are unable to learn or be comfortable with mathematics. (p. 91) Since math anxiety has been shown to be both real and measurable, researchers have been
able to utilize various quantitative tests to research and study it. The two most commonly utilized
tests for measuring math anxiety are the Mathematics Anxiety Rating Scale (MARS), whose
focus is on testing adult math anxiety, and the Scale for Early Mathematics Anxiety (SEMA),
created by Young et al. (2012), to address issues of math anxiety in youth and to expand upon
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the MARS. Shorter versions of the MARS, including the Abbreviated MARS (AMARS) focused
on undergraduate students (Richardson & Suinn, 1972), and the 10-point Revised Mathematics
Anxiety Scale (RMAS) focused on college/university students (Betz, 1978), have also been
created to study math anxiety in greater detail. While these tools are helpful in giving us a
starting place to understanding math anxiety, they do still have some inherit flaws and may not
show a complete picture of math anxiety (Cipora et al., 2019). In addition, math anxiety, and its
influence on math performance, can also be affected by a number of factors.
[…] the influence of [math anxiety] on math performance depends on certain conditions, for instance, it increases with time pressure (see Ashcraft & Moore, 2009). After establishing that individuals can master some easy problems when tested in a relaxes and untimed condition (Faust, Ashcraft, & Fleck, 1996), introducing time pressure leads to a drop in performance for those with medium or high MA (see Ashcraft & Moore, 2009). (Cipora et al, 2019, p. 29) However, even with measurable data and research, the causes of math anxiety are still not
fully understood. Finlayson (2014) proposes a number of possible causes of math anxiety
including: the classroom environment; teaching style of math instructors; lack of self-confidence;
and fear of failure. These causes can also be intensified by attitudes and beliefs of teachers
(Archambault et al., 2012; Beilock et al., 2010), parents (Soni & Kumari, 2017), and the general
classroom environment (de Beer, 2015). Studies also suggest that the method of instruction
further plays a large role in how students relate to the subject of math in general (Liu, 2017).
Further complicating the issue is that research suggests that there does not seem to be a
consistent reason behind the development of math anxiety in students. Some students develop it
when parents and educators tried to be supportive and “gave the impression that math was easy,
when the student was struggling with math” (Finlayson, 2014, p. 107) and yet others would
receive the message from parents that math was difficult and “if a parent could not do math, then
15
the student would probably not be good in math either” (Finlayson, 2014, p. 108). Research such
as this clearly indicates that math anxiety is a deeply complex issue that is not limited to one
group or practice, and continues to be the subject of much study, interest, and impact for
researchers, and the general population, alike.
2.2.1 Teacher Beliefs and Influences
Students begin to develop math anxiety in elementary school, and they are very often
influenced by their teachers (Finlayson, 2014). Stoehr (2017) states that “[m]athematics
educators agree elementary teachers should possess confidence and competence in teaching
mathematics” (p. 69). Since one of the defining features of math anxiety is a fear, and, by
extension, an avoidance of mathematics, it can be surmised that elementary teachers with math
anxiety will most likely not possess these traits. Stoehr (2017) also found that “[m]any
prospective elementary teachers (particularly women) pursue careers in elementary teaching
despite personal repeated experiences of mathematics anxiety” (p. 69). Lake and Kelly (2014)
further this research by showing “[e]arly childhood PSTCs routinely state that they choose to
teach young children (birth–age 8) because they do not like mathematics, are not good at
mathematics, or will not have to know or teach a lot of math” (p. 262).
Teacher beliefs and attitudes have been found to have a profound effect on the future
attitudes of students regarding math. Boyd, Foster, Smith, and Boyd (2014) found that both
positive and negative experiences of teacher candidates could be linked back to their own
personal early teacher experiences. Positive experiences could be traced back to “having a
teacher who was passionate about teaching mathematics, provided hands-on activities, made it
fun and consequently engaged the students contributed to positive learning of mathematics” (p.
16
211). While negative experiences had similar links back to:
[…] teaching approaches, use of resources and the teacher’ classroom management style as contributing to negative attitudes to mathematics. Teaching approaches included poor explanations, teaching as too fast a pace and could not keep up, a lack of hands-on experiences, and teaching directed approaches that did not support the students’ learning. (Boyd et al., 2014, p. 212)
Furthermore, when math anxious teachers teach mathematics in their own classrooms,
they tend to utilize a number of methods to overcome their own anxieties including: relying on
curriculum, finding mentors or guides, or simply hoping that their desire to become a teacher and
do a good job would be enough to overcome their anxiety (Gresham, 2007; Stoehr, 2017).
Shields (2005) sums up this concept nicely:
In order to mathematically encourage students to pursue math courses and math-related careers, teachers need to exude confidence and excitement for the subject. They also need to portray positive attitudes, design effective math curricula, implement effective pedagogy, create classrooms focused on inquiry and discovery, and assess fairly. Teachers who can do this have the power to alleviate math anxiety in the classroom. (para 14)
As one can see the attitudes and beliefs of teachers towards mathematics can have a profound
and long-lasting impact on their students and that, in order to address math anxiety in students, it
must first be addressed in their teachers.
2.2.2 Gender Differences
While gender differences are not a focus of this study, it is important to make note that
they do exist since there is a common stereotype that math, and math related subjects, are male-
dominated and therefore that boys are more suited / better than girls at mathematics (Nosek et al.,
2002). These beliefs are nothing new as, according to Gunderson et al. (2012), “[i]ssues of
gender equity in math achievement, course-taking, and careers have been of concern at least
since the 1970s” (p. 153). Of interest, however, are findings by Wigfield and Meece (1988), who
17
wrote:
[i]n regard to gender differences, there were no differences in the structure of boys’ and girls’ responses to the MAQ, which indicates that they were answering the items in similar ways. Boys and girls also did not differ in their reports of math worry, which indicates that they were equally concerned about doing well in mathematics. However, girls reported experiencing more negative affective reactions to math than did boys. (p. 215) This indicates that, while there may be similar levels of anxiety between genders, the
impact of math anxiety is often felt more strongly by female students. Gunderson et al. (2012)
indicate that there are many, and various, influences that female students encounter when
studying math that affects their level of math anxiety and overall performance. These include
such influencers as parents, teachers and general societal views. When gender stereotypes, and
various influences, are taken into account, they found that female students with math anxiety are
more likely to make long term decisions on math-related careers and studies by relating it more
strongly to their math anxiety and not necessarily to their math ability than boys.
Furthermore, while math anxiety is evident in both male and female students, studies
have shown that female teachers with math anxiety carry a greater chance of passing along their
math anxiety to their female students (Beilock et al., 2010). According the OECD Indicators:
On average across OECD countries 70% of teachers are women in all levels of education combined. The greatest concentration of female teachers occurs in the earlier years of schooling, and the share shrinks at each successive level of education. While women represent 97% of the teaching staff at pre-primary level and 83% at primary level, they make up 60% at upper secondary and only 44% at tertiary level on average across OECD countries. (Organization for Economic Co-operation and Development, 2019, p. 436)
According to OECD.Stat (2020) data from 2018, 75% of elementary teachers in Canada are
female, and since math anxiety has been shown to be particularly prevalent in females (Nosek et
al., 2002), this issue has the potential for a significant compounding effect. Beilock et al. (2010)
18
also found that boys and girls performed equally in math achievement and held similar beliefs at
the beginning of the school year regarding math ability and gender, yet they had divergent beliefs
by the end of the school year. This further suggests that gender differences in mathematics have
a more social construct to them than a biological basis.
However, since researchers such as Ashcraft and Kirk (2001) have seen negligible
differences with respect to gender in studies on math anxiety in university level participants, this
means that levels of math anxiety have the potential to be seen in both male and female teacher
candidates. Since this study focuses on university level participants, gender differences will be
outside of the scope of this paper.
2.2.3 Mathematics and Equity
One further consideration of math anxiety on students deals with mathematics as a
gatekeeper subject, and one that has the potential to have a profound impact on issues related to
equity. As students in Ontario enter grade nine, they must make a decision that will have a deep
and lasting impact on their futures. Currently, in Ontario, students in secondary school are part of
a streamed program. This means that, when they enter secondary school, they choose one of two
main streams: Academic (a post-secondary track) or Applied (a more hands-on, college-based
track). This means that, as early as grade 9, students are making a decision as to where their
further education, and by extension, career choices will lead. Most universities in Ontario require
at least one University level (Academic stream) mathematics course to apply; even if you are
applying for a Bachelor of Arts (Ontario Universities’ Info, 2020). This means that students who,
in grade nine chose the Applied stream, will have severely limited options for higher education.
Boaler (2016) speaks of the high degree of inequality these types of streaming systems
19
have in the United States and how students of colour are affected more greatly. Both streaming
of students and weaker math skills in general have been shown to have a great effect on equity
and access to higher education and opportunities for students in later life.
Mathematics is a subject that is critical for all students’ futures, as it is a prerequisite for college and many fields. This should mean that mathematics teachers have additional responsibilities - and opportunities - to make mathematics equitably accessible to all. (Boaler, 2016, p. 110)
Furthermore, Ahmed (2018) found that race/ethnicity had a strong correlation with math anxiety
and its trajectory though adolescence and that the students in their study with high anxiety levels
tended to avoid STEM careers and fields in future. This is also supported by the fact that
[m]ath anxious students are more likely to avoid math-related courses and activities (Ashcraft, 2002). Unfortunately, this avoidance behavior is likely to have a snowballing effect— avoidance leads to less skill development which erodes confidence thereby increasing anxiety. This process may be one mechanism through which high math anxiety is maintained overtime. (Ahmed, 2018, p. 164)
2.3 Strategies to Address Math Anxiety
Over the past 100 years, the mathematics curriculum has changed dramatically. Topics
covered have shifted, methods of instruction have been altered, new technologies have been
utilized, and studies have been implemented to understand the effectiveness of various types of
math instruction. Generally speaking, strategies for addressing math anxiety are almost as broad
as the research undertaken to understand the issues themselves. Researchers have focused on
everything from student perspectives of math anxiety (Stoehr, 2017), to the neurobiological
aspects (Young et al., 2012), to its effect on working memory in relation to math studies
(Ashcraft & Krause, 2007). Wigfield and Meece (1988) also investigated when interventions to
math anxiety would be most effective with the suggestion that:
[i]ntervention programs to alleviate the negative effects of math anxiety must deal with
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both affective and cognitive aspects of math anxiety. These programs should be implemented during the elementary school years, before children’s anxiety about math becomes strongly established. (p. 215)
Many of the strategies researched suggest a more holistic approach to math anxiety,
dealing with both the cognitive and physical aspects that those with math anxiety may
experience. Tok (2013) suggests that more specific, targeted approaches to deal with math
anxiety such as know-want-learn (KWL) strategies derived from language learning. Fiore (1999)
used one-page written accounts for students to describe their math background and feelings
towards math in order to develop an understanding of the issues these students face and to
provide a starting point for the instructor when teaching and interacting with these students.
There are many politicians and decision makers in Ontario who have promoted a “return to rote”
(The Star Editorial Board, 2018) method of mathematics instruction that focuses on improving
math assessment scores, without necessarily addressing the underlying issues of math anxiety
itself.
A wide range of research has been undertaken to understand math anxiety, and its root
causes, and to develop strategies to address it on many different levels (Finlayson, 2014; Liu,
2017; Tok, 2013), including research specifically on the causes of math anxiety in teacher
candidates (Harper & Daane, 1998). There has been significant research undertaken on strategies
to address math anxiety (Beilock & DeCaro, 2007; Ramirez et al., 2016; Tok, 2013). Yet, with
all of these different studies, methods, strategies, and opinions on the issue, the one thing that has
been consistent between them is the fact that math anxiety has significant, measurable impacts
on the ways that students process mathematics, and this is quite often reflected in their attitudes,
mathematical achievement scores, and future educational and career prospects (Ahmed, 2018;
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Ramirez et al., 2016; Young et al., 2012).
2.3.1 Self-Efficacy and Learning to Fail
Bandura (1994; 1997) speaks of self-efficacy as a person’s perceived ability to have an
influence or effect over their own lives. A high level of self-efficacy often leads to the person
being successful and confident and improves a person’s overall personal well-being. A low level
of self-efficacy often leads a person to avoid what they see as difficult tasks and may even view
those tasks as “personal threats”.
Gürefe and Bakalım (2018) investigated how issues of self-efficacy (confidence) and
learned helplessness can have an effect on mathematics anxiety. Referencing work by
Nicolaidou and Philippou (2003), they state: “[i]n mathematics, affective factors are considered
to be just as important as cognitive variables in learning and teaching processes” (Gürefe &
Bakalım, 2018, p. 154). Boyd et al. (2014) further expand on the idea that “[t]eaching
mathematics confidently is associated with teachers’ beliefs about their mathematical ability,
which is their mathematical self-efficacy” (p. 207).
To address math anxiety, Boaler (2016, 2019) recommends looking at the connections
between brain science and learning mathematics. Much of her work focuses on the idea of a
growth (instead of fixed) mindset and its effect on learning mathematics effectively. Her research
also focuses heavily on the idea of failure in learning and how true learning can only take place
amongst failure. This research suggests that, if we allow students to fail in mathematics, we are
encouraging them to develop new neural pathways for building knowledge and understanding
and that this new learning is not limited by age.
Research by Lake and Kelly (2014), which references previous work of theirs in the same
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field, along with Geist (2010), have shown that constructivist methods of instruction have had
positive effects on decreasing math anxiety and improving self-efficacy. One proposed method
of improving teacher self-efficacy is through the development of math content knowledge (Reid
& Reid, 2017)
2.3.2 Math Content Knowledge
Reid and Reid (2017) define math content knowledge as “the basic math knowledge
possessed by individuals considered to be mathematically literate” (p. 853). The development of
this knowledge has been shown to increase feelings of self-efficacy and reduce math anxiety
(Reid et al, 2018; Reid & Reid, 2017; Vinson, 2001). Building math content knowledge typically
focuses on the process of scaffolding. This process involves building a framework of
mathematics skills from basic, concrete, foundational concepts to more abstract, complicated
materials - checking and assessing at each level to ensure solid foundational information has
been acquired before building to the next, more complicated, level (Dehaene, 2011).
A report by the Fields Institute for Mathematical Sciences that focused on
recommendations to the Ontario government for improving teacher education in mathematics
and recommendations for changes to the Education Act in Ontario included a recommendation
for a requirement that:
[t]eachers admitted to Primary/Junior/Intermediate teacher education must have at least one undergraduate course in mathematics, but preferably two in the areas of curriculum-related mathematics concepts and “profound understanding of elementary mathematics” (Ma, 1999). Teachers need to learn the underlying concepts, models and connections supporting each developmental stage of mathematics knowledge. (Kajander et al., 2013, p. 3)
The reason for this suggested requirement stems from the fact that the researchers involved in the
writing of this report felt that “[p]re-service teachers need more preparation in mathematics
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education” (p. 4) and that “[t]eachers need to have a “profound understanding of fundamental
mathematics” (Ma, 1999) in order to make sound pedagogical decisions to support learning
(Ball, Hill, & Bass, 2005)” (p. 2). Ball, Hill and Bass (2005) further expand upon this concept in
that “[h]ow well teachers know mathematics is central to their capacity to use instructional
materials wisely, to assess students’ progress, and to make sound judgments about presentation,
emphasis, and sequencing” (p. 14).
The MCK course was developed specifically to address issues of low math content
knowledge in incoming teacher candidates at the elementary level (grades K - 8) in Ontario.
Initially, these candidates were provided with self-study resources to address gaps in their
content knowledge, yet most failed to improve by the end of their program. Upon seeing research
gathered from diagnostic and summative math content tests of these teacher candidates, a new
course was proposed where the focus was on teaching these teacher candidates the basics of
math content knowledge, with a focus on numeracy skills.
2.3.3 Numeracy as a Foundation for Math
According to (Reid et al., 2018), “[a]ll five math strands (i.e., number sense and
numeration, measurement, geometry and spatial sense, patterning and algebra, and data
management and probability) are interwoven and interconnected, however, number sense and
numeration is the strand that is deeply embedded throughout” (p. 14). It is for this reason that a
focus on numeracy has been shown to be effective in building overall foundational math skills in
students. Dehane (2011) postulates that humans are not born with an innate ability to do
mathematics, nor are those who excel at mathematics “endowed with an exceptional
neurobiological structure” (p. xxi). Instead, he surmises that passion (or hatred) of the subject
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encourage students to rigorously pursue (or avoid) it. He uses neurobiology to show that a strong
foundation in numeracy, understanding numbers and how they relate, is required to both
understand, and enjoy, more complex mathematical computations. Boaler (2016) describes it as:
[t]he high-achieving students solved the question by using what is known as number sense - they interacted with the numbers flexibly and conceptually. The low-achieving students used no number sense and seemed to believe that their roles was to recall and use a standard method even when this was difficult to do. (p. 35)
By focusing on building a strong sense of numbers and numeracy, students are given the tools to
approach mathematical problems from a less complex perspective; and therefore achieve greater
rates of success.
2.3.4 Teaching Methods
Research has also shown that the way in which a mathematics course is taught and the
methods and pedagogy employed can have a significant effect on math anxiety and the retention
of mathematics content knowledge. In fact, a substantial portion of math anxiety can come from
how the materials are taught. Methods such as instructor-led lessons, timed tests, memorizing
formulas and algorithms, and a focus on the application of specific rules can all increase math
anxiety in students (Harper & Daane, 1998; Tobias, 1998). Alternatively, when the focus of
mathematics instruction is on understanding mathematics, such as through the use of
manipulatives and other ways to make abstract concepts more concrete, including the use of
multiple teaching strategies, math anxiety shows a marked decline (Gresham, 2007; Tobias,
1998; Vinson, 2001).
Shulman (1986, 1987) addresses this issue by focusing on the concept of pedagogical
content knowledge (PCK) and how it seeks to address the fact that teachers need more than just
content knowledge in order to be successful.
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Pedagogical content knowledge is knowledge that lies at the confluence of content knowledge, knowledge of students’ thinking (the understandings they bring to a particular class or lesson and how it can be capitalized upon), and knowledge of mathematics education and pedagogy (e.g., curriculum, particularly difficult concepts, and effective images and instructional aids). (Silverman & Thompson, 2008, p. 499)
These ideas are further built upon by Hill et al. (2008) in relation to mathematics knowledge
specifically.
By “mathematical knowledge for teaching,” we mean not only the mathematical knowledge common to individuals working in diverse professions, but also the subject matter knowledge that supports that teaching, for example, why and how specific mathematical procedures work, how best to define a mathematical term for a particular grade level, and the types of errors students are likely to make with particular content. (p. 431)
Furthermore,
When Shulman (1986) proposed the distinction between subject matter knowledge (SMK) and pedagogical content knowledge (PCK), he emphasised the importance that teachers’ knowledge should include knowing that as well as knowing why. In mathematics, this knowledge extends beyond definitions, theorems and algorithms to understanding the structure of mathematics as a discipline, the relative importance of particular mathematical ideas within the discipline and the principles of mathematical inquiry – how new ideas are added to the body of knowledge and erroneous ideas are rejected. (Pournara, 2016, p. 137)
These researchers focused on the fact that teachers need to be taught the content of
mathematics (the how) and the context of what they are learning (the why) in order to be truly
effective in addressing both gaps in math content knowledge and math anxiety.
2.4 Elementary Teacher Candidate Requirements
The Ontario College of Teachers requires teachers to register with their organization in
order to be certified to teach in Ontario. Currently, applicants must hold the equivalent of a four-
year (full time study) degree in addition to a four-semester teacher education program. The
requirements for an acceptable teacher education program include a 10% focus on educational
26
foundations; 20% focus on teaching methods for two grades or subjects; 20% practice teaching
(minimum 80 days of supervised practicum); 50% in other areas of education (methodology,
classroom management, technology support, etc). There are slightly different requirements for
teachers educated outside of Ontario/Canada but these are the general requirements to qualify for
certification to teach in Ontario.
Applicants must also meet language proficiency requirements in either English or French
and, as of March 2020, are required to write a math proficiency test and achieve a score of 70%
or greater. Finally, they are required to show "professional suitability" (Ontario College of
Teachers, 2020, p. 5) through a current Canadian criminal record check and Applicant
Declaration. Once registered, certified teachers pay an annual membership fee to remain
registered and those teaching in the school system must fill out an annual declaration form
stating they are still "professionally suitable". There are no formal requirements mentioned for
math courses for elementary teacher candidates as this seems to be left to the individual teacher
education programs to determine. The only mention of math training is the newly added
requirements for the math proficiency test (Ontario College of Teachers, 2020, p. 28).
To gain a clearer picture of how institutions receive approval to provide teacher education
programming, I have included the current program requirements as gained from a recent audit of
the Brock University program (Ontario College of Teachers Accreditation Committee, 2019),
made available on the OCT website. These audits outline the requirements for a program to be
accredited and seem to have the following requirements: that the institution is a permitted
institution (i.e. a university or other accredited and registered institution) and must "preserve the
integrity of student records relating to the program" (p. 36). The institution must also show
27
"continuous improvement and quality assurance of the program" (p. 37) along with a Teacher
Education Advisory Committee (or similar); the program is four semesters in length and contains
the appropriate number of practicum hours (note this was a recent change to the requirements,
previously a program was only required to be two semesters in length); the program needs to be
built upon a clearly delineated conceptual framework; the program also needs to be consistent
with, and reflect, the Ontario College of Teachers "Standards of Practice for the Teaching
Profession" and the "Ethical Standards for the Teaching Profession", have links to current
research in education and integrate both theory and practice in teacher education.
It also requires that the program meet the requirements of Schedule 1 as set out by the
Ontario government (more on this below); the program must also be current and include
references to the current Ontario curriculum and the application of a wide knowledge base of
current research in teacher education; the program must also include theory, method and
foundation courses and have an appropriate format and structure for the course content.
The teaching methods course must be related to the divisions the teacher candidates will
be teaching (such as Primary/Junior or Intermediate/Senior). These courses must also contain
content on human development and learning, and information on government policies and
legislation related to education; teacher candidates in the program must also be assessed and
informed of program on an ongoing basis and teacher candidates must be provided with, and
required to participate in, a practicum of a minimum of 80 days in length that includes both
observation and practice teaching in an area relevant to the division and subject area of the
teacher candidates, supervised by an experienced teacher who assesses their practicum. Teacher
candidates must also be appointed a faculty member as an advisor. Finally, the faculty who teach
28
the courses must be defined as appropriate with academic qualifications, practice in the field and
appropriate expertise in the various divisions (Ontario College of Teachers Accreditation
Committee, 2019).
The programs must also follow the Ontario Ministry of Education laws as set out by
Schedule 1 and include content on Curriculum Knowledge (current Ontario curriculum and
policy documents - lesson planning and design, special education, equity and diversity, and
assessment and evaluation); Pedagogical and Instructional Strategies and Knowledge (using
research and data, technology, classroom management, child/adolescent development, current
strategies in student observation, assessment and evaluation - including subject specific
assessment, teaching/learning theories and differentiated instruction, teaching ESL or FSL
students, providing special education supports); and Teaching Context Knowledge (mental
health issues, ethical standards, transitions, Ontario school system context, professional
relationships) (Ontario Ministry of Education, Schedule 1, 2016).
The Ontario Ministry of Education and the Ontario College of Teachers are responsible
for defining the policies and procedures of teacher certification and overseeing the institutions
that instruct future teachers. However, neither of these organizations provide guidelines on the
mathematics requirements, or instructional methods, for elementary teacher candidates. This is
especially concerning given the Ontario government’s recent addition of the Provincial Math
Teacher Qualification test, which became a requirement for all graduating teacher candidates as
of March 2020. A report by the Fields Institute for Research in Mathematical Sciences (Kajander
et al., 2013) found that:
there is much variability in the level of preparation teachers in elementary and secondary panels receive during their teacher education program and prior to it. For example, while it is
29
recommended that future elementary teachers have some undergraduate mathematics courses, it is not presently a mandated requirement in the Province of Ontario. Additionally, some pre-service programs in Ontario provide as little as 30 hours of instruction in mathematics education […]. Evidence across numerous sources suggests that this is well below what is recommended by researchers and practitioners for the preparation of teachers, and that 100 hours or more of specific instruction are required for P/J/I teachers. (p. 2)
2.4.1 Provincial Math Teacher Qualifying Test
As of March 2020, all newly graduated teacher candidates are required to take a
mathematics proficiency test in order to be certified to teach in Ontario. Bill 48 (Ontario Ministry
of Education, 2019), passed on April 3, 2019, adds a clause related to registration with the
Ontario College of Teachers (the governing body of teacher registration in the province) that
states registrants must “successfully completes any prescribed examinations relating to
proficiency in mathematics that are required for the issuance of the certificate” (p. 6). The test,
developed in collaboration with the EQAO, has 50 math content questions and 21 math
pedagogy questions. An applicant is required to score 70% or higher on both portions of the test
in order to qualify to register as a teacher in Ontario (EQAO, 2019).
An extensive literature review, released by the EQAO offices, details “the current
research and evidence available on the topic of the compulsory standardized testing of teachers
and the relationship between these tests and student achievement” (EQAO, 2019, p. 3). In the
paper, they state:
[c]urrent available evidence shows that two fundamental types of a teacher’s subject matter knowledge directly influence student outcomes. The first is good content knowledge (i.e., a foundation in and understanding of the subject) that would be familiar to anyone working in the subject area. The second is pedagogical content knowledge, which requires an understanding of how to teach the subject matter effectively. (p. 3)
It is also important to note that the “student outcomes” to which the report refers are student test
scores on standardized tests as measured both internationally and in Ontario.
30
In general, the report found a positive correlation “[…] between teachers’ pedagogical
mathematical knowledge and student outcomes […]” yet there was no clear conclusion for a
similar correlation “[…] between mathematical content knowledge and student outcomes (as
measured by standardized student tests […]” (p. 4). The report acknowledges that there are many
additional factors to be taken into consideration when looking at student achievement such as:
[…] curriculum materials (Shechtman et al., 2010), socioeconomic measures (Baumert et al., 2010), academic/non-academic tracks (Baumert et al., 2010) and the school or district environment (Corey, Phelps, Ball, Demonte, & Harrison, 2012; Petscher & Logan, 2014; Tighe & Schatschneider, 2014). (p. 4)
It also states that there are mixed outcomes when looking at the effectiveness of teacher
competency scores and student outcomes, where the general consensus on this issue is “weak
and not universal” (p. 4). The report also makes reference to the fact that teacher competency
testing, such as the ones being implemented in Ontario:
[…] impacts more than just student outcomes and can influence the teacher recruitment pipeline and the curriculum taught to trainee teachers. Changes in these factors can cause a decrease in the enrolment of pre-service teachers from minority ethnic groups (Cobb, , Shaw, Millard, & Bomotti, 1999; Graham, 2013; Nettles, Scatton, Steinberg, & Tyler 2011; Petchauer, 2012; Shuls & Trivitt, 2015). (p. 4)
It is interesting to see such an analysis coming from the organization responsible for creating and
implementing this test in Ontario and, while questions related to the Ontario Math Proficiency
Test and its impacts on teacher candidates are included in this research, a deeper understanding
of teacher competency testing and student outcomes is outside of the scope of this paper.
2.4.2 Building and Evaluating Math Content Knowledge
Given that the provincial requirements for mathematics are not laid out in the teacher
qualifications programs, how do various institutions ensure their graduates have the tools
required to teach math effectively to their students? An internet search of multiple faculties of
31
education located in Ontario, Canada revealed a wide range of systems and programs in place to
either address this issue directly or to simply test teacher candidates to see if they meet the
expectations.
Faculties of Education in Ontario seem to divide their programs into one of three main
streams: a graduate level degree (a Masters of Teaching or Master of Arts in Child Study); an
undergraduate degree (B.Ed.) where the teacher candidates gain their teaching degree alongside
their undergraduate degree (such as a B.Sc. or B.A.); or a teaching degree (B.Ed.) obtained after
completing their undergraduate degree (such as a B.Sc. or B.A.). In order to meet ministry
requirements, however, all programs are offered as at least a four-semester (typically two-years)
program.
A search of Ontario faculties of education found that most faculty of education programs
in Ontario had a mathematics course aimed at teaching basic curriculum content and pedagogy to
elementary teacher candidates. Some had two courses with one focusing more on reviewing the
themes and content of the mathematics curriculum and the other focusing on teaching styles and
methods of instructing mathematics (pedagogy). Others had a single course focused on
combining both areas of mathematics instruction. In addition, the content courses seemed to
cover the full spectrum of mathematics curriculum (covering all five strands across K-6 grade
levels). Other schools had courses that combined math instruction and content with other STEM
areas, and some had courses that focused on building mathematical thinking and encouraging
alternative methods to think about and teach math.
Some universities required their teacher candidates to write a comprehensive mathematics
competency exam that they must pass within three attempts to receive their degree and be put
32
forward to be certified by the Ontario College of Teachers (Lakehead University, 2020).
However, the majority of courses seemed to focus on a brief introduction to concepts with most
of the focus being on teaching methods and mathematical pedagogy.
While this search of faculty of education programs in Ontario seems to indicate that the
MCK course being studied by this paper is unique in its design and implementation, it must be
noted that this search only focused on Ontario universities and was based on what was publicly
available on their websites; programs similar to the MCK may exist in other areas or may not be
accurately reflected in the materials presented. What appears to make the MCK program unique
is its sole focus on developing numeracy skills in teacher candidates and containing no pedagogy
content. Its sole purpose is to provide teacher candidates enrolled with instruction and practice
with basic, foundational mathematics skills.
2.4.3 Teaching Teacher Candidates Mathematics
Reid et al. (2018) acknowledges the link between math content knowledge and math
anxiety. In their research, they find a direct correlation between math performance and math
anxiety and investigate multiple methods for addressing this in teacher candidates. Such methods
include various high stakes tests like those required for admission to the program, those required
to graduate from the program, and those, such as the Provincial Math Proficiency Test, that are
required for certification. Their research shows how these tests can “disadvantage specific
populations or discourage some students from applying to teacher education programs” (p. 13).
They also make note of how many of these tests potentially “[…] amplifies already existing
[Math Anxiety] in [Teacher Candidates]” (p. 13). Other programs that have teacher candidates
take a test and then require those who fail the test to take a remedial course can serve to further
33
discourage those teacher candidates and brings up issues surrounding failure (Boaler, 2016).
So how can we effectively teach teacher candidates mathematics? What strategies have
already been tried and what were the results? Research by Daniel’s et al. (2011) looked at the
effectiveness of various aspects of teacher education from a broad perspective and found that:
improvements to initial teacher education must affect outcomes that we know are beneficial for teachers and students. There is general consensus that teachers with lower anxiety and more efficacy and commitment tend to have more positive outcomes than their colleagues who lack these psychosocial characteristics. Thus one logical way to examine the effectiveness of program changes is to investigate their effectiveness in reducing anxiety and maximizing efficacy and commitment. (p. 98)
In their study, Daniels et al. (2011) found that the “ethics of teaching was the largest predictor of
each outcome. This dimension involved three discrete components: moral reasoning,
professional judgment, and appreciating multiple perspectives on complex issues” (p. 102).
Van der Sandt and O’Brien (2017) investigated a MCK course similar to the one in this
study but that approached the content in a more advanced way and focused on a broader range of
topics. In their research, they found that different teaching approaches and styles had a
statistically significant impact on lowering anxiety of teacher candidates. They found that those
teacher candidates with instructors who followed a problem-based teaching style (with
manipulatives and real-world applications) had a significantly lower math anxiety level versus
those with a more direct teaching method (“instructor driven and blackboard and textbook
dominated”) (p. 101).
Matthews et al. (2010) investigated the math content teacher candidates were taught. The
study involved teacher candidates in three groups: those who had never taken a math course
targeted to elementary teachers (the control group); and those enrolled in at least one math
course for elementary teachers - with one of these groups in a course “designed to enhance the
34
teacher’s mathematical knowledge of numbers, place value, fractions, and number sense” (p. 4)
and the other “included topics in geometry” (p. 4).
Both mathematics courses focused on instruction that would increase the future teachers’ ability to fully explain mathematics and to develop a thorough understanding of these topics. The content courses also modeled some pedagogical strategies (such as the use of manipulatives) that might also be used in their later instruction of elementary students. (p. 4)
It is important to note that the teacher candidates in both math content classes had similar initial
math scores and that:
among this sample, the attitudes evidenced by all participants, were overall neutral. There was no significant difference between the measures of attitudes toward mathematics; therefore enrollment in the mathematics for elementary teachers course did not impact the prospective teachers’ attitudes toward mathematics. (p. 7)
Overall, their study:
[…] appear[s] to indicate that [math] content courses are indeed an effective way of enhancing the mathematical knowledge that elementary teachers might require for their own classroom instruction of mathematics […] These results would seem to indicate that teacher preparation programs without such courses should seriously consider adding some specialized mathematics content courses […] to their elementary teacher preparation programs. (p. 7)
2.5 Summary
Teachers with math anxiety, who are required to teach math concepts that they may never
have been exposed to in their own teachings, could have a potentially terrifying personal
experience and are highly likely to have a significant impact on their students. Furthermore, the
focus on standardized math scores in students (and now for teachers to receive certification) adds
further pressure to both students and teachers to the pressures of succeeding in math. It has also
become a platform for some politicians as a way to meet their claims of raising standards and
preparing our students for the future.
35
This study seeks to provide insight on how one university is approaching this issue from a
different angle. By working with teacher candidates in a structured, guided way, and providing
them with scaffolded learning of basic math concepts, they are seeking to build their math
content knowledge and to build their self-efficacy and confidence to be able to attempt further
math on their own. By focusing on teaching them the skills they may have missed while going
through the education system, or refreshing them on concepts since forgotten, the MCK course in
this study is seeking to provide teachers with the tools, and the confidence, to teach their
students. This study provides an in-depth analysis of this course, how it could be improved, and
how it could be used as a template for other institutions to build teacher confidence in this vital
subject area. It is also hoped that this research will provide insights to administrators and
governments on the different approaches that can be taken to ensure our elementary students are
getting the highest quality math education from our teachers.
36
Chapter Three: Methodology
3.1 Introduction
This study provides insight into the development, implementation, and evaluation of a
Math Content Knowledge course in a graduate teacher education program at a Faculty of
Education in a large urban city. The participants were the professors who created the course, the
teaching assistants who taught it, and the elementary teacher candidates who took it. Quantitative
data collected from the course were utilized in order to provide a fuller picture as to the impact of
the course. Two surveys were used: the Revised Mathematics Anxiety Scale (RMAS) (Betz,
1978) and the Math Content Knowledge assessment (Reid & Reid, 2017).
This research is a mixed methods study of how one Faculty of Education is approaching
issues of math content knowledge gaps and math anxiety in a graduate level pre-service teacher
program. By interviewing the three different levels of participants involved in the course, this
research provides an in-depth look at the intentions, expectations, and effects of this course. It
uncovers some of the more personal stories of those affected by math anxiety in an effort to stem
the historical passing of math anxiety from teacher to student. By helping elementary teacher
candidates uncover their relationship with the subject of mathematics, and their exposure to a
course designed to address their ability in mathematics, it is hoped that systemic issues found in
pre-service teacher programs can begin to be addressed.
By undertaking this research, I gained a better understanding of the topics proposed and
built upon my own background and history with the subject. The purpose of this research was to
further develop knowledge and resources that participants, the researcher, and course developers
in other institutions can apply to their own work and will allow mathematics education to have a
37
positive influence on society as a whole. This chapter outlines the methodology undertaken
within this study.
3.2 Research Design
This study follows a mixed methods research design as it utilizes qualitative data
collected through interviews with course participants along with quantitative data obtained and
analyzed from diagnostic and summative content tests and pre- and post-anxiety scales. While a
mixed-methods methodology is often seen as additional work “given the added resources, time,
and expertise required to conduct a mixed methods study” (McKim, 2017, p. 202), it also allows
for researchers to “not only to validate their findings through triangulation, but also often to give
a deeper, broader and more illustrative description of the phenomenon” (Hurmerinta-Peltomäki
& Nummela, 2006, p. 452). Additionally, they found that this type of research sometimes created
contradictory information that created “puzzles” that needed to be solved and this “offered
rewards in terms of the creation of new knowledge, which in turn increased the theoretical
contribution of their studies” (p. 453). McKim (2017) also found that:
[g]raduate students scored mixed methods higher with regard to perceived value and further explained that when done correctly, mixed methods has something for all readers, regardless of their philosophical worldview. They also stated that mixed methods is more rigorous than quantitative and qualitative methods. (p. 213) Since the course being studied in this paper is complex in both its design and
implementation, along with the expectations of the course and the potential for differing
viewpoints from those being interviewed (teacher candidates vs instructors vs course creators), a
mixed methods study seems like a valid option. It is also assumed that there will be “puzzles”
that will arise when comparing the qualitative data to quantitative data collected from the course
as a whole and a mixed-methods methodology provides a framework for addressing and solving
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these puzzles.
Finally, the qualitative portion of this study focuses on only 14 members of the course,
yet over 400 teacher candidates were involved in the course over the two-year pilot.
Incorporating the quantitative data provides a fuller story and incorporates those additional
teacher candidates. The qualitative data will be described using cases. This mixed methods study
focuses on the course as a whole and is bound by the first two years of the course; considered the
pilot years. In addition, the teacher candidates chosen for the study come from a cross-section of
participants in the course and were matched to quantitative data in order to ensure teacher
candidates from different math content knowledge and math anxiety levels were included.
3.2.1 The Math Content Knowledge Course
The Math Content Knowledge (MCK) course in the Faculty of Education (FoE) was
created by Lila and Henry, two faculty members within the Faculty. The idea for the creation of
the course came from Lila’s research during her Ph.D. studies along with her work as a faculty
member. Lila’s research, and her own experiences working as an instructor in the program, had
shown that teacher candidates participating in her math pedagogy course, where she includes an
emphasis on math content knowledge in addition to pedagogy, showed improvement in their
math skills (through diagnostic and summative math content test results) and showed a
significant decrease in their math anxiety levels. Henry, working in administration in the faculty
at the time, saw these results and Lila recalls him remarking to her that “we need to do
something like this, but scale it up. It cannot just be within your own classes that you teach”
(Lila, interview, April 17, 2020).
The MCK course is a compulsory, non-credit mathematics course in a graduate level
39
teacher qualification program at a Faculty of Education at a Canadian university. It is required
for teacher candidates enrolled in the Primary/Junior (PJ) and Junior/Intermediate (JI) divisions
of the program. The course was designed to be a required part of the pre-service program; a two-
year graduate level program leading to a provincial teaching certification. It is a pass/fail course
taken for credit only and teacher candidates may choose to write an exemption test to determine
if their math skills are strong enough to be given credit without having to take the course. The
course was scheduled for three hours per week and took place over 12 weeks.
Based on a report submitted to the Faculty of Education by Lila and Henry requesting
permission to run the course, the stated goal is to “ensure that all of our elementary teacher
candidates have a satisfactory grasp of math concepts up to a middle-school level.” The course
consists of twelve, three-hour classes and is taken by teacher candidates in the first year of their
program. The course does allow teacher candidates who score 90% or above on a math test at the
grade 8/9 level to receive credit for the course and be exempt from taking it. Their report also
outlines other methods Lila and Henry attempted in order to address the issue including
“providing teacher candidates with individualized diagnostic reports that identify the math
concepts where they require additional study”, directing them to “online tutorials and exercises
designed to help them develop an understanding of these concepts”, and providing the teacher
candidates with additional information on math supports provided by the Faculty of Education.
While all teacher candidates were afforded these resources, many did not take advantage of them
and follow-up assessments with these teacher candidates in Year 2 of their program indicated
that “of those who score less than 70% on their first-year test, almost three-quarters remain
below 70% on a follow-up test a year later.”
40
As such, the MCK course was created to address these issues related to the limited math
content knowledge of incoming teacher candidates. Teacher candidates were tested for their
math content knowledge at the start of the course and these test results showed that many of the
teacher candidates struggled with basic math concepts such as fractions and decimals,
percentages, order of operations, and other numeracy concepts up to the sixth-grade level.
Previously, these issues were addressed by providing the teacher candidates with detailed
feedback on their results and then directing them to self-directed online tutorials and other
mathematics supports offered by the university. However, follow-up assessments of these
teacher candidates in their second year showed little to no improvement of their math content
knowledge.
In their original proposal to the administration of the FoE, the creators of the course cite
the following reasons to create the MCK course:
[…] teachers with math anxiety tend to avoid engaging in further professional development, provide less time in class for the study of math, and can transmit their anxiety to their students. The literature gives evidence that when teacher candidates improve their math content knowledge, their anxiety diminishes and math efficacy increases. Thus, it is important that we find ways to help all our teacher candidates become successful at mathematics. We need to help them develop their foundational math skills and support their math confidence. Moreover, we need to do so in a way that is supportive, non-stigmatizing, and sensitive to our teacher candidates’ math-related concerns and uncertainties. (Faculty of Education, 2017)
Information from the province’s EQAO math testing was used to further justify the creation of
such a course and cites a steady decline in large-scale math assessment results between 2009 and
2017 in grade 3 and 6 students in the province of Ontario. Utilizing incoming teacher candidate
math content knowledge tests collected since 2014, Reid and Reid (2017) found that, after their
first year in the program, “almost three-quarters of the teacher candidates (TCs) who scored less
41
than 70% on the pre-test, remained below 70% on the post-test” (p. 866) providing further
justification of the need for a different approach to building math content knowledge in teacher
candidates.
The MCK course utilizes a combination of instruction, practice, math games, peer
tutoring and other, research based, pedagogical methods of effective math instruction. There are
no high-stakes exams, and instead the focus is on weekly quizzes, two projects, in-class
participation, and weekly online assignments utilizing the online platform Khan Academy. The
course also runs in conjunction with a math pedagogy course that focuses on teaching methods in
mathematics.
3.2.1.1 Structure
The MCK course was designed as a 12-week course with two hours spent in class and
most teacher candidates spent about one hour of time on independent homework. The teacher
candidates were expected to start the class with a short 10-mark quiz on the concepts learned in
the previous week. Initially, these quizzes took about 40 minutes to complete but as the course
progressed the time required reduced to about 20 minutes. They were then provided with in-
class, instructor-led instruction on a specific math topic. These topics changed each week and the
instructors were provided with all of the materials relevant for the lessons each week. These
materials were created by Rashida, a head Teaching Assistant, with input from Lila on various
aspects of the course, especially the quizzes.
The teacher candidates were assigned homework each week and the teacher candidates
were required to use an online, free tool known as Khan Academy for their homework. Khan
Academy was chosen for its ease of use, cost (free), and its tracking ability for instructors to see
42
when teacher candidates completed their homework, how many attempts it took, and their
overall understanding / score on the work.
In the first class of the course, the teacher candidates wrote a diagnostic test. This test
included questions on basic numeracy including operations (add, subtract, multiply, divide,
exponents, BEDMAS), along with fractions, decimals, and percentages. The results were
gathered for research purposes only as the teacher candidates were informed that their grade on
the test would not be counted. The teacher candidates were not given their tests back (so the test
could be reused for subsequent years to ensure data consistency), however, they were provided
with a detailed breakdown of their scores along with resources they could access related to areas
where they were weak. Instructors for the course were not involved in the marking of the
diagnostics and the results were not released to them. When asked the reason for this, both Lila
and Rashida responded that it was to prevent the instructors from developing bias towards the
teacher candidates in their class.
The next 10 weeks were focused on learning concepts related to numeracy. The focus on
numeracy was due to the fact that “research shows good foundation in number sense sets you up
better for other areas in math” (Lila, interview, October 28, 2020). During the first year of the
course, the focus was mostly on numeracy skills. In year two, some additional content in
geometry and algebra was added. In addition, one of the weekly quizzes was replaced with an
assignment as the teacher candidates in the class had a significant break over the December
holidays and many came back having forgotten the materials from before the break. Therefore,
an assignment was designed for teacher candidates to work on during the break to reinforce the
skills learned without utilizing the quiz format.
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3.3 Participants
Participants for the study were drawn from the MCK course and include: two course
creators (Lila and Henry), five instructors (Julia, Justin, Isabel, Rashida and Gerdie) and seven
teacher candidates (Zed, Jack, Kira, Juno, Lisa, Maya, and Francis). The course creators and
instructors represent nearly all of the members of these categories that have been involved in the
course over its two-year pilot (one instructor from year one of the course was not available to
participate). The teacher candidates interviewed represent a cross-section of teacher candidates
enrolled in the course. They were recruited using a general call for participants from both Year
One and Year Two of the course.
Permission was then obtained from all but one teacher candidate participant in order to be
able to view their individual quantitative results from their diagnostic tests and math anxiety
scales. This was done for two reasons: one was to ensure that enough teacher candidates were
interviewed to obtain a relevant cross-section of teacher candidates from the class (including
from both the first and second year of the course, from Primary/Junior and Junior/Intermediate
levels, high, middle, and low math content scores, and from different instructor classes); and the
second was to create a profile of each teacher candidate interviewed to give context to their
various responses.
3.4 Data Collection
The data were collected using semi-structured interviews to gather personal narratives of
the participants and by compiling survey and diagnostic test results.
3.4.1 Interviews
The participants were asked questions related to their general experiences with
44
mathematics and math anxiety, their experiences with the MCK course, their experiences with it,
suggestions for improvements, expectations from the course, and any benefits they felt they
obtained from the course (Appendices E, G, and H). The teacher candidate participants were also
asked some questions regarding the recently implemented Provincial Math Proficiency Test
(MPT) and if they felt that the MCK course had any impact on either their writing of the test or
their feelings towards having to write the test in the near future (Appendix H). All interviews
were done via video conferencing software (Zoom) and recorded and transcribed.
3.4.2 Surveys
The quantitative data collected from the two years of the course covered diagnostic (pre-
course) and summative (post-course) test scores measuring math content knowledge and pre- and
post-course math anxiety scores obtained using the AMARS. Permission from all but one teacher
candidate in this study was obtained in order to utilize their personal scores to create a
personalized profile for each teacher candidates; giving a broader context and establishing their
overall place within the course. These data were used to provide a generalized overview of the
effectiveness of the course and to provide insights into overall effectiveness of the course from a
quantitative viewpoint.
3.5 Data Analysis
The interviews were transcribed using the software Otter.ai. The software Scrivener, a
writing tool, was then used to separate each question from the interview into various individual
sections. These sections were then populated with responses from individual participants from
their transcripts. Once complied, the data were investigated to see if similar statements or themes
appeared between participants. Each theme was reviewed for commonalities and to search for
45
relevant pieces of information and quotes that could be utilized in this paper. These individual
themes and areas were then used to address the research questions and to look for additional
themes not present in the original proposal.
The quantitative data utilized in this study were analyzed using Microsoft Excel. In order
to determine whether the course had a significant impact on teacher candidate performance, or if
the observed difference in means (pre- vs. post-) could be readily explained by sample variation
or other non-controlled variables, T-tests focusing on mean scores, number of blanks, and
anxiety scores - grouped by aggregate (all teacher candidate) data - were performed. Further
analysis was then conducted to determine if the patterns observed in the changes (pre- vs post-
course) could, in part, be explained by tiers of achievement determined by diagnostic scores.
Analyses were also performed on a per-subject, and per-question, basis to look for trends and
patterns in how teacher candidates improved in individual topic areas on the diagnostic versus
summative tests. Finally, a regression analysis was performed in order to assess the strength and
nature of the relationships between scores, anxiety, and blanks (pre-/post-course) as well as the
change in scores, anxiety and number of blanks.
3.6 Ethical Considerations
Ethics approval was obtained from the appropriate office at the university to utilize such
data (including qualitative data that was provided by the course creators). Approval was also
obtained from the Office of the Vice-President, Research and Innovation, Human Research
Ethics Programs at the University of Toronto. Participants in the study were all above the age of
majority and were informed that their participation in the study was voluntarily and that every
effort would be made to protect their identity. They were provided with a consent letter that
46
specifically stated that they were free to withdraw from the study at any time and there would be
no penalty for withdrawing from the study (Appendices A, B, C and D). If they wished to
withdraw from the study, all data related to their interviews (including any transcripts and
video/audio files) would be removed from the study’s data set and thus will not be included into
the data to be analyzed. They were also informed that this withdrawal process was available up
to the point where the thesis was submitted. There were no participants who withdrew from the
study.
Some members who were interviewed, such as instructors and course creators, were
informed that their identities could not be guaranteed to be completely anonymous due to their
unique position within the course and, in some cases, qualifications that were unique to them and
their background and skills. All participants were provided with opportunities to remove
statements from the record or to choose not to answer specific questions.
The video interviews were performed using the software Zoom, which utilizes encryption
protocols for data in transit, and recordings were stored on a local drive, also encrypted. Third-
party software / services were utilized for transcription purposes but were encrypted during
transit and removed from the service after the transcription was complete. Finalized transcripts
were stored locally on a secured computer and were anonymized.
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Chapter Four: Findings
4.1 Introduction
This chapter contains profiles and information on both the MCK course and the
participants in this study. It is a compilation of both qualitative data gathered from the interviews
with the course designers, instructors, and teacher candidates, and quantitative data collected
from the course over a two-year pilot period. The teacher candidates interviewed for this study
represent a cross-section of the class as a whole, and they come from a diverse range of
backgrounds. Some scored high on the initial diagnostic math content test and some scored near
the bottom of their class. Some were pursuing a Primary/Junior (PJ) qualification, while others
were focused on the Junior/Intermediate (JI) qualifications. Furthermore, profiles on the
instructors for the course are also included in this chapter in order to provide an overview of not
only the MCK course but also those who taught it in order to provide further context for some of
their responses.
4.2 The MCK Course
About five years prior to this study, members of the Faculty of Education program
noticed that the incoming elementary teacher candidates’ math skills were lacking. Henry, a
Professor in the faculty and the program coordinator, noticed that incoming candidates were
lacking foundational math skills. He approached various members of the department in an
attempt to address this and found Lila, a new addition to the department, who was discovering a
similar pattern in the teacher candidates in her pedagogy course, and had started collecting data
on the issue for her Ph.D. degree. Building upon this work, Lila and Henry began working on
different ways to address this issue. They researched what other faculties of education were
48
doing such as entrance requirements (where teacher candidates would need to write and pass a
math test to be accepted into the program), and exit tests (requiring them to pass to graduate).
They even created an independent study course but noticed that the teacher candidates did not
retain the information after the course ended. After researching different options, Henry and Lila
decided they did not want to implement these for a number of reasons:
In part, we worried that the entry and exit test might disadvantage certain populations. And we also felt that having [teacher candidates] cram for a math test was not a good way to really learn and understanding the mathematics. We felt that [teacher candidates] would have a much deeper understanding of the mathematics if they could learn it in a more gradual and structured fashion – i.e., in a course. (Henry, 2021)
So they proposed the MCK course to their faculty on a pilot basis to see if they could
address these issues in a different way. However, this decision still left them with many
questions such as how to justify such a course in a graduate-level program. They struggled with
the issue of requiring all teacher candidates in the program having to take the course; and what
about teacher candidates who were already proficient in math, would they have to take it, or
could they be exempt? And if those teacher candidates were allowed to be exempt from the
course, would this then create a stigma for those who did have to take it? After consultation with
other faculty and those in administration in the Faculty of Education, it was decided that the
course would be a required course and it would be pass/fail, non-credit.
We made the course “required” because we felt it was necessary for [teacher candidates] to demonstrate they possessed the necessary math knowledge. At the same time, we made it a zero-credit course, because we didn’t feel we could justify awarding a Masters-level half-credit for learning elementary-level mathematics. (Henry, 2021)
Even the teacher candidates who wrote and passed the exemption test were required to
have the MCK course on their transcript; they simply earned the credit by writing and passing an
exemption test. In this way, the MCK course became required for every elementary teacher
49
candidate in the program, which removed the stigma from the course and made it “a normal
requirement for everyone” (Henry, 2021). And so the MCK course was proposed, approved, and
implemented.
During the first two years of the course, data was collected on participants. This data was
collected via a math content knowledge diagnostic test developed based on the work of Reid and
Reid (2017) and the RMAS (Betz, 1978) to test for teacher candidate anxiety levels associated
with mathematics. The teacher candidates were required to complete both assessments at the start
and end of the course. This data was utilized to evaluate the effectiveness of the course over the
pilot two-years and is included in this study.
4.2.1 The Course Structure
In the first year, the course was structured such that teacher candidates had two-hours of
in-class instruction each week, followed by one-hour of online homework. The course took place
over 12 weeks for a total of 24 hours of in-class instruction. The course started with the teacher
candidates writing a diagnostic test that evaluated their understanding in a variety of different
topics in mathematics (most involving number sense and numeracy). This diagnostic test was
repeated at the end of the course. This post-course diagnostic test will be referred to in this paper
as the summative test for ease of comparison.
Each class, other than the first and last, started with a quiz on the previous lesson, and
proceeded to a lesson on a single topic (such as division, fractions, ratios, area and perimeter,
etc). This was followed by individual practice by the teacher candidates utilizing the online
learning tool, Khan Academy. All instructors were provided with detailed handouts and materials
and, while they were given some freedom as to general teaching style, all sections of the course
50
were designed to have the exact same content taught from the same general pedagogical
standpoint. This content was developed by a single instructor and all instructors were provided
with detailed handouts each week. Teacher candidates were required to attend all classes and a
portion of their mark (20%) was determined by participation in the class. The class itself was
pass/fail for credit but teacher candidates must obtain a minimum of 70% in the course in order
to receive the credit.
Teacher candidates who felt they were strong in mathematics were able to request and
write a math exemption test. Each year, out of about 380 teacher candidates, about 60 teacher
candidates requested the exemption test and, of these, about 50% (or 30 teacher candidates) pass
the test and are given a credit for the course without having to take it. The remaining teacher
candidates in the program are required to take the course, and pass, in order to meet the overall
requirements of the Masters of Teaching program.
4.3 The Course Creators
In order to gain a deeper understanding of why the course was created and the course
expectations, it is important at this point to describe the course creators, their backgrounds, and
their motivation to create the course. This helps us to build context as to why this was considered
an area of importance for both of them along with allowing us to understand what expectations
they had of the course and why.
4.3.1 Henry
Henry has a math and computer science background and is currently an Associate
Professor in the Faculty of Education. During the time of the pilot project for the MCK course,
Henry was also a Program Coordinator, which gave him a unique view into the teacher
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candidates coming into the Masters of Teaching program. He noticed that some teacher
candidates coming into the program had very poor science skills and he wanted to investigate
whether their math skills were poor as well.
I remember being at the time unsettled at how little science elementary science [the teacher candidates] knew. And later, when I became program coordinator for the program, I became worried about their math skills too. One of the things I wanted to do as a program coordinator was figure out just how bad are their math skills were; I wanted to test all the elementary teacher candidates coming into the program. (interview, May 8, 2020)
When asked why the focus was on math specifically, Henry noted:
[…] the disparity between our teacher candidates’ math ability, and what was expected of them, was the greatest if nothing else. There are very few teacher candidates who cannot read at a grade six level, but there are a lot of teacher candidates who cannot do math at a grade six level. (interview, May 8, 2020)
He also felt that gaps in other subjects, such as science, could also begin to be addressed
by improving the math skills of elementary school students, and the best way to do this was
through their teachers. In addition, Henry hoped that, if a math content course could be shown to
be successful, perhaps content courses in other subjects, like science, could be an area for future
focus.
However, in order to affect any change in this area, Henry realized that he was going to
need assistance and data to support any initiatives. So he “went to the math [pedagogy]
instructors at the time to talk to them about this, and they were very resistant” (Henry, interview,
May 8, 2020). When asked about this resistance, he said he thought it most likely had to do with
the instructors of those courses not wanting to have teacher candidates in their class start the year
off with a math test, setting the tone of the course in a negative way and perhaps reflecting
poorly on the skills of the instructors of those courses. He thinks there was also “[…] a lot of
52
suspicion about tests and testing and whether this is a good measurement of anything” (Henry,
interview, May 8, 2020).
Then I went to a new instructor, Lila, who had just joined the program. Lila was the one person who was supportive because she had done her under her undergrad doctoral degree on this subject of teacher candidates’ knowledge of mathematics. (Henry, interview, May 8, 2020)
Over the next three years, Lila collected data on the teacher candidates in her own math
pedagogy courses. Henry was then able to use that data to advocate, with Lila, for the MCK
course within the Faculty. They wanted to propose a course that was not “[…] concerned too
much with how to teach something, [the course] is more concerned with helping our teacher
candidates actually learn content of the math” (Henry, interview, May 8, 2020). They designed a
one-year, non-credit, pass/fail course designed to help teacher candidates to learn math content.
When asked why not just combine it with the already running math pedagogy course, Henry felt
this was in some ways already being done, but not effectively. The instructors of the pedagogy
course were having to go back and fill in content gaps before they could teach them different
ways to teach the content. However, it was not being done “systematically”. Henry also felt it
would have required doubling the length of the pedagogy course from 36 hours to 72 hours,
which “did not seem feasible in some ways” (Henry, interview, May 8, 2020).
4.3.2 Lila
Lila began teaching at the University in the B.Ed. program as a seconded instructor from
the local school district. Before her secondment, she was a principal in an elementary school. At
that time, she realized that “one of my assumptions was that everyone knew their math really
well. As I started teaching, I realized that I did not know the needs of my students” (Lila,
interview, April 17, 2020). Based on this understanding, Lila decided to pursue a Ph.D. degree
53
with a focus on the math content knowledge of teacher candidates. “Since then, math, that area
of research, the math content knowledge of teacher candidates, has been my focus, my passion,
and my everyday kind of work” (Lila, interview, April 17, 2020).
Utilizing the data she collected in her math pedagogy classes during her Ph.D. work, she
was able to see statistically significant increases in the math content knowledge, and decreases in
the math anxiety, of her teacher candidates; and this was without a formal, separate course
focused on math content. Working with Henry, she was able to turn this research into a formal
proposal for the MCK course. In order to ensure the course was approaching the issue from a
unique perspective, Lila compared the idea for a MCK course to what other institutions were
doing, including entrance tests, exit tests, and courses that were taught by the math department
(not the faculty of education) and saw that those were “a disaster because they are taught by the
math department which created more algorithm dependency” (Lila, interview, April 17, 2020),
something both she and Henry wanted to avoid.
Lila’s personal motivations for the MCK course come from her passion about math
education and from her general approach to education, which she sees “through a lens of equity
and to ensure that as instructors, we are meeting the needs of our students” (Lila, interview, April
17, 2020). She strongly believes that “attitudes and values and beliefs towards math, to me is just
as important as your knowledge and skill level, they go hand in hand” (Lila, interview, April 17,
2020). Lila also believes that attitudes towards math play a huge factor as well.
What my research shows is that anxiety and negative attitudes and negative experiences can really be a compromising factor in how we learn math and how we feel about ourselves in mathematics. It is so important that we take that into consideration and in order to foster and ensure that those attitudes are positive we've got to make the math activities and our lessons engaging and fun. And at the just right level where [teacher candidates] feel challenged, but not frustrated, and not bored, you know, that zone of
54
proximal development. (interview, April 17, 2020) When asked why she thought there should be a course focused on building math skills,
Lila referred to equity and how math is seen as a “gatekeeper subject” and how we need to
encourage more students to see themselves in mathematics. She said we need to broaden that
view of who can be a mathematician and a lot of that comes from their teachers; so we need
them to see their diverse teachers as competent mathematicians as well:
[u]nfortunately, math is seen as a gatekeeper. If you think about it, look at all the people who are math professors who are in STEM fields, your engineers, your computer analysts, your, you know, technology people, scientists, and quite often, those intersectionalities do not include women of color, women and women of color. It is a very white male dominated profession, the STEM field. (Lila, interview, April 17, 2020) Based on her research and teaching experiences in the Faculty of Education, Lila saw a
need to address math anxiety and math content from a different perspective. Based on data
collected from her math pedagogy course, she proposed a different way to teach teacher
candidates math content knowledge as a way to decrease math anxiety and improve the quality
of teachers in the teacher qualification program. For this reason, she worked towards creating the
MCK course, whose focus is on building math content skills and building confidence in teacher
candidates.
4.4 Course Expectations
When Lila and Henry first envisioned the MCK course, they were looking for a way to
boost the math skills of elementary teacher candidates and to change their attitudes and
viewpoints on mathematics. Using research-based methods and focusing on building skills in
numeracy, their expectations for creating the course ranged from issues of equity to increasing
overall teacher candidate math proficiency. Henry did worry about how the course was going to
55
be seen as:
it was kind of a weird thing to stick into a graduate program. It is a very traditional form of instruction where you teach a concept and then you rehearse it, and then you teach another concept. […] I was ready for the possibility that there could be a wholesale rebellion against it, you know, the [teacher candidates] signing petitions and all that stuff. (interview, May 8, 2020)
But the course was well received and Henry credits that to the work of Lila and Rashida.
The teacher candidates taking the course also had expectations on what the course was
going to be. Some thought it was going to be “a refresher on the math concepts” (Zed, interview,
May 4, 2020) while others thought “I need my toolbox refilled on how to do all these things
[…and an expectation to…] drill all these math facts that I need to know and all these methods
and ways of solving problems that I know I need to know that I would forgotten” (Jack,
interview, May 4, 2020). Other teacher candidates noted that “we were not told much at all” and
expected “they were going to make us practice or brush up on our math skills” (Kira, interview,
May 5, 2020). While others stated they were
scared again because I am not used to, and I do not like doing, math and I thought it was going to be this really like daunting experience of relearning math and everyone in my course in my classes are already knows these things and I am like the only one who does not understand certain math concepts. (Juno, interview, May 6, 2020)
There were some teacher candidates who expected to “just to go through the content from
grades one to eight” (Maya, interview, May 7, 2020). One teacher candidate thought “we would
be starting more at middle school math. I did not think it would be so back to basics” (Francis,
interview, May 20, 2020). However, this teacher candidate also acknowledged that this was a
good thing:
even the basics were challenging for me [… and …] I thought I would be drowning more. I thought it would be way more out of outside of my realm of ability, and I thought I was worried it would be too fast paced and that it would be similar to my experience at school
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where once you fallen behind, it is like so hard to catch up. (Francis, interview, May 20, 2020)
4.5 Distribution of Data and Initial Findings
The MCK course is offered in the first year of the Education program. This research
looks at the first two years of offering this course, where there were only a few minor changes in
both the course content and its instructors between Year One and Year Two. In both Year One
and Year Two of the course, the average math content knowledge scores, as assessed by the
diagnostic and summative tests, which focused on basic numeracy math skills, were collected. In
addition, anxiety scores were collected from pre- and post-course anxiety scales (the RMAS).
4.5.1 Diagnostic and Summative Test Results
Table 1 shows the sample size distribution of the results from 483 teacher candidates
combined from Year One and Year Two of the course and includes both Primary/Junior (PJ) and
Junior/Intermediate (JI) candidates. Initially, data on approximately 580 teacher candidates was
collected. However, this data was cleaned to include only teacher candidates who wrote both the
diagnostic and summative tests, as well as the pre- and post-course anxiety scales; removing
teacher candidates who dropped out of the course for various reasons throughout the year or did
not complete all four tests. Initially, there were 315 teacher candidates in Year One, of which 57
were removed from this study (52% were PJ and 48% were JI), and in Year Two 43 teacher
candidates were removed (63% were PJ and 37% were JI) for the reasons stated above. Year
Two the removed data roughly aligns with that year’s remaining distribution.
For Year One, a slightly higher ratio of JI candidates were removed from the analysis. It
should also be noted that, in Year Two, Intermediate/Senior teacher candidates were allowed to
take the course. While they completed the diagnostic test, they did not complete the summative
57
and have therefore been removed. Table 1 and Figure 1 show the distribution of the remaining
teacher candidates.
Table 1
Sample Size Distribution (Total Sample Size = 483)
Year 1 PJ 163 (63.2%) Year 2 PJ 153 (68.0%) JI 95 (36.8%) JI 72 (32.0%)
Total 258 Total 225 Figure 1
Sample Size Distribution
The distribution of teacher candidates across Year One and Year Two of the course was
fairly consistent. There were significantly more PJ than JI candidates in both years of the course
and this is consistent with the Masters of Teaching program overall.
Teacher candidates in the course were required to write a diagnostic math content test,
developed by Lila, which focused on basic numeracy skills and was pulled mostly from grade
three and six EQAO tests at the start of the course. They were then required to repeat the same
test at the end of the course, referred to as the summative. Table 2 shows both initial diagnostic
and final summative math content scores for both Year One and Year Two of the course for both
PJ and JI candidates.
Year 1
PJ JI
Year 2
PJ JI
Combined
Year 1 Year 2
58
Table 2
Diagnostic vs Summative Test Score (out of 36 points)
Diagnostic Summative
SS M SD SE CI M SD SE CI Year 1 All 258 16.75 8.9 0.55 16.20-17.30 27.66 6.55 0.41 27.25-28.07
PJ 163 15.4 8.5 0.67 14.73-16.07 27.44 6.5 0.51 26.93-27.95 JI 95 19.06 9 0.93 18.13-19.99 28.04 6.66 0.68 27.36-28.72 Year 2 All 225 18.19 8.5 0.56 17.63-18.75 27.95 6.54 0.44 27.51-28.39
PJ 153 17.39 8.4 0.68 16.71-18.07 27.41 6.58 0.53 26.88-27.94 JI 72 19.89 8.4 0.99 18.90-20.88 29.1 6.35 0.75 28.35-29.85
Combined 483 17.42 8.7 0.40 17.02-17.82 27.79 6.54 0.30 27.09-28.09 Change
SS M SD SE CI
Year 1 All 258 10.91 6.2 0.38 10.16-11.66
PJ 163 12.04 6.2 0.48 11.10-12.98
JI 95 8.98 5.7 0.59 7.83-10.13
Year 2 All 225 9.76 6.2 0.41 8.95-10.57
PJ 153 10.01 6.1 0.49 9.04-10.98
JI 72 9.21 6.4 0.75 7.74-10.68
Combined 483 10.37 6.2 0.28 9.82-10.92
SS = Sample Size; M = Mean; SD = Standard Deviation; SE = Standard Error; CI = 95% Confidence Interval
This shows that the average change in scores was between 8.98 and 12.04 points. It also
appears that PJ teacher candidates in Year One of the course seemed to have improved the most
overall. The final summative scores were also quite consistent across both years and levels of the
course. Since there is no overlap in the diagnostic or summative scores at the 95% confidence
interval, we can assume the results are statistically significant. Figure 2 shows these results
graphically.
59
Figure 2
Diagnostic vs Summative Test Score Averages (out of 36) and Overall Change in Scores
4.5.2 Anxiety and Confidence
Teacher candidates in the course were also assessed for their level of math anxiety. This
was done using the RMAS assessment scale where a higher score indicates a lower level of
anxiety. Table 3 shows changes in teacher candidates’ scores in Year One and Year Two of the
course, along with distributions in PJ and JI candidates and the average changes in those scores
between pre- and post-course assessments. On average, teacher candidates in the course
improved by between 4.75 and 6.74 points on the anxiety scale. Furthermore, the results were
mostly consistent between Year One and Year Two, with the Year One PJ candidates showing
the most significant improvements, which is also consistent with the change in math content
knowledge scores for this group. Figure 3 shows these results graphically.
048
12162024283236
All PJ JI All PJ JI
Year 1 Year 2 Combined
Diagnostic Summative
02468
101214
All PJ JI All PJ JI
Year 1 Year 2 Combined
60
Table 3
Pre-Anxiety and Post-Anxiety Scales (out of 50)
Pre-Anxiety Score Post-Anxiety Score
SS M SD SE CI M SD SE CI Year 1 All 258 25.06 9.55 0.59 23.90-26.23 31.07 9.41 0.59 29.92-32.21
PJ 163 24.09 9.67 0.76 22.60-25.57 30.82 9.05 0.71 29.43-32.21 JI 95 26.74 9.16 0.94 24.89-28.58 31.48 10.03 1.03 29.47-33.50 Year 2 All 225 25.36 8.89 0.59 24.19-26.52 30.76 9.02 0.60 29.58-31.93
PJ 153 24.75 8.57 0.69 23.39-26.10 30.05 9.26 0.75 28.58-31.52 JI 72 26.65 9.45 1.11 24.47-28.84 32.25 8.36 0.99 30.32-34.18
Combined 483 25.20 9.24 0.42 24.37-26.02 30.92 9.22 0.42 30.10-31.74 Change
SS M SD SE CI
Year 1 All 258 6.00 6.74 0.42 5.18-6.83
PJ 163 6.74 6.86 0.54 5.68-7.79
JI 95 4.75 6.39 0.66 3.46-6.03
Year 2 All 225 5.40 6.04 0.40 4.61-6.19
PJ 153 5.31 6.33 0.51 4.30-6.31
JI 72 5.60 5.42 0.64 4.35-6.85
Combined 483 5.72 6.43 0.29 5.15-6.30
SS = Sample Size; M = Mean; SD = Standard Deviation; SE = Standard Error; CI = 95% Confidence Interval Figure 3
Pre-Anxiety vs Post-Anxiety Scales (out of 50) and Overall Changes
0
20
40
All PJ JI All PJ JI
Year 1 Year 2 Combined
Pre-Anxiety Score Post-Anxiety Score
0.00
2.00
4.00
6.00
8.00
All PJ JI All PJ JI
Year 1 Year 2 Combined
61
Overall, the anxiety scores in the sample ranged between 9 and 50 with 9 considered to
be extremely anxious about math and 50 considered to have virtually no self-reported math
anxiety. Teacher candidates in both Year One and Year Two showed a marked decrease in math
anxiety (as indicated by a higher score on the math anxiety scale).
In addition, the number of blank (unanswered) questions declined significantly from the
diagnostic to the summative test (Table 4 and Figure 4). An analysis of the number of blanks was
undertaken as it was considered to be related to anxiety as teacher candidates were not docked
marks for incorrect answers; incorrect answers were given a mark of zero, yet incomplete (or
blank) answers were not awarded any marks, leaving the final score unaffected.
Table 4
Diagnostic vs Summative Number of Blank Responses
SS Diagnostic Summative Change Year 1 All 258 8.68 2.48 6.20
PJ 163 9.89 2.39 7.50 JI 95 6.60 2.62 3.98
Year 2 All 225 6.65 0.89 5.76 PJ 153 6.72 1.01 5.71 JI 72 6.50 0.64 5.86
Combined 483 7.73 1.74 6.00
62
Figure 4
Diagnostic vs Summative Number of Blank Responses and Changes
This data shows a significant drop in the number of questions left blank from the
diagnostic test to the summative. Part of this is because teacher candidates were able to answer
significantly more questions on the summative test (as shown in the increases in their summative
scores). However, this does not account for the entire difference. In many cases, teacher
candidates still got the answer wrong but were willing to attempt an answer. On the diagnostic
test, 42% of incorrect answers were due to blank responses, while only 21% of incorrect answers
on the summative were due to blank responses. It was hypothesized that, if a teacher candidate
was not confident in their abilities (ie. had a lower score on the anxiety scale), they would be
more likely to not attempt a question even if they knew they would not be penalized for an
incorrect response. To determine if this correlation was random or not, a regression analysis was
performed on number of blanks vs. math anxiety scores and compares the pre-course anxiety
scale to the post-course anxiety scale (Figure 5).
0.002.004.006.008.00
10.0012.00
All PJ JI All PJ JI
Year 1 Year 2 Combined
Diagnostic Summative
0.001.002.003.004.005.006.007.008.00
All PJ JI All PJ JI
Year 1 Year 2 Combined
63
Figure 5
Regression Analysis of Blanks vs Anxiety
By comparing the number of blanks (questions not attempted) and anxiety using
regression analysis on the diagnostic/pre-course anxiety scale (left) and the summative/post-
course anxiety scale (right), we can estimate the relationship between the blanks and anxiety.
For the diagnostic/pre-course chart, we find that, for someone with an exceptionally high anxiety
score of 9 (the highest amongst the sample), this regression model would predict that a teacher
candidate would not attempt approximately 13.5 questions on their diagnostic test and anyone
with an anxiety score greater than 46.5 would attempt all questions (y = -0.35x + 16.64 (y =
blanks, x = anxiety score); coefficient of determination (R-squared) equal to 0.21).
The R-squared value of 0.21 implies that approximately 21% of the difference between a
teacher candidate with 13.5 blanks and a teacher candidate with no blanks can be attributed to
each teacher candidates’ anxiety level. At the end of the course, this inverse relationship between
the number of blanks and anxiety score still remains (y = -0.16x + 6.60 (y = blanks, x = anxiety
y = -0.3536x + 16.642
0
5
10
15
20
25
30
35
0 10 20 30 40 50
# of
Bla
nks
Anxiety
y = -0.1573x + 6.6025
0
5
10
15
20
25
30
35
0 10 20 30 40 50#
of B
lank
s
Anxiety
64
score); coefficient of determination (R-squared) equal to 0.16). This means that, at the end the
course, a high-anxiety teacher candidate is predicted to still not attempt 5.2 questions (a change
of 8.3 points) and any teacher candidate with an anxiety score of 42 or greater is predicted to
attempt all questions (a change of 3.5 points). The R-squared value of 0.16 now implies that
approximately 15.5% (a change of 5.5%) of the differences between the teacher candidate with
5.2 blanks and no blanks could be explained by the differences in each teacher candidates’
anxiety level.
This echoes the sentiments expressed by a number of the teacher candidates interviewed
for this study, who reported their own levels of confidence in the subject, and their willingness to
try questions they would have previously avoided, did in fact increase due to their participation
in the MCK course.
4.5.2.1 Correlation Between Math Content Knowledge and Math Anxiety
Based on the findings of other researchers (Daniels et al., 2011), and the apparent initial
correlation of data in this study, a regression analysis was performed to determine if there was a
positive correlation between math content knowledge increases and decreases to math anxiety
(Figure 6).
65
Figure 6
Regression Analysis of Math Content Scores vs Math Anxiety
This regression model confirms that a strong positive correlation between math content
score and math anxiety exists both before (above left, R-squared 0.31) and after (above right, R-
squared of 0.29) the course. Based on the data, we would predict that, before the course, an
exceptionally high anxiety teacher candidate would have a score of approximately 9 on the
diagnostic test, while, after the course, a teacher candidate with the same anxiety level would
have an estimated summative test score of approximately 19.5. This regression model indicates
that, prior to the course, 31.5% of the differences in scores between low scoring teacher
candidates and high scoring teacher candidates could be attributed to differences in anxiety; after
the course this reduces to 29%. This tells us that, of all the factors affecting math content
knowledge scores, math anxiety contributes to approximately 30%.
While the regression models confirm the positive correlation between math content and
y = 0.5257x + 4.1693
0
5
10
15
20
25
30
35
40
0 20 40 60
Dia
gnos
tic S
core
Anxiety Score (Pre-Test)
y = 0.381x + 16.011
0
5
10
15
20
25
30
35
40
0 20 40 60
Sum
mat
ive
Scor
e
Anxiety Score (Post-Test)
66
math anxiety, the diagnostic appeared to have a high degree of predictability as no scores on the
diagnostic achieved the ceiling score of 36. However, for the summative, a higher proportion of
teacher candidates were either at or approaching the ceiling score; lowering the model’s overall
predictability.
4.5.3 Distribution of Scores by Topic
The diagnostic and summative tests assessed teacher candidates in seven different, mostly
numeracy-related, mathematical topics (Table 5). As each topic area was marked out of a
different score, percentages have been used to indicate improvement. The topic areas included:
(1) Addition and Subtraction; (2) Multiplication and Division; (3) Understanding Fractions,
Percent, Decimals, and Ratios; (4) Operations of Fractions and Decimals; (5) Solving Word
Problems; (6) Order of Operations; and (7) Integers.
Table 5
Diagnostic vs Summative Test Score Averages by Topic
Topics (1) (2) (3) (4) (5) (6) (7) Questions Per Topic 4 4 12 6 4 3 3 Year 1 (PJ) Diagnostic 70.1% 23.5% 43.7% 34.5% 41.3% 46.0% 43.8% Summative 93.7% 62.4% 77.5% 69.4% 70.2% 84.3% 79.6% Change 23.6% 39.0% 33.8% 35.0% 29.0% 38.2% 35.8% Year 1 (JI) Diagnostic 71.1% 21.8% 56.8% 46.5% 56.6% 58.6% 57.5% Summative 94.7% 62.4% 82.1% 71.9% 69.5% 81.1% 79.3% Change 23.7% 40.5% 25.4% 25.4% 12.9% 22.5% 21.8% Year 2 (PJ) Diagnostic 75.5% 21.7% 53.6% 37.6% 44.0% 52.5% 47.7% Summative 87.9% 50.2% 80.7% 68.1% 73.2% 88.7% 84.1% Change 12.4% 28.4% 27.1% 30.5% 29.2% 36.2% 36.4% Year 2 (JI) Diagnostic 75.3% 29.9% 60.4% 47.2% 50.3% 65.3% 54.2% Summative 86.8% 53.1% 85.2% 82.2% 81.9% 93.5% 90.3% Change 11.5% 23.3% 24.8% 35.0% 31.6% 28.2% 36.1%
67
Figure 7
Diagnostic vs Summative Test Score Changes by Topic (Percent Improvement)
This data shows that teacher candidates had the most significant increase in
Multiplication and Division (2) skills, with the least amount of improvement in Addition and
Subtraction (1). However, it must be noted that the average diagnostic score on Addition and
Subtraction was about 73%, whereas, on Multiplication and Division, it was 25%, leaving more
room for improvement in this area to start with. This is illustrated in Figure 8, where the average
of Year One, Year Two, PJ and JI are shown in a more proportional way.
Figure 8
Combined Topic Score Changes and Initial Scores by Topic (Including Proportional Ratios)
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
(1) (2) (3) (4) (5) (6) (7)
Y1 PJ Y1 JI Y2 PJ Y2 JI
0%
20%
40%
60%
80%
100%
(1) (2) (3) (4) (5) (6) (7)
Summative (Y1 PJ)Summative (Y1 JI)Summative (Y2 PJ)Summative (Y2 JI)Diagnostic (Y1 PJ)Diagnostic (Y1 JI)Diagnostic (Y2 PJ)Diagnostic (Y2 JI)Av. DiagnosticAv. Summative
68
This illustrates the diagnostic score (bottom) as well as the summative scores in the same
topic area (top) giving us a more proportional representation of how each of the cohorts (PJ and
JI) did in Year One and Year Two in each of the topic areas. This allows us to see that
multiplication and division (2), operations of fractions and decimals (4), order of operations (6),
and integers (7) had similar percentage changes (~33%), yet in multiplication and division (2),
the change was proportionately larger due to the lower initial diagnostic scores. Yet, it still
remains the lowest scoring topic overall. In addition and subtraction (2), there was a higher
overall summative score, but the initial diagnostic score was already quite high, leaving a smaller
proportionate difference.
4.5.4 Individual Question Analysis
To investigate this further, each individual question was analyzed from both the
diagnostic and summative tests. For this analysis, the data from Year 1 and Year 2 (PJ and JI)
has been combined as the test used for both years was identical and there was no significant
variation between these groups in terms of scores. The numbers shown represent the proportion
of teacher candidates who got each individual question correct on the diagnostic and summative
tests. In the table:
• Correct represents the proportion of teacher candidates who got the question correct.
• Max Room for Improvement indicates is the proportion of teacher candidates who did not get the question correct on the diagnostic.
• Actual Improvement is the proportion of teacher candidates who got the question incorrect on the diagnostic but correct on the summative.
• Improved is the proportion of teacher candidates who were able to improve and showed an improvement from the diagnostic to the summative.
All values are a proportion of the aggregate data.
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Table 6
Individual Question Analysis
Addition and Subtraction A1 A2 B1 B2 Correct (Diagnostic) 0.78 0.83 0.68 0.61 Correct (Summative) 0.94 0.95 0.87 0.86 Max Room for Improve 0.22 0.17 0.32 0.39 Actual Improve 0.16 0.12 0.19 0.25 Improved 0.73 0.71 0.60 0.65 Multiplication and Division C1 C2 D1 D2 Correct (Diagnostic) 0.29 0.44 0.11 0.08 Correct (Summative) 0.72 0.87 0.39 0.31 Max Room for Improve 0.71 0.56 0.89 0.92 Actual Improve 0.43 0.43 0.28 0.22 Improved 0.61 0.77 0.31 0.24 Understanding Fractions, Percents, Decimals, and Ratios E FF1 FF2 FP1 FP2 FD1 Correct (Diagnostic) 0.16 0.51 0.47 0.51 0.64 0.44 Correct (Summative) 0.34 0.85 0.84 0.86 0.87 0.84 Max Room for Improve 0.84 0.49 0.53 0.49 0.36 0.56 Actual Improve 0.18 0.34 0.37 0.35 0.22 0.40 Improved 0.21 0.70 0.70 0.71 0.62 0.71 Understanding Fractions, Percents, Decimals, and Ratios (cont) FD2 G1 G2 G3 G4 H Correct (Diagnostic) 0.64 0.73 0.42 0.73 0.46 0.42 Correct (Summative) 0.90 0.86 0.82 0.87 0.82 0.73 Max Room for Improve 0.36 0.27 0.58 0.27 0.54 0.58 Actual Improve 0.26 0.13 0.40 0.14 0.36 0.31 Improved 0.73 0.49 0.69 0.52 0.66 0.54 Operations of Fractions and Decimals I1 I2 J1 J2 K1 K2 Correct (Diagnostic) 0.54 0.26 0.52 0.47 0.29 0.24 Correct (Summative) 0.83 0.60 0.77 0.79 0.69 0.56 Max Room for Improve 0.46 0.74 0.48 0.53 0.71 0.76 Actual Improve 0.29 0.34 0.26 0.32 0.40 0.32
70
Improved 0.63 0.46 0.53 0.60 0.57 0.42 Solving Word Problems L1 L2 L3 L4 Correct (Diagnostic) 0.25 0.44 0.52 0.63 Correct (Summative) 0.70 0.72 0.70 0.79 Max Room for Improve 0.75 0.56 0.48 0.37 Actual Improve 0.44 0.28 0.19 0.16 Improved 0.59 0.50 0.39 0.43 Order of Operations M1 M2 M3 Correct (Diagnostic) 0.50 0.44 0.65 Correct (Summative) 0.86 0.83 0.88 Max Room for Improve 0.50 0.56 0.35 Actual Improve 0.36 0.40 0.23 Improved 0.72 0.70 0.66 Integers N1 N2 N3 Correct (Diagnostic) 0.48 0.49 0.49 Correct (Summative) 0.76 0.79 0.91 Max Room for Improve 0.52 0.51 0.51 Actual Improve 0.28 0.30 0.42 Improved 0.54 0.59 0.82
Table 7
Average improvements for teacher candidates who could improve
Improv Max Addition and Subtraction 0.72 0.19 Subtraction 0.62 0.36 Multiplication 0.69 0.63 Division 0.28 0.90 Understanding Fractions, Percents, Decimals, and Ratios 0.61 0.49 Operations of Fractions and Decimals 0.53 0.61 Solving Word Problems 0.48 0.54 Order of Operations 0.70 0.47 Integers 0.65 0.51
71
Figure 9
Difficult Level of Individual Questions by Proportion Correct (Diagnostic and Summative)
Figure 9 shows the proportion difference per question on the diagnostic versus summative
tests. It indicates that all questions showed improvements in their scores, yet some topics had a
higher improvement proportion than others.
Figure 10
Overall Improvement by Teacher Candidates Who Could Improve
0.000.100.200.300.400.500.600.700.800.901.00
A1 A2 B1 B2 C1 C2 D1 D2 EFF
1FF
2FP
1FP
2FD
1FD
2 G1 G2 G3 G4 H I1 I2 J1 J2 K1 K2 L1 L2 L3 L4 M1
M2
M3 N1 N2 N3
Diagnostic Summative
0.000.100.200.300.400.500.600.700.800.901.00
A1 A2 B1 B2 C1 C2 D1 D2 EFF
1FF
2FP
1FP
2FD
1FD
2 G1 G2 G3 G4 H I1 I2 J1 J2 K1 K2 L1 L2 L3 L4 M1
M2
M3 N1 N2 N3
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Figure 10 indicates the overall improvement in individual question scores by those
teacher candidates who answered incorrectly (or left blank) a question on the diagnostic test yet
scored correctly on the summative. The line at 0.60 is the average improvement across all
questions.
4.5.5 Distribution of Teacher Candidates into Tiers
In order to more effectively understand the true impact of the course on teacher
candidates’ math content knowledge and math anxiety improvements, a tiered approach to
analyzing the data was undertaken. This was done in order to determine the impact that the
course had on four different levels (or tiers) of teacher candidates. The tiers were determined by
the overall diagnostic test scores.
• Tier 1: those teacher candidates who scored 75% or above • Tier 2: those teacher candidates who scored between 50% and 74% • Tier 3: those teacher candidates who scored between 25% and 49% • Tier 4: those teacher candidates who scored less than 24%
Table 8
Distribution of Teacher candidates in Each Tier
#
Students Tier 1
(>=75%) Tier 2
(<=50 % < 75%) Tier 3
(<=25% < 49%) Tier 4
(<=24%) Year 1 All 258 51 68 83 56
PJ 163 23 44 55 41 JI 95 28 24 28 15
Year 2 All 225 45 72 74 34 PJ 153 23 51 52 27 JI 72 22 21 22 7
Combined 483 96 140 157 90
The preceding table indicates the number of students assigned to each Tier based on their
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diagnostic test scores. Using these tiered categories, a teacher candidates’ overall change in math
content knowledge, their topic area scores, and their math anxiety scores were then analyzed.
The first stage of this analysis was dividing the teacher candidates into the various tiers by their
overall diagnostic test scores.
The following charts show the distribution of teacher candidates through the various
Tiers. Figure 11 is a combined graph for comparison across years and cohorts (PJ and JI). The
remainder are pie charts that show the individual distributions for comparison.
Figure 11
Distribution of Teacher candidates in Each Tier
04080
120160
All PJ JI All PJ JI
Year 1 Year 2 Combined
Tier 1 (>=75%)
Tier 2 (<=50 % < 75%)
Tier 3 (<=25% < 49%)
Tier 4 (<=24%)
19.9%
29.0%32.5%
18.6%
1 2 3 4
19.8%
26.4%32.2%
21.7%
Year 1
1 2 3 4
14.1%
27.0%
33.7%
25.2%
Year 1 (PJ)
1 2 3 4
29.5%
25.3%
29.5%
15.8%
Year 1 (JI)
1 2 3 4
20.0%
32.0%32.9%
15.1%
Year 2
1 2 3 4
15.0%
33.3%34.0%
17.6%
Year 2 (PJ)
1 2 3 4
30.6%
29.2%
30.6%
9.7%
Year 2 (JI)
1 2 3 4
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This data allows us to see how the teacher candidates in each year, and in each cohort (PJ
or JI), were distributed across the various tiers. In general, the JI candidates had a significantly
higher proportion of teacher candidates in the Tier 1 category than the PJ candidates, with Tiers
2 and 3 being somewhat similar, and accounting for over 50% of the teacher candidates in each
case. In addition, the distribution between Year One and Year Two was mostly similar, with
some slight changes in Tiers 1 and 4. Year One PJ also showed the highest proportion of Tier 4
teacher candidates.
This tiered analysis was further examined to look for patterns in topic specific areas,
allowing for the analysis of each topic within each tier. Teacher candidates were first assigned to
a tier depending on their diagnostic score (as seen above) and then each tier was analysed on a
per topic basis. Figures 12 to 15 show the distribution of changes for each of the 4 tiers as
defined in the following way:
• Significant Improvement: > 9 points improvement on score, > 10 points on anxiety, > 50% per topic area
• Modest Improvement: 0 > 9 points improvement on score, 0 > 10 points on anxiety, 0 > 49% per topic area
• No Change: no change in score, anxiety, or topic area • Remained Perfect: same as no change but initial score was 100% • Modest Decline: 0 < -9 points decline on score, 0 < -10 points on anxiety, 0 < -50%
per topic area • Significant Decline: < - 9 points decline on score, < -10 points on anxiety, < -50% per
topic area
The above values were chosen as a teacher candidate improving 9 points on the
summative test from their diagnostic would move from the bottom of one tier up into the next
highest tier. For anxiety, the lowest recorded score was 9 points, leaving 41 possible points of
movement in anxiety levels. These were broken down into four sections to match the content
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scores, providing a measure of +/- 10 points for significant changes. For topics, as each topic has
a different possible score, a standardized measure of 50% was considered to be a significant
improvement in those areas.
4.5.5.1 Tier 1 Analysis
For Tier 1 teacher candidates, the majority showed only modest improvements in their
overall math content scores (Figure 12). In most topic areas, across Year One and Year Two (PJ
and JI), nearly a quarter of teacher candidates showed no change. In each cohort, there were
teacher candidates who had modest or significant declines across most areas. Year Two PJ
candidates had the highest proportion of declines. The area of most improvement across all years
and cohorts was within Multiplication and Division (2), which is consistent with findings across
all tiers.
Figure 12
Tier 1 (>=75%) Changes (Content Score, Anxiety, and by Topic Area) - Years 1 & 2 (PJ & JI)
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 PJ
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In Year Two, 25% of the PJ candidates saw a modest decline in Operations of Fractions
and Decimals (4). In terms of anxiety, most showed a significant to moderate improvement.
However, there were a significant number of teacher candidates who showed a moderate decline
in their anxiety scores across three of the four cohorts.
4.5.5.2 Tier 2 Analysis
The Tier 2 teacher candidates across all four categories showed significant improvement
in their overall score and a significant to modest improvement in their anxiety scores (Figure 13).
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 JI
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 PJ
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 JI
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They also showed a modest improvement in their multiplication and division skills (3). There
was also a modest decline seen in addition and subtraction (1) for all four groups. There were
also very few ‘remained perfect’s in this group, yet there was a significant number of ‘no
change’s seen across all four categories.
Figure 13
Tier 2 (<=50 % < 75%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI)
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 PJ
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 JI
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4.5.5.3 Tier 3 Analysis
This Tier shows a significantly high percentage of significant improvements across both
years and cohorts, with the highest level being seen in the Year One PJ candidates (Figure 14).
Additionally, there were very few declines seen in this group as well, and even the no change
category was quite low compared to Tier’s 1 and 2. The highest no changes were seen in
Addition and Subtraction (1), which is consistent with previous findings in this paper.
0%
25%
50%
75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 PJ
0%
25%
50%
75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 JI
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Figure 14
Tier 3 (<=25% < 49%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI)
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 PJ
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 JI
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 PJ
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4.5.5.4 Tier 4 Analysis
This tier has a similar profile to Tier 3, yet there is a higher proportion of no change, and
declines (both moderate and significant) (Figure 15). The anxiety scores for these teacher
candidates also showed only a moderate improvement, which is consistent with the other tiers as
well. PJ candidates in both years showed the highest levels of overall score improvements and JI
candidates showed an unusual improvement in Addition and Subtraction (1), not consistent with
the other tiers.
Figure 15
Tier 4 (<=24%) Changes (Total and by Topic Area) - Years 1 & 2 (PJ & JI)
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 JI
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 PJ
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4.5.5.5 Summative Tier Distribution
By the end of the course the Tier distribution for the combined (Y1, Y2, PJ/JI) data was:
• Tier 1: 336 out of 483 (69.57%) • Tier 2: 110 out of 483 (22.77%) • Tier 3: 29 out of 483 (6.00%) • Tier 4: 8 out of 483 (1.66%)
As the distribution of Tiers was fairly consistent across Years 1 and 2 and the cohorts
(PJ/JI), the summative results have been combined.
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y1 JI
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 PJ
0%25%50%75%
100%
Rem Perfect Sig Imp Mod Imp No Change Mod Dec Sig Dec
Y2 JI
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Table 9
Distribution of Teacher candidates in Each Tier at End of Course
#
Students Tier 1
(>=75%) Tier 2
(<=50 % < 75%) Tier 3
(<=25% < 49%) Tier 4
(<=24%) Year 1 All 258 175 64 16 3
PJ 163 109 42 10 2 JI 95 66 22 6 1
Year 2 All 225 161 46 13 5 PJ 153 102 36 12 3 JI 72 59 10 1 2
Combined 483 336 110 29 8
Figure 16
Summative Tier Distributions
0100200300400
All PJ JI All PJ JI
Year 1 Year 2 Combined
Tier 1 (>=75%)
Tier 2 (<=50 % < 75%)
Tier 3 (<=25% < 49%)
Tier 4 (<=24%)
69%
23%
6% 2%
1 2 3 4
68%
25%6% 1%Year 1
1 2 3 4
67%
26%6% 1%
Year 1 (PJ)
1 2 3 4
70%
23%6% 1%
Year 1 (JI)
1 2 3 4
72%
20%6% 2%
Year 2
1 2 3 4
67%23%
8%2%
Year 2 (PJ)
1 2 3 4
82%
14%
1% 3%Year 2 (JI)
1 2 3 4
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Table 9 and Figure 16 show that, by the end of the course, the majority of teacher candidates in
Tiers 2, 3, and 4 moved to a higher Tier. A small percentage of students (less than 1%) dropped a
Tier, and 24.64% of teacher candidates remained at the same Tier (including Tier 1, which
represents 19.47% of this value). Table 10 and Figure 17 outline these changes. Initial Tier
placements are based on diagnostic results, final Tier placements are based on summative results.
Table 10
Diagnostic to Summative Changes in Student Tiers
Tier 1 (Initial) Tier 2 (Initial) Tier 3 (Initial) Tier 4 (Initial) Final # Final # Final # Final # Tier 1 94 Tier 1 126 Tier 1 100 Tier 1 16 Tier 2 2 Tier 2 13 Tier 2 51 Tier 2 44 Tier 3 0 Tier 3 1 Tier 3 5 Tier 3 23 Tier 4 0 Tier 4 0 Tier 4 1 Tier 4 7
Figure 17
Changes in Tiers
The following figure shows how students moved from their diagnostic Tier (left) to their
summative Tier (right).
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These figures show that the majority of teacher candidates moved up at least one Tier
from their diagnostic to their summative tests, with the majority ending the course in Tier 1. It is
important to note that 7.66% of teacher candidates in the course still did not achieve a grade of
50% or higher on their summative and nearly 2% of teacher candidates remained in Tier 4. A
small percentage of students (less than 5%) also showed a modest to significant decline in their
score change.
4.5.5.6 Individiual Question Analysis by Tier
In order to more fully understand the effect of Tiers on the individual questions, and by
extension the validity of the diagnostic/summative test, a Tiered analysis of the individual
questions was undertaken. Figure 18 shows the actual improvement on the summative, divided
by the maximum possible improvement from the diagnostic to the summative. A value of 1
indicates that all teacher candidates in that Tier got that question correct on the summative. A
negative value indicates a decline on that question from the summative to the diagnostic. The
figures (left to right, top to bottom) show Tiers 1, 2, 3, and 4 respectively.
As Figure 18 shows, teacher candiates in Tier 1 had a number of questions where there
was no further room for improvement (indicated by a 1) or that some teacher candidates got
questions on the summative incorrect that they had gotten correct on the diagnostic. This effect is
not seen for the other three Tiers.
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Figure 18
Tiers 1 – 4 Individual Question Analysis
4.6 Data Confidence and Course Effectiveness
The data shown indicates that the MCK course showed an increase on math content
knowledge as well as a decrease in math anxiety at a significant level for both sections of the
course (PJ and JI) over the two years. Given our observed t-stat and critical values shown in
Table 11, we can be confident that observed differences in the mean were not due to random
variation and can be considered to be statistically relevant.
-0.2
0
0.2
0.4
0.6
0.8
1
A1 B2 D1 FF1
FP2 G1 G4 I2 K1 L2 M1 N1 0
0.2
0.4
0.6
0.8
1
A1 B2 D1 FF1 FP2 G1 G4 I2 K1 L2 M1 N1
0
0.2
0.4
0.6
0.8
1
A1 B2 D1 FF1 FP2 G1 G4 I2 K1 L2 M1 N10
0.2
0.4
0.6
0.8
1
A1 B2 D1 FF1 FP2 G1 G4 I2 K1 L2 M1 N1
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Table 11
T-stat Tables
Correct Answers Number of Blanks Anxiety Scales
Diagnostic Summative Diagnostic Summative Pre-
Course Post-
Course Mean 17.44008 27.80785 7.716942 1.733471 25.24174 30.96074 Variance 75.69205 42.80979 50.88245 13.63689 86.12985 85.6527 Observations 484 484 484 484 484 484 Hyp Mean Diff 0 0 0 Df 897 725 966 t Stat -20.953 16.38819 -9.59962 P(T<=t) two-tail 1.16E-79 1.39E-51 6.64E-21 t Critical two-tail 1.962612 1.963241 1.962423
Furthermore, an analysis of the effectiveness of the course in terms of increased math
content scores and decreased math anxiety scores was performed. It should be noted with this
data that, even teacher candidates who self-reported less than 10% decrease (including those who
indicated an increase in math anxiety), showed around a 10-point average increase in their score
and around a 6 point decrease in the number of blank responses (Table 12).
Table 12
Summative and Anxiety Changes of > 10%
Summative Score > 50%
Score Change > 10%
Anxiety Change > 10%
Both Score and Anxiety > 10%
Tier 1 0 (0.0%) 44 (45.8%) 44 (45.8%) 23 (24.0%) Tier 2 1 (0.7%) 9 (6.4%) 49 (35.0%) 3 (2.1%) Tier 3 6 (3.8%) 3 (1.9%) 68 (43.3%) 1 (0.6%) Tier 4 30 (33.3%) 6 (6.7%) 49 (54.4%) 4 (4.4%) Total 37 (7.7%) 62 (12.8%) 210 (43.5%) 33 (6.8%)
Percentages above indicate percent of category (Total = 483, Tier 1 = 96, Tier 2 = 140, Tier 3 = 157, Tier 4 = 90)
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4.7 The Instructors
The instructors for this course came from a wide variety of backgrounds. Most of them
had a background in mathematics, but many were not education teacher candidates and had little
to no training in math pedagogy methods. Initially, Lila and Henry wanted to hire I/S level
teacher candidates from the teacher education program, but administratively this was not
permitted (Lila, interview, April 17, 2020). Instead, they shifted their focus to “making sure that
we had these math majors, somebody with a good math background” (Lila, interview, April 17,
2020). The other instructors were math education students (either former teacher candidates or
graduate level students). “What we look for a TA is definitely the content knowledge and having
the credentials of that math content knowledge, and then also the ability to understand teaching,
at least a little bit” (Lila, interview, April 17, 2020).
The creators of the course thought that, if they had the math content knowledge then they
had support through Lila and Rashida (one of the instructors who was also responsible for
content creation) for the pedagogy parts. No formal training in this area was provided to the
instructors, however, they were provided with all the materials they could need: “We gave them
everything, the lessons, down to the script, almost” (Lila, interview, October 28, 2020). The
focus was on the math content from grades 6 to 9 as it was felt that the more important part was
focusing on ensuring that those instructing the course were strong in that over other areas (such
as pedagogy).
The instructors interviewed for this study included all but one of the instructors from both
Year One and Year Two. Most of the instructors taught both years of the course and taught a
combination of both PJ and JI classes. General profiles on each instructor are provided here to
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give context to their interview responses. In addition, a tiered breakdown of their class structure
and a summary of their math content and math anxiety scores are also provided to give context to
the types of teacher candidates in each of their classes. These are compared at the end of the
profiles.
4.7.1 Julia
Julia instructed the course for both Year One and Year Two and focused on teaching JI
candidates in Year One and PJ candidates in Year Two. She had an undergraduate background in
mathematics and previous experience in teaching math to younger students. She was also
younger than many of the teacher candidates in the course and did not have a graduate level
degree or background in education. When asked why she decided to teach the MCK course, she
responded:
I love that feeling of just being able to help teachers teach and help teachers inspire their kids as well, because I know that doing this review was very good for them. And so that is why I came back the second year as well. (Julia, interview, April 21, 2020)
4.7.2 Justin
Justin taught both PJ and JI candidates in both years and, like Julia, was significantly
younger than the teacher candidates taking the course. He also did not have a graduate level
degree or a background in education.
I have been teaching math for a while now, and I have been teaching mostly younger students, […] students who are in elementary school, so from grade one to grade eight, and I have been teaching a lot of math curriculum before teaching [in the MCK course] and I found I wanted to get a change in environments. I wanted to teach a different group […] I wanted to see if there was a difference in teaching younger students compared to master students. And I also wanted to help teachers because I know a lot of teachers are struggling with math and I found this whole idea really, really interesting. (Justin, interview, April 22, 2020)
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4.7.3 Isabel
Isabel taught at both the PJ (Year 1) and JI (Year 2) levels. She has a background in
education and a number of years’ experience in the classroom teaching. While teaching the MCK
course, she was pursuing a graduate level degree in education. When asked why she decided to
teach in the MCK course, she responded
[…] one of the biggest reasons I think is because I love teaching mathematics, I really do. I love teaching my grade nine to twelve math. […] But I have always wanted to teach teaching programs, or even the graduate programs. How to teach. In this case, you cannot teach pedagogy, you can only teach the actual content base. I just thought it would be a really good experience to even teach adult learners because I have always taught students, and this would be a different type of teaching because now you are teaching basic mathematics to adult learners. It is a different dynamic, and I thought it was really interesting. And so I decided, why not? (Isabel, interview, April 27, 2020)
4.7.4 Gerdie
Gerdie was a teacher candidate in Year One of the MCK course. She transferred into a
different degree program and so was provided with the opportunity to teach the course for Year
Two. This gave her a unique perspective on the course, its structure, its teacher candidates, and
effective ways to present the information. She also had the unique opportunity of teaching a
group of Intermediate/Senior (IS) candidates who wanted to take the course as well, even though
it was not required for their course. She was also the only instructor who did not have a formal
background in mathematics.
[When] I applied for the [MCK course] position I thought why not? Cause I remember [taking] that course and thinking I could teach this, this would be cool to teach right? And most of my tutoring is in math. It is funny, it is like I do not actually have a math background. I feel like the [teacher candidates] are always surprised. They are like, wait, you are not a math major? I just like math. (Gerdie, interview, May 13, 2020)
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4.7.5 Rashida
Rashida had a combined role of being both a course instructor and assisting with the
creation of most of the content for the course. This included handouts for the lessons, selecting
materials from Khan Academy to match with lessons, the creation of the quizzes and
assignments, and the general overseeing of the implementation of the course. She is a secondary
school science and mathematics teacher, who was pursuing a Ph.D. degree in math education at
the time of this study. She was extremely passionate about the course and her passion for
teaching came through clearly in her interview and she touched on a number of big issues even
just in the initial question of “why did you decide to teach the MCK course?”
We have this persistent issue of [teacher candidates] coming who hate math, and do not know parts of math, and I do not know where that comes from. […] But, on top of that, for me, I want them to know math, of course, and I want them to like it and be able to teach it. But I want them to be really empathetic because early on, I kind of realized […] one of our biggest problems in this course is not going to be whether or not they know how to do math. It is going to be the fact that they feel really ashamed of this. There is this huge psychological toll. If I go up there and I am like Yeah, I have this math background, and I am a high school math teacher and all these things, that is not comforting necessarily, especially because so many of them, as I learned over time, their problems in math begin in high school. So being ‘I am a high school math teacher’ is the opposite of what is useful for them to know. (Rashida, interview, April 30, 2020)
4.7.7 Comparing Instructor Distributions and Scores
Figures 19 and 20 show that, when scores are normalized for expected scores in terms of
tier distribution patterns for each class, there is some variance in the various instructors.
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Figure 19
Normalization of Instructor Content Score Changes (Expected – Actual)
Figure 20
Normalization of Instructor Anxiety Score Changes (Expected – Actual)
However, the overall amount of variance is low (less than 2.5 points for score and 3.5 points for
anxiety) and many of these variations occurred not across instructors but across classes (so the
same instructor could have two widely different classes, as shown by Justin, Isabel, and
Rashida). However, Julia appeared to have an overall lower than expected effect on both scores
and anxiety.
4.7.8 Instructor Perspective
Overall, the instructors felt that there were definite benefits to the course, and they could
see the impact of the course on their teacher candidates.
I felt there was a significant [number] of [teacher candidates] in those classes, in the first
-3.0
-1.0
1.0
3.0
Bob Gerdie Isabel Julia Justin Rashida
-4
-2
0
2
4
Bob Gerdie Isabel Julia Justin Rashida
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year of my teaching, that said it was very good review for them especially because a lot of them have not done math in a long time. And just reviewing even the basics was helpful for them when they went into their practicums. (Julia, interview, April 21, 2020)
4.7.8.1 Benefits
In general, the structure of the course was well received by the instructors. They
appreciated the ready-made materials, yet some noted that it limited their ability to be creative in
their teaching. They also found it rewarding to see their teacher candidates improve as the course
progressed and enjoyed the opportunity to have an impact on that. The instructors liked the
inclusion of Khan Academy for the homework as it gave the teacher candidates the opportunity
to try multiple times to get their homework correct and often provided the teacher candidates
with both a review of the concept and, sometimes, a slightly different way to approach it (though
this did cause some issues in at least one case). It was also easy for the instructors to check the
answers for completion each week.
Isabel also recalled the positive atmosphere at the end of the course and how:
they were very positive, and they were just thankful that they could take the course, even the ones that were not supposed to be in my course because they passed the [exemption test], they came and thanked me for being in the course. (interview, April 27, 2020) Justin, Gerdie, and Rashida also shared some selected feedback from their teacher
candidates who found the course, and their instructors, to be helpful and positive. Justin, Rashida
and Isabel also made comments about how their teacher candidates appreciated when they stayed
back to help them or made themselves available outside of class time to assist. Many of the
instructors, and the teacher candidates, also spoke of the ways in which the class was made more
inviting (such as potlucks and treats brought in to share) and that it helped to create an
environment where the teacher candidates felt more comfortable asking questions and taking
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chances with their work.
4.7.8.2 Challenges and Suggestions for Improvements
When asked if they felt there could be improvements to the course, Justin noted that he
felt the course would benefit from being longer but realized that there were logistical constraints
that prevented that. Isabel also noted that a longer course would be beneficial. Isabel really
enjoyed using Khan Academy “[b]ut sometimes the questions are difficult in sense of the
problem-solving questions [were] sometimes are worded in a difficult way” (Isabel, interview,
April 27, 2020) and recommended investigating a different platform, or creating a custom,
tailored one for the course, in future.
Rashida and Isabel had concerns about the focus of the course on quizzes and the
diagnostic test. Both felt it was counter to the idea of learning math for the sake of learning math
and instead reinforced ideas of learning math for the test. The time constraints placed on the
teacher candidates when writing quizzes also ran counter to the idea that “math is not a speed
sport” instead of promoting the idea that “math is a social endeavor” (Rashida, interview, April
30, 2020). Rashida also wanted to incorporate more teaching styles and tools into her teaching.
They did like having handouts, but I really wanted to do a lot more of the conceptual work through manipulatives […] I desperately wanted to use integer chips, but there just were not enough for everybody to have. You have to teach people how to use integer chips, teachers included. I really would like to have more manipulatives, especially because the people this course is intended for are elementary. I want to demonstrate more to them how you can do group work with all these things, how you can do a lot of discovery in mathematics, as opposed to me saying, here is a thing. (Rashida, interview, April 30, 2020)
Rashida did note as well that, in order for an instructor to be effective in teaching the MCK
course with different pedagogical strategies (such as group work, manipulatives, etc), they would
need to have a strong math background and an understanding of math teaching methods as well,
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which many of the instructors for the course did not have.
Julia suggested that stronger teacher candidates (such as those who would fall into the
Tier 1 category) found the three-hour long course to be boring, yet weaker teacher candidates
were looking for additional assistance from the instructor. While she understands that there are
issues with streaming, she did recognize that the course was not equally effective for all of her
teacher candidates. Isabel also felt that breaking up the teacher candidates into different groups
based on their diagnostic might be helpful as “some of the weaker [teacher candidates] felt that it
was going too fast and if I did not stay after the course to help them, they would have been really
lost” (Isabel, interview, April 27, 2020). However, she also noted that that there are issues with
“ability grouping” and that perhaps it was not the best possible solution.
4.7.8.3 Changes in Teacher Candidate Performance and Evaluating Success
The instructors for the course were asked how they evaluated success in the course and in
their teacher candidates. Most said what was important to them went beyond the quantitative
marks the teacher candidates were earning in the course (through quizzes, homework,
assignments, and the summative test) and instead their participation and attitudes towards the
subject were a far greater indicator of success. In fact, the instructors were never privy to the
summative scores of their teacher candidates.
My purpose in the course, and I tried to convey this to the [other instructors] as well, was I just wanted people to leave the course feeling more confident in their abilities to do math than they did at the beginning. Point blank, that is all I cared for. For that, I think we have had a resounding success. I think even in the classes where sometimes people were like, I do not care for this or whatever, I think in general, people [improved] where they started and where they ended; even just in their attitudes towards math. (Rashida, interview, April 30, 2020) This sentiment was shared by most of the instructors. While success was measured in a
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formal way through quiz scores, homework marks, and participation in class, the instructors also
looked for markers of increased confidence in their teacher candidates. Julia also found state of
mind of the teacher candidates to be important as well.
I think some of the [teacher candidates] came in with an open mind, and they saw they had to do something, they had improvements to make. And they tried. And at the end, they were more comfortable with the math and they were more willing. [Although] 12 weeks is not enough time to completely change someone's outlook on math […], I do think that, if they leave with an open mind, and they have the resources that they need, [they can succeed]. I know even some [teacher candidates] were asking me, can I have this handout so I can use it? Or can I have this game that I can use it for my class? I think that would be a successful or a positive outcome for a [teacher candidate]. (Julia, interview, April 21, 2020) Finally, Isabel remembers seeing a dramatic change in her teacher candidates from the
first class to the last.
[…] attitude wise, I remember the first day, and the first day is kind of scary because the first day it is like, ‘Hello everybody’ and introductions, but then it is like, here is your diagnostic assessment. Here's an hour of just cruelty because [they] do not remember how to divide, they do not remember ‘What are exponents?’ so their faces are just like ghost white and I was like, ‘Oh, no, you poor things’. But that was the structure of the course. So I could not change it. (interview, April 27, 2020)
When asked how she addressed this, how she assisted these “poor things” while still maintaining
the requirements of the course, she said she told her teacher candidates “do not worry next week
will be better” (Isabel, interview, April 27, 2020). She then spent the rest of the course working
on creating a positive environment and that:
[…] by the end the course, they were all like high fiving. They were so happy when they can get things right. And they were just so proud of themselves that I was like super. […] By the end of it, they were doing such a great job and they felt like they were doing a great job, which was the most important part of the course, I think. (interview, April 27, 2020)
4.7.9 Advice to Future Instructors
The instructors were also asked what advice they would give to future instructors of the
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course, based on what they themselves had learned. Julia recommended teaching slowly and
carefully and trying your “best to be non-judgmental when it comes to some student’s learning”
(Julia, interview, April 21, 2020) and recognizing that the teacher candidates may be hesitant to
ask for help. Justin also recommended making it fun. He found that, in his first year, he was a bit
stricter and, in the second year, he had more fun and found the teacher candidates benefited more
from this. Gerdie also echoed this sentiment by suggesting that the classroom be a more relaxed
environment where the focus is not just on the content, but also on building confidence in the
teacher candidates taking the course, and that this was just as important.
In addition, Justin recommended reviewing all the material before teaching it, and being
“very familiar, especially if it is your first time teaching or if you get nervous in front of a class”
(Justin, interview, April 22, 2020). While Isabel was conscious of the fact that these are adult
students who are learning to become teachers, they cannot be taught like elementary students.
Her recommendation was to teach them by reminding them that this is what they are going to be
teaching eventually.
4.8 The Teacher Candidates
The teacher candidates interviewed for this study came from a range of backgrounds and
represent a cross-section of math abilities, levels of math anxiety, and general feelings towards
the MCK course. They represent teacher candidates from both years of the course and were
instructed by a cross-section of the instructors for the course. They represent both Primary/Junior
(Grades K - 6) (PJ) and Junior/Intermediate (Grades 4 - 10) (JI) teacher candidates.
4.8.1 Jack (Year One, PJ)
Jack started the course as a Tier 3 teacher candidate (at the very bottom of the tier) and
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scored extremely high on the anxiety scale (between 0 and 10 points). By the end of the course,
Jack was a Tier 1 teacher candidate with a dramatic 22-point increase in his math content scores,
and a stunning 26-point decrease in anxiety. Compared to his peers, Jack’s scores put him well
above the average for both score changes. Overall, Jack’s comments about the course were very
positive:
seeing that, even in one semester, I could go from failing a diagnostic to confidently doing well on a diagnostic. It taught me that I am not bad at math, I just need to be doing this more often and refresh these skills and keep them fresh. (interview, May 4, 2020) Jack described himself as a former professional actor who fell in love with teaching after
“teaching drama as sort of a side counterpart to that profession as an artist” (Jack, interview, May
4, 2020). When it comes to math anxiety, Jack did not initially know of the term but identified
with it as he came to learn more about it. He stated “I am not confident in my ability with math,
and because of that, I was really scared how I was going to be able to teach it” (Jack, interview,
May 4, 2020). Jack was expecting the course was going to be a refresher of math facts and
getting his “toolbox refilled” (Jack, interview, May 4, 2020).
4.8.2 Lisa (Year One, PJ)
Lisa was educated outside of Canada where she feels her math and science courses were
strong and prepared her well. This was supported by her math content knowledge and math
anxiety scores where her score on the diagnostic test placed her in Tier 2 of the course, and her
pre-course anxiety score shows her as having moderate anxiety (between 20 and 30 points). Lisa
referred to herself as an average student who described herself as strong in math and science.
When asked if she considered herself to have math anxiety, she stated “No, but I would say [I
have] a lack of interest” (Lisa, interview, May 6, 2020). She had a somewhat negative attitude
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towards the MCK course and felt she was not in need of the course and did not find it to be of
direct benefit to her. By the end of the course, she had a Tier 1 score and lower anxiety (between
30 and 35 points).
4.8.3 Zed (Year One, JI)
Zed, a JI candidate from Year One of the course, described himself as a high-level math
student who took the course because he wanted to experience the course, but was not required to
take it due to his passing score on the exemption exam. He did not describe himself as having
math anxiety and stated “my experience with math has been a very positive one” and says he
likes that “there is one right answer. And for me that that is simplicity” (Zed, interview, May 4,
2020). Zed was educated in a number of countries and followed the American school system up
to about grade 11 when he came to Canada. He found a love of teaching while working in the
finance field and decided to pursue a teaching degree. When asked why he opted into taking the
MCK course, he said:
the reason I opted in is because I was actually really curious […] I have not taught math to little kids in forever, and actually never. And so I was like, I am actually curious how it is going to be taught.” (interview, May 4, 2020)
Zed’s expectations of the course included gaining “a refresher on the math concepts” along with
“some of that discovery math, exploration math, concepts” learned in other courses.
By the end of the course, Zed did not describe his scores as changing much but he did
comment on the ways in which it was taught and had some suggestions for improvement. Zed
described himself as highly familiar with math concepts and that he felt confident in his math
skills prior to the course; he did not describe himself as having high levels of math anxiety.
Based on his self-reported scores, this would place him in Tier 1 with low anxiety (between 40
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and 50). Zed did not give permission for me to get his exact scores.
When asked about his past experiences with math teachers, Zed seemed mostly
unimpressed. He remembers that “sometimes I would feel the teacher was all over the place that
I really could not follow” (Zed, interview, May 4, 2020) while other teachers would specifically
comment on how answers he got wrong were, in fact, “easy”. He felt that “the instructors
themselves, some were great, but unfortunately it was hit or miss” (Zed, interview, May 4,
2020). He also recalls one instance where a student pointed out to the teacher that the answer was
incorrect and the teacher “[s]he goes, "Oh, really? Well, if it is wrong, you take over" and, you
know literally gives them the chalk, and she sits back down” (Zed, interview, May 4, 2020). Zed
believes that these early influences by teachers are part of what influenced him to become a math
teacher, and to try and improve on what he had experienced in his past.
4.8.4 Kira (Year One, JI)
Kira scored average on her diagnostic and showed medium levels of anxiety. Her
diagnostic scores place her in Tier 2 with moderate anxiety (between 20 and 30 points). By the
end of the course, her score had increased 10 points (a significant improvement), which placed
her in Tier 1. Her anxiety also decreased slightly by 4 points (moving her into the 30 to 40
range).
The majority of Kira’s elementary education took place outside of Canada and both of her
parents were teachers who supported her in mathematics throughout her schooling. Kira had a
strong background in math having studied related subjects in biology and physics; though she did
recall that “I would say [I had] a pretty bad experience with math as a student” (Kira, interview,
May 5, 2020).
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Kira also speaks fondly of her past teachers, including her parents, and credits these early
experiences with eventually pursuing a science degree. She did recall some teachers who “were
not very nice. I can tell you that, if you did not do well in math, they would just kind of
discourage you completely from doing math and tell you to just drop it completely” (Kira,
interview, May 5, 2020). These experiences were not enough to discourage her from pursuing
sciences and teaching though and she found “being in Teachers College, kind of changed that for
me, just seeing how wonderful it can be to make math meaningful for students” (Kira, interview,
May 5, 2020).
Kira expected the course was going to be a “brush up” of her math skills and that there
was going to be “a lot of math thrown at me” (Kira, interview, May 5, 2020). Yet, after the
course, she felt that the course had been a “waste of two hours in the morning” and that “I could
probably learn that stuff two hours before teaching it and probably be fine” (Kira, interview,
May 5, 2020). When asked if the course provided her with any strategies, Kira did admit that:
for some certain things yeah, for sure. Some of the worksheets that they were given to us had different strategies that I probably would have come across while teaching but it was good having it in my hands and being ready to get into practicum and I just like here, I have it already, let's use that. (Kira, interview, May 5, 2020)
She also said she had used the worksheets during her practicum placement.
4.8.5 Juno (Year Two, PJ)
Juno started the course in Tier 4 with an anxiety score in the 30 to 40 point range (low to
moderate). By the end of the course, Juno had increased to the very bottom of Tier 3 in her math
content knowledge. Surprisingly, her anxiety levels increased by 9 points placing her in the
moderate to high anxiety range (20 to 30 points). No reason was given for this shift in anxiety
during her interview. However, the number of blanks left on her summative test (which, as
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shown previously in this paper, could be considered an increase in confidence) were reduced by
half from the diagnostic to the summative tests.
When asked if she found a benefit to the course, Juno did respond saying “[t]his course
has truly given me so much confidence and, I know what my weaknesses are and I know what I
need to work on when it comes to my math skills” (Juno, interview, May 6, 2020). It is possible
that Juno’s initial anxiety scales did not reflect her true levels of anxiety (an issue commonly
found with subjective anxiety scales).
Juno describes herself as being passionate about teaching and she has a background in
psychology. She described herself as having some statistics courses but remembers dreading
them. She tells a powerful story of her relationship with mathematics describing herself as “bad
at math” and that she had “[…] always been that one kid who just does not get it” (Juno,
interview, May 6, 2020). She further elaborates with a story from her family:
I remember this one time sitting with my entire family. It was my mom, my dad, my grandpa and they were like ‘Are you kidding me? This is so simple. We are just moving it to the other side of the equation and this is what is going on.’ And it finally clicked after three days, but it was just, continually not understanding and not being confident. (interview, May 6, 2020)
When asked about how she dealt with math in school, Juno replied that:
I used to cheat so much during math in elementary school. I would just copy the answer off the person next to me and not know why I was doing it [and] I went with math tutoring in high school because I was not competent. (Juno, interview, May 6, 2020)
This lack of confidence led her to not only avoid answering questions in class but to also
physically hide from the teacher:
[…] in grade four, it was a four or five split, the class would have carpet time to talk about math. I was in the grade four class and, instead of coming to sit at the carpet, I used to hide under my desk and because [the teacher] would not see me. […] I kind of slipped through the cracks, which now I look at and [it was] definitely is one of the reasons why I
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was not confident in my math skills. (interview, May 6, 2020)
It also caused her to question how others in her class always seemed so confident and assured
about their answers, something she herself did not feel:
I was not confident, even if I knew the answer. I just did not put my hand up because I was so scared that it was wrong. That was the number one thing in my head, like this answer is wrong, your answer is wrong; because of my experiences. And it just amazed me to see people next to me in my class, like in elementary school and in middle school, and I guess probably even in high school. I distinctly remember […] looking around and being like, how are these people so confident and just raising their hand up and saying ‘the answer is five’. I would never be able to do that. (interview, May 6, 2020) Despite her confidence issues and lack of math content knowledge, Juno knew she
wanted to become a teacher:
I love teaching and I love being around students and kids. And I have learned so much from students and different ways of learning. I think that is one reason and I am just passionate about education and learning new perspectives of things. (interview, May 6, 2020)
She also spoke of tapping into the creativity of young minds and being less focused on marks
with her eventual students, and “foster[ing that] so that when they get older they can be critical
thinkers” (Juno, interview, May 6, 2020).
Juno spoke about how a teacher’s approach to math could have a profound impact on her
as a student and how important the idea of a “welcoming space” was for her to be able to
effectively learn. She spoke of the importance of making mistakes in mathematics and how
teachers that “were very rigid in the way that were they were teaching math that, kind of like
held me back” (Juno, interview, May 6, 2020). Juno recalls being scared when she was informed
that she was going to have to take the MCK course and feeling like “it was going to be this really
daunting experience of relearning math and everyone in my class already knows these things and
I am the only one who does not understand certain math concepts” (Juno, interview, May 6,
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2020).
4.8.6 Maya (Year Two, PJ)
Maya started the course as a Tier 2 teacher candidate and, by the end, saw a 10-point
increase in her score that placed her in the Tier 1 category. Her anxiety score pre-course was in
the low to moderate range (20 – 30 points) and saw an increase of 7 points that placed her in the
low anxiety category (30 – 40 points).
She was educated mostly in the elementary Montessori system and credits this with
giving her a different attitude towards mathematics, and the MCK course, than many of her
classmates. She had spent time as a Montessori instructor and entered the Faculty of Education’s
program in order to broaden her skills and teaching opportunities. Maya was actually a bit
shocked by the concept of math anxiety and states “I did not actually realize how many people
were afraid of math until I joined this program, and it became abundantly clear that almost half
of my cohort is terrified of math” (Maya, interview, May 7, 2020). She states she was always
good at math and, going through a Montessori program, credits it with learning the basics “in a
very hands-on way”. She also credits her math ability “partially because I have been blessed, you
know, to be pretty smart” (Maya, interview, May 7, 2020). She expected the course would
simply go through math concepts from grades one to eight.
4.8.7 Francis (Year 2, JI)
Francis began the course at the bottom of Tier 3 and ended with a 15-point increase
placing her squarely in the middle of Tier 2. Her anxiety scores went from high anxiety (10 – 20
points) to moderate (20 – 30 points) over the course of the course. Francis’ attitude towards the
MCK course was overwhelmingly positive. One of her first comments during her interview was
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“I will just say this, right off the bat, I loved [the MCK course]” (Francis, interview, May 20,
2020). She spoke of her struggle with mathematics and the development of significant math
anxiety from an early age. Francis started off strong in math but, due to “many personal
problems and stuff going on” in middle school, she “just stopped doing work in math class, I
would just draw” (Francis, interview, May 20, 2020).
While Francis struggled from grades 6 to 8 in mathematics, she does recall some positive
interactions with teachers from her past. Specifically, she remembered her grade 9 teacher who
attempted to help her fill in the gaps and built her confidence up to where she was willing to try
the grade 10 academic stream. Sadly though, she describes her grade 10 math teacher as “not the
kindest teacher” (Francis, interview, May 20, 2020) and so she dropped from Academic to
Essential Math in grade 10, limiting many of her future options and nearly limiting her from
pursuing her educational goals as well.
Francis spoke of how now, as a teacher candidate herself, that she can “kind of
sympathize with the teachers that I did not jive with in high school” (Francis, interview, May 20,
2020). When asked about how she felt when she learned she had to take the MCK course, she
said:
I just felt so much panic and I came and spoke to [Lila] actually after her talk and I was like, ‘is there anything I can do to prepare for [the MCK course] because I cannot express enough how behind I am and I am like, very concerned. (interview, May 20, 2020) She also reported “stomach dropping fear like, I just completely freaked out. I mean I was
in denial about the fact that I would be teaching math eventually” (Francis, interview, May 20,
2020). Francis also expected that the course would be mostly middle school math concepts and
was surprised when it was so “back to basics”, yet she appreciated that it was.
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4.8.8 Teacher Candidate Perspectives
The teacher candidates interviewed for this study had mixed reactions to the course.
While most found it to be highly beneficial to them, a few found it did not meet their
expectations and that perhaps they would have been better off with an independent study instead
of the formal course. In this section, I will describe some of the benefits, issues and suggestions
for change in the course identified by the teacher candidates.
4.8.8.1 Benefits
While he was not required to take the course, Zed found the structure of the course to be
beneficial. Jack found the idea of an “unabashful […] let us just go back and review; yes, this is
a grade two concept, but let us really take our time with it so that you understand it” (Jack,
interview, May 4, 2020) approach helpful. There was a “sort of really open approach, non-
judgmental, just element of like, no, let us not just skirt over this. Maybe we do not know how to
do long division but let us just talk about it” (Jack, interview, May 4, 2020). He also liked the
idea of being in a classroom where “everyone was kind of in the same boat for the most part”
(Jack, interview, May 4, 2020).
Maya enjoyed the refresher aspect of the MCK course, how it helped her to be “more
confident in my own knowledge base” and how it related to other courses where she felt “like for
things like reading and science, you do not really have to do that” (Maya, interview, May 7,
2020). Finally, Francis found it helped her greatly in her practica where she was placed in a
grade eight class and had to teach the math lesson, which she asked to do, she found that:
even though we had not covered eighth grade material in [the MCK course], I [had] already built my confidence so much […and…] I asked for math classes. I asked to teach math lessons, because at that point I felt confidence to at least give it a try. (interview, May 20, 2020)
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It is important to remember that Francis was the teacher candidate who felt “panic” when she
learned she had to take the MCK course in the first place.
It is also interesting to note that these benefits were mentioned by most of the teacher
candidates who took the course in its first year when it came time to write the Provincial Math
Proficiency Test (MPT). While the MCK course was created prior to the announcement of the
MPT, most of the teacher candidate participants in this study indicated that the MCK course had
a positive impact on either their writing of the MPT (for those who took the MCK course in Year
One) or preparing to write it (for those who took the MCK course in Year Two). One of the most
dramatic comments about this issue came from Francis (who was initially panicked to take the
MCK course) who, when asked about the MPT, stated:
Oh, it is going to be no issue, I feel completely prepared. Obviously, I will study [but] I do not feel particularly stressed about it. I think it is going to be fine. As I mentioned, I will study because it goes up to ninth grade math, and then there are math pedagogy questions [but] yeah, with [the MCK course] and then practicum experience on top of that, no issues. (Francis, interview, May 20, 2020)
However, Zed and Lisa, who both scored high on their diagnostic tests, felt the MCK
course had little impact on the MPT for them. In fact, Lisa believes that, without the MPT, the
MCK course had little to no value and that teacher candidates requiring additional help in math
should just seek it from other sources instead of requiring all teacher candidates to take a
mandatory math content knowledge course.
There was also a marked increase in confidence overall with some teacher candidates
showing a dramatic increase in their confidence and anxiety.
I feel like I could teach myself the ninth-grade curriculum in its entirety; if I needed, not that I would need to. If I found out I was teaching ninth grade next year, I could spend a summer studying the curriculum and be totally fine because I have that literacy, that math
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literacy. I have the building blocks to learn other math skills, because I have the basics which I did not have before. (Francis, interview, May 20, 2020)
This boost of confidence and building of math skills can also be seen by Jack, who scored quite
low on his diagnostic test, yet stated that he passed the MPT and got about 90% on the math
content portion.
4.8.8.2 Challenges and Suggestions for Improvement
The challenges raised by teacher candidates mostly related to the initial diagnostic test
and not receiving specific feedback on their responses. When asked about feedback, Lila stated
that the teacher candidates were informed that marked tests were not returned due to the use of
the same test from year to year to ensure research validity. Furthermore, both Lila and Henry
confirmed that teacher candidates received detailed feedback on their scores with a breakdown of
how they did on each section of the diagnostic. Still, some teacher candidates felt there was a
degree of “secrecy” (Zed, interview, May 4, 2020) regarding the test results and other teacher
candidates felt that “I would prefer having more hard stats” (Kira, interview, May 5, 2020) in
terms of having the pass/fail of the course be based on the diagnostic and summative tests alone.
It should be noted that this was not the case; the diagnostic test was not included in teacher
candidate scores and the summative only counted for 5% of their final marks.
Other suggestions included those for course content. Juno suggested having something
similar to Khan Academy for the homework but “that was catered just to our course, like if it
could be created just for this math course” (Juno, interview, May 6, 2020). Though she did admit
to finding the Khan Academy lessons and exercises useful, Maya also described a few small
issues that came up with Khan Academy. For example, a topic being taught in a slightly different
way in class than taught on the platform, but that overall it did not impact the course in any
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significant way.
Conversations around the Provincial Math Proficiency Test (MPT) also arose and many
of the teacher candidates suggested having more targeted content associated with the test as part
of the course. However, all but one of the teacher candidates interviewed for this study found the
MCK course useful regardless of the MPT requirements.
4.8.8.3 A Magic Wand for the Course
When the teacher candidate participants were asked, “if you had a magic wand and could
redesign the course from scratch, what would you do?” most suggested minimal changes. Zed
suggested a more general focus on supporting the teacher candidates in terms of emotions and
feelings associated with math and math anxiety and building “[…] an environment in the class,
that we can easily close the door and say, listen, guys, whatever happens in this class stays in this
class, do not worry that you are making mistakes” (Zed, interview, May 4, 2020). He also
suggested doing “a lot more questionnaires on feelings and emotions” (Zed, interview, May 4,
2020) in order to gain a deeper understanding of how the teacher candidates were reacting to the
materials and how it was influencing them on a more reflective or personal level.
Kira also wished for a bit more focus on content up to the grade 8 level and focusing
more to “include more of the huge units kids have trouble with […] from EQAO results or
experience from others” (Kira, interview, May 5, 2020). Kira also had the suggestion to focus
more on:
fractions, percentages and decimals. That is huge in math. If we can build on that for teachers, just to get more comfortable converting between all that stuff. I would have that for everyone to do and not just a one day focus. So, I would not do traditional math like here convert this this this. No, I would give them more like meaningful projects kind of thing like we do with kids. (Kira, interview, May 5, 2020)
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Jack, Lisa, Maya, and Francis all commented on how the link between the math pedagogy
course and the MCK course could be strengthened and how it would have been nice to bring
concepts from the pedagogy course into the MCK course. Francis commented on how she would
have appreciated “using manipulatives more” (Francis, interview, May 20, 2020) and how, in the
math pedagogy course, the focus on using manipulatives made concepts easier to understand.
Francis also had the unique perspective where her pedagogy class was taught by Lila, so the
content in each course lined up and this was seen as a definite positive when taking the MCK
course. One area many of the teacher candidates requested was the addition of different teaching
styles and manipulatives in the course and to be taught using more of the methods they were
learning themselves in their pedagogy classes.
4.8.9 Teacher Candidate Expectations of Instructors
When asked for feedback on their instructors, the teacher candidates had generally
positive remarks. One comment that came up in the instructor interviews was the age difference
between the teacher candidates and two of the instructors, Julia and Justin. Both were
undergraduate students and the teacher candidates in the course were at the Masters level. When
the teacher candidates in this study were asked if the age difference caused any issues for them,
none of the teacher candidates in their classes stated this was an issue for them, and that both did
a fine job teaching the course. There were comments about how the age difference was kept
somewhat as a secret until the end of the course and that teacher candidates in one section found
out Justin’s age through his social media account. However, Kira relayed that: “it was a weird
dynamic there for sure, but he was very knowledgeable teaching the math” (Kira, interview, May
5, 2020).
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In addition, neither Julia nor Justin had an education background and this came up as a bit
of concern for some of the teacher candidates. Kira remarked:
We had asked him about a question about how to show division or multiplication of fractions using shapes, or squares, or rectangles or whatnot; shaded areas. And he told us ‘oh, sorry, I cannot help you with that, you have to go ask your math instructor, the pedagogy instructor’. We were kind of, like, okay, he cannot help us, we have to ask someone else […] he is not a math teacher. He is an economics guy. He is going to be a banker. He is not going to be a math teacher. It makes sense. He cannot explain it but we are trying to understand how to do it here. (Kira, interview, May 5, 2020)
Zed made similar comments about Julia when he remarked:
[she] had tons of passion for math and I could see that in her eyes. When she talks about math, she genuinely likes [it] […] but I just feel that there was a big disconnect between her passion, her excitement, and what was on the other end that the [teacher candidates] were checked out. (Zed, interview, May 4, 2020)
Maya also found that Julia:
taught sort of how I imagined teaching happened like 15 years ago, like how she probably learned math, which for me was fine, because I already understood most of the concepts. But the thing was, we had been learning in our math classes [the math pedagogy class], how to do things differently. (Maya, interview, May 7, 2020)
Maya felt that this disconnect confused some of the teacher candidates in the class.
At this point, it should also be noted that the instructors were specifically told that they
were responsible for teaching math content, and not pedagogy. They were given freedom to
teach in their own style but were not provided with manipulatives and were informed not to teach
multiple methods for various concepts. This information was not provided to the teacher
candidates and so may have had an impact on their assessment of the instructors.
When asked to evaluate the instructors overall, the teacher candidates were mostly
positive. They found them all passionate about the subject and that they all created a warm, safe
and welcoming environment for learning. Lisa even recalled Rashida:
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turning the class into like, an everlasting safe environment. I remember she used to bring with her cookies and Timbits. And one of our colleagues baked a cheesecake so it was like this friendly environment. (Lisa, interview, May 6, 2020)
Francis recalled Gerdie:
had a very low-key, laid-back attitude that made everyone so comfortable. Like, chill is not the most professional word to describe an instructor, but she had an extremely chill [attitude]. She made the class feel very low stakes. Like this is nothing to worry about. Everyone is going to be fine and no one is going to fall behind in this class. And she created a really fun atmosphere as well. She always started with a game or something to get everyone kind of, you know, loosened up. (Francis, interview, May 20, 2020) This idea of a safe, welcoming environment was important to all of the teacher candidate
participants and it seems to be an area all of the instructors seemed to have succeeded. Zed
would have liked to see a high school, or preferably a grade 8 teacher, brought in to teach the
course. Kira agreed about bringing in:
someone who had already taught math, as a high school teacher or elementary school teacher; knowing what to expect when we are going to classroom ourselves. And they have already probably experienced so many ways that students would react to the way they are teaching and how they would have to change it for next time. So, someone who was in the field in real life, coming back and teaching us how to teach math because content anyone can teach just how to teach it that Is different. (Kira, interview, May 5, 2020) Gerdie, who had the unique position of both taking the course as a teacher candidate and
instructing it, also felt that:
I guess the main thing is [the instructor] needs to feel that the courses is important, and I guess kind of realize that confidence is the biggest thing. I think as long as you are kind of aware of what you are trying to do, like you are teaching them concepts but like, really, we have got to just make them feel comfortable. (Gerdie, interview, May 13, 2020)
4.9 A Response from Lila and Henry
When asked about teacher candidate feedback on various areas of concern, Lila took the
time to respond to many of them in a second, follow-up, interview (Appendix F). When asked
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about the inclusion of quizzes over assignments, she found that “the quizzes were just really the
most efficient way to get a snapshot of their ability to demonstrate what they learned that week
from the modules. And it is easy on behalf of instructors because they can mark it quickly” (Lila,
interview, October 28, 2020). However, she was aware of the anxiety these types of quizzes
(even though they were short) could induce and mentioned that some of the quizzes had already
been substituted for assignments and this was a consideration for future.
She then commented on the idea of combining the MCK course with the math pedagogy
course and indicated that from an administrative level (mostly budgetary) this wouldn’t be
feasible. She also commented that perhaps:
not all faculty would be comfortable teaching a math only content knowledge [course]. Or maybe they would, but would feel it is not in their realm of scholarship to teach grade six to nine math […] I do not think it would be the best way to maximize the potential from faculty. (interview, October 28, 2020) A further area of discussion came from the idea of streaming the classes somehow; as
both teacher candidates and instructors noted that teacher candidates that would fall in the Tier 1
category, would often find themselves bored or disengaged from the course and those in Tier 4
needed additional supports. Lila noted “you have got some [teacher candidates] who almost
passed [the exemption test] and they are at the very beginning [feeling] regret, feeling a grudge,
that they should have passed and this is too easy for them” (Lila, interview, April 17, 2020). She
did note that this was more of an issue in Year One of the course and that, in Year Two, they
started adding more challenge questions into the content to keep these teacher candidates more
engaged. In terms of streaming, however, this idea was discussed and researched but ultimately
not implemented due to the “stigma” attached to that type of system. Instead, she focused on
keeping the course within the cohorts (PJ and JI) and that:
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all the instructors were encouraged to talk about that [with their class]: ‘We are community and we help each other. So some [teacher candidates] [may] require more time and you are there to support them. Right? If you know the math a bit better than they do.’ (interview, October 28, 2020)
Henry also re-iterated this fact in a follow-up discussion stating:
one important thing that most MCK course instructors emphasize in their courses is the role of community. [Teacher candidates] are expected to mentor each other and help everyone in the class master the math content. In some sense, I think they view their work in the MCK course as necessary preparation for teaching. This type of friendly, collegial and supportive culture goes a long way toward reducing people’s math anxieties and increasing their math proficiencies. (Henry, 2021)
This fact was emphasised in the classes so that teacher candidates who were stronger in the
subject (the Tier 1 teacher candidates) would feel a sense of community to help their peers out
and develop their teaching skills.
The final area Lila and I discussed surrounded the issue of who was teaching the class,
specifically those who did not have an education background, and some who were still working
on their undergraduate degrees; as this was brought up as a major issue by some of the teacher
candidates, and even some of the instructors as well. When asked about the idea of bringing in
retired teachers or others to teach the course, Lila remarked how bringing in a sessional lecturer
or a seconded or retired teacher had two issues: the first being budgetary as the amount required
to pay a lecturer is significantly higher than a that of a Teaching Assistant. The second issue was
based around the expectations of a lecturer or teacher and what the purpose of the MCK course
was compared to the math pedagogy course. “To have a sessional lecturer teach that course does
not really maximize their full potential of sharing their scholarship. This is a math course, it is
grade six to nine math” (Lila, interview, October 28, 2020). As the purpose of the course was to
teach math concepts, the creators found that it was more beneficial to have subject experts
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teaching over pedagogy or teaching method experts.
4.10 Seeing the Future
When asked what they would like to see for the future of the MCK course, and of math
education in general, both Lila and Henry saw one where the MCK course was widespread and
implemented in teacher education programs across the country. Henry recognizes that:
[a] lot of the [teacher candidates] going into elementary to become elementary teachers have backgrounds in the arts and humanities and are not particularly strong in math. I think this would be a good thing to add to every teacher education program across Ontario. (Henry, interview, May 8, 2020)
Lila seconded this thought with “I want every teacher education program across this province,
and across our nation, to have a version of [the MCK course] within their [teacher education
program]” (Lila, interview, April 17, 2020). It should also be noted that both Lila and Henry
were strongly opposed to the idea of a math qualifying test for teachers and both saw an ideal
future where a MCK course replaced the idea of a qualifying test to ensure teachers were well
prepared for their profession.
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Chapter Five: Discussion
5.1 Introduction
In this chapter, I answer the research questions posed and report on the major findings of
the study; linking them back to the literature and providing deeper insights into the data. It is
then followed by a discussion of the major findings from the study, along with implications for
future research. The chapter then concludes with some final thoughts on the MCK course, the
Math Proficiency Test, and the impacts such a course could have on the future of teacher
education programs.
5.2 Research Questions
This research focused on the following five questions:
1. What is the purpose of a mathematics content course in a teacher education program? 2. What was the focus of the course? 3. What are the perceived benefits and challenges of the MCK course? 4. What effect did the course have on math content knowledge and math anxiety? 5. How was success in the course evaluated and did the course meet expectations? 5.2.1 Research Question 1: What is the purpose of a mathematics content course in a
teacher education program?
The MCK course was designed to address issues of math content knowledge in teacher
candidates and to address issues of equity and access to future studies, careers, and opportunities
for elementary level students by improving the math skills of their teachers. Building upon
previous research related to teacher influences on their students as it relates to mathematics
(including issues related to gender) and the importance of mathematics as a “gatekeeper subject”
(Lila, interview, April 17, 2020), this course was implemented to provide elementary teachers
candidates with basic numeracy skills. The course was created following both observations and
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quantitative data collected by Lila from her math pedagogy course that showed the lack of
mathematics skills in her elementary teacher candidates was a real and measurable issue. With
this data in hand, Lila and Henry proposed a course where the intention was to build the basic
fundamental math skills of teacher candidates in a highly prescribed method.
By providing teacher candidates with basic skills in mathematics with a focus on
numeracy, the goal of the course was not to teach them all math concepts from grades one to
eight, but instead was focused on building their ability and confidence to teach themselves
further mathematics skills, thereby improving their self-efficacy which, as shown by Gürefe and
Bakalım (2018), has a positive effect on math anxiety.
A secondary outcome of the course was to help elementary teacher candidates decrease
their math anxiety by focusing on building this confidence through their improved self-efficacy
with mathematics skills. It was believed that the course would provide these teacher candidates
with the skills and confidence to be able to learn and teach mathematics more effectively in their
own classrooms following graduation from the program. This relates to work by Boyd et al.
(2014) that the key to confident math teachers is building their own beliefs in their math abilities.
The MCK course was designed to do this and was quite effective at it across a number of
measurements.
The intended effect of the MCK course was to address many of the well-researched
detrimental effects that teachers with weak mathematics skills and high levels of math anxiety
can have on their students, such as those studied by Stoehr (2017), Finlayson (2014) and Beilock
et al. (2010). An unintended side effect of the course was that it also helped prepare the teacher
candidates for the implementation of the Provincial Math Proficiency Test (MPT) and helped
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build the skills and the confidence to attempt, and pass, this new qualification requirement.
5.2.2 Research Question 2: What was the focus of the course?
Following on research by Reid et al. (2018) and Dehane (2011), this course focused on
issues of math content knowledge, and numeracy specifically, as these were areas that were seen
as foundational to building the tools for other areas in the curriculum, such as science and
technology courses. There was also a feeling that teacher candidates, through their previous
schooling, and even through the application process to university, would already have strong
literacy and language skills.
Furthermore, building upon this previous research, the course creators felt that a focus on
numeracy would be the most effective use of the limited time and resources available and would
help teacher candidates build a framework for other areas of mathematics. While some of the
teacher candidates and instructors indicated a desire to have other math subjects included in the
course, the focus on numeracy was decided due to limited time available for the course along
with previous research indicating a strong numeracy foundation could help them understand
other math concepts on their own. This fact was backed up by interviews with teacher candidates
in this study who commented on being able to study higher level math concepts independently
because they had both the confidence and the foundational skills to do so.
The participants of this study also saw the value of building the mathematics skills of
teacher candidates as they saw the course as addressing both math anxiety and responding to
political issues surrounding math education in the province; including some who saw it as being
created as preparation for the Provincial Math Proficiency Test (which was a coincidental impact
of the course and not a reason for its creation). Other teacher candidates realized the significant
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effect their math anxiety could have on their future students and saw the MCK course as being
created as a way to address that.
The creators and instructors of the course saw the focus on mathematics as a way to
narrow the gap in what math teachers are expected to do and what their capabilities entering the
teacher education program were; thereby attempting to provide them with the tools and
confidence to effectively teach their future students mathematics and address long-standing
issues related to mathematics anxiety and poor performance in the subject in the future.
5.2.3 Research Question 3: What are the perceived benefits and challenges of the
MCK course?
Overall, the MCK course was well received, with the majority of teacher candidates
showing statically relevant improvement in areas of mathematics content knowledge and a
decrease in math anxiety. The Tier distribution changes from the diagnostic test to the
summative test illustrates how dramatic those changes were. However, the course had a high
degree of variability in its effects both across topic areas and across tiers of teacher candidate
achievement; with nearly 13% of the teacher candidates in the course over the two years seeing
little to no improvement (and some even declining in scores). The overall benefits and issues
found in this study are listed in the next two sections.
5.2.3.1 Benefits
The teacher candidates interviewed found the materials useful, the use of Khan Academy
beneficial, the quizzes appropriate and informative of their progress, and the overall instruction
and attitudes of their instructors to be beneficial. They found the pacing of the course to be
mostly good, though there were some requests for the focus on certain concepts to either be
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lessened or increased (such as too much time spent on multiplication and division but not enough
on ratios, percentages, and fractions, which they felt were rushed). This fact seems to be
supported by the diagnostic and summative test data, which showed significant improvement for
teacher candidates in multiplication and division, but only modest improvement in some of these
other areas.
Instructors also felt that they were able to create a safe and welcoming environment for
their teacher candidates and this was felt, and appreciated, by the teacher candidates who
commented on the fact that they were encouraged to try concepts and to not be afraid of failure.
This is a concept that Boaler (2016) shows is instrumental to building confidence in students.
The course was also quite successful in affecting the scores of the majority of teacher
candidates in the course and had the most profound effects on Tiers 2 and 3, which accounted for
more than 50% of the teacher candidates enrolled. Teacher candidates in Tier 4 (the lowest
scorers on the diagnostic test) also had dramatic improvements in their scores and attitudes
towards mathematics by the end of the course. The teacher candidates interviewed for this study
who fell into Tiers 3 and 4 had dramatic changes in their test scores on the summative and found
their attitudes and confidence towards mathematics had shifted substantially to the positive and
they had more confidence to learn new math concepts independently.
5.2.3.2 Challenges
While many teacher candidates improved in their overall scores, there was a significant
proportion who did not show an increase in their scores (and some who showed a decline). Many
of the teacher candidates, who showed little to no improvements, resided in the Tier 1 (highest)
category, indicating that the course may not be beneficial to teacher candidates who already have
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a strong initial foundation in mathematics. The instructors noted that teacher candidates in Tier 1
often seemed bored with the content, while teacher candidates at lower tiers often needed
significant additional help.
There were also issues associated with the teacher candidates in Tier 4 in that, while they
improved significantly in their overall math skills by the end of the course, over 30% of them
still scored below 50% on the summative test. Finally, the overall changes to anxiety scores for
almost 50% of the teacher candidates in the course improved by less than 10% (or 4 points).
The content of the course was also an issue for some with teacher candidates who were
asking for a broader base of concepts to be covered rather than basic numeracy. Some teacher
candidates wanted the course to be two years in length with the second year focusing on more
complex concepts, building upon the numeracy foundations. However, most understood that the
second year of the program (with practicum placements and other course work) might be too
intense for the insertion of an additional math course.
The usage of Khan Academy for the homework also caused some issues for teacher
candidates as the content and methods used on this platform sometimes did not match what they
had been taught in class and this caused some confusion. Some of those interviewed suggested
things like streamlining the homework to be more in line with in-class materials, and creating a
course specific online module for homework instead of utilizing Khan Academy. This idea was
considered by the course creators, however, the difference in the amount of work required to
create such a course rather than working with the pre-established Khan Academy was not seen to
be significantly beneficial.
There were also issues concerning the instructors and instruction methods of the course.
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Many of the teacher candidates, and even some of the instructors, commented on the lack of
hands-on learning, and the focus on a more instructor-centered method of teaching. There were
also concerns that some of the instructors for the course (Justin and Julia) did not have education
backgrounds and were not only unaware of different teaching styles but, like the other
instructors, were required to teach in a specific way and only specific methods. There were also
some concerns raised about the age difference between some of the instructors and their teacher
candidates.
Therefore, it is important to note that various biases may have come into play when
evaluating the effectiveness of the instructors. Sprinkle (2008) notes that:
[r]esearch suggests that student perceptions of college professor and adjunct instructor effectiveness are influenced by a variety of fixed and dynamic professor/instructor-held traits. Fixed traits such as personality, age, and gender (Arbuckle & Williams, 2003; Amin, 1994; Freeman, 1994), mainly beyond the control of professors and instructors, can sway student perceptions of effectiveness and subsequent evaluation ratings. (p. 276)
However, Sprinkle’s research shows that, younger, college-aged students found younger
(under fifty-five) professors/instructors to be more effective. However, their research also found
that “students were most likely to deem instructors/professors as effective if they disseminated
course information in a manner congruent with the respondents’ learning style” (p. 286). As
stated previously, most of the teacher candidates interviewed for this study found the teaching
style of the instructors in the course to be different from what they were learning in their
pedagogy courses, particularly in the teaching style of Julia, with whom there were also age
concerns.
This could mean that the teacher candidates were more concerned with the teaching style
of the various instructors and their approach to the materials rather than their age differences. As
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all of the materials were provided to the instructors (such as handouts, quizzes, and even scripts
for lessons), this ensured that all teacher candidates were receiving the same content but left little
room for alternative learning methods, which was considered a major issue for teacher
candidates and instructors alike and further supports the work of Sprinkle (2008). This
observation is also supported by the work of Van der Sandt and O’Brien (2017), who found
teaching style had a profound effect on teacher candidates in a math content knowledge course.
Finally, the focus on assessment marks from quizzes and homework was also a concern
and alternative methods of assessment were requested. There were also concerns by the teacher
candidates that they were unable to see their diagnostic tests and did not receive adequate
feedback on where they were weak, as this would have provided them with valuable insights as
to where to focus their attention. The instructors also had concerns that there was too much of a
focus on speed and accuracy with these quizzes and this was not only an additional source of
stress and anxiety on the teacher candidates but also focused on areas that they themselves, as
future teachers, were learning in other courses to avoid.
5.2.4 Research Question 4: What effect did the course have on math content
knowledge and math anxiety?
In the context of this study, “effect” is measured both by quantitative data collected on
each teacher candidate and on the class as a whole. A positive effect would be an increase in a
teacher candidate’s math content test scores (as measured by their diagnostic and summative
tests) and/or a decrease in their overall anxiety levels (as measured by the RMAS). The
qualitative interviews also provided some insight as to the effect of the course on individual
teacher candidates in a more subjective way. A positive effect in this case would be self-reported
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increases in confidence and an ability or belief that teacher candidate could now attempt more
difficult math questions and/or reported some direct benefit to themselves after taking the course.
Most of the teacher candidates interviewed for this study had strong memories and
feelings towards math that they felt affected their current attitudes and abilities in math. Based on
both the quantitative data collected from Year One and Year Two of the course and the
qualitative data collected from interviews, the course had a statically significant effect on both
math content knowledge and math anxiety of teacher candidates enrolled in the course. However,
this effect was not universal in its distribution was and highly dependent on the initial tier
placement of the teacher candidate. An analysis of the relationship between content scores and
math anxiety did show that an inverse relationship existed; teacher candidates with higher
content knowledge tended to have lower anxiety and vice versa. This is supported by the work of
Daniels et al. (2011), who also used regression analysis to show similar patterns existed.
What is interesting to note is that, in this study, 43% of participants indicated minimal
positive changes in anxiety or increased anxiety, while only 13% of participants showed a less
than 10% improvement in their math content scores. This may be due to the fact that the anxiety
scale is subjective and self-reporting and may be influenced by factors outside of the classroom
and outside of math anxiety. This could also include the timing of when the scale was
administered (directly after the diagnostic and summative tests), or a lack of interest in filling out
the scale accurately (some tests were filled in such a way that teacher candidates chose all 1’s or
all 5’s for their responses). Additionally, some teacher candidates may have felt shame about
their math anxiety and filled the scale in inaccurately as they were not anonymous. These issues
of reliability in self-reported anxiety were also found in research by Daniels et al. (2011).
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By dividing the teacher candidates in the course up into four tiers, another interesting
trend appeared. While all tiers showed significant improvement overall, it seems as though
teacher candidates who scored >75% on the initial test (those in tier 1) showed overall less
improvement in specific subject areas than those in the other tiers. This can be most likely
attributed to the fact that their potential for improvement was more limited than those in other
tiers (based on their initial scores and the total points available). This supports other data
collected in this study that shows that the course had a less relevant impact on these teacher
candidates than others.
Zed, who, due to passing the exemption test, could be classified as a Tier 1 teacher
candidate, felt he personally did not receive much from the course. However, Lisa, who felt she
was a high-level teacher candidate but fell into Tier 2 initially and felt she did not improve at all
and that the course was not helpful for her, saw an improvement of over 20% from diagnostic to
summative test results; which falls in line with the overall data observed for this tier.
When the quantitative data is combined with the interviews, it shows that the course did
have a measurable effect on both math content knowledge and math anxiety. By improving the
math content knowledge of course participants, there was a measurable effect on their math
anxiety levels. However, this effect was somewhat moderated at the upper end of the scale
(teacher candidates with very low levels of initial anxiety) as there was little room for
improvement to begin with.
However, this course was not truly designed for Tier 1 teacher candidates. It was
designed to help teacher candidates with significant gaps in their math content knowledge (such
as those in Tiers 3 and 4 particularly – who scored below 50% on their diagnostic tests). For
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teacher candidates in those tiers, only 2% (Tier 3) and 7% (Tier 4) did not show at least a 10%
improvement in their scores. It should also be noted that, by the end of the course, less than 8%
of the teacher candidates had a score of less than 50% on their summative test. This is compared
to 51% who scored less than 50% on the diagnostic test.
5.2.5 Research Question 5: How was success in the course evaluated and did the
course meet expectations?
Evaluating a course like this must incorporate a few different perspectives. The first
involves those who created the course and understanding the impact of the course from an
administrative level; is the course meeting its stated goals of improving math content of teacher
candidates and is it worth the investment by the Faculty of Education? This data is mostly
collected from the diagnostic and summative tests along with the math anxiety scales.
Further details can also be collected from end of course evaluations, however, these must
be tempered with the fact that it is a skewed sample (those who feel they want to say something
about the course / have strong feelings towards the course) and may not be representative of all
teacher candidates in the class. However, since it is used at an administrative level as an internal
evaluation of success, this information should be taken into account. While, due to
confidentiality, these evaluations were not made available for this study, Lila did indicate that
evaluations received had been mostly positive of the course, its content, and its instructors and
that exit surveys of teacher candidates enrolled in the program spoke highly of the MCK course
specifically.
The instructor’s evaluation of the course is gained from the teacher candidates’
participation in class, assignment and homework marks, and independent conversations between
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teacher candidates and instructors. The instructors also gain access to course evaluations, but
again these must be tempered. The instructors found the course to be effective and met many of
their expectations. Furthermore, the instructors used a more subjective evaluation of success that
included ensuring teacher candidates felt safe and supported in the class and that, by the end of
the course, they felt confident in their abilities and willing to try math on their own. From the
instructor interviews it seems like, from this evaluative standpoint, the course was successful.
Finally, the course can be evaluated from the teacher candidate’s perspective, which takes
in a number of different aspects. These include quantitative data, such as scores from the
diagnostic and summative tests and anxiety scales, along with weekly quizzes, assignment,
homework, and class participation. It also includes more subjective data like conversations with
peers and personal feelings towards the course and their view of their own personal
improvements (or failures) in the course.
From the data gathered in this study, it seems that the course was successful for many
teacher candidates and it met their expectations. However, it is important to note that the teacher
candidates interviewed for this study expected vastly different outcomes from the course. Some
expected a refresher of concepts, while others were more interested in seeing how these types of
topics could be taught. Some were terrified that the course was going to be difficult and fast-
paced (something they would not be able to keep up with) and some felt it would be a waste of
their time. While some teacher candidates had their expectations met, finding new confidence
and a refresh of concepts they had forgotten, others found they got very little out of the course
and wished it had been presented differently.
From an administrative point of view, this course met the stated expectations and
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requirements for success. From a quantitative perspective, the overall averages of the teacher
candidates enrolled in the class improved significantly, and their anxiety scores decreased as
well. While there were some outliers, the general improvement of the teacher candidates enrolled
in the class as a whole were positive and measurable. From a more subjective level, the course
creators (Lila and Henry) felt that the course had been successful. They also stated that they had
relied more on the input from the instructors to determine success than just the raw data. The
instructors felt the course had been successful as they felt they could see their teacher candidates
improving on weekly assignments and quizzes, but they were also able to see the teacher
candidate’s confidence grow throughout the course.
Finally, most of the teacher candidates interviewed found the course to be beneficial and
useful. Even teacher candidates like Zed and Lisa, who were somewhat negative about the
impacts of the course on them personally, agreed that the course was useful in helping them
develop some skills and they saw the positive impact it was having on their classmates who they
reported were weaker in the subject than they were at the beginning of the course. There were
some teacher candidates, like Jack, Francis, and Maya, who had only positive comments to make
about the course and felt it improved their math content skills and had a dramatic effect on the
way they approached mathematics as a whole.
5.3 Major Findings
This research had the following major findings:
1. The MCK course had varying effectiveness for teacher candidates based on their initial tier distribution and across different topic areas.
2. The standardization of course content moderated the differences in instructor backgrounds.
3. Course content was not taught in a manner consistent with the math pedagogy course.
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4. The anxiety scales utilized may not be fully indicative of the actual changes in anxiety of teacher candidates in the course.
5. Utilizing tier and topic specific data gathered from diagnostic tests may be helpful for both teacher candidates in the course and the creation of targeted course content.
5.3.1 Effectiveness Across Tiers and Topics
On average, the content knowledge scores for all teacher candidates in the course showed
significant improvements; from being able to successfully answer only about half of the
questions (48.39%) on the diagnostic to 77.20% on the summative. Additionally, the greatest
improvements in specific skill areas were seen in: Multiplication and Division; Order of
Operations; and Integers. Reasons for this could be attributed to lessons that focused on those
specific skill areas but could also be attributed to the fact that initially the teacher candidates may
have not seen such concepts since they themselves were in elementary school and the refresh of
the concepts was significant enough to show an improvement in their marks – in other words,
they were not lacking the knowledge, they just had not encountered it/used it in so long that it
was temporarily forgotten.
Furthermore, an analysis of the individual tiers showed that, overall, all four tiers of
teacher candidates showed an increase in math content knowledge across all subject areas.
Overall, Tiers 2, 3 and 4, those scoring less than 75% on the pre-test showed the most significant
improvements, while those in Tier 1 showed mostly moderate or no change. This is most likely
due to the fact that teacher candidates in Tier 1 were already strong in this area and had little
room for further improvement.
Tier 1 teacher candidates did see a uniform improvement of around 10% across most
topics and on their overall scores. Furthermore, their anxiety scores improved by a similar
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percentage (11.5%). Improvements were highest in the multiplication and division topic area
(22.4%) and this is in line with the 2nd and 3rd tier teacher candidates as well. It is important to
note that, while improvements in the multiplication and division area were well above
improvements seen in other subjects, the overall scores in these areas were still significantly
lower than other topic areas.
5.3.2 Standardization and Instructor Backgrounds
The instructors for this course came from a variety of backgrounds with some working on
graduate degrees in education, some with in-class experience, others coming from a pure
mathematics background and working on an undergraduate degree, and one without a formal
math background. The one consistent factor between all of the instructors is that none held
graduate degrees and none were considered formal instructors in the program; they were all
classified as teaching assistants within the Faculty of Education. This, along with some
administrative decisions, caused some issues in the way the materials were presented to the
teacher candidates. The instructors were all provided with standardized handouts and materials
and were instructed to teach in a more teacher-centered manner. This allowed for a
standardization of the content and ensured that all teacher candidates, regardless of class or
instructor, would receive the same materials. On evaluation of class/instructor specific data, it
appears that this method had the intended effect as there were very little variations on average
between instructors in terms of content and anxiety scores.
One area of note though was how the instructors, regardless of background, worked to
make the learning environment safe and inviting for their teacher candidates. Rashida
specifically spoke on how she felt that an instructor’s attitude towards their teacher candidates
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was just as important as their math knowledge and background. This sentiment was echoed by
the teacher candidates interviewed who felt the ideal instructor for the course should be someone
who knew the math content but also created a safe and welcoming environment and encouraged
teacher candidates to not be afraid to fail. Building this type of environment is supported by
researchers such as Boaler (2016), whose research deals with encouraging students to fail as a
way to learn.
Another area of significant interest in this study was a lack of significant difference in
overall scores between classes and instructors and even between years of the course. This most
likely indicates that the approach and materials created in the first two years of the course were
effective and were not significantly changed between year one and two of the course (with the
exception of the replacement of a quiz with an assignment and some minor changes to overall
content).
In addition, the lack of significant difference in teacher candidate performance across
instructors / classes shows that providing detailed, structured handouts to the instructors had the
effect of providing all participants in the course with an equal opportunity for success. This also
indicates that the focus on content, more than the exact method of instruction, was more
important to outcomes. However, it should be noted that instructors were all required to follow
the provided materials precisely and changes to teaching style or content were actively
discouraged. If instructors have been able to be more flexible with these aspects, different results
may have been seen.
5.3.3 Course Content Presentation
A number of teacher candidates interviewed asked for closer ties of the MCK course to
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their math pedagogy course. While it must be acknowledged that the teaching of mathematics
contains two portions (that the content and the pedagogy both are important pieces), it is also
important to note that there is a distinction between the two. Researchers such as Shulman (1996,
1998), Hill et al. (2008), Pournara (2016), and Silverman and Thompson (2008) have shown that
teaching math content knowledge separate from pedagogy can lead to increases in math anxiety.
Since this course was taught separate from the math pedagogy course, and with methods not
congruent with what was being taught in the pedagogy course (such as instructor-led, a lack of
manipulatives, and a focused on timed tests and assessments), it could lead to questions as to if
the course on its own was effective in addressing math anxiety or did the pedagogy course the
teacher candidates were taking at the same time also have an effect?
The course was kept separate from the pedagogy course as Lila and Henry wanted a
course that focused strictly on math content knowledge. However, because there was such a
disconnect in how the teacher candidates in the course were being taught to how they were being
taught to teach in their math pedagogy course, many of the teacher candidates (and some of the
instructors) interviewed felt the course was not as effective as it could have been. In fact, many
of the teacher candidates interviewed felt that a retired or a seconded classroom teacher with in-
school experience might make a better instructor for the course rather than undergraduate
mathematics students. This theory is supported by Van der Sandt and O’Brien (2017) who found
that instructor teaching style (specifically problem-based learning versus direct teaching) had a
significant effect on pre-service teacher outcomes in a math content course. Further research by
Lake and Kelly (2014) and Geist (2010) also supports the idea that constructivist methods in
teaching mathematics can have positive effects on building math efficacy.
132
While the teacher candidates interviewed understood that the focus of the course was to
teach math content, and the course creators stated they wanted to hire those who had strong math
skills, there did seem to be a disconnect between the instructor’s math content skills and their
math pedagogy abilities. Even if they were given the freedom to teach in a different manner, they
would not have had the background or skills to do so. In addition, the lack of manipulatives and
being taught various approaches to the math concepts was frustrating for some teacher
candidates, and for some of the instructors who did have math education backgrounds.
There were also concerns with the usage of timed and graded assessment methods (such
as quizzes and the diagnostic/summative test), due to their impact on math anxiety and
questionable usefulness for accurately evaluating math content knowledge, especially in math
anxious individuals. This correlation is supported by the research of Cipora et al (2019).
5.3.4 Effectiveness of Anxiety Scales
There was a marked decrease in anxiety levels (where a higher score indicates lower
anxiety levels). Interestingly, this improvement was roughly uniform across the 4 observed Tiers.
Overall, the data shows that we would expect reported anxiety scores to improve by
approximately 5 points (10%) for all teacher candidates who took the class. While this does
appear to be true for the average of anxiety scores, further investigation shows that about 43% of
teacher candidates who took the course self-reported either an increase in math anxiety or a
decrease of less than a 10%.
While math anxiety scales are a well-respected and validated test of math anxiety (Hopko
et al., 2003), these scales can have inherent issues. One of these relates how “they can be
manipulated consciously like in selection situation, but also unconsciously (social desirability
133
effects)” (Cipora et al., 2019, p. 28). It could perhaps have been hypothesized that the teacher
candidates who answered the questionnaire initially did not want to be seen as having math
anxiety, yet, by the end of the course, they were more confident in answering more truthfully. Or
perhaps the timing of the scale had an effect on the teacher candidate’s answers (the scale was
filled in immediately following the diagnostic and summative tests).
Furthermore, the teacher candidates were informed that the diagnostic content test was
not counted towards their marks in the course but that the summative was required for them to
pass the course and counted towards their final mark. As Cipora et al. (2019) notes, additional
pressures from assessments, such as timed and being weighted, can have a negative effect on
anxiety scores. Additionally, the summative test was the last required portion of the course and
perhaps teacher candidates did not take the anxiety scales seriously and rushed to complete them
and ‘be done with the course.’ Regardless of the reasons the anxiety scale changes in this course
are not necessarily indicative of changes seen in math scores, which measured math content
knowledge from test scores, or the number of blanks, which could be correlated to overall
confidence in answering math question and which were shown to be related to math anxiety. It is
also important to note that, while the majority of teacher candidates showed marked
improvements in both math content knowledge and decreases in math anxiety, the relationship
between these two concepts may have remained constant. These concepts should, therefore, be a
consideration for the MCK course in future.
5.3.5 Targeted Course Content
When looking at the tiers of teacher candidates and the overall impact of the course on
the various levels, a significant difference can be seen between Tier 1 (those scoring great than
134
75% on the diagnostic) and the other tiers. While the idea of streaming the course was brought
up to the course creators, it was felt that the negative impacts of a streamed course (such as
separating teacher candidates and causing some to feel as if they were in a remedial or
punishment class) were not worth the potential benefits. However, perhaps altering the
exemption test or requiring teacher candidates who scored high on the diagnostic to be exempt
but requiring specific skills be practiced independently, might be worth considering.
It was noted, however, that having those higher tier teacher candidates in the class did
have a positive impact on the rest of the teacher candidates as they were often able, and
encouraged to, help their classmates understand concepts. Perhaps ways could be found to more
appropriately utilize such teacher candidates in order to make the MCK course experience more
impactful for them as well. Perhaps specific, targeted, materials (based on diagnostic test results)
could be provided to teacher candidates to assist them in areas where they are weak; allowing
them to more directly focus their attention on needed skills.
Additionally, one area of concern and frustration for many teacher candidates came from
the fact they did not received their marked diagnostic tests back. This did not allow for teacher
candidates to gain valuable insights into their areas of weakness and where they could improve.
Furthermore, the diagnostic tests were not used to create targeted plans for teacher candidates. In
fact, the instructors for the course never saw their teacher candidate’s diagnostics and were not
provided with feedback on their grades. This was done in order to not influence the instructors as
to their classes weaknesses; however, they could provide valuable data to instructors and teacher
candidates for targeted materials on areas needing improvement. Instead, a standardized
approach to teaching all teacher candidates, all topics, were chosen.
135
Finally, when we look at how teacher candidates did on individual questions, we see that
questions on addition and subracrtion had little room for improvement, and teacher candidates
may have encountered a ceiling effect. However, most other topics had a maximum proportion of
room for improvement of between 0.47 to 0.63 and an actual proportion of improvement of
between 0.48 and 0.70. The one outlier to this was division, which had a maximum improvement
proportion of 0.90, yet the actual average improvement proportion was only 0.28. This indicates
that, between 48% and 70% of students who did not get the question correct on the diagnostic,
were able to correctly answer it on the summative; with the exception of division questions.
However, it is important to note that this analysis is an average across all students in the
course. When a Tiered analysis of the data were performed, we can see that students in Tier 1
encountered a significant ceiling effect for multiple questions. They even showed declines in
their scores on a few topics. This further indicates that the MCK course requires a different
approach for students in Tier 1.
This also links back to the analysis of content scores versus anxiety, in which teacher
candidates may have encountered another ceiling effect on their summative maximum score.
This tells us that, while there is a strong correlation between math anxiety and math content
scores, the overall change in the relationship between the two may not have shifted as strongly as
the regression model shows, meaning the MCK alone may not have caused the shifts in anxiety
post-course. Perhaps the addition of harder questions to the diagnostic / summative tests (in order
to limit the ceiling effect) would help to alleviate this phenomenon. However, these results can
still assist in providing guidance on both the math content knowledge tests and in the creation /
inclusion of targeted course materials that could be beneficial in addressing problem areas for
136
specific teacher candidates, while not focusing on areas where the teacher candidates are already
strong.
5.4 Implications for Future Research
While this research is limited to one course at a single university, it has potentially wide-
reaching implications. By showing the effectiveness of such a course on both increasing teacher
candidate math content knowledge and decreasing math anxiety through a standardized course, it
shows the potential to be extended to other institutions. By showing little variation in results
between instructors, it can be surmised that the course could be taught elsewhere with similar
results. A further study could be implemented at a different university utilizing the same
materials and methods to determine if the MCK course could achieve similar levels of success in
other environments.
A further area of study related to this research could be an in-depth comparison of this
MCK course to other courses and programs at other institutions and also comparing it to the
effectiveness of the Provincial Math Proficiency Test. Studies could be undertaken to see how
teacher candidates taking a MCK course compare to those not provided with such a course when
taking the MPT. There could also be a comparison between courses and programs at other
faculties of education and the MCK course to determine differences in MPT scores. A further
study on the MPT and its effect on math anxiety could also be undertaken (as this was an area of
discussion that came up in the interviews for this research and would make a fascinating area of
further investigation). This could also be expanded into a longitudinal study that investigates
whether the concepts learned in this course were retained by the teacher candidates once the
course ended. While the teacher candidates interviewed for this study, who took the course in
137
Year 1 of the program, found they retained the knowledge and skills for a year after the course
and were able to use it on their MPT, further follow-up on these teacher candidates was not done.
One additional area for study would be to determine if a MCK course affects gender and
minorities differently. As there are already studies linking mathematics to issues of gender and
equity, it would be interesting to see how this course relates to those issues as well.
Finally, a study on how the changes proposed to this course, when implemented, affect
the results could be undertaken. Changes such as allowing teacher candidates to view their
diagnostic test results and targeting individual, specific content based on these results; allowing
instructors to teach in a different, more student-centered / problem-based manner with the
addition of manipulatives; or changes in the qualifications of instructors teaching the course.
These changes could then be compared using this study as a base model for comparison. There is
also potential for looking at how different teaching styles utilizing technology (such as adaptive
learning, AI, and micro-teaching) could have an effect on the effectiveness of this course;
especially in addressing the impact of the course on the different Tiers.
5.5 Conclusion
Overall, this research found that the MCK course was well received by participants and
provided a positive benefit to nearly all of those interviewed. While there were some issues
raised and suggestions made by teacher candidates and instructors to improve the course, the
general consensus from the participants in this study was that the benefits of the MCK course
were significant. This study has provided some insights into the effectiveness, as well as the
necessity for a course specifically focused on math content knowledge in a pre-service
elementary teacher program.
138
The MCK course was established with the goal of increasing math content knowledge
and decreasing math anxiety in teacher candidates at the elementary level. The teacher
candidates interviewed provided insights into how the course assisted them and also suggestions
as to how the course could be improved. They provided positive feedback about the course and
its effect and this was supported by the quantitative data collected. While there were some
outlying data, overall, the majority of teacher candidates in the course showed positive (and
statistically relevant) improvements in both their math content knowledge and math anxiety.
A tiered analysis of the teacher candidates, however, showed that these improvements
were not universal across the teacher candidates and some benefited far greater than others.
There were also some teacher candidates who showed decreases in their math anxiety and nearly
45% showed a decrease of less than 10%. This information was not in line with previous studies
and further investigation of the reasons is warranted. Additionally, only 12.8% of teacher
candidates showed a less than 10% increase in their math content scores, with the majority
(~70%) coming from the Tier 1 category.
The instructors and course creators also had generally positive remarks about the course
and its impacts. However, there was a difference in how each group evaluated success in the
course. Many of the instructors were more concerned about creating a positive, welcoming
environment where the teacher candidates felt safe to both learn and fail. They often evaluated
the success of their teacher candidates based on how the teacher candidates reported feeling to
them. For the creators though, much of the success of the course was based on diagnostic versus
summative tests on mathematics content and pre- and post-course anxiety scales.
The materials were created by the course creators and the lead instructor and each class
139
had access to the same materials. This was reflected in the quantitative data with very little
differences being noticed between cohorts / instructors. This indicates that things like instructor
background and teaching style may have had little influence on the achievement of teacher
candidates in this course and was instead more closely related to the materials provided. It could
also be an indication that the instructors themselves had a commonality among them (such as
creating a safe and welcoming environment for their teacher candidates) that was reflected in the
scores as well.
The following recommendations for the MCK course are suggested:
• Materials should be targeted to the different Tiers of teacher candidates as this would allow for teacher candidates at different levels to feel more engaged in the content and may be more effective in addressing gaps in knowledge at the different levels. Teacher candidates in lower Tiers finish the course with significant gaps in their knowledge and need a different experience than those in the higher Tiers. This would also allow for a broader base of knowledge (outside of numeracy) for teacher candidates who improved their numeracy skills early in the course.
• An ‘in-house’ or tailored online platform for homework and practice of concepts should be used for all teacher candidates.
• The instructional methods of the course should be adjusted to be more in line with concepts being learned in the math pedagogy course. This would also involve having instructors skilled in different pedagogical methods of math instruction and not just skilled at mathematics.
• Alternative methods of assessment should be used in addition to quizzes, assignments, homework marks, and the summative content test.
As research shows, strong math teachers at the elementary level will go a long way to
ensuring we have strong future math students; opening up career and educational opportunities
for them in the future (Finlayson, 2014; Lake & Kelly, 2014; Stoeher, 2017; Vinson, 2001). This
is even more prominent for female students and their teachers (Beilock et al., 2010; Gunderson et
al., 2012; Nosek et al., 2002), of which most elementary teachers are female (OECD.Stat, 2020)
as so the effect is even more profound.
140
5.6 Researcher Reflection
While the MCK course still has some room for improvements, it is overall a good, solid,
template for education programs looking to improve the math skills of elementary teacher
candidates and to provide them with the tools to be effective teachers in the future. It is my belief
that, rather than stressful qualifying tests, a focus on teaching fundamental math skills and
building confidence (something the MCK course has proven itself capable of) is a far more
effective course of action to create the teachers we are seeking for our schools.
Furthermore, I believe, like Lila, that mathematics education is one of the keys to equity
in our schools and in our world. Many of the teacher candidates I interviewed for this study came
from different backgrounds, and most were women. If these are the people our students are
seeing as their early role models, then enabling them to be confident in mathematics, can help us
start to change the way students see those in STEM and create strong, confident math students,
encourage a more equitable education system and, by extension, a more equitable world. If we
can create strong, elementary math teachers now, then hopefully we can create strong math
elementary students in the future and eventually, hopefully, we will not need a MCK course,
because our students will already know the solutions.
141
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Appendices
Appendix A: Letter of Consent – Course Creators
Thank you for your consideration in assisting with this research project for my PhD study. This research will be a case study of a Math Content Knowledge course, which you are involved in, and will focus on the questions: How has one faculty of education attempted to address the lack of mathematics content knowledge in PSTCs through the development of a specific MCK course? How was it developed and introduced, and how has it been experienced by its designers, instructors, and students? The research will consist of an interview with yourself in a one-on-one context. The interview will either be in-person (at a mutually agreed upon location) or online using Zoom and will be recorded for transcription purposes only. The recordings will not be viewed by anyone except the researcher and will be deleted upon transcription completion. The video, and accompanying transcript, will be kept on a secure hard drive and password protected for security. Video recordings will be deleted after transcription is complete, however anonymized transcripts may be kept on file by the researcher for further research at a later date. Should your interview questions contain personally identifiable information this will either be replaced with a pseudonym (such as names, locations, programs, etc) or, if this is not possible, such information will be removed from the transcript. To further protect your privacy the name of the University and the name of the program you created will be replaced in the report and a pseudonym will be utilized instead of your real name. Furthermore, faculty administrators, other faculty and those involved in supervisory roles for you will not be informed of your participation and this will not have an effect on your current employment / researcher status. Additionally, non-identifiable information may be shared with members of the thesis committee upon request. The interview should take approximately 1 hour and will be scheduled at a time that is convenient for both yourself and the researcher sometime within the next 6 weeks. During this time it is hoped you will have a chance to share your experiences with the MCK course. Your participation in this study is purely voluntary, you will receive no direct compensation for your time. You are under no obligation to participate and may withdraw your consent at any point prior to the thesis being submitted for final consideration; after this point you will be unable to withdraw consent for materials to be included in the thesis, but you may request any additional information be excluded from future research. Should you choose not to participate there will be no negative consequences to you. Should you choose to withdraw your consent after the interview is complete your responses will not be included in the report and your video and transcript will be deleted immediately. You may also choose to omit specific sections of the interview as well; in this case the specific responses will be deleted from the transcript as well. You may also choose not to answer any questions you do not feel comfortable with or you may answer a question at a later time or change a response. You may view a copy of the transcript at any time to review your answers. You will be presented with a final copy of the thesis once completed and you will be free to share it, or information gained from it, as you please. The paper itself will be submitted for
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consideration as a final PhD thesis and may be published, in all or part, in various journals, online resources and other repositories. Identifying information on participants will not be included. Please note: based on the fact that you are one of two course creators for this course, should you both chose to participate it will be impossible to anonymize your responses from the other course creator. Furthermore, based on specific roles, responsibilities, and expertise you may have complete anonymity cannot be strictly guaranteed from other participants and you may be identified by those familiar with the course, or with the study. All efforts will be taken to generalize your responses and ensure that specific responses cannot be linked back to a specific person but, given the nature of this study, and your role in it, this issue needs to be considered by the participant (yourself). However, by participating in this study you will have a chance to contribute not just to my research but also to have a say in the future of the MCK course along with other courses that may choose to use this case study to help design, develop, and improve their courses. You will also be able to gain valuable insights into the course you have created, its implementation, and effectiveness. Should you have any question you may contact the researcher (Pamela Brittain) at [email protected]. You may also contact the Research Oversight and Compliance Office - Human Research Ethics Program at [email protected] or 416-946-3273, if you have any questions about your rights as participants The research study you are participating in may be reviewed for quality assurance to make sure that the required laws and guidelines are followed. If chosen, (a) representative(s) of the Human Research Ethics Program (HREP) may access study-related data and/or consent materials as part of the review. All information accessed by the HREP will be upheld to the same level of confidentiality that has been stated by the research team. Informed Consent I agree to take part in this study, which has been explained to me. I have been given an opportunity to ask questions about the study. I understand that I will be interviewed along with another colleague and that my interview will be video recorded for transcription purposes only. I understand that my identity will be known only to the researcher and my fellow interviewee. I also understand that my participation is completely voluntary, and I may withdraw from the study at any time. I am 18 years old or over, and am legally able to provide consent. _______________________________________ _____________ Name (Print) Date _______________________________________ Signature Please return signed letter to Pamela Brittain at [email protected]
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Appendix B: Letter of Consent - Instructors
Thank you for your consideration in assisting with this research project for my PhD study. This research will be a case study of a Math Content Knowledge course, which you were involved in, and will focus on the questions: How has one faculty of education attempted to address the lack of mathematics content knowledge in PSTCs through the development of a specific MCK course? How was it developed and introduced, and how has it been experienced by its designers, instructors, and students? The research will consist of an interview with yourself in a one-on-one context. The interview will either be in-person (at a mutually agreed upon location) or online using Zoom and will be recorded for transcription purposes only. The recordings will not be viewed by anyone except the researcher and will be deleted upon transcription completion. The video, and accompanying transcript, will be kept on a secure hard drive and password protected for security. Video recordings will be deleted after transcription is complete, however anonymized transcripts may be kept on file by the researcher for further research at a later date. Should your interview questions contain personally identifiable information this will either be replaced with a pseudonym (such as names, locations, programs, etc) or, if this is not possible, such information will be removed from the transcript. To further protect your privacy the name of the University and the name of the program you instructed will be replaced in the report and a pseudonym will be utilized instead of your real name. Furthermore, course creators and others in a supervisory role for you will not be informed of your participation and will not have an effect on your current employment status. Additionally, non-identifiable information may be shared with members of the thesis committee upon request. The interview should take approximately 1 hour and will be scheduled at a time that is convenient for both yourself and the researcher sometime within the next 6 weeks. During this time it is hoped you will have a chance to share your experiences with the MCK course. Your participation in this study is purely voluntary, you will receive no direct compensation for your time. You are under no obligation to participate and may withdraw your consent at any point prior to the thesis being submitted for final consideration; after this point you will be unable to withdraw consent for materials to be included in the thesis, but you may request any additional information be excluded from future research. Should you choose not to participate there will be no negative consequences to you. Should you choose to withdraw your consent after the interview is complete your responses will not be included in the report and your video and transcript will be deleted immediately. You may also choose to omit specific sections of the interview as well; in this case the specific responses will be deleted from the transcript as well. You may also choose not to answer any questions you do not feel comfortable with or you may answer a question at a later time or change a response. You may view a copy of the transcript at any time to review your answers. You will be presented with a final copy of the thesis once completed and you will be free to share it, or information gained from it, as you please. The paper itself will be submitted for consideration as a final PhD thesis and may be published, in all or part, in various journals, online resources and other repositories. Identifying information on participants will not be included.
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Please note: based on specific roles, responsibilities, and expertise you may have complete anonymity cannot be strictly guaranteed and you may be identified by those familiar with the course, or with the study. All efforts will be taken to generalize responses and ensure that specific responses cannot be linked back to a specific person but, given the nature of this study, this issue needs to be considered by the participant (yourself). However, by participating in this study you will have a chance to contribute not just to my research but also to have a say in the future of the MCK course along with other courses that may choose to use this case study to help design, develop, and improve their courses. Should you have any question you may contact the researcher (Pamela Brittain) at [email protected]. You may also contact the Research Oversight and Compliance Office - Human Research Ethics Program at [email protected] or 416-946-3273, if you have any questions about your rights as participants The research study you are participating in may be reviewed for quality assurance to make sure that the required laws and guidelines are followed. If chosen, (a) representative(s) of the Human Research Ethics Program (HREP) may access study-related data and/or consent materials as part of the review. All information accessed by the HREP will be upheld to the same level of confidentiality that has been stated by the research team. Informed Consent I agree to take part in this study, which has been explained to me. I have been given an opportunity to ask questions about the study. I understand that I will be interviewed along with another colleague and that my interview will be video recorded for transcription purposes only. I understand that my identity will be known only to the researcher and my fellow interviewee. I also understand that my participation is completely voluntary, and I may withdraw from the study at any time. I am 18 years old or over, and am legally able to provide consent. _______________________________________ _____________ Name (Print) Date _______________________________________ Signature Please return signed letter to Pamela Brittain at [email protected]
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Appendix C: Letter of Consent – Teacher Candidates
Thank you for your consideration in assisting with this research project for my PhD study. This research will be a case study of a Math Content Knowledge course, which you were involved in, and will focus on the questions: How has one faculty of education attempted to address the lack of mathematics content knowledge in PSTCs through the development of a specific MCK course? How was it developed and introduced, and how has it been experienced by its designers, instructors, and students? The research will consist of an interview with yourself in a one-on-one context. The interview will take place online using Zoom and will be recorded for transcription purposes only. The recordings will not be viewed by anyone except the researcher and will be deleted upon transcription completion. The audio recording, and accompanying transcript, will be kept on a secure hard drive and password protected for security. Video recordings will be deleted immediately after the interview and will not be kept for any purpose, only the audio files will be retained. However anonymized transcripts may be kept on file by the researcher for further research at a later date. Should your interview questions contain personally identifiable information this will either be replaced with a pseudonym (such as names, locations, programs, etc) or, if this is not possible, such information will be removed from the transcript. To further protect your privacy the name of the University and the name of the program you attend will be replaced in the report and a pseudonym will be utilized instead of your real name. Furthermore, course instructors, TAs, and creators will not be informed of your participation and participation will have no impact on your grade in the course (if you are a current student). Non-identifiable information may be shared with members of the thesis committee upon request. The interview should take approximately 1 hour and will be scheduled at a time that is convenient for both yourself and the researcher. During this time it is hoped you will have a chance to share your experiences with the MCK course. Your participation in this study is purely voluntary, you will receive no direct compensation for your time. You are under no obligation to participate and may withdraw your consent at any point prior to the thesis being submitted for final consideration; after this point you will be unable to withdraw consent for materials to be included in the thesis, but you may request any additional information be excluded from future research. Should you choose not to participate there will be no negative consequences to you. Should you choose to withdraw your consent after the interview is complete your responses will not be included in the report and your audio file and transcript will be deleted immediately. You may also choose to omit specific sections of the interview as well; in this case the specific responses will be deleted from the transcript as well. You may also choose not to answer any questions you do not feel comfortable with or you may answer a question at a later time or change a response. You may view a copy of the transcript at any time to review your answers. You will be presented with a final copy of the thesis once completed and you will be free to share it, or information gained from it, as you please. The paper itself will be submitted for consideration as a final PhD thesis and may be published, in all or part, in various journals, online resources and other repositories. Identifying information on participants will not be included.
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By participating in this study you will have a chance to contribute not just to my research but also to have a say in the future of the MCK course along with other courses that may choose to use this case study to help design, develop, and improve their courses. Should you have any question you may contact the researcher (Pamela Brittain) at [email protected]. You may also contact the Research Oversight and Compliance Office - Human Research Ethics Program at [email protected] or 416-946-3273, if you have any questions about your rights as participants The research study you are participating in may be reviewed for quality assurance to make sure that the required laws and guidelines are followed. If chosen, (a) representative(s) of the Human Research Ethics Program (HREP) may access study-related data and/or consent materials as part of the review. All information accessed by the HREP will be upheld to the same level of confidentiality that has been stated by the research team. Informed Consent I agree to take part in this study, which has been explained to me. I have been given an opportunity to ask questions about the study. I understand that I will be interviewed along with another colleague and that my interview will be video recorded for transcription purposes only. I understand that my identity will be known only to the researcher and my fellow interviewee. I also understand that my participation is completely voluntary, and I may withdraw from the study at any time. I am 18 years old or over, and am legally able to provide consent. _______________________________________ _____________ Name (Print) Date _______________________________________ Signature Please return signed letter to Pamela Brittain at [email protected]
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Appendix D: Additional Letter of Consent – Teacher Candidates
Thank you for taking the time to contribute to my thesis research, your time and assistance are greatly appreciated. As part of my research I would like to include generalized information about your personal scores on your diagnostic test and math anxiety scale (both your pre- and post-test results). To protect your identity exact scores would not be included in the final paper but they would instead be used to place you in a category of student compared to others in the course. Furthermore, I have obtained permission to utilize the entire data set for the course so I would be the only person looking up your score so as to ensure no one else was informed of your participation, thereby maintaining confidentiality. This letter serves to inform you of this additional of personal information and seeks your consent to include it in my research. Should you have any question you may contact the researcher (Pamela Brittain) at [email protected]. You may also contact the Research Oversight and Compliance Office - Human Research Ethics Program at [email protected] or 416-946-3273, if you have any questions about your rights as participants The research study you are participating in may be reviewed for quality assurance to make sure that the required laws and guidelines are followed. If chosen, (a) representative(s) of the Human Research Ethics Program (HREP) may access study-related data and/or consent materials as part of the review. All information accessed by the HREP will be upheld to the same level of confidentiality that has been stated by the research team. Informed Consent I agree to allow the researcher, Pamela Brittain, to include generalized information regarding my pre- and post-diagnostic test and pre- and post-anxiety scale tests in her research. This is in additional to my previously signed informed consent to participate in the study. I also understand that my participation is completely voluntary, and I may withdraw from the study at any time. I am 18 years old or over, and am legally able to provide consent. _______________________________________ _____________ Name (Print) Date _______________________________________ Signature Please return signed letter to Pamela Brittain at [email protected]
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Appendix E: Interview Questions – Course Creators
• Tell me a bit about yourself • What is your approach to teaching (math specific or education in general)? • What formally lead to the creation of this course? • How is the course structured? • Are there any similar programs you’ve seen? • How are the student’s evaluated? How do you evaluate the course? • What type of feedback do you get from the instructors? • Who are the instructors for this course? What are some of the things you look for in an
instructor? • What types of materials are the instructors provided with? • What results have you seen from the course? What were you expecting to see? Was there
anything that surprised you? • Would you consider the course to be successful? • Have there been any major changes to the program from year one to year two? • What type of feedback have you gotten from various parties? • Why math literacy? (why not science or reading) • How does this course differ from the math pedagogy course? • If you had to redesign the course from the start, would you? And if so what would you
change? • Is this course having an effect on passing the math proficiency test? Do you see that as
being part of the course? • If you had a crystal ball and could see into an ideal future, what would you love to see? • Anything else to add?
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Appendix F: Interview Questions – Course Creator (Follow-up with Lila)
• Why a 24-hour course? Why not 36-hours? • Student’s really liked the assignments; why not all assignments instead of quizzes? • Did you ever consider combining it with the math pedagogy course? • Why Khan Academy? • Did you ever consider a flipped-classroom model? • Why is the course not just fully-independent? • Did you consider streaming the students? • Why were the instructors TAs? Why not faculty or lecturers or seconded teachers? • Why focus on numeracy specifically? • Why did the students not receive their diagnostic tests back? • The diagnostic was marked by an external person and not the instructor, why was that? • Was the OCT resistant to this idea? • Do you think other universities might want to model this program?
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Appendix G: Interview Questions – Instructors
• What is your background? (Tell me about yourself) • What is your general approach to education? • Why did you apply to be an instructor for this course? • Did your age/background/qualifications cause any issues? • How is the course structured? • What materials were you provided with? Do you feel they enough? • What materials (if any) did you make yourself? • Why do you think the focus was on math and not another subject? • How does it compare to the math pedagogy course? Should they be combined? • Who are the students for this course? What are the requirements for taking the course? • How do you evaluate student success in the course? What success have you seen? • Have there been any logistical challenges you’ve seen? • Any advice you’d give to future instructors of this course? • Do you feel this course effectively prepares its students for teaching math? • Math curriculum in Ontario has been highly debated for many years. Recently the
government has really focused on math education, including testing for teacher candidates. Have you seen an impact of this in this course?
• Where do you think the course could be improved? If you could redesign it from the start, what would you change (if anything)?
• Any other comments you’d like to make?
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Appendix H: Interview Questions – Teacher Candidates
• What is your background? (Tell me about yourself) • Why are you interested in being a teacher? • What level are you certified to teach? Why that level? • Why the MT program? • Tell me a bit about your previous math experience. • In general, how do you feel about math? What types of experiences have you had with it
in the past? • What influence do you think math teachers have on their students? • Tell me a bit about the initial and final diagnostic test. What were you told about in
advance of the test/class? • What class were you in (course code / TA name if possible)? • How was the course structured? • How were the weekly quizzes (timing, structure, questions)? • How did you find Khan Academy? Did you find it was a reasonable supplement to the
course? • How the instruction method (lecture, group work)? • What about your course instructor stood out to you? • Did the age difference (if any) between you and the instructor cause any issues? • Who do you see as an ideal instructor for the course? • How did it compare to the pedagogy course? How did it ‘pair up’? • Why do you think there was a focus on a math content course? (Why not reading or
science?) • Compared to your other courses how valuable did you find this course? • Did you have a lot of group work / teach each other options / use or apply the pedagogy
content? • Do you feel this course helped you improve your math skills / make you more prepared to
be a math teacher? • Do you feel this course had an effect on your level of math confidence / anxiety? • How many hours did you spend on this course in a typical week? Was this more or less
than other courses (in general)? • Do you feel it prepared to you to learn math on your own? Did it build your confidence to
learn it? • What did you expect to get out of the course? • How does this compare to what you feel you got at the end of the course? • How do you evaluate success in the course? • Was there anything you found particularly beneficial about this course? • Were there been any logistical challenges you saw in the class? Anything that could be
improved? • Who would be the ideal instructor for this course? • How would you explain this class to a potential MT student?
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• Was there anything that surprised you about this course? • What materials were you provided with? Do you feel they were enough? • Did you find the quizzes helpful or stressful? How about the timing of the quizzes? • Math curriculum in Ontario has been highly debated for many years. Recently the
government has really focused on math education, including testing for teacher candidates. Do you feel this course was effective in preparing you for these new requirements?
• Do you think it would be better in 2nd year or over 2 years? • Without the MPT do you still think there is value in the MCK course? • Where do you think the course could be improved? If you could redesign it, what would
you change (if anything)? • Any other comments you would like to make?