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Running Head: Elementary Matheamtics Teachers Matter: The Journey to Becoming a

Great Elementary Mathematics Teacher

Elementary Mathematics Teachers Matter:

The Journey to Becoming a Great Elementary Mathematics Teacher

By

Bektjona Zaimi

A research paper submitted in conformity with the requirements

For the degree of Master of Teaching

Department of Curriculum, Teaching and Learning

Ontario Institute for Studies in Education of the University of Toronto

This work is under a Creative Commons Attribution-

Non-Commercial-No Derivs 4.0 International License

ELEMENTARY MATHEMATICS TEACHERS MATTER 2

Abstract

Canadian students’ enjoyment of and achievement in mathematics are decreasing.

Meanwhile, the number of jobs requiring mathematics and their wages are increasing. A

major factor in a students’ success in mathematics is teacher quality. Thus, my study

examines the following question: What are teachers’ perspectives about increasing the

quality of Junior/Intermediate mathematics teachers? Sub-questions that follow from the

main question are: What changes, if any, would improve the current process of becoming

a mathematics teacher? Where should these changes happen along the timeline of

becoming a mathematics teacher? My participants were two elementary teachers that

taught mathematics and other subjects. My findings suggest that there is a disconnect

between the characteristics of a successful mathematics teacher and those of current

mathematics teachers. While both my participants agree that the ideal mathematics

teacher has sufficient knowledge of mathematics content and pedagogy, the current

process of becoming mathematics overly prioritizes pedagogy and assumes content is

known. Possible changes to remedy this situation included increasing requirements for

undergraduate education, and the inclusion of entry and exit tests to and from teacher

education programs.

Keywords: mathematics, teacher education, elementary, pedagogical content knowledge,

teacher evaluation


Acknowledgements

I wish to acknowledge and thank my research supervisor, Limin Jao, for her

wisdom, guidance and patience throughout the process of writing this research paper.

Next, I want to recognize and thank my participants for allowing me to interview and

learn from their years of experience. Also, I would like to express my gratitude to my

parents, sister, and boyfriend for their support and encouragement.


TABLE OF CONTENTS

Chapter Page

Abstract 2

Acknowledgements 3

1. INTRODUCTION 6

Introduction to the research study 6

Purpose of the study 8

Research questions 10

Background of the researcher 11

Overview 13

2. LITERATURE REVIEW 15

Introduction 15

Mathematics pedagogy 15

Mathematics content knowledge 17

Mathematics pedagogical content knowledge 19

Evaluations 21

3. METHODOLOGY 25

Introduction 25

Instruments of data collection 25

Participants 26

Data collection and analysis 28

Ethical review procedures 30

4. FINDINGS 31

Theme 1: Ideal mathematics teacher 31

Content knowledge 32

Pedagogy 33

Theme 2: Before teacher education programs 35

Undergraduate education 36

Entry test 36

Teacher education programs’ application and interview 38

Theme 3: During teacher education programs 38

Content knowledge 39

Pedagogy 40

Theme 4: After teacher education programs 41

Exit test 42

School boards’ application and interview 43


5. DISCUSSION 45

Introduction 45

Relating findings to current literature 46

Pedagogical content knowledge 46

Evaluations 47

Evaluation and discussion 48

Implications and recommendations 50

Limitations and next steps 51

REFERENCES 54

APPENDICES 58

Appendix A: Interview Questions 58

Appendix B: Letter of Consent for Interview 60


Chapter 1: INTRODUCTION

Introduction to the research study

“Canadian university students are not choosing careers in mathematics, science,

technology and engineering” (“Selling science to students”, 2012, para. 1). This was the

headline in the Globe and Mail newspaper that caught my attention at the beginning of

the first year of my Master of Teaching program and inspired my Master of Teaching

Research Project (MTRP) topic. The article stated that even though Canadian students

recognize that most future jobs will require mathematics, they are apprehensive and

fearful of mathematics. The rest of the article discussed the findings from a study by

Amgen Canada and Let’s Talk Science. Amgen Canada is a global human therapeutics

company in the biotechnology industry. “[It] contributes to the development of new

therapies or new uses for existing medicines in partnership with many of Canada’s

leading health-care, academic, research, government and patient organizations” (“About

Amgen”, 2013, para. 2). Let’s Talk Science is an “award-winning, national, charitable

outreach organization” whose goal is to “prepare youth for their future careers and roles

as citizens in a rapidly changing world” by providing programs in science, technology,

engineering and mathematics (STEM) (“We’re losing out”, 2013, para. 8).

Understanding the value of science, Amgen Canada has partnered with Let’s Talk

Science to do research in order to raise awareness of STEM education importance. Their

newest report, “Spotlight on Science Learning 2013: The high cost of dropping science

and math”, stated that Canadian students’ lack of interest in senior level STEM courses is

costing Canada’s economy (Amgen Canada Inc. & Let’s Talk Science, 2013). This

report was a follow-up to another research report, “2012 Spotlight on Science Learning:

A benchmark of Canadian talent”, which found that the performance of Canada’s talent


pool for science and math-based careers is strong, but its size is small (Amgen Canada

Inc. & Let’s Talk Science, 2012).

In order to understand the cause of this recent decrease in student enjoyment of

and success in mathematics, I began by researching mathematics anxiety and learned that

a great majority of the population suffers from mathematics anxiety and have a strong

belief that they were born unable to do mathematics (Small, 2013). Small explained that

this anxiety can be partially caused by the incorrect view of mathematics as a “black and

white” or “right and wrong” subject. For example, 72 subtract 46 results in only one

outcome: 26. However, the ways of arriving to this conclusion are endless, for example

via the use of a number line, base ten blocks, money, rounding-subtracting-adding,

carrying, and adding enough to the smaller number to get the bigger one, just to name a

few strategies. From early on, children hear parents and teachers making negative

remarks about mathematics, which enables them to “attribute their lack of success to their

lack of innate ability”, instead of working on getting better (Small, 2013, p. 66). Many of

my peers in the Master of Teaching program also mentioned past teachers as the cause of

their mathematics anxiety.

In President Obama’s words “The single most important factor in determining

[student] achievement is not the color of their skin or where they come from. It’s not who

their parents are or how much money they have. It’s who their teacher is” (Obama, 2007,

para. 15). The Globe and Mail article also echoed the role that teachers play in students’

enjoyment of STEM subjects, stating that they “are not always taught effectively in high

school, while primary school teachers often don’t have a science background” (“Selling

science to students”, 2013, para, 4).


Mathematics education researchers have pointed to incompetent mathematics

teachers as one of the predominant factors (e.g., Ball & Rowan, 2004; Shulman, 1986).

Wu (2011) stated, “Mathematics education cannot improve so long as educators and

administrators remain mathematically ill-informed” (p. 382). As a result, I decided to

examine the following question: What are teachers’ perspectives about increasing the

quality of Junior/Intermediate mathematics teachers? Sub-questions that follow from this

research question are: What changes, if any, would improve the current process of

becoming a mathematics teacher? Where should these changes happen along the timeline

of becoming a mathematics teacher?

Purpose of the study

The purpose of this study is to investigate and collect teachers’ opinions on the

current quality of Junior/Intermediate mathematics teachers. This issue is important to

several people for many reasons. Firstly, it is important to Canadian citizens because,

when compared to students in countries like China, Canadian students are falling behind

in STEM subjects: science, technology, engineering, and mathematics (Amgen Canada

Inc. & Let’s Talk Science, 2013). President of Let’s Talk Science, Bonnie Schmidt, Ph.D,

asserts, “Canada must focus on building a strong STEM talent pool with the skills to

contribute to our country’s long-term prosperity” (“We’re losing out”, 2013, para. 3). The

PISA 2012 results confirmed a clear declining trend in Canadian students’ performance

in mathematics, as well as an increase in the number of countries outperforming Canada

(Brochu, Deussing, Houme, & Chuy, 2013). Concretely, it was stated, “The performance

of Canadian 15-year-olds has declined by 14 points in the past nine years, a decline that

is statistically significant” (Brochu et al., 2013, p. 30). I am choosing to focus specifically

on mathematics out of the four STEM subjects because it is the common denominator


across all four. Without a strong base in mathematics, one cannot be successful in the

other three, evidenced by the fact that in order to specialize or major in science,

technology or engineering, students must successfully complete a mathematics course in

the first year (“Program requirements”, 2014).

Secondly, the issue of improving teacher quality is important to teachers.

Improvements in the current process of becoming a mathematics teacher will give all

teachers a higher status and the respect they deserve. Perhaps, the higher status will

attract more mathematics-interested people to pursue a career in teaching. Another result

could be that with stricter admission requirements, teachers will no longer come from the

bottom two-thirds of their undergraduate cohort (Auguste, Kihn, & Miller, 2010).

Personally, when I share my undergraduate background with people and tell them that I

am pursuing teaching, I am always asked why I chose teaching when I could go into

other, in their opinion, more esteemed and monetarily rewarding professions. Popular

belief has it that teaching, especially in elementary grades, is a last resort career.

Rasmussen Reports (2013) found that while 67 percent of adults say teaching is one of

the most important jobs in the United States today, only 24 percent believe teaching is a

desirable profession. Coupled with my personal experience, the Rasmussen Reports study

gives me the sense that a teaching career is viewed as unattractive and a last resort.

Improvements in the process of becoming a mathematics teacher may be a contributing

factor in changing the status of teaching in our society.

Most importantly, any improvements to the process of becoming a mathematics

teacher will benefit the students. There exists a negative view of mathematics among the

majority of Canadian students today (Amgen Canada Inc. & Let’s Talk Science, 2012).

This is unfortunate and disadvantageous because in the 21st century, an era of


technological innovation, many careers require an increasing amount of mathematics

knowledge. In addition, according to Statistics Canada, jobs requiring STEM skills are on

the rise (as cited in Amgen Canada Inc. & Let’s Talk Science, 2012). Also, wages in

these jobs are also rising at a faster rate than inflation (Amgen Canada Inc. & Let’s Talk

Science, 2012).

Furthermore, during my undergraduate education, I noticed that many students in

my mathematics program and courses had completed part or all of their previous

education outside of Canada, predominately in Asian countries. In addition, the ones who

had completed their earlier education in Canada, myself included, had a lot of catching up

to do and struggled more. With better qualified elementary teachers, we will be able to

level the playing field for Canadian students in higher level mathematics courses. This is

supported by research on Teach For America, which recruits top graduates with other

desirable qualifications, which suggests that “its teachers are more effective on average

than other teachers of similar experience levels, with the largest impact on achievement

in mathematics” (Auguste et al., 2010, p.10).

Taking into account the above points, we can see that there is room for

improvement with regards to the teaching of mathematics. These improvements will

benefit students, teachers and Canada’s economy as a whole.

Research questions

The research question that I set out to answer is: What are teachers’ perspectives

about increasing the quality of Junior/Intermediate mathematics teachers? Sub questions

that I investigated based on my overall research question are: What changes, if any,

would improve the current process of becoming a mathematics teacher? Where should

these changes happen along the timeline of becoming a mathematics teacher?


Background of the researcher

I have wanted to become a teacher ever since I can remember and having a

younger sister, teaching was a skill I developed at a very early age. Recognizing my

ability to teach, my father enticed us to hold tutoring sessions after school by giving us an

allowance for every hour that we played our roles. I used to pretend I was the teacher and

would even write on the inside of my closet door with chalk as if it was a blackboard. All

the extra-curricular activities I have participated in since then strongly reflect my desire

to teach. For example, I was a volunteer tutor for the Leading to Reading program at the

Toronto Public Library, where I helped grade one to four students with reading, writing,

and oral communication. I was also a Youth Assisting Youth (YAY) mentor for an

eleven year old girl who was being bullied due to being new to the country.

The focus of my study in mathematics was influenced by the fact that

mathematics has always been my favorite subject. Prior to starting grade one, my parents

taught me basic mathematics at home. This gave me a head start as compared to my

peers, and I benefited greatly from this. My parents’ strong mathematics background,

high expectations and involvement in my education worked in my favour. My earlier

education, up to grade six, took place in Albania. There, mathematics was taught

differently, specifically, at a faster pace. Also, rotary started in grade six which meant

teachers were specialized in their subject areas much earlier than they typically are here

in Canada. All of my mathematics teachers were women and very knowledgeable.

When I moved to Canada, I found mathematics classes easy in comparison to my

mathematics classes in Albania and I was not being taught any new material. In grade

eight, as a result of my outstanding achievement in mathematics, my teacher selected me

as the recipient of the mathematics award for my cohort’s graduation ceremony. This


might not mean much now, but at the time it gave me the confidence to pursue and look

forward to mathematics learning at a higher level. Throughout my four years in

secondary school, I was lucky to have some great teachers that I still look up to. I also

had some not so great teachers that clearly did not have the ability or desire to teach.

Being interested in teaching, I often found myself asking how it was possible for teacher

quality to vary so much, despite the fact that they all held the title of mathematics teacher.

I earned an Honours Bachelor of Science from the University of Toronto in 2012,

with a specialist in Mathematics and its Applications: The Teaching Concentration, a

major in Economics, and a minor in Statistics. This firm content background has shaped

me into a teacher who not only has a deep knowledge of mathematics, but also has a

passion for it. I hope to be a role model for all my students, especially girls, who tend to

be less confident with their mathematics skills as compared to boys. In addition, having

taken mathematics courses alongside economics, statistics and computer science courses,

I will be able to share many real life connections of mathematics with my students.

Currently, I am completing my 2nd

year in the Master of Teaching program, a

two-year teacher education program, at the Ontario Institute for Studies in Education of

the University of Toronto. I was drawn to this program by the virtue that it is research-

driven as there is a research component in all of its classes. This program has fueled my

desire to educate myself through different channels and evolve into a superior teacher. In

addition to reading research, this program gives me the opportunity to do some of my

own research on a topic of interest to me. I have learned a lot during the process and feel

that it has had implications on how I will approach teaching mathematics. Specifically, it

has made me aware that despite having studied mathematics in university, I should

continually update my mathematics content knowledge as well as my mathematics


pedagogy skills. Also, the four one-month long practice teaching blocks were appealing

to me as a developing teacher as compared to the two found in the Bachelor of Education

program. Looking back, I definitely do not think I would have been ready at the end of

my first year of the Master of Teaching program. The extra two teaching experiences

provided me with more opportunities to develop my teaching style and confidence in

front of a classroom, making me a better teacher.

Furthermore, as a future Junior/Intermediate teacher with mathematics as my

teachable subject, I wanted to select a topic that is personally connected to my experience

with the Master of Teaching program. Junior/Intermediate teachers are required to have a

background in at least one subject and end up graduating from the Master of Teaching

program and other teacher education programs qualified to teach all subjects, except

French and music. I feel this had the biggest impact on my MTRP topic because I noticed

that many of my colleagues are not prepared to teach mathematics and have expressed

having mathematics anxiety. On one hand, this reassured me that my strong mathematics

background has better prepared me to teach mathematics. On the other hand, after

reading research about Canadian students’ mathematics achievement declining (e.g.,

Amgen Canada Inc. & Let’s Talk Science, 2012; Brochu et al., 2013), I wondered

whether teacher education programs need to increase the requirements for becoming a

Junior/Intermediate mathematics teacher.

Overview

Chapter 1 includes the introduction and purpose of the study, the research

questions, as well as my personal connection to the topic of my study. Chapter 2 contains

a literature review where I set out to compare and contrast existing research on teacher

content knowledge and pedagogy in mathematics. Chapter 3 provides the method and


procedure that were used in my study including information about my two participants

and data collection instruments. Chapter 4 narrates the findings without interpreting,

explaining or evaluating them. Specifically, the findings were organized under four

categories with two or three sub-categories each: the ideal mathematics teacher

(pedagogy, content knowledge), requirements to get into teacher education programs

(undergraduate education, entry test, application/interview process), teacher education

programming (pedagogy, content knowledge), and requirements to get into school boards

(exit test, additional qualifications/professional development, application/interview).

Lastly, Chapter 5 develops an argument by analyzing and interpreting my findings with

reference to the literature review. I conclude my MTRP by highlighting areas for further

research with respect to optimizing the quality of our teacher education.


Chapter 2: LITERATURE REVIEW

Introduction

Mathematics education has always been a cause for debate and most recently

there has been an increase in school improvement discussions. Some argue that the focus

should be curricula, class size, and better student support, while others strongly maintain

that our attention should be on teacher quality (Winters, 2012). Furthermore, there lies a

much deeper division among those who agree on the importance of teacher quality

(Winters, 2012). The two opposing approaches to teacher quality improvement are

making sure we are hiring the kind of teachers we need versus trying to make already-

hired teachers into what we need (Winters, 2012). Furthermore, due to the intricacy of the

teaching profession, it has been difficult for research to pinpoint which teacher traits

make the biggest difference for student achievement in mathematics (Small, 2013). In

this literature review, I will be addressing two such traits: mathematics pedagogy and

mathematics content knowledge. I then bring these two themes together to discuss a

teacher trait that draws on both: pedagogical content knowledge. Lastly, I argue the

importance and current lack of evaluations of the previously mentioned teacher traits.

Mathematics pedagogy

Since the mid-1980s, increasing importance has been placed on ensuring that

teachers understand students’ perspective of learning mathematics (Small, 2013). A

major research study by Carpenter and Moser (1984) brought about a movement called

Cognitively Guided Instruction (CGI), which argued that superior mathematics teachers

understand children’s thinking and hence are able to better help their students

comprehend the mathematics. A decade later, a study by Fennema et al. (1996) confirmed

that students with CGI teachers improved considerably in terms of mathematics


achievement. CGI teachers exhibited the following instructional strategies: facilitating

longer work period with more complex problems, allowing discussion about mathematics

topics, adjusting their teaching to fit their students differing levels (Fennema et al., 1996).

Similarly to Carpenter and Moser (1984) and Fennema et al. (1996), Gearhart et al.

(1999) showed that students benefit from a teacher who understands and focuses on

students’ thinking, in addition to using manipulative materials to bridge the gap between

concrete and symbolic. This is important because mathematics easily lends itself for

symbolic representations, but students respond better when shown concrete materials

(Gearhart et al., 1999)

The amount of time spent on pedagogy in mathematics courses in teacher

education programs is considered a benefit for teachers. Research shows that

mathematics methods courses in teacher education programs are more beneficial than a

content course for pre-service teachers who are mathematics anxious but will have to

teach mathematics due to the fact that they will most likely be generalists once they enter

the workforce (Gresham, 2007; Tooke & Lindstrom, 1998). Tooke and Lindstorm’s

(1998) study used the Mathematics Anxiety Rating Scale for Adults to compare the

effectiveness of three courses in reducing mathematics anxiety. The first was a

mathematics course for elementary teachers taught using a traditional approach. The

second was a mathematics course for elementary teachers taught using recommendations

of the National Council of Teachers of Mathematics. The third was a methods course

covering the same content as the first two in addition to pedagogy. Tooke and Lindstorm

(1998) found that both mathematics courses did not reduce mathematics anxiety much.

On the other hand, the mathematics anxiety of participants in the method course reduced

considerably. Gresham’s (2007) study spanned six semesters in length and had 246


participants. The methods course in the study incorporated manipulative materials,

journal writing, group presentations, and literature-based mathematical tasks. About

eighty-seven percent of the participants said that the use of manipulatives contributed to

their reduction in mathematics anxiety. Other factors mentioned were journal writing and

the passionate course professor.

Other studies that point to the benefits of pedagogy rely on a hypothesis called the

“expert blind spot” (EBS). In education, the EBS is when subject-matter expert teachers

are unable to empathize with students’ challenges when learning a new concept

(Lombardi, 2007). Nathan and Petrosino (2003) studied 48 secondary school teacher

candidates in order to establish a relationship between teacher content knowledge in

mathematics with how they viewed students’ difficulty with algebra problem solving.

The study revealed a contrast between how students learn and do mathematics and how

subject-matter expert teachers think students learn and do mathematics (Nathan &

Petrosino, 2003). Specifically, while teachers with a high level of content knowledge tend

to emphasize symbolic representations and equations when solving story problems,

students understand mathematics best when concepts are presented using more concrete

materials (Nathan & Petrosino, 2003).

Mathematics content knowledge

While some research shows that a focus on mathematics pedagogy is beneficial

for generalist teachers, a substantial amount of research in the field has shown that

teacher content knowledge is just as important. Small (2013) states, “There has always

been a feeling that teachers need math content background” (p.12). Shulman (1986)

recalled fondly a time when content knowledge received the attention it deserves in all

subjects and writes about the recent gravitation towards mainly pedagogy. Research by


Ball (2000) attests to the importance of content knowledge instruction in addition to

pedagogy, specifically in mathematics. She asserted that many teachers do not have the

particular mathematics content knowledge that they need to know (Ball, 2000).

Furthermore, both Shulman (1986) and Ball (2000) believe that teacher education

programs focus too heavily on pedagogy and lament that almost no content knowledge is

taught. These results are also echoed by other scholars in the field (e.g., Bates, Kim, &

Latham, 2011; Thames & Ball, 2010)

Ma (1999) noticed that U.S. teachers lacked deep content knowledge in

mathematics as compared to Chinese teachers. Ma explained, “Given a topic, a [good]

teacher tends to see other topics related to its learning” (Ma, 1999, p. 118). She uses the

term “knowledge package” when referring to teacher content knowledge of the given

concept and its related topics. She asserted that teachers who lack knowledge packages

find it difficult to deliver unified mathematics lessons. In addition, Ma (1999) presented

the idea of Profound Understanding of Fundamental Mathematics (PUFM) clarifying, “A

teacher with PUFM goes beyond being able to compute correctly and to give a rationale

for computational algorithms” (p. xxiv). A PUFM teacher is more likely to make

connections between mathematical concepts, present many ways of arriving to an answer,

and have mathematical knowledge that extends beyond the grade they are teaching (Ma,

1999).

Recommendation 17 in the report by the National Mathematics Advisory Panel

(2008) stated, “Research on the relationship between teachers’ mathematical knowledge

and students’ achievement confirms the importance of teachers’ content knowledge. It is

self-evident that teachers cannot teach what they do not know” (Flawn, 2008, p. xxi).

However, according to Wu (2011), currently most teachers do not have enough content


knowledge, because education systems are not insisting that they do. The premise of

Wu’s (2011) paper, “The Mis-Education of Mathematics Teachers”, is that elementary

teachers need to know K-12 level mathematics in order to effectively teach it effectively.

He stated,

[T]here is no such thing as knowing too much mathematics in mathematics

education. Every bit of mathematical knowledge will help in the long run.

However, faced with the almost intractable problem of improving the education of

all math teachers, it is only proper that we focus on a modest and doable first step:

make sure that mathematics teachers all know the mathematics of K–12. Let us

get this done. (p. 383)

Furthermore, the lack of teacher content knowledge seems to be a purely American

phenomenon, reflecting Ma’s (1999) motivation for her comparative study between

teachers in the United States versus China. According to the Trends International

Mathematics and Science Study (TIMSS, 1995), 71% of grade eight mathematics

teachers from all participating countries had a bachelor's degree in mathematics as

compared to only 41% in the United States (Gardner, 2005).

Mathematics pedagogical content knowledge

Ma’s (1999) work adds to Shulman’s (1986) paper, which originally introduced

the idea of pedagogical content knowledge (PCK). Shulman (1986) was motivated to

come up with PCK by the fact that often pedagogy and content knowledge were referred

to as separate entities in the teaching profession. Not agreeing with this dichotomy, he

came up with PCK, which marries the two ideas. Essentially, PCK refers to “teachers’

knowledge of the mathematics they are teaching in light of how students might think

about the mathematics” (Small, 2013, p. 12). Shulman’s (1986) paper compared the then

current teacher standards with those previously followed by the United States. He

presented categories covered by the California State Board’s examination for elementary


school teachers in March 1875, which include the following: written arithmetic, mental

arithmetic, algebra, theory and practice of teaching school law of California, etc. One

century later, at the time of Shulman’s paper, the subject matter and content was no

longer part of examinations. As written by Shulman (1986),

Perhaps Shaw [an Irish critic, who wrote the well-known saying “He who can,

does. He who cannot, teaches.”] was correct. He accurately anticipated the

standards for teaching in 1985. He who knows, does. He who cannot, but knows

some teaching procedures, teaches. (p.5)

Shulman’s (1986) main idea was “the ultimate test of understanding rests on the ability to

transform knowledge into teaching” (p. 14). This suggested that a high level PCK is

interwoven with a high level of content knowledge.

Research by Ball, Hill, and Bass (2005) added to Ma (1999) and Shulman’s

(1986) work by further researching the idea of pedagogical content knowledge. Ball et al.

(2005) were concerned with why a student may make mistake, how to present multiple

ways of solving a problem, and how to react when a student arrives to a correct answer

using an unforeseen method, all of which require substantial PCK. They have been

developing instruments to assess teachers’ mathematics PCK (Ball et al., 2005). With

other colleagues, Ball found that teachers’ PCK is an effective tool to improve student

achievement in mathematics (Hill, Rowan, & Ball, 2005). Furthermore, a Canadian study

by Bruce and Ross (2008) looked into strategies for improving teacher efficacy in

mathematics. One effective strategy was to deliver PCK to teachers via peer coaching

(Bruce & Ross, 2008). This study reflects recent trends in Canadian provinces pushing

for more collaborative inquiry in mathematics (Small, 2013). Specifically, elementary

school teachers have been working together to build on their PCK (Small, 2013).


Evaluations

Gardner’s (2005) article on teaching and teacher training provides staggering

statistics that are very telling of the importance of teacher evaluations. She writes about

recent tests by the National Assessment of Educational Progress (NAEP) being too easy

and inflating student achievement. This was attributed to elementary teachers not having

sufficient knowledge and skills to teach mathematics (Gardner, 2005). Winters (2012)

asserts that there is room for teacher improvement in terms of training, hiring,

compensation, and evaluation. Given the extensive job description of a teacher, Winters

calls for greater accountability for teachers, based on rigorous qualitative and quantitative

evaluations.

Since there is extensive research on the benefits of both mathematics content

knowledge and pedagogy, both should be evaluated. Our current system of evaluation

fails to identify effective teachers because nearly all practicing teachers receive the same

positive rating on their official evaluations (Winters, 2012). Duncan (2009) illustrated

this well when he said,

In California, they have 300,000 teachers. If you took the top 10 percent, they

have 30,000 of the best teachers in the world. If you took the bottom 10 percent,

they have 30,000 teachers that should probably find another profession, yet no

one in California can tell you which teacher is in which category. Something is

wrong with that picture. (as cited in Winters, 2012, p. 27)

A strategy that has become popular recently for evaluating teacher efficiency is

value-added assessment. This approach uses a statistical model to approximately

calculate a particular student’s performance at the end of the year if assigned to one

teacher instead of another, accounting for the student’s backgrounds and school resources

(Winters, 2012). Using value-added assessment, it was estimated that a teacher ranked in

the twenty-fifth percentile in quality would help students increase their proficiency by


about half a grade level, while a teacher in the seventy-fifth percentile led to a gain of a

grade-and-a-half worth of proficiency (Winters, 2012). This signifies the great impact

teachers alone can have on their students. In addition, this increased rigorousness to

teacher evaluations will lead to a need for better mathematics teacher education.

There is a general consensus that knowledge of content knowledge, pedagogy,

ethics, and psychology are important for teachers. However, there is ongoing debate on

which is least likely to be affected by the time constraints faced by educators. Basile and

Kimbrough (2008) stated the importance of sufficient teacher content knowledge in

addition to pedagogy, real world connections and assessment. They deem this impossible

and unrealistic due to the fact that the amount of knowledge a teacher needs is increasing

while the structure of the educational system is remaining constant (Basile & Kimbrough,

2008). If all essential components of an effective teacher cannot be taught during teacher

education programs due to time constraints, an alternative would be to ensure that the

content knowledge is acquired before acceptance into the program.

Recommendation 17 in the National Mathematics Advisory Panel (2008) report

attested to the importance of teacher content knowledge and its benefits on student

achievement. The recommendation states,

[B]ecause most studies have relied on proxies for teachers’ mathematical

knowledge (such as teacher certification or courses taken), existing research does

not reveal the specific mathematical knowledge and instructional skill needed for

effective teaching, especially at the elementary and middle school level. Direct

assessments of teachers’ actual mathematical knowledge provide the strongest

indication of a relation between teachers’ content knowledge and their students’

achievement. More precise measures are needed to specify in greater detail the

relationship among elementary and middle school teachers’ mathematical

knowledge, their instructional skill, and students’ learning. (as cited in Flawn,

2008, p. xxi)


Currently, in order to be accepted into the Ontario Institute for Studies in Education’s

(OISE) teacher education programs (Bachelor of Education and Master of Teaching) for

the subject of French or International Languages (German, Italian, Spanish or other), one

must pass a proficient test (“French and International Languages Proficiency Testing”,

2014). As presented on the OISE website, the proficiency test is made up of the following

three components:

1. Comprehension: Listening to a tape, reading text and answering questions

2. Written: Cloze test (filling in the blanks with the right word, composing short

memos, faxes, notes, reports, letter to a parent)

3. Oral: Interview with a teacher, may include two other students or one-on-one

dialogue, (approx. 15 minutes duration) (“French and International Languages

Proficiency Testing”, 2014, para. 6)

In order to pass the test, you must score 70% or higher in each component (French and

International Languages Proficiency Testing, 2014). Some school boards also require

applicants to pass an additional French proficiency test prior to being hired (e.g., TDSB).

Specific to mathematics, in order to take the Mathematics Intermediate Basic

Additional Qualification (AQ) course at OISE, one must pass a mathematics proficiency

test if they not have at least two full university degree courses in mathematics

(“Mathematics Pre-Admission Testing”, 2014). The test level is comprised of mainly

grades 7-10 level word problems and 70% is needed to pass (“Mathematics Pre-

Admission Testing”, 2014). By contrast, no mathematics tests or university requirements

exist for enrolling into the Primary/Junior division of both teacher education programs at

OISE (Bachelor of Education and Master of teaching) (“Teaching Subjects”, 2014). The

same applies to Junior/Intermediate candidates, unless the candidate would like to select

mathematics as his/her teachable subject (“Teaching Subjects”, 2014).


In this chapter, I have discussed the importance of both content knowledge and

pedagogy for mathematics teachers, as well as the importance and current lack of

evaluating the two. Research shows that effective mathematics teachers have knowledge

of both mathematics content and pedagogy. Furthermore, they are able to merge the two

into what is called pedagogical content knowledge. In order to ensure that these traits are

present in teachers, different evaluation strategies can be employed.


Chapter 3: METHODOLOGY

Introduction

This study explores the question: What are teachers’ perspectives about increasing

the quality of Junior/Intermediate mathematics teachers? To answer this research

questions, I investigated two sub-questions. First, I wanted to find out if any specific

changes can be made to the current course of action for becoming an elementary

mathematics teacher that may lead to better quality elementary mathematics teachers. The

second sub-question aims to place any suggested changes in the most ideal place along

the timeline of becoming an elementary teacher.

This chapter presents the method and procedure that was used in this study

including information about the data collection instruments, sample participants, data

collection and analysis, ethical review procedures and limitations. Using a qualitative

methodology, to expand on ideas gained from reviewing current literature, I collected my

data by recording face-to-face semi-structured interviews, guided by a carefully crafted

set of questions. My participants are elementary school teachers teaching mathematics

and other subjects. The data was transcribed, reviewed and analyzed for themes. The rest

of this chapter describes in more detail instruments of data collection, criteria for

selecting participants, data collection and analysis methods, and ethics review procedures

of the study.

Instruments of data collection

As previously stated, I collected my data through face-to-face semi-structured

interviews. The interviews were mainly guided by a carefully crafted set of questions

(Appendix A), but discussion was not restricted if the interviewee wanted to clarify or

elaborate. This type of interview most resembles a Standardized Open-Ended Interview


(Turner, 2010). I selected this mode of interview because, although structured in terms of

wording, it still “allows the participants to contribute as much detailed information as

they desire and it also allows the researcher to ask probing questions as a means of

follow-up” (Turner, 2010, p. 2). I pilot-tested the interview questions with someone of a

similar background to my participants in order to determine if revisions were necessary

prior to the real interviews (Kvale, 2007). I interviewed both teachers in their classroom

during times of their convenience. Interviewees were provided with the list of questions

(Appendix A) a few days in advance of the interview so that they could become familiar

with the questions. Due to the complex nature of some of my interview questions, I felt

that I would be able to capture better quality data if my participants had time to

thoroughly think about their answers. Also, on the day of the interview, I presented my

questions verbally and visually via a printout. Furthermore, the interviews were about 40

minutes long. Both interviews were digitally recorded. I began my interviews by thanking

the participants, reminding them about my topic of study, explaining the interview

process, and asking them if they had any questions that I could clarify before the

interview began.

Participants

I interviewed two participants, both of whom are currently teaching mathematics

and other subjects at the elementary level in different Toronto District School Board

(TDSB) schools and each has been teaching for more than 10 years. Having both

participants in the Greater Toronto Area allowed for face-to-face interviewing. I wanted

participants to have at least 10 years of teaching experience because they would have

more examples and insight to share. In addition, I ensured that both exhibited

characteristics of an exemplary mathematics teacher, in terms of knowledge of content


and pedagogy. To ensure this last criterion, I chose two of my Associate Teachers as my

participants, since I had watched them teach mathematics.

I will introduce my participants, Ben and Liz, in more detail now. At the time of

the interview, Ben was teaching mathematics and science to students from grade four to

grade eight in a TDSB school. Before going entering his teacher education program, he

worked for Cops and Civilians Employee Police, as a rescue driver, and was briefly

enlisted in the air force. He also had his own business teaching swimming and first aid.

When these avenues “didn’t pan out as well as [he] had hoped, teaching sort of popped

in.” Ben has been teaching for twelve years and has taught physical education, science

and mathematics.

At the time of the interview, the second participant, Liz, was teaching

mathematics, English, and art to students in grade six and family studies to students in

grade seven and eight in a TDSB school. She has been teaching for eleven years and has

experience teaching language arts, mathematics and social studies, art and family studies.

I e-mailed my participants with the details of the interview such as topic and

length and they agreed to the interview. As previously mentioned, my participants were

my former Associate Teachers, and this was a thoughtful decision for my study as I

already had a lot of background knowledge about them. Dilley (2000) states that having

background information about interviewees is “an important first step in interview

preparation” because it leads to better understanding of the cultural context and makes

new researchers feel more comfortable (p. 131-132). In the initial phases of my research,

I was very interested in interviewing an elementary teacher with mathematics as their

teachable subject. However, I was unsuccessful finding such a participant for my study.


Data collection and analysis

I captured my data by digitally recording the interviews. Transcribing of the

recording followed, prior to analysis. Initially, during the coding phase, I read and re-read

my transcript several times in order to “immerse [my]self in the details, trying to get a

sense of the interview as a whole before breaking it into parts” (Creswell, 2013, p. 150).

After getting an understanding of the interview as a whole, I chose to use a more

traditional approach to coding. In this method, similar to Creswell’s (2013) description,

the researcher aims to form a description of the data and relate this description to the

literature. In analyzing my data, I followed the following four steps. First, I highlighted

information that was answering my research question and information that kept coming

up. Second, I re-read these codes and categorized them even further. Third, I arranged

and color-coded all categories and codes in a table format to make it easier for an outsider

to look at and analyze my coding, and finally, I organized data under code headings by

copy-pasting.

Furthermore, I analyzed my data using both the a priori and emergent analysis

approaches (Falk & Blumenreich, 2005). After writing my literature review and

formulating my interview questions, I decided on some categories that would help

organize my research. Since my area of inquiry is increasing the quality of

Junior/Intermediate mathematics teachers and more specifically improving the current

process of becoming a mathematics teacher, it made sense that three of my categories

would be the three stages of the process displayed below.


2 1

q

3

q

Teacher education program

Hiring point

Figure 1. The process of becoming a mathematics teacher. This figure illustrates the

process of becoming a mathematics broken up into three stages: (1) Before entering a

teacher education program; (2) During a teacher education program; and (3) After

completing a teacher education program, but before being hired by a school or school

board.

During the interviews, I gave my participants the above figure in order to

structure the flow of questions and answers. Therefore, categories “before the teacher

education program”, “during the teacher education program” and “after the teacher

education program”, from the above figure, came about using a priori emergent analysis

approach (Falk & Blumenreich, 2005). On the other hand, the theme “ideal mathematics

teachers” became an “emergent” category (Falk & Blumenreich, 2005) as it arose from

my reading of the transcripts. This theme emerged as I noticed a lot of talk about what

characteristics the ideal teacher should have. Looking back, this was an important

addition to my list of categories because it helped me see what qualities teachers should

have or develop though teacher education programs. As such, I decided to use this theme

as my overarching theme, before the themes relating to teacher education programs, in

order to provide a contrast. This theme would provide the context for what kind of

teachers my participants believed that we need and want versus the kind of teachers that

we currently have and are getting with the current process of becoming a teacher. During

the process of data analysis, I asked for my supervisor’s help to eliminate biasness and

over-analyzing (Turner, 2010).


Ethical review procedures

I followed the ethical review procedures for the Master of Teaching program. I

contacted my participants via e-mail, providing them with a brief overview of my topic

and requesting an interview based on their time and place preferences. In my e-mail, I

attached a copy of the consent letter (Appendix B), which outlines the purpose of the

study, the length of the interview, how the data would be used, who would see the data,

and their rights in terms of withdrawing from the study at any point. Also, during the

interview, I advised my participants that they may choose to not answer any question that

they wish. Participants were assured that their name and personal information would not

be revealed and the information provided would be used by only myself, my supervisor

and the course instructor of my research methods course. Lastly, participants were told

the recording would be kept for a maximum of five years.


Chapter 4: FINDINGS

In this chapter, my aim is to narrate the findings without interpreting, explaining

or evaluating them. It will follow from the research questions and methodology and set

the stage in sufficient detail for the discussion and analysis in Chapter 5. I will be

organizing this chapter by themes and include what each participant said about each

theme.

Before I begin, I will briefly re-introduce my participants, Ben and Liz. Ben

completed his undergraduate education with a degree in science and it was consequently

his teachable during his teacher education program. He has twelve years of teaching

experience in the following subjects: physical education, science and mathematics. At the

time of the interview, Ben was teaching mathematics and science to students from grade

four to grade eight in a TDSB school. Liz completed her undergraduate education with a

degree in English and it was also her teachable during her teacher education program.

She has eleven years of teaching experience in the following subjects: language arts,

mathematics, social studies, art and family studies. At the time of the interview, Liz was

teaching mathematics, English, and art to students in grade six and family studies to

students in grade seven and eight in a TDSB school.

Theme 1: Ideal mathematics teacher

A couple of my interview questions aimed to capture my participants’ vision of an

“ideal” mathematics teacher. It was important to establish this from the beginning in

order to understand the foundation from which they made their decisions for subsequent

questions. One question was very broad and open, asking “What makes a good

mathematics teacher?” This was followed by the question: “How important is content

knowledge versus pedagogy for good math teachers?” This subsequent question


narrowed the focus to two sub-concepts: content knowledge (knowledge of subject-

specific material) and pedagogy (teachers’ rationale for their used of instructional

methods and practices).

Content knowledge

Ben described an ideal mathematics teacher to be someone who has content

knowledge as well as the ability to teach it. He stated that a good mathematics teacher is

someone who “knows their stuff well and can answer pretty much any question on it”. He

shared the following personal experience, which speaks to the importance of teacher

content knowledge:

We’re struggling right now with the fact that we have grade 6, 7, and 8 students

who don’t know their multiplication tables because even though they’re supposed

to understand multiplication, the go to for teachers that don’t understand the math

or how to deliver it properly is to pull out a calculator and do it. So, when you

start trying to do mental math, mental estimation with numbers in the thousands,

or two or three digit multiplication or division, these kids have no idea of how to

even do single digits, so how are they supposed to do it with two or three digit.

He added that the importance of content knowledge is especially evident when a student

does not understand a concept after it is explained via one method. Ben said that a good

mathematics teacher knows the material well enough to be able to look at concepts from

multiple angles in order to reach all students by presenting different approaches to the

same problem at a moment’s notice. Based on this, he believes, “You should start having

a specialist in certain subjects: math, science, English as early as grade five or six.” His

rationale for this, with respect to mathematics specifically, is because in “grade four,

you’re still sort of figuring out [the basics] and in grade five, you start introducing

multiplication and dealing with BEDMAS rule, order of operations and your really need

to know your content.”


Ben said that having rotary specialists make for better well-rounded students as

compared to students with generalist teachers. He gave the following example:

As a former high school teacher, I could tell which students came out of a middle

school and which were coming out of a K-8 school because the kids coming out

of the middle school had specialists and they tended to excel a little better at

certain subjects.

He made the distinction that a good teacher is not “someone who just aced mathematics

because it comes easy to them”, but someone who is strong in and passionate about the

subject.

Liz shared Ben’s view of the important role content knowledge plays. She said, “I

strongly believe a good math teacher has to have some form of grasp of the math”. Liz

took mathematics until grade twelve, but believes that the good grasp she has of it,

especially of number sense and numeration, plays a huge role in her teaching

mathematics. Liz shared implications that ill-prepared mathematics teachers can have on

students. She stated that if adults are afraid to talk about being uncomfortable about

mathematics and then they go out and teach it, it translates to the students that you are

uncomfortable. She shared what a student may be thinking about the issue: “So if you are

uncomfortable as my math teacher, how do I trust what you are teaching me is valid?”

She added, “Once in a blue moon, you get a student that will really challenge you, what

do you do? You lose credibility because the child stumps you every time and you don’t

want that.”

Pedagogy

Ben stressed the importance of pedagogy by stating that a good teacher is able to

communicate ideas to diverse students effectively via “manipulatives for those hands on,

kinesthetic and visual learners” and technology. He added that delivery is detrimental in a


student’s enjoyment of mathematics and has the potential to “turn kids off”. He used his

grade ten teacher as an example of poor delivery saying, “She probably knew the math

inside out and backward”, but elaborated that the teacher did not have the ability to

deliver it effectively. He said, “She had a very heavy thick accept and was too busy

slapping rulers on desks to get you to work and that’s just not good delivery in my

books.” Furthermore, he made a connection between content knowledge and delivery

when he said, “A good mathematics teacher should know enough that they can deliver it

in different ways.”

Liz also identified pedagogy as a key ingredient of a good mathematics teacher.

She stated, “What you don’t know, you can always learn and there are a ton of lessons

and resources out there for us.” She said that a good mathematics teacher should be able

to look at “what they have in front of them and determine which is the most effective and

efficient way of using that information and translating it clearly to student.” She added

that, from the perspective of the students,

It’s not so much you knowing [the content], because the kids perceive, as soon as

you walk into the classroom that any math teacher was an intelligent teacher.

Even a supply teacher is an intelligent teacher. But what will set the tone is what

will come out from the teacher – how you are explaining the question, how you

are explaining the problem, how we’re allowing them to experience the problem.

Lastly, when asked “How important is content knowledge vs. methodology for good

mathematics teachers?” Liz replied, “Both have equal merit.”

In summary, both my participants believed that a good mathematics teacher is

someone who has mathematics content knowledge as well as pedagogy skills. More

specifically, Ben felt that pedagogy is more important and makes a connection between

the two, whereas Liz thought they are equally important.


Equipped with my participants’ vision of an “ideal” mathematics teacher, I took

them through three stages of becoming a teacher: before, during and after teacher

education programs. I asked them to discuss the strengths and weaknesses of each stage,

followed by any changes they feel would improve the process in order to yield better

mathematics teachers.

Theme 2: Before teacher education programs

I asked my participants questions aimed to capture their experiences prior to

entering their teacher education program. One question was very broad and open, asking

“Could you please describe any post-secondary (undergraduate) training you had in

math?” This was followed by an attempt to link their post-secondary mathematics

training to their professional practice by asking, “In what ways, if any, did it inform your

current professional practice?” The next two questions dealt with the process of getting

into a teacher education program: “Could you describe the process to get into teachers’

college?” and “What were the requirements in terms of grade point average (GPA), tests,

experience, involvement, etc.?” It was important to capture their experiences with this

stage in the process of becoming a teacher from the beginning in order to understand the

foundation from which they made their decisions for subsequent questions.

Next, I asked them: “What are some strengths and/or weaknesses of the

requirements to get into teachers college in terms of undergraduate background, entry

test, etc?” and “What changes should be made?” Their answers led me to the following

subtopics: undergraduate education requirements, entry test, and teacher education

programs’ application and interview process. The sub-category “undergraduate education

requirements” refers to requirements a teacher education program may require applicants

to have in order to be accepted into the program. Entry test refers to a test administered


prior to the start of one’s teacher education program which determines whether an

applicant is successful or not. Lastly, teacher education programs’ application and

interview process refers to any application package and/or interview required by teacher

education programs in order to determine whether an applicant will be admitted into the

program.

Undergraduate education

Ben shared the following regarding his application to teacher education programs,

“[The application] was loosely based on math marks and I think it is a weakness.” He

added that taking some credits in university does not mean that you necessarily

understand the content. He also restated, “You should start specializing in about grade six

and up.”

Liz recalled taking a psychology course which included some statistics, but did

not take any mathematics courses during her undergraduate education. When asked

whether her lack of undergraduate mathematics courses had an effect on her practice as a

mathematics teacher, Liz replied that she did not think so because she had a good grasp

of the mathematics concepts that she learned in elementary and secondary school.

Summing up, Ben thought teachers teaching mathematics in grade 6 and up

should have a strong undergraduate background in the subject, not just a few credits.

Meanwhile, based on her experience, Liz thought that knowledge of secondary level

mathematics is enough to teach mathematics in the elementary grades.

Entry test

After I asked for strengths and/or weaknesses of undergraduate degree

requirements for teacher education programs, Ben asked whether there currently was an

entry test required by universities for admission into teacher education programs. After I


informed him that there is an entry test only for the subject of French, he replied, “[That]

kind of makes sense, but I think that should be across the board, I think no matter what

subject you want to get into.” He said that an entry test would be good because it would

“weed people out who don’t really get the material”.

Specifically, Ben suggested that a test be taken before teacher education programs

in the subject areas that you will be teaching. He said, “For example, if you wanted to

become Intermediate/Senior, I think you need to know your subject, otherwise what’s the

point?” Ben also stated that he would support a process where someone can bypass the

specific undergraduate education/ course requirements to get into teacher education

programs by passing the entry test. He used his neighbour as an example: “She’s never

really taken French outside of high school, but because she is Francophone, she was able

to write the test and she got herself into [a teacher education program], and I think that

was good.” When asked to clarify what kind of testing he is in favour of, he replied,

“Knowledge based testing.”

When asked to describe the process she followed to get into her teacher education

program, Liz mentioned that she did not have to write an entry test. Furthermore, she

acknowledged knowing that French teachers are required to pass a French test prior to

entering teacher education programs. When asked how she would feel about requiring

mathematics teachers in the Junior/Intermediate division to pass a test on mathematics

content, she replied: “I don’t feel the need for it because what you’re teaching in

elementary school is not beyond the high school math content that you know.”

My participants had opposing views on the topic of entry tests. Ben felt that it

would make a valuable addition to the current process, preferring entry tests over


undergraduate course requirements. On the other hand, Liz did not feel that an entry test

was necessary to ensuring a high quality of teacher candidates.

Teacher education programs’ application and interview

When asked about his opinion about the application process to enter teacher

education programs in Ontario, Ben said: “Too much of the requirements to get into

[teacher education programs] right now are based on the personality profile section about

yourself, whatever the essay question may be.” He found it unfair that the focus was on a

candidate’s written communication skills. He said, “All you need to do, really, is find

someone that could write well, ask them to help you out with it or edit is, or even write it

for you and that basically got you the interview or in.”

Liz described her application and interview process to get into York University’s

Faculty of Education as being, “Not just the written component, your marks, and that’s

it.” Liz continued, “What I truly appreciated about York’s process was their interview

because that was their way of meeting us face to face.” Liz described this interview as the

opportunity where she was able to give “a personal piece of [herself] in the process.”

To summarize my participants’ perspectives on the phase prior to entering a

teacher education program, Ben felt that the focus was wrongly placed on written

communication skills, which could have easily been falsified by applicants. Liz

appreciated that York University (where she did her teacher training) looked beyond the

written application component and conducted applicant interviews.

Theme 3: During teacher education programs

This theme refers to the phase of a teacher’s development during which the

teacher is in a teacher education program, which includes courses and in-class teaching

experiences. For this stage, I focused on two subtopics: content knowledge and pedagogy


because the two were the most commonly debated aspects of teacher education programs

when I was doing my literature review. I wanted to see where my participants placed

themselves in this debate. Content knowledge refers to the knowledge of subject-specific

material. Pedagogy refers to teachers’ rationale for use of specific instructional methods

and practices. This theme has the same subtopics as Theme 1: The ideal mathematics

teacher because I wanted to be able to compare and contrast between the teachers we

need and should have and the teachers we have and are producing based on current

teacher education programs.

Content knowledge

Ben recalled that during his teacher education program a lot of focus was on

ensuring familiarity with the curriculum expectations and how to use them. Content was

not a focus under the assumption that teacher candidates already knew it. He felt that this

assumption is an issue when it comes to mathematics. He shared, “Most of the grade six

teachers here are English specialists and when it comes to math, certain areas are weaker

than others.” He suggests that the entry test will alleviate the debate of whether content

knowledge should be taught during teacher education programs or not.

Similar to Ben, Liz shared that content knowledge was not covered during her

teacher education program. She stated, “What you’re teaching in elementary school is not

beyond the high school math that you know.” With this statement, Liz alluded to the

assumption that all elementary teachers who are teaching mathematics took and/or feel

comfortable with secondary mathematics. She wished that “bigger and more realistic

conversations…about how comfortable [the teacher candidates were] about mathematics”

occurred in her teacher education program, “instead of a small snippets of conversations”.

Furthermore, she asserted that, in terms of mathematics content knowledge, “If there is a


concern then obviously that’s where your professors are signalling that alarm bell.” What

she means by this is that, instead of having a test to determine if a teacher candidate has

enough mathematics content knowledge to teach mathematics, professors at teacher

education programs should be responsible to identify these students and take appropriate

action to alleviate the problem.

Pedagogy

Ben recalled that the majority of the time in his teacher education program was

spent on pedagogy, such as “[How to] meet [curriculum] expectations, creating rubrics

and checklists.” In his opinion, delivery is the most important thing in teaching. He

asserted, “[Teaching] is like performing, you’re an actor really, you’re on stage.”

Therefore, when asked what changes he would make during teacher education programs,

Ben stated, “The focus needs to be more on developing a teaching style and your own

personal pedagogy. I think what you need to do is have more in class experience.” As a

good example, he used the Master of Teaching (MT) program, a two-year teacher

education program which includes a qualitative research paper requirement as well as

four in-class teaching experiences. He said,

I think [the MT program] is great because you get four practicums and I think by

the end of your fourth one you’ve pretty much nailed it. Whereas I think, if you’re

in the one year program, depending on what school you’re at, you’ve only get two

[practica].

Overall, Ben felt that the current focus in teacher education programs is where it should

be – pedagogy, as long as the content knowledge had been accounted for prior to being

accepted into the program.

Liz shared, “The biggest concern you have when you’re in [a teacher education

program] is getting over being nervous and teaching properly,” not whether or not you


knew the content. She stated, “I really liked that the focus [in my teacher education

program] was majorly about how we were teaching and how clear we were teaching.”

In conclusion, both participants were on the same page when it came to where the

focus should be during teacher education programs. Both participants believed that

teacher education programs should focus on pedagogy and the teaching practices and

strategies that teachers should use in their teaching. In terms of content knowledge, Ben

felt that as long as a teacher candidate’s content knowledge is tested before entering the

teacher education program, it does not need to be part of the teacher candidate’s training.

Liz felt that knowledge of secondary level mathematics was more than enough

knowledge for teaching elementary level mathematics. However, she indicated that this

was only the case if the elementary teacher did well in secondary level mathematics.

Theme 4: After teacher education programs

I broke this theme down into the following subtopics: exit test, additional

qualifications, and school boards’ application and interview process. The exit test refers

to a test administered after the end of one’s teacher education program and determines

whether an applicant is prepared to be a certified teacher. Additional qualifications refers

to any additional mathematics courses taken after completion of the teacher education

program as a requirement for elementary teachers who plan to teach mathematics at the

elementary level before they are hired by school boards. Lastly, the school boards’

application and interview process refers to any application package and/or interview

required by school boards in order to determine whether an applicant will be admitted

into the board.

Some of my interview questions aimed to capture my participants’ experiences

after completing their teacher education program. One question asked, “Did you


complete any additional qualification courses and/or professional development sessions

in math after your teacher education program?” This was followed by the following two

questions on exit testing: “Did you have to write any tests/exams after teachers college?”

and “What do you know about and how did you feel about the Ontario Teacher Testing

Program which was established, and administered between 2000 and 2005?” The next

question dealt with the process of getting into a school board: “What was the hiring

process for your past and current teaching position(s)?” Again, it was important to

capture their experiences with this stage in the process of becoming a teacher from the

beginning in order to understand the foundation from which they made their decisions for

subsequent questions. Next, I asked them: “What are some strengths and/or weaknesses

of the requirements to get certified by the Ontario College of Teachers (OCT) and hired

by school boards in terms of exit tests, additional qualifications, school boards’

applications and interview process?”

Exit test

Ben said that he sees merit in an exit test, but is not sure what it would look like.

He stated:

I don’t know if I believe in a paper and pen test at that point or whether you

should come up with a well rounded lesson plan and get evaluated by your peers

on how well you do on the delivery of that lesson.

He concluded that he would more readily support the idea of an evaluation of a lesson

plan and delivery, assuming the content knowledge was tested before entry to a teacher

education program.

Liz recalled having to write the Ontario Teacher Qualifying Test after completing

her teacher education program at York University. The Ontario Teacher Qualifying Test

was established and administered between 2000 and 2005 with the hopes of improving


teacher quality. Liz remembers it being a multiple choice test and said, “It was not very

useful at all.” She stated that the test focused on content, rather than teaching practice.

Liz added, “It was not that difficult” and that no one in her class failed it. She does not

recall there being any mathematics content knowledge on the test. She suggested, “If we

are assessing teachers for efficacy, then you need to see me in action teaching, just like a

performance appraisal.” In terms of mathematics content knowledge testing, she asserted,

“If there is a concern then obviously that’s where your professors are signalling that

alarm bell, but otherwise, I don’t see why they would let you graduate and then you have

to write a test.”

In summary, Ben would support an exit test after the completion of a teacher

education program that evaluates a teacher’s teaching skills. Liz, on the other hand, is not

in favour of any testing after graduation and thinks professors should facilitate formative

evaluations during the program.

School boards’ application and interview

Ben recalled his application to the school board as being a “long and grueling”

process of providing paperwork, such as Ontario Teachers’ College (OCT) certification,

transcripts, associate teachers’ evaluations, etc. He expressed that he wished that the

board application and interview process would have been faster. Furthermore, similar to

how he felt about his application and interview process to get into his teacher education

program, Ben found that the school board application and interview was evaluating

mainly ones pedagogy and English written and oral communication skills. He felt this

unfairly gave language-based majors (e.g., English, history) an advantage over the

science-based majors (e.g., mathematics, science, technology).


Liz described the board application and interview process she went through in

detail as follows:

You just had to hand in your resume and [the Toronto School Board (TDSB)]

would contact you and you go into a hotel and they have a sectioned area where

you go in, they call you and a couple of principals would interview you. One

takes notes and one asks questions. And when you pass that, they send you a

letter, you go into the TDSB building, you sign your contract and then they let

you know, “Okay, now, you are okay to apply to the various jobs that come out.”

When asked what kind of questions she was asked during the interview, she replied that

the questions were about a range of topics including: differentiated instruction,

behavioural management, and community building. When asked if mathematics was part

of any of the questions, she replied, “No, not at all.” Liz added, “What they are looking

for, unfortunately, is something more general. They don’t look for something as specific

as your content knowledge in math.” Overall, unlike Ben, she found the board application

and interview process to be easy and not tedious.

To sum up, once again, my participants had opposing views of the school boards’

application and interview process based on their own experiences. Liz thought it was

simple and quick, while Ben felt it was demanding and long.


Chapter 5: DISCUSSION

In this chapter, my aim was to develop an argument by analyzing and interpreting

my findings from Chapter 4, while making connections to the theoretical and research

base from Chapter 2. I related back to my initial research questions, in terms of whether

findings confirm or disconfirm expectations and the extent to which the questions are

answered. Also, I critically evaluated my methodology and analysis. Lastly, I made the

significance of my findings clear and presented where I think research should go next.

Introduction

The main purpose of the study was to examine the following question: What are

teachers’ perspectives about increasing the quality of Junior/Intermediate mathematics

teachers? My sub-questions stemming from the main question are: What changes, if any,

would improve the current process of becoming a mathematics teacher? Where should

these changes happen along the timeline of becoming a mathematics teacher?

My findings suggested that there is a disconnection between the characteristics of

a successful mathematics teacher and those of current mathematics teachers. While my

participants asserted that pedagogy is the most important aspect of a successful teacher,

they also emphasize the importance of teacher content knowledge in subjects such as

mathematics, science and French. However, their experiences as teacher candidates and

teachers, reveal that in reality the focus in teacher education programs is

disproportionately skewed, prioritizing pedagogy. They stated that it was assumed that

you have the content knowledge and it was rarely brought up in classes. Ben saw this as a

weakness of the current system of becoming a mathematics teacher because of the

implications it is having on students’ learning. Specifically, he expressed how teachers

who do not have enough content knowledge tend to stick to one way of explaining


concepts, give shallow instruction, lack real world mathematics connections and overly

rely on technology such as calculators and computers. He gave examples of how all of

the former clearly hinder students by leading to a decline in student interest and learning

in mathematics. Some changes were suggested at different stages of the process, such as a

content knowledge entry tests before teacher education programs and a pedagogy exit test

after teacher education programs. On the other hand, Liz was not be in favour of any

entry or exit testing, but called for more open discussions during teacher’s college about

how comfortable teacher candidates feel with mathematics. She also stated that if content

knowledge was a concern, professors should step in and take action.

Relating findings to current literature

Pedagogical content knowledge

I found that both my participants did not feel inclined to pick a side in the content

knowledge versus pedagogy debate. Overall, they described the ideal teacher to have both

of these kinds of knowledge. This falls in contrast with some of the research articles

where these two concepts were put up one against the other. For example, Gresham

(2007) and Tooke and Lindstrom (1998) compared mathematics methods courses in

teacher education with content courses for pre-service teachers who are mathematics

anxious but will have to teach mathematics because they will most likely be generalist

teachers. Both studies concluded that methods courses are more beneficial than content

courses (Gresham, 2007; Tooke & Lindstrom, 1998). My findings harmonized with

previous research that advocated for pedagogical content knowledge, a combination of

pedagogy and content knowledge (e.g., Ball, Hill, & Bass, 2005; Bates, Kim, & Latham,

2011; Bruce & Ross, 2008; Hill, Rowan, & Ball, 2005; Shulman, 1986; Small, 2013;

Thames & Ball, 2010).


In contrast to current practices, my participants both felt that the emphasis of the

process to become an elementary mathematics teacher greatly favours pedagogy, at the

cost of ignoring content knowledge. Furthermore, in terms of requirements to get into a

teacher education program, Ben revealed that most mathematics teachers in the school in

which he teaches have completed undergraduate degrees in English and have

subsequently chosen it as their teachable subject. This aligns with the North American

statistics from the Trends International Mathematics and Science Study (TIMSS, 1995),

which found that only 41% of American grade eight mathematics teachers held

bachelor’s degrees in mathematics, which is significantly less compared to the 71% in all

participating countries (Gardner, 2005). While both Ben and Liz agree that the time

available during teacher education programs should be dedicated to pedagogy, they

disagree on how content knowledge should be addressed.

Evaluations

Both my participants held that there is not enough time during teacher education

programs to cover mathematics content knowledge and pedagogy. Likewise, as presented

in my literature review, Basile and Kimbrough (2008) state that the current stagnant

structure of the educational system makes it impossible for teachers to gain the increasing

amount of necessary knowledge in terms of subject-matter content, pedagogy, real world

connections and authentic assessment. To ameliorate this problem, one of my

participants, Ben, suggested administering content knowledge evaluation prior to entry to

the teacher education program. Research has shown that teacher assessment is beneficial

for students’ learning (Winters, 2012). My second participant, Liz, did not feel the need

to test elementary mathematics teachers’ content knowledge because the content is not

harder than the mathematics they learned in secondary school, assuming they understood


it. My literature review disagrees with the fact that elementary teacher have adequate

content knowledge (Ma, 1999; Wu, 2011)

Furthermore, my participants were in agreement that a content knowledge exit test

administered after teacher education programs would not be appropriate; rather it would

be a good idea to facilitate an observation of a lesson. This aligns with Winters’ (2012)

opinion that a teacher’s job description is so extensive and that calls for more frequent

regulated observations. A study demonstrated the correlation between teacher evaluations

and student success when it found that students of teachers ranked in the seventy-fifth

percentile in quality gained one grade level more in proficiency than students of teachers

in the twenty-fifth percentile (Winters, 2012). Both my participants also often brought up

implications the quality of mathematics teachers can have for students. For example, Ben

shared that at the moment grades six, seven, and eight students are struggling with

multiplication due to teachers that do not understand the mathematics or how to deliver it

properly.

Evaluation and discussion

At the start of my research paper, I expected to hear that there is a need to

improve the quality of Junior/Intermediate mathematics teachers. I also anticipated

hearing some potential changes that would improve the current process of becoming a

mathematics teacher. Specifically, I expected to hear about the merits of a mathematics

content test, but also the disadvantages of one. I also expected rotary to come up, even

though none of my questions were targeting it. In general, I found that my findings did

support these initial expectations. Although both participants shared different strategies to

improve the current process of becoming a mathematics teacher in order to improve the

quality of the teachers, the idea of rotary was only brought up by Ben.


Once I picked my participants, Ben, a science and mathematics teacher with a

science undergraduate degree and teachable, and Liz, a generalist teacher with an English

undergraduate degree and teachable, I speculated that their perspectives would be widely

different. Specifically, from having been in his classroom for a month, I expected Ben to

put more emphasis on content knowledge than pedagogy, as well as more supportive of

the idea of content knowledge test. I learned that he holds pedagogy as more important

than content knowledge and would be in favour of content knowledge testing. On the

other hand, I expected Liz to view pedagogy as more important than content knowledge,

but she stated that they have equal merit. Also, as I anticipated, she was not supportive of

any type of testing.

Another aspect that came up was rotary specialists versus generalists. While, both

participants agreed that the ideal mathematics teacher needs to have the mathematics

content knowledge as well as pedagogy, it was never directly stated by neither of the

participants that generalist teachers should be equally qualified in all the subjects they

teach. It leads me to wonder whether this is unattainable. Furthermore, most generalist

teachers in Ben’s school have English backgrounds and he said they lack content

knowledge in some areas of mathematics. Holding their pedagogy skills constant, it is

fair to say that these teachers are better at teaching English language based subjects such

as language arts and social studies than they are at teaching mathematics. Thus, there is

reason to believe that in this particular school, the majority of students’ have superior

English teachers and less than ideal mathematics teachers. It comes as no surprise then

that these students’ mathematics achievement and enjoyment suffers more in comparison

with other school subjects.


Ben bypassed the idea of having well-rounded generalists by saying that another

option would be to take the rotary route, where teachers teach only the subject they are

specialized in. I expected Liz to bring up rotary as well because at the middle school in

which she teaches, many subjects are taught on rotary by teachers with the appropriate

subject of specialization. Language arts and mathematics were the only two subjects

taught by homeroom teachers, regardless of their teachable subject. However, Liz did not

mention rotary as a solution to improving mathematics education.

Implications and recommendations

This study has implications for me as a researcher. From doing my literature

review, I learned a lot about the history of the teaching profession. I also learned about

current research on mathematics education as well as mathematics teacher education. The

study has implications for me as a teacher because it has taught me that it is not enough

for me to have the content knowledge; pedagogy is just as important. I will make sure to

continue expanding and updating my pedagogy knowledge throughout the years. In

addition, it makes me realize that I have a disadvantage when teaching other subjects that

are not my teachable due to my lack of content knowledge. For example, my science

content knowledge is limited, having taken general science until grade ten and physics

and chemistry only in grade eleven. I was able to test this out when teaching science to

grade four through seven students on rotary in Ben’s class. I only felt comfortable

teaching the grade six students because they were doing the space unit and I had taken an

astronomy course while completing my undergraduate degree. For example, I struggled

teaching the Pullies and Gears unit to the grade four students and the Heat in the

Environment unit to the grade seven students. I was not familiar with the content and

often found myself memorizing material the night before. During the lesson, I was not


equipped to answer students’ questions that were beyond what I had prepared for at

home. Being aware of this deficit, I will continue to expand my science content

knowledge and seek help from science specialists in the school. On the other hand, this

study makes me realize the advantage I have in teaching mathematics. I will make myself

available to colleagues at my school who might need help preparing for mathematics

lessons.

Most importantly, this study has implications for the education community,

specifically the current process of becoming a mathematics teacher. Stemming from my

research, I have the following recommendations:

Adding an entry mathematics content knowledge test before acceptance to teacher

education programs in the Junior/Intermediate division in order to qualify teacher

candidates to teach mathematics

Continuing to focus on pedagogy during teacher education programs as time is

limited and this may be the first and only time teacher candidates are taught it in a

formal setting

Continuing to do lesson observation evaluations during in-class teaching

experiences

Consider making mathematics a rotary subject taught by a teacher with a

mathematics background and teachable to ensure mathematics content knowledge

and pedagogy in order to improve students’ mathematics achievement and

enjoyment

Limitations and next steps

The major limitations of this study were due to the time constraints in my

program. With a full course load, two month long teaching experiences and numerous job


applications, the number of literature sources reviewed, as well as my sample size of two

participants were both limited. Time also limited the number of interview sessions I was

able to conduct with each participant. It would have been beneficial to have a second

interview with my participants after I transcribed and coded the first interview in order to

seek clarification or to ask for further elaboration for specific questions or topics.

Furthermore, although I tried my best to pick two diverse participants, it was highly

selective, making the research somewhat biased and systematic.

Another major limitation is the fact that both participants were practicing teachers

who teach mathematics, which makes the subject matter of my research personal for

them. There is a conflict of interest because while certain changes may positively impact

students, they may negatively impact them. For example, it is in Liz’s best interest to say

that secondary level mathematics is enough to teach elementary school mathematics,

since she did not take any mathematics courses during her undergraduate education. It is

important to recognize that while teacher satisfaction is vital, changes in education should

be made with student’s best interest in mind, not teachers’. A possible next step would be

to interview, observe, and/or test students with teachers of varying mathematics content

knowledge.

Likewise, I recognize that it would be in a mathematics specialist’s best interest to

say that secondary mathematics is not enough to teach elementary school mathematics.

This brings me to another limitation. The fact that I myself have a mathematics

undergraduate background and will soon be a certified Junior/Intermediate teacher makes

this research somewhat biased.

Lastly, during my interview, when I asked about entry or exit tests, both

participants expressed uncertainty about what the test would include. Once and if it is


established that a test is needed, the next step would be to determine its level of difficulty.

How much mathematics content should a teacher know in order to be able to effectively

teach mathematics? Should it be based on the grades you wish to teach or should it be

universal for any teacher who is teaching mathematics? Another possible next step would

be to determine the optimal grade to start rotary?


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APPENDICES

Appendix A: Interview Questions

Firstly, I would like to thank you for taking out some time from your busy schedule to

help me with my research by participating. As mentioned in my e-mail, my study will

examine the following question: What are teachers’ perspectives about increasing the

quality of Junior/Intermediate mathematics teachers? My sub-questions are: What

changes, if any, would improve the current process of becoming a mathematics teacher?

Where should these changes happen along the timeline of becoming a mathematics

teacher?

Background information

1. Why did you choose to pursue this career? How long have you been teaching? What

subjects have you taught?

2. Could you please describe yourself as a K-12 mathematics student? (enjoyment,

parents, teachers)

3. Could you please describe any post-secondary (undergraduate) training you had in

math?

a) In what ways, if any, did it inform your current professional practice?

4. Could you describe the process to get into teachers’ college?

a) What were the requirements? (GPA, tests, experience, involvement,

etc.)

b) Could you describe your mathematics experiences while in teachers

college? (difficulty, assignments, focus (pedagogy vs. content

knowledge), how materials were presented)

5. Did you complete any Additional Qualifications and/or Professional Development

courses in mathematics after teachers college?

6. Did you have to write any tests/exams after teachers college?

a) What do you know about and how did you feel about the Ontario

Teacher Testing Program which was established, and administered

between 2000 and 2005?

7. What was the hiring process for your past and current teaching position(s)?

Understanding of the topic

8. What is the most important factor for student success in mathematics (teacher,

curriculum, family, society/community)?

9. What makes a good mathematics teacher?

a) How important is content knowledge vs. pedagogy for good

mathematics teachers?

Strengths and Weaknesses

10. What are some strengths and/or weaknesses of each of the following:


1

q

3

q

Teacher education Program

Hiring point 2

a. Pre-teachers’ college: requirements to get into teachers college

(undergrad, entry test)

b. Teachers college (focus: pedagogy vs. content knowledge)

c. Post-teachers’ college: requirements to get certified by the Ontario

College of Teachers and hired by school boards (Additional

Qualifications, exit tests)

Next Steps

11. What changes, if any, do you think should be made to the current process of

becoming a mathematics teacher? (GPA, undergraduate courses, experience, tests?)

a) Do you know that French teachers are required to pass a proficiency

French test prior to entering teachers college? How would you feel

about requiring mathematics teachers to pass a mathematics test? How

about other subjects?

12. Where should these changes happen along the timeline (before, during, and/or after

teachers college)?


Appendix B: Letter of Consent for Interview

Date: ___________________

Dear ___________________,

I am a graduate student at OISE, University of Toronto, and am currently enrolled as a

Master of Teaching candidate. I am exploring the question: How can Ontario policy

makers improve mathematics education in Ontario through the betterment of mathematics

teachers’ quality by learning from other highly respected professions? for the purposes of

a investigating an educational topic as a major assignment for our program. I think that

your knowledge and experience will provide insights into this topic.

I am writing a report on this study as a requirement of the Master of Teaching Program.

My course instructor who is providing support for the process this year is

Dr.__________________. My research supervisor is _____________________. The

purpose of this requirement is to allow us to become familiar with a variety of ways to do

research. My data collection consists of a 40 minute interview that will be tape-recorded.

I would be grateful if you would allow me to interview you at a place and time

convenient to you. I can conduct the interview at your office or workplace, in a public

place, or anywhere else that you might prefer.

The contents of this interview will be used for my assignment, which will include a final

paper, as well as informal presentations to my classmates and/or potentially at a

conference or publication. I will not use your name or anything else that might identify

you in my written work, oral presentations, or publications. This information remains

confidential. The only people who will have access to my assignment work will be my

research supervisor and my course instructor. You are free to change your mind at any

time, and to withdraw even after you have consented to participate. You may decline to

answer any specific questions. I will destroy the tape recording after the paper has been

presented and/or published which may take up to five years after the data has been

collected. There are no known risks or benefits to you for assisting in the project, and I

will share with you a copy of my notes to ensure accuracy.

Please sign the attached form, if you agree to be interviewed. The second copy is for your

records. Thank you very much for your help.

Yours sincerely,

Researcher name: _________________________________

Phone number, email: ______________________________


Instructor’s Name: ____________________________________________

Phone number: _________________ Email: _______________________

Research Supervisor’s Name: ___________________________________

Phone #: ______________________ Email: _______________________

Consent Form

I acknowledge that the topic of this interview has been explained to me and that any

questions that I have asked have been answered to my satisfaction. I understand that I can

withdraw at any time without penalty.

I have read the letter provided to me by Bektjona Zaimi and agree to participate in an

interview for the purposes described.

Signature: ________________________________________

Name (printed): ___________________________________

Date: ______________________

elementary mathematics teachers matter: the …...however, the ways of arriving to this conclusion...

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