Contributed by:

This study critically examined the math content knowledge (MCK) of teacher candidates (TCs) enrolled in a two-year Master of Teaching (MT) degree. Teachers require a solid math knowledge base in order to support students’ achievement.

1.
International Electronic Journal of Elementary Education, 9(4), 851-872, June 2017.

Learning to be a Math Teacher:

What Knowledge is Essential?

Mary REID a Steven REID a

a University of Toronto, Canada

Received: 23 January 2017 / Revised: 17 March 2017 / Accepted: 15 May 2017

This study critically examined the math content knowledge (MCK) of teacher candidates (TCs)

enrolled in a two-year Master of Teaching (MT) degree. Teachers require a solid math knowledge

base in order to support students’ achievement. Provincial and international math assessments

have been of major concern in Ontario, Canada, due to declining scores. Research aimed to

investigate the development of TCs’ math capacities for effective teaching is important to teachers,

school districts, universities, professional learning associations, and policy makers. The

researchers of this study analyzed the basic numeracy skills of 151 TCs through pre- and post-tests.

In addition, eight TCs took part in semi-structured interviews and shared their experiences in the

MT math program. Test results indicated improvements in many areas, however, not all numeracy

skills improved significantly. Interviews revealed TCs’ perceptions of the math test, courses, and

instructors, as well as the importance of teaching math during their practicum placements. The

researchers made recommendations to teacher education programs in areas such as: establishing

minimum math competency standards, enhancing coherence between MT math courses and

practicum placements, and providing additional support for TCs with low math proficiency.

Keywords: Elementary math teacher education, Teacher candidates, Pre-service teacher education,

Math content knowledge, MCK, Teacher learning

Large-scale math assessment results have become a matter of great concern in Ontario,

Canada, due to declining scores over the years. In Ontario, provincial assessment scores

have revealed a steady decrease in grades 3 and 6 students’ math achievement over the

previous eight years. During the period between 2009 and 2016, the percentage of grade 6

students who achieved at or above the provincial standard in math declined by 13

percentage points (63% to 50%); while the percentage of grade 3 students at or above the

provincial level dropped by seven points (70% to 63%) (Education, Quality and

Accountability Office [EQAO], 2016a). Additionally, the latest math results from the 2015

Program for International Student Assessment (PISA) show Ontario scored statistically

lower than Canada as a jurisdiction, as well as the province of Quebec (EQAO, 2016b).

PISA assesses the achievement of 15-year-old students; in Ontario, this involves students

Corresponding author: Mary REID, Assistant Professor, Toronto University, 252 Bloor Street

West, Toronto, ON, Canada, M5S 1V6. Phone: (416) 978 0045, E-mail: mary.reid@utoronto.ca

Copyright © IEJEE

Learning to be a Math Teacher:

What Knowledge is Essential?

Mary REID a Steven REID a

a University of Toronto, Canada

Received: 23 January 2017 / Revised: 17 March 2017 / Accepted: 15 May 2017

This study critically examined the math content knowledge (MCK) of teacher candidates (TCs)

enrolled in a two-year Master of Teaching (MT) degree. Teachers require a solid math knowledge

base in order to support students’ achievement. Provincial and international math assessments

have been of major concern in Ontario, Canada, due to declining scores. Research aimed to

investigate the development of TCs’ math capacities for effective teaching is important to teachers,

school districts, universities, professional learning associations, and policy makers. The

researchers of this study analyzed the basic numeracy skills of 151 TCs through pre- and post-tests.

In addition, eight TCs took part in semi-structured interviews and shared their experiences in the

MT math program. Test results indicated improvements in many areas, however, not all numeracy

skills improved significantly. Interviews revealed TCs’ perceptions of the math test, courses, and

instructors, as well as the importance of teaching math during their practicum placements. The

researchers made recommendations to teacher education programs in areas such as: establishing

minimum math competency standards, enhancing coherence between MT math courses and

practicum placements, and providing additional support for TCs with low math proficiency.

Keywords: Elementary math teacher education, Teacher candidates, Pre-service teacher education,

Math content knowledge, MCK, Teacher learning

Large-scale math assessment results have become a matter of great concern in Ontario,

Canada, due to declining scores over the years. In Ontario, provincial assessment scores

have revealed a steady decrease in grades 3 and 6 students’ math achievement over the

previous eight years. During the period between 2009 and 2016, the percentage of grade 6

students who achieved at or above the provincial standard in math declined by 13

percentage points (63% to 50%); while the percentage of grade 3 students at or above the

provincial level dropped by seven points (70% to 63%) (Education, Quality and

Accountability Office [EQAO], 2016a). Additionally, the latest math results from the 2015

Program for International Student Assessment (PISA) show Ontario scored statistically

lower than Canada as a jurisdiction, as well as the province of Quebec (EQAO, 2016b).

PISA assesses the achievement of 15-year-old students; in Ontario, this involves students

Corresponding author: Mary REID, Assistant Professor, Toronto University, 252 Bloor Street

West, Toronto, ON, Canada, M5S 1V6. Phone: (416) 978 0045, E-mail: mary.reid@utoronto.ca

Copyright © IEJEE

2.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

in grade 10. When comparing Canada versus the top scoring Singapore, the math results

are telling. In Canada, 85.6% students were able to at least employ basic algorithms (level

2 questions), and in Singapore, 92.3% could do the same. As questions became more

complex on the PISA assessment, the gap between Canada and Singapore increased. For

example, when students were asked questions that required reasoning skills and the

ability to communicate their reasoning (level 5 questions), only 15.1% of Canadian

students could achieve this level, however, 34.8% of Singapore students could achieve

success. These recent findings reveal a current situation of major concern in math

performance across Canada and the province of Ontario.

Ontario Ministry of Education documents and provincial targets include the concepts of

raising student achievement for all students and closing student achievement gaps. Using

district and provincial data, ambitious targets have been proposed for improved numeracy

outcomes for elementary students. As part of setting the stage for a renewed vision for

Ontario’s drive to achieve excellence in education, the ministry acknowledges the

challenges observed in the area of math achievement … “Like many other jurisdictions

across Canada and around the world, Ontario has also seen a decline in student

performance in mathematics.” (Ontario Ministry of Education, 2014, p. 2). The ambitious

provincial goal of 75% of students achieving levels three or four on the EQAO is stated

clearly in its vision, level three being the provincial standard and level four being the

highest level of attainment. In 2016-17, the ministry commenced a renewed math strategy

to provide differentiated levels of supports to schools based on student achievement data.

For example, this involved professional learning opportunities for educators, as well as

math lead teachers in elementary schools. The ministry has also implemented a

monitoring approach to identify impact across the province and will share results in the

years to come.

The Ontario education system starts with two years of full day kindergarten that immerses

students aged four and five in an inquiry and play-based program. Kindergarten

educators are encouraged to create learning environments that entice students to build

upon their natural curiosities in math through the presentation of intentional

provocations. Students then continue with their elementary education through grades 1 –

8, approximately ages 6 – 13. When students enter secondary school in grade 9, they must

choose between applied and academic math curriculums. Historically, there has been a

tremendous gap in achievement on the grade 9 EQAO math assessment, between students

in applied and academic courses. Most recently in 2016, only 45% of students in applied

math courses achieved the provincial standard, whereas 83% of those in academic met the

same standard. The grade 9 EQAO math assessment is considered low stakes, that is,

passing the assessment is not a requirement to graduate secondary school. The only

requirement for secondary students in the area of math involves passing three math

courses, at least one of the courses must be at the grade 11 or 12 level; grade 12 being the

final year of secondary school. Nevertheless, the EQAO scores are vital pieces of data, as

the choice between applied and academic math courses can affect future educational and

career opportunities (Parekh, 2014).

Initial teacher training in Ontario recently transformed from a two-semester program to a

four-semester program in September 2015; a semester involving four months of study.

This study took place in a graduate teacher education program in which TCs earn a Master

of Teaching degree as well as certification to teach in Ontario. Because the Master of

Teaching program is a graduate degree, it involves five semesters (20 months in total),

which goes beyond the minimum undergraduate teacher training program. In Ontario,

teacher candidates must participate in a minimum of 80 days of practicum teaching;

observing a qualified teacher in the classroom and teaching in the classroom under the

852

in grade 10. When comparing Canada versus the top scoring Singapore, the math results

are telling. In Canada, 85.6% students were able to at least employ basic algorithms (level

2 questions), and in Singapore, 92.3% could do the same. As questions became more

complex on the PISA assessment, the gap between Canada and Singapore increased. For

example, when students were asked questions that required reasoning skills and the

ability to communicate their reasoning (level 5 questions), only 15.1% of Canadian

students could achieve this level, however, 34.8% of Singapore students could achieve

success. These recent findings reveal a current situation of major concern in math

performance across Canada and the province of Ontario.

Ontario Ministry of Education documents and provincial targets include the concepts of

raising student achievement for all students and closing student achievement gaps. Using

district and provincial data, ambitious targets have been proposed for improved numeracy

outcomes for elementary students. As part of setting the stage for a renewed vision for

Ontario’s drive to achieve excellence in education, the ministry acknowledges the

challenges observed in the area of math achievement … “Like many other jurisdictions

across Canada and around the world, Ontario has also seen a decline in student

performance in mathematics.” (Ontario Ministry of Education, 2014, p. 2). The ambitious

provincial goal of 75% of students achieving levels three or four on the EQAO is stated

clearly in its vision, level three being the provincial standard and level four being the

highest level of attainment. In 2016-17, the ministry commenced a renewed math strategy

to provide differentiated levels of supports to schools based on student achievement data.

For example, this involved professional learning opportunities for educators, as well as

math lead teachers in elementary schools. The ministry has also implemented a

monitoring approach to identify impact across the province and will share results in the

years to come.

The Ontario education system starts with two years of full day kindergarten that immerses

students aged four and five in an inquiry and play-based program. Kindergarten

educators are encouraged to create learning environments that entice students to build

upon their natural curiosities in math through the presentation of intentional

provocations. Students then continue with their elementary education through grades 1 –

8, approximately ages 6 – 13. When students enter secondary school in grade 9, they must

choose between applied and academic math curriculums. Historically, there has been a

tremendous gap in achievement on the grade 9 EQAO math assessment, between students

in applied and academic courses. Most recently in 2016, only 45% of students in applied

math courses achieved the provincial standard, whereas 83% of those in academic met the

same standard. The grade 9 EQAO math assessment is considered low stakes, that is,

passing the assessment is not a requirement to graduate secondary school. The only

requirement for secondary students in the area of math involves passing three math

courses, at least one of the courses must be at the grade 11 or 12 level; grade 12 being the

final year of secondary school. Nevertheless, the EQAO scores are vital pieces of data, as

the choice between applied and academic math courses can affect future educational and

career opportunities (Parekh, 2014).

Initial teacher training in Ontario recently transformed from a two-semester program to a

four-semester program in September 2015; a semester involving four months of study.

This study took place in a graduate teacher education program in which TCs earn a Master

of Teaching degree as well as certification to teach in Ontario. Because the Master of

Teaching program is a graduate degree, it involves five semesters (20 months in total),

which goes beyond the minimum undergraduate teacher training program. In Ontario,

teacher candidates must participate in a minimum of 80 days of practicum teaching;

observing a qualified teacher in the classroom and teaching in the classroom under the

852

3.
Learning to be a Math Teacher / Reid & Reid

supervision of a qualified teacher. Teacher education program providers are accredited

by the Ontario College of Teachers to offer courses that lead to Ontario teaching

certification. TCs earn qualification in one of three divisions: primary/junior (P/J),

kindergarten – grade 6; junior/intermediate (J/I), grades 4 – 10; and intermediate/senior

(I/S), grades 7 – 12. Further, to be qualified to teach J/I, a teacher must have one

teachable such as English, math, science, or social science/humanities. This requires the

TC to have taken four or more full undergraduate or graduate courses in the area of the

teachable. For the I/S certification, the teacher must have two teachable subjects; the first

teachable must be supported by five or more full undergraduate or graduate courses and

the second teachable must be supported by three or more undergraduate or graduate

courses. As this current study focused on P/J and J/I, most of the TCs did not have a

background in math; instead, most had an undergraduate focus in the social sciences.

The basic math competencies of teacher candidates have received much attention in the

last few decades. Math content knowledge (MCK) refers to the basic math knowledge

possessed by individuals considered to be mathematically literate. Researchers

emphasize that content knowledge in math is an important construct that can either

support or hinder progress toward exemplary classroom instruction (Philipp et al., 2007;

Thames & Ball, 2010). Ball, Thames, and Phelps (2008) suggest that the absence of

improved math instruction is resultant from teachers’ lack of content knowledge within

this subject area. “Teachers who do not themselves know a subject well are not likely to

have the knowledge they need to help students learn this content” (p. 404). Unfortunately,

a report conducted by Amgen Canada Incorporated and Let’s Talk Science (2013)

indicated that more than 50% of Canadian high school students drop science and math as

soon as they can, thereby only taking the minimal compulsory courses to grade 10 or 11.

This underscores the need for teacher education math programs to foster TCs’ math

competencies so they are better equipped to meet their students’ needs through effective

math pedagogy. The researchers of this current study readily note that more than MCK is

required to successfully teach math in the elementary grades, hence, this research also

examined the perceptions of TCs’ self-efficacy in teaching math in the classroom.

Importantly, self-efficacy has been identified as a predictive factor in student achievement

in math studies (Tella, 2008)

The National Council of Teachers of Mathematics (NCTM) established standards to engage

math teachers in reform, that is, learning math for deep understanding through

meaningful problem solving contexts (1989, 2000). More recently, NCTM published

Principles to Action (2014) outlining the guiding principles for school math. Ontario’s

2005 math curriculum is aligned with the NCTM’s standards (McDougall, Ross, & Jaafar,

2006). The math curriculum highlights the importance of a balanced pedagogy in which

operational skills are not in any way ignored, and still an important component of the

curriculum. Further, it is acknowledged that operational and higher-order thinking skills

are attained differently by students as they develop conceptualizations of math around

them. NCTM’s standards are often referred to as reform math in comparison to traditional

math instruction which relies heavily on rote learning and memorization. A major tenet of

reform math is the notion of constructivism, in which knowledge is actively created by the

learner. In a constructivist class, teachers guide and support their students’ math ideas

rather than transmit procedural knowledge. This approach to math instruction is complex

and it requires teachers to have a deep knowledge of the subject area in order to pose

appropriate tasks, explain models, and ask effective questions. However, a number of

researchers have pointed out that teachers teach much in the same way they were taught

(Ball, Lubienski, & Mewborn, 2001; Kagan, 1992; Tabachnik & Zeichner, 1984). Tabachnik

and Zeichner (1984) assert that constructivist modes of teaching tend to conflict with TCs’

previous ideas about good teaching, and that they are inclined to maintain old

853

supervision of a qualified teacher. Teacher education program providers are accredited

by the Ontario College of Teachers to offer courses that lead to Ontario teaching

certification. TCs earn qualification in one of three divisions: primary/junior (P/J),

kindergarten – grade 6; junior/intermediate (J/I), grades 4 – 10; and intermediate/senior

(I/S), grades 7 – 12. Further, to be qualified to teach J/I, a teacher must have one

teachable such as English, math, science, or social science/humanities. This requires the

TC to have taken four or more full undergraduate or graduate courses in the area of the

teachable. For the I/S certification, the teacher must have two teachable subjects; the first

teachable must be supported by five or more full undergraduate or graduate courses and

the second teachable must be supported by three or more undergraduate or graduate

courses. As this current study focused on P/J and J/I, most of the TCs did not have a

background in math; instead, most had an undergraduate focus in the social sciences.

The basic math competencies of teacher candidates have received much attention in the

last few decades. Math content knowledge (MCK) refers to the basic math knowledge

possessed by individuals considered to be mathematically literate. Researchers

emphasize that content knowledge in math is an important construct that can either

support or hinder progress toward exemplary classroom instruction (Philipp et al., 2007;

Thames & Ball, 2010). Ball, Thames, and Phelps (2008) suggest that the absence of

improved math instruction is resultant from teachers’ lack of content knowledge within

this subject area. “Teachers who do not themselves know a subject well are not likely to

have the knowledge they need to help students learn this content” (p. 404). Unfortunately,

a report conducted by Amgen Canada Incorporated and Let’s Talk Science (2013)

indicated that more than 50% of Canadian high school students drop science and math as

soon as they can, thereby only taking the minimal compulsory courses to grade 10 or 11.

This underscores the need for teacher education math programs to foster TCs’ math

competencies so they are better equipped to meet their students’ needs through effective

math pedagogy. The researchers of this current study readily note that more than MCK is

required to successfully teach math in the elementary grades, hence, this research also

examined the perceptions of TCs’ self-efficacy in teaching math in the classroom.

Importantly, self-efficacy has been identified as a predictive factor in student achievement

in math studies (Tella, 2008)

The National Council of Teachers of Mathematics (NCTM) established standards to engage

math teachers in reform, that is, learning math for deep understanding through

meaningful problem solving contexts (1989, 2000). More recently, NCTM published

Principles to Action (2014) outlining the guiding principles for school math. Ontario’s

2005 math curriculum is aligned with the NCTM’s standards (McDougall, Ross, & Jaafar,

2006). The math curriculum highlights the importance of a balanced pedagogy in which

operational skills are not in any way ignored, and still an important component of the

curriculum. Further, it is acknowledged that operational and higher-order thinking skills

are attained differently by students as they develop conceptualizations of math around

them. NCTM’s standards are often referred to as reform math in comparison to traditional

math instruction which relies heavily on rote learning and memorization. A major tenet of

reform math is the notion of constructivism, in which knowledge is actively created by the

learner. In a constructivist class, teachers guide and support their students’ math ideas

rather than transmit procedural knowledge. This approach to math instruction is complex

and it requires teachers to have a deep knowledge of the subject area in order to pose

appropriate tasks, explain models, and ask effective questions. However, a number of

researchers have pointed out that teachers teach much in the same way they were taught

(Ball, Lubienski, & Mewborn, 2001; Kagan, 1992; Tabachnik & Zeichner, 1984). Tabachnik

and Zeichner (1984) assert that constructivist modes of teaching tend to conflict with TCs’

previous ideas about good teaching, and that they are inclined to maintain old

853

4.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

conceptions. Specifically, in the discipline of math there is research that suggests TCs

enter teacher education programs with predetermined ideas on how to teach math based

on the way they were instructed (Ball, Sleep, Boerst, & Bass, 2009; Hill & Ball, 2004, 2009).

Overall findings from these studies reveal how prospective teachers are more likely to

replicate teaching approaches that were modeled to them as students of math. Moreover,

Hill and Ball (2004) state that these approaches derive from years of personal experiences

of traditional math teaching, where the teacher is holder of all knowledge, with an

emphasis on memorization of facts and procedures. It appears that math education is

caught in a vicious cycle. Prospective teachers tend to hold oversimplified beliefs about

classroom practice and pre-existing ideas of how to teach math based on their experiences

in traditional math classrooms (Ball, 1996; Ball, Lubienski, & Mewborn, 2001). In order to

break this cycle, teacher education math courses must give meaning to the content and

pedagogy TCs need to know (Thames & Ball, 2010). This requires considerable math

content knowledge and a wide range of pedagogical skills to implement math programs

that promote authentic problem solving, reasoning, and communication.

In this article, researchers examined a two-year Master of Teaching (MT) degree program

that promotes varied instructional strategies beyond a transmissive approach to teaching

math. This study focused on the following questions: What were the basic numeracy skills

of TCs upon entering the MT program? What changes in basic numeracy skills of TCs

occurred after a year of the MT program? And what changes occurred in TCs’ math beliefs

and confidence as learners and teachers after completing two MT math courses and three

practice teaching placements? Through pre- and post-tests of basic math skills, as well as

semi-structured interviews, the researchers analyzed the growth of TCs in their MCK and

their conceptions and confidence in teaching math in the classroom. This research

identified key findings to support the improvement of teacher education programs in

meeting the needs of TCs’ math development and teaching capacities.

Literature Review

A number of research studies have raised serious concerns about the depth of math

content knowledge (MCK) in teacher candidates (TCs) enrolled in elementary teacher

education programs (Ball, 1990a; Grover & Connor, 2000; Hill & Ball, 2004, 2009; Ma,

1999; Philipp et al., 2007; Thames & Ball, 2010). In general, the literature pertaining to

MCK of TCs overwhelmingly supports the need for conceptual understanding of the

subject matter, and specialized math knowledge for teaching in order to implement

effective teaching strategies.

Math Content Knowledge (MCK) of Teachers Candidates (TCs)

The math content knowledge (MCK) of teacher candidates (TCs) continues to be identified

as an important component of effective math teaching in the classroom. Although the

possession of strong content knowledge in of itself is not enough to ensure a person will

be an effective math teacher, it is difficult to help students to acquire deep math

understandings when the teacher has inadequate content knowledge (Ponte & Chapman,

2008). Philipp et al. (2007) and Thames and Ball (2010) strongly suggest it is necessary

for teachers to possess conceptual math knowledge in order to effectively explain

algorithms and concepts, as well as describe connections between concepts. Number

sense and numeration in the Ontario curriculum is an essential foundation of MCK to

successfully build sophisticated math conceptualizations (Biddlecomb & Carr, 2011).

Unfortunately, teachers of math can be deficient in understanding numeracy concepts,

specifically in understanding how to develop numeracy skills in their students beyond rote

memorization (Yackel, Underwood, & Elias 2007).

854

conceptions. Specifically, in the discipline of math there is research that suggests TCs

enter teacher education programs with predetermined ideas on how to teach math based

on the way they were instructed (Ball, Sleep, Boerst, & Bass, 2009; Hill & Ball, 2004, 2009).

Overall findings from these studies reveal how prospective teachers are more likely to

replicate teaching approaches that were modeled to them as students of math. Moreover,

Hill and Ball (2004) state that these approaches derive from years of personal experiences

of traditional math teaching, where the teacher is holder of all knowledge, with an

emphasis on memorization of facts and procedures. It appears that math education is

caught in a vicious cycle. Prospective teachers tend to hold oversimplified beliefs about

classroom practice and pre-existing ideas of how to teach math based on their experiences

in traditional math classrooms (Ball, 1996; Ball, Lubienski, & Mewborn, 2001). In order to

break this cycle, teacher education math courses must give meaning to the content and

pedagogy TCs need to know (Thames & Ball, 2010). This requires considerable math

content knowledge and a wide range of pedagogical skills to implement math programs

that promote authentic problem solving, reasoning, and communication.

In this article, researchers examined a two-year Master of Teaching (MT) degree program

that promotes varied instructional strategies beyond a transmissive approach to teaching

math. This study focused on the following questions: What were the basic numeracy skills

of TCs upon entering the MT program? What changes in basic numeracy skills of TCs

occurred after a year of the MT program? And what changes occurred in TCs’ math beliefs

and confidence as learners and teachers after completing two MT math courses and three

practice teaching placements? Through pre- and post-tests of basic math skills, as well as

semi-structured interviews, the researchers analyzed the growth of TCs in their MCK and

their conceptions and confidence in teaching math in the classroom. This research

identified key findings to support the improvement of teacher education programs in

meeting the needs of TCs’ math development and teaching capacities.

Literature Review

A number of research studies have raised serious concerns about the depth of math

content knowledge (MCK) in teacher candidates (TCs) enrolled in elementary teacher

education programs (Ball, 1990a; Grover & Connor, 2000; Hill & Ball, 2004, 2009; Ma,

1999; Philipp et al., 2007; Thames & Ball, 2010). In general, the literature pertaining to

MCK of TCs overwhelmingly supports the need for conceptual understanding of the

subject matter, and specialized math knowledge for teaching in order to implement

effective teaching strategies.

Math Content Knowledge (MCK) of Teachers Candidates (TCs)

The math content knowledge (MCK) of teacher candidates (TCs) continues to be identified

as an important component of effective math teaching in the classroom. Although the

possession of strong content knowledge in of itself is not enough to ensure a person will

be an effective math teacher, it is difficult to help students to acquire deep math

understandings when the teacher has inadequate content knowledge (Ponte & Chapman,

2008). Philipp et al. (2007) and Thames and Ball (2010) strongly suggest it is necessary

for teachers to possess conceptual math knowledge in order to effectively explain

algorithms and concepts, as well as describe connections between concepts. Number

sense and numeration in the Ontario curriculum is an essential foundation of MCK to

successfully build sophisticated math conceptualizations (Biddlecomb & Carr, 2011).

Unfortunately, teachers of math can be deficient in understanding numeracy concepts,

specifically in understanding how to develop numeracy skills in their students beyond rote

memorization (Yackel, Underwood, & Elias 2007).

854

5.
Learning to be a Math Teacher / Reid & Reid

In her studies, Ball (1990a, 1990b) examined math conceptual content knowledge through

responses to questionnaires and interviews by 252 prospective teachers. Findings

revealed that the subject knowledge held by prospective teachers remains inadequate for

teaching math successfully. These findings are congruent with those of other studies. For

example, Tirosh (2000) demonstrated how elementary TCs were overly dependent on

computational algorithms for multiplication and division structures, resulting in

procedural dependency and limited conceptual guidance. Another study by Bartell, Webel,

Bowen, and Dyson (2013) concluded that MCK is necessary but insufficient in supporting

the assessment of children’s conceptual understanding of math. More specialized

understanding of math is required to understand the complexities behind children’s

mathematical thinking.

Procedural and Conceptual Knowledge of Math Content

Both Hiebert (1992) and McCormick (1997) describe procedural knowledge as applying a

sequence of actions to find answers. These actions, also known as algorithms, follow a set

of rules that students repeatedly practise to reinforce the algorithm. The National Council

of Teachers of Mathematics (NCTM) (2000) defines conceptual knowledge as a rich

understanding of the relationships among math concepts. This involves solving problems

through reasoning, communicating, and justifying. Merely memorizing computational

procedures without understanding them will not develop the capacity to reason about the

type of calculations needed. Thus, procedural skills that are not accompanied by some

form of conceptual understanding are weak and easily forgotten (Hiebert et al., 2003). A

major aspect of the NCTM’s (2000) standards calls for a balance between conceptual and

procedural knowledge of math. Unfortunately, without this balance in place, students

often do not know when to implement procedures and the learning is often “fragile” (p.

Research reports that teachers with weak math competencies cannot be flexible with their

math instruction and this may result with an emphasis on procedural knowledge, in which

teachers deliver curriculum in a repetitive, undemanding, and non-interactive fashion

(Frykholm, 1999; Hiebert et al., 2003; Stigler & Hiebert 1997; Thames & Ball, 2010). Little

attention is given to the development of conceptual ideas or making connections between

procedures and mathematical concepts. Furthermore, the prominence of procedural

knowledge in schools is most likely the type of math preparation experienced by teacher

candidates (TCs) (Hill & Ball, 2004, 2009; Kajander, 2010; Thames & Ball, 2010).

Consequently, the challenge for teacher education programs is to unpack the math

knowledge of prospective teachers in order to develop deeper conceptual understanding

(Adler & Davis, 2006).

The intersection of procedural and conceptual knowledge is of utmost importance when

examining the content knowledge of math teachers (Ambrose, 2004; Hiebert, 1999; Hill &

Ball, 2004, 2009; Rittle-Johnson & Kroedinger, 2002). It is imperative that math teacher

education programs emphasize procedural skills and conceptual understanding as

interconnected, so students have the capacity to understand why and how algorithms

work and thereby grasp the underlying mathematical concepts (Ambrose, 2004; Kajander,

2010; Reid, 2013). Research has demonstrated that if students repeatedly practice

algorithms before understanding them, they often struggle with making sense of why and

how the formula works (Hiebert et al., 2005). Conversely, when conceptual understanding

is the sole focus of instruction, then learners are likely to struggle with procedural

competency (Kajander, 2010). Without any emphasis on computational algorithms,

procedural knowledge can be negatively impacted (Alsup & Sprigler, 2003). Instructional

practice that over emphasizes only one component, either conceptual or procedural, will

result in limited math understanding. Some researchers argue that teaching algorithms

855

In her studies, Ball (1990a, 1990b) examined math conceptual content knowledge through

responses to questionnaires and interviews by 252 prospective teachers. Findings

revealed that the subject knowledge held by prospective teachers remains inadequate for

teaching math successfully. These findings are congruent with those of other studies. For

example, Tirosh (2000) demonstrated how elementary TCs were overly dependent on

computational algorithms for multiplication and division structures, resulting in

procedural dependency and limited conceptual guidance. Another study by Bartell, Webel,

Bowen, and Dyson (2013) concluded that MCK is necessary but insufficient in supporting

the assessment of children’s conceptual understanding of math. More specialized

understanding of math is required to understand the complexities behind children’s

mathematical thinking.

Procedural and Conceptual Knowledge of Math Content

Both Hiebert (1992) and McCormick (1997) describe procedural knowledge as applying a

sequence of actions to find answers. These actions, also known as algorithms, follow a set

of rules that students repeatedly practise to reinforce the algorithm. The National Council

of Teachers of Mathematics (NCTM) (2000) defines conceptual knowledge as a rich

understanding of the relationships among math concepts. This involves solving problems

through reasoning, communicating, and justifying. Merely memorizing computational

procedures without understanding them will not develop the capacity to reason about the

type of calculations needed. Thus, procedural skills that are not accompanied by some

form of conceptual understanding are weak and easily forgotten (Hiebert et al., 2003). A

major aspect of the NCTM’s (2000) standards calls for a balance between conceptual and

procedural knowledge of math. Unfortunately, without this balance in place, students

often do not know when to implement procedures and the learning is often “fragile” (p.

Research reports that teachers with weak math competencies cannot be flexible with their

math instruction and this may result with an emphasis on procedural knowledge, in which

teachers deliver curriculum in a repetitive, undemanding, and non-interactive fashion

(Frykholm, 1999; Hiebert et al., 2003; Stigler & Hiebert 1997; Thames & Ball, 2010). Little

attention is given to the development of conceptual ideas or making connections between

procedures and mathematical concepts. Furthermore, the prominence of procedural

knowledge in schools is most likely the type of math preparation experienced by teacher

candidates (TCs) (Hill & Ball, 2004, 2009; Kajander, 2010; Thames & Ball, 2010).

Consequently, the challenge for teacher education programs is to unpack the math

knowledge of prospective teachers in order to develop deeper conceptual understanding

(Adler & Davis, 2006).

The intersection of procedural and conceptual knowledge is of utmost importance when

examining the content knowledge of math teachers (Ambrose, 2004; Hiebert, 1999; Hill &

Ball, 2004, 2009; Rittle-Johnson & Kroedinger, 2002). It is imperative that math teacher

education programs emphasize procedural skills and conceptual understanding as

interconnected, so students have the capacity to understand why and how algorithms

work and thereby grasp the underlying mathematical concepts (Ambrose, 2004; Kajander,

2010; Reid, 2013). Research has demonstrated that if students repeatedly practice

algorithms before understanding them, they often struggle with making sense of why and

how the formula works (Hiebert et al., 2005). Conversely, when conceptual understanding

is the sole focus of instruction, then learners are likely to struggle with procedural

competency (Kajander, 2010). Without any emphasis on computational algorithms,

procedural knowledge can be negatively impacted (Alsup & Sprigler, 2003). Instructional

practice that over emphasizes only one component, either conceptual or procedural, will

result in limited math understanding. Some researchers argue that teaching algorithms

855

6.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

and conceptual understanding should not be viewed as dichotomous extremes. Rather,

procedural skills and problem solving skills are intertwined (Hiebert, 2013; Wu, 1999).

Ultimately, math teachers will require a deep understanding of both conceptual and

procedural knowledge in order to support a balanced approach in the classroom.

Math Content Knowledge (MCK) and Pedagogical Instruction

Grover and Connor (2000) argue for content knowledge as a key characteristic of effective

pedagogical instruction, and this should be a central focus in teacher education math

courses. During their study of teacher education courses, they found that a critical aspect

of reaching course objectives is to recognize the important interaction between teaching

and subject content knowledge. The authors discussed the need for teacher candidates

(TCs) to not only understand math, but to understand the concepts in ways that will

support effective instruction and assessment of the discipline. These claims suggest that

math content knowledge (MCK) is directly connected to pedagogical styles of teaching.

Correspondingly, other researchers such as Hill and Ball (2004), Ma (1999), Shulman

(1987), and Thames and Ball (2010), also advocate for deep subject matter knowledge and

its subsequent positive influence on instructional techniques. Philipp et al. (2007)

similarly confirm that teachers who achieve greater math knowledge are more capable of

the conceptual teaching than their counterparts, who implement procedural based

instruction. All studies propose that teachers’ decisions rely on their understanding of

math subject matter. Hence, research concludes that the deeper MCK that a teacher holds,

the better equipped they are to communicate with students about mathematical concepts,

models, and representations.

In her study of elementary TCs, Kajander (2010) observed how prospective teachers’

conceptual knowledge and beliefs about reform math teaching changed over the

progression of a math methods course. Similar to Ma (1999), and Ball (1990a), Kajander’s

(2010) findings illustrated TCs’ inadequate understandings of math for teaching. Most TCs

entered their teacher preparation programs with limited conceptual proficiency in how to

represent mathematical concepts, explain their thinking, and justify mathematical

procedures. However, after completing a math methods course that focused on

developing conceptual knowledge for teaching, improvements in conceptual knowledge

were examined based on the comparisons of pre- and post-tests. This study endorses Hill

and Ball’s (2004) findings that content knowledge can be positively increased by a single

course experience. Kajander (2007, 2010) posited that due to the increase in conceptual

knowledge of math, TCs shifted their pedagogical beliefs about teaching math, and they

were more focused on problem solving and understanding, and less focused on traditional

learning methods. Hence, these findings suggest how content knowledge and pedagogical

beliefs are linked.

Understanding Math for Teaching

Shulman’s (1986, 1987) notion of pedagogical content knowledge gives attention to the

role of content when teachers make pedagogical decisions. Pedagogical content

knowledge recognizes that teaching requires a unique specialized knowledge of content.

Over the last three decades, considerable research has gone into developing Shulman’s

notion of pedagogical content knowledge through the lens of math teaching. Drawing

upon Shulman’s conceptualizations of pedagogical content knowledge, several researchers

have identified and described a unique understanding of math knowledge required for

teaching (Ball, 1990b; Ball, 1996; Ball, Hill, & Bass, 2005; Ball et al., 2008; Hill, Rowan &

Ball, 2005; Thames & Ball, 2010). Subsequently, Ball et al. (2008) developed a practice-

based theory of math knowledge for teaching (MKT) that include the following domains

empirically generated through factor analysis: 1) common content knowledge (CCK) is the

856

and conceptual understanding should not be viewed as dichotomous extremes. Rather,

procedural skills and problem solving skills are intertwined (Hiebert, 2013; Wu, 1999).

Ultimately, math teachers will require a deep understanding of both conceptual and

procedural knowledge in order to support a balanced approach in the classroom.

Math Content Knowledge (MCK) and Pedagogical Instruction

Grover and Connor (2000) argue for content knowledge as a key characteristic of effective

pedagogical instruction, and this should be a central focus in teacher education math

courses. During their study of teacher education courses, they found that a critical aspect

of reaching course objectives is to recognize the important interaction between teaching

and subject content knowledge. The authors discussed the need for teacher candidates

(TCs) to not only understand math, but to understand the concepts in ways that will

support effective instruction and assessment of the discipline. These claims suggest that

math content knowledge (MCK) is directly connected to pedagogical styles of teaching.

Correspondingly, other researchers such as Hill and Ball (2004), Ma (1999), Shulman

(1987), and Thames and Ball (2010), also advocate for deep subject matter knowledge and

its subsequent positive influence on instructional techniques. Philipp et al. (2007)

similarly confirm that teachers who achieve greater math knowledge are more capable of

the conceptual teaching than their counterparts, who implement procedural based

instruction. All studies propose that teachers’ decisions rely on their understanding of

math subject matter. Hence, research concludes that the deeper MCK that a teacher holds,

the better equipped they are to communicate with students about mathematical concepts,

models, and representations.

In her study of elementary TCs, Kajander (2010) observed how prospective teachers’

conceptual knowledge and beliefs about reform math teaching changed over the

progression of a math methods course. Similar to Ma (1999), and Ball (1990a), Kajander’s

(2010) findings illustrated TCs’ inadequate understandings of math for teaching. Most TCs

entered their teacher preparation programs with limited conceptual proficiency in how to

represent mathematical concepts, explain their thinking, and justify mathematical

procedures. However, after completing a math methods course that focused on

developing conceptual knowledge for teaching, improvements in conceptual knowledge

were examined based on the comparisons of pre- and post-tests. This study endorses Hill

and Ball’s (2004) findings that content knowledge can be positively increased by a single

course experience. Kajander (2007, 2010) posited that due to the increase in conceptual

knowledge of math, TCs shifted their pedagogical beliefs about teaching math, and they

were more focused on problem solving and understanding, and less focused on traditional

learning methods. Hence, these findings suggest how content knowledge and pedagogical

beliefs are linked.

Understanding Math for Teaching

Shulman’s (1986, 1987) notion of pedagogical content knowledge gives attention to the

role of content when teachers make pedagogical decisions. Pedagogical content

knowledge recognizes that teaching requires a unique specialized knowledge of content.

Over the last three decades, considerable research has gone into developing Shulman’s

notion of pedagogical content knowledge through the lens of math teaching. Drawing

upon Shulman’s conceptualizations of pedagogical content knowledge, several researchers

have identified and described a unique understanding of math knowledge required for

teaching (Ball, 1990b; Ball, 1996; Ball, Hill, & Bass, 2005; Ball et al., 2008; Hill, Rowan &

Ball, 2005; Thames & Ball, 2010). Subsequently, Ball et al. (2008) developed a practice-

based theory of math knowledge for teaching (MKT) that include the following domains

empirically generated through factor analysis: 1) common content knowledge (CCK) is the

856

7.
Learning to be a Math Teacher / Reid & Reid

math knowledge used in a wide variety of settings that is not exclusive to teaching; 2)

specialized content knowledge (SCK) involves knowledge that goes beyond a conceptual

understanding of mathematical ideas. It constitutes the knowledge and skills that are

unique to math teaching as it requires teachers to understand math content with a

strategic focus on pedagogy; 3) knowledge of content and students (KCS) comprises of

teachers’ knowledge about students as well as math content. Understanding common

errors and misconceptions made by students, and interpreting students’ mathematical

thinking are all key aspects of KCS; and 4) knowledge of content and teaching (KCT)

involves the combination of pedagogical knowledge and math content. This requires

teachers to understand instructional design, such as how to represent mathematical

concepts, sequence content, select examples, and explain methods and procedures.

An underpinning of MKT is the notion that teachers require a specialized kind of

knowledge to teach math successfully. A teacher’s MKT influences pedagogical decisions

such as when to interject and redirect students, when to pose questions to further

students’ learning, and how to respond to students’ mathematical remarks. Ball et al.

(2008) assert that solid MKT enables teachers to develop and demonstrate mathematical

models based on students’ levels of understanding, and explain why a method works and

whether it is generalizable to other problems. Furthermore, the authors claim that MKT is

uniquely different from being a student of math. Specifically, MKT requires a conceptual

knowledge base to promote discussions about models and connections between concepts

and procedures. These types of interactions are often done immediately on the spot,

during teachable moments in response to students’ needs. Unfortunately, research

suggests that math teachers are limited in their MKT, which poses many challenges for the

implementation of reform math (Lo & Luo, 2012).

Math Content Knowledge (MCK) and Its Impact on Student Achievement

The content knowledge of math teachers and its relationship with student success in math

has been of interest to researchers (Conference Board of Mathematical Sciences [CBMS],

2012). Sowder (2007) stated that in order to increase math knowledge and achievement,

all math classrooms require teachers with in-depth knowledge of math. Rowan, Chiang,

and Miller (1997) identified teachers’ math content knowledge (MCK) as a predictor of

student achievement in grade 10 math. In their quantitative study, they found that

students produced higher levels of achievement if they were taught by teachers who also

scored higher on math test themselves. Furthermore, students who were taught by

teachers who held a math degree also earned higher levels of test scores (Rowan, Chiang,

& Miller, 1997). However, Darling-Hammond and Youngs (2002) reviewed various

research studies of teacher education programs and concluded that although the MCK of

teachers can contribute to student achievement, other aspects of a teacher candidates’

(TCs’) education are equally important such as math methods courses and practice

It is important to note that higher levels of math courses do not automatically equate to

better math teaching. Although it could be speculated that having a major in math should

increase a teachers’ capacity in successfully teaching math, there is no evidence that this is

true for the elementary grades, and for the secondary grades, it is not a consistent

predictor (CBMS, 2012). In order to make math meaningful, Sowder (2007) argued for

TCs to become immersed in learning mathematical concepts and have opportunities in

their courses to make connections between representations and applications, rules and

algorithms. Unfortunately, considerable evidence suggests that many math teachers can

apply the rules and procedures required to do math but lack conceptual knowledge and

reasoning skills to teach for deep understanding (CBMS, 2012; Ma, 1999).

857

math knowledge used in a wide variety of settings that is not exclusive to teaching; 2)

specialized content knowledge (SCK) involves knowledge that goes beyond a conceptual

understanding of mathematical ideas. It constitutes the knowledge and skills that are

unique to math teaching as it requires teachers to understand math content with a

strategic focus on pedagogy; 3) knowledge of content and students (KCS) comprises of

teachers’ knowledge about students as well as math content. Understanding common

errors and misconceptions made by students, and interpreting students’ mathematical

thinking are all key aspects of KCS; and 4) knowledge of content and teaching (KCT)

involves the combination of pedagogical knowledge and math content. This requires

teachers to understand instructional design, such as how to represent mathematical

concepts, sequence content, select examples, and explain methods and procedures.

An underpinning of MKT is the notion that teachers require a specialized kind of

knowledge to teach math successfully. A teacher’s MKT influences pedagogical decisions

such as when to interject and redirect students, when to pose questions to further

students’ learning, and how to respond to students’ mathematical remarks. Ball et al.

(2008) assert that solid MKT enables teachers to develop and demonstrate mathematical

models based on students’ levels of understanding, and explain why a method works and

whether it is generalizable to other problems. Furthermore, the authors claim that MKT is

uniquely different from being a student of math. Specifically, MKT requires a conceptual

knowledge base to promote discussions about models and connections between concepts

and procedures. These types of interactions are often done immediately on the spot,

during teachable moments in response to students’ needs. Unfortunately, research

suggests that math teachers are limited in their MKT, which poses many challenges for the

implementation of reform math (Lo & Luo, 2012).

Math Content Knowledge (MCK) and Its Impact on Student Achievement

The content knowledge of math teachers and its relationship with student success in math

has been of interest to researchers (Conference Board of Mathematical Sciences [CBMS],

2012). Sowder (2007) stated that in order to increase math knowledge and achievement,

all math classrooms require teachers with in-depth knowledge of math. Rowan, Chiang,

and Miller (1997) identified teachers’ math content knowledge (MCK) as a predictor of

student achievement in grade 10 math. In their quantitative study, they found that

students produced higher levels of achievement if they were taught by teachers who also

scored higher on math test themselves. Furthermore, students who were taught by

teachers who held a math degree also earned higher levels of test scores (Rowan, Chiang,

& Miller, 1997). However, Darling-Hammond and Youngs (2002) reviewed various

research studies of teacher education programs and concluded that although the MCK of

teachers can contribute to student achievement, other aspects of a teacher candidates’

(TCs’) education are equally important such as math methods courses and practice

It is important to note that higher levels of math courses do not automatically equate to

better math teaching. Although it could be speculated that having a major in math should

increase a teachers’ capacity in successfully teaching math, there is no evidence that this is

true for the elementary grades, and for the secondary grades, it is not a consistent

predictor (CBMS, 2012). In order to make math meaningful, Sowder (2007) argued for

TCs to become immersed in learning mathematical concepts and have opportunities in

their courses to make connections between representations and applications, rules and

algorithms. Unfortunately, considerable evidence suggests that many math teachers can

apply the rules and procedures required to do math but lack conceptual knowledge and

reasoning skills to teach for deep understanding (CBMS, 2012; Ma, 1999).

857

8.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

Study Design and Method

The theoretical methodology for this research study was underpinned by the theory of

math knowledge for teaching (MKT) and specific components of this theory, mainly, math

content knowledge (MCK) and the relationships between procedural and conceptual

knowledge. This selected theory was deemed to be highly pertinent to the phenomena of

teacher candidates’ (TCs’) math development and drew upon a range of theorists such as

Ball, Thames, and Phelps (2008) and Heibert (2013). MKT requires a specialized

understanding of content that is interwoven with knowledge of students, pedagogical

strategies, and curriculum (Ball, Thames, & Phelps, 2008). In order for teachers to develop

effective MKT, MCK is a required rudimentary foundation. Ball et al. (2005) refers to MCK

as common content knowledge (CCK) and is considered one of the domains of MKT. The

authors describe CCK as the basic math knowledge and proficiency necessary to be

considered a mathematically literate person.

A primary purpose in this current study was to measure and critically examine TCs’ basic

MCK as an underpinning of deeper math concepts, as well as their experiences in

developing as math learners and teachers. The MCK of TCs was measured prior to the

start of their two-year graduate program as participants were not expected to have

developed the other domains of MKT. TCs took classes in different cohorts, either

primary/junior (P/J) or junior/intermediate (J/I). TCs in the P/J division earn

qualifications to teach kindergarten to grade 6, whereas their counterparts in the J/I

division earn qualifications to teach grades 4 to 10. Researchers and teacher education

math instructors discussed what could be reasonably expected from TCs to already know

in math before commencing their teacher education program, that is, to successfully

develop the MKT for kindergarten to grade 8 classrooms. Although J/I students could

teach grades 9-10 math, in such cases they would usually have completed university math

courses in their undergraduate degree, with additional math courses in their teacher

education program. Researchers also reviewed interview transcripts to determine aspects

of the math program that supported or hindered the development of MKT. Based on these

data, the researchers recommended modifications to the teacher education program to

better support TCs’ learning and teaching of math.

This study focused on TCs enrolled in a two-year Master of Teaching (MT) graduate

degree in a large urban southern Ontario university. TCs enrolled in the MT program take

a 36-hour math methods course in year one, and an 18-hour issues in numeracy course in

year two. There were 89 TCs in the primary/junior program (kindergarten to grade 6)

and 62 junior/intermediate TCs (grades 4 to eight) who completed both the pre- and post-

tests, totalling to 151 participants. An additional 30 students did not complete the pre-

and/or post-tests; their results were not utilized in this study.

The methodology used in this research focused on a pragmatic paradigm through a mixed

methods approach. The quantitative data comprised of pre- and post-tests that assessed

participants’ numeracy operation skills: addition, subtraction, multiplication, division,

fractions, percent, decimals, ratio, order of operations, and integers. The questions on the

test essentially measured MCK in the area of numeracy. This assessment was

administered at the beginning of the program prior to the start of classes. A slightly

modified version of the test was administered at the beginning of year two. Achievement

results for each question were analyzed and the data was further examined to identify

significant changes in TCs’ achievement. Mid-way through the second year, eight TCs took

part in semi-structured interviews. To gain a deeper understanding of how TCs

experienced the math test and MT math classes, “purposeful sampling” was used to

intentionally select participants (Creswell & Clark, 2007, p. 112). The purpose for

targeting specific TCs included the researchers’ desire to gather narratives from students

858

Study Design and Method

The theoretical methodology for this research study was underpinned by the theory of

math knowledge for teaching (MKT) and specific components of this theory, mainly, math

content knowledge (MCK) and the relationships between procedural and conceptual

knowledge. This selected theory was deemed to be highly pertinent to the phenomena of

teacher candidates’ (TCs’) math development and drew upon a range of theorists such as

Ball, Thames, and Phelps (2008) and Heibert (2013). MKT requires a specialized

understanding of content that is interwoven with knowledge of students, pedagogical

strategies, and curriculum (Ball, Thames, & Phelps, 2008). In order for teachers to develop

effective MKT, MCK is a required rudimentary foundation. Ball et al. (2005) refers to MCK

as common content knowledge (CCK) and is considered one of the domains of MKT. The

authors describe CCK as the basic math knowledge and proficiency necessary to be

considered a mathematically literate person.

A primary purpose in this current study was to measure and critically examine TCs’ basic

MCK as an underpinning of deeper math concepts, as well as their experiences in

developing as math learners and teachers. The MCK of TCs was measured prior to the

start of their two-year graduate program as participants were not expected to have

developed the other domains of MKT. TCs took classes in different cohorts, either

primary/junior (P/J) or junior/intermediate (J/I). TCs in the P/J division earn

qualifications to teach kindergarten to grade 6, whereas their counterparts in the J/I

division earn qualifications to teach grades 4 to 10. Researchers and teacher education

math instructors discussed what could be reasonably expected from TCs to already know

in math before commencing their teacher education program, that is, to successfully

develop the MKT for kindergarten to grade 8 classrooms. Although J/I students could

teach grades 9-10 math, in such cases they would usually have completed university math

courses in their undergraduate degree, with additional math courses in their teacher

education program. Researchers also reviewed interview transcripts to determine aspects

of the math program that supported or hindered the development of MKT. Based on these

data, the researchers recommended modifications to the teacher education program to

better support TCs’ learning and teaching of math.

This study focused on TCs enrolled in a two-year Master of Teaching (MT) graduate

degree in a large urban southern Ontario university. TCs enrolled in the MT program take

a 36-hour math methods course in year one, and an 18-hour issues in numeracy course in

year two. There were 89 TCs in the primary/junior program (kindergarten to grade 6)

and 62 junior/intermediate TCs (grades 4 to eight) who completed both the pre- and post-

tests, totalling to 151 participants. An additional 30 students did not complete the pre-

and/or post-tests; their results were not utilized in this study.

The methodology used in this research focused on a pragmatic paradigm through a mixed

methods approach. The quantitative data comprised of pre- and post-tests that assessed

participants’ numeracy operation skills: addition, subtraction, multiplication, division,

fractions, percent, decimals, ratio, order of operations, and integers. The questions on the

test essentially measured MCK in the area of numeracy. This assessment was

administered at the beginning of the program prior to the start of classes. A slightly

modified version of the test was administered at the beginning of year two. Achievement

results for each question were analyzed and the data was further examined to identify

significant changes in TCs’ achievement. Mid-way through the second year, eight TCs took

part in semi-structured interviews. To gain a deeper understanding of how TCs

experienced the math test and MT math classes, “purposeful sampling” was used to

intentionally select participants (Creswell & Clark, 2007, p. 112). The purpose for

targeting specific TCs included the researchers’ desire to gather narratives from students

858

9.
Learning to be a Math Teacher / Reid & Reid

who considered themselves competent in math, as well as those who felt that math was a

struggle. Four TCs with low confidence in math and four TCs with high confidence in math

were invited to participate. These qualitative data focused on exploring TCs’ feelings

toward the math test and observations of their year one and two math classes. The

combination of the quantitative and qualitative data allowed for the researchers to

identify common errors made by TCs on the math tests, as well as identify attitudes and

beliefs toward their math learning and teaching in the program.

Instrument Development and Administration

The math pre- and post-tests were collaboratively developed by a committee of math

teacher education instructors. The committee met several times over a six month period

to discuss and create questions. The goal for the pre-test was to determine the entry

points of teacher candidates’ (TCs’) numeracy operation skills. The goal of the post-test

was to determine gains in math content knowledge (MCK) that TCs achieved. Committee

members carefully based test questions on the Number Sense and Numeration strand in

The Ontario Curriculum, Grades 1-8: Mathematics (2005), mostly at grades 5 and 6, with

approximately 10% of the questions at the grade 7 level focused on integers. The tests

comprised of several questions in the following numeracy areas: addition, subtraction,

multiplication, division, fractions, percent, decimals, ratio, order of operations, and

integers. Calculators were not permitted and the format of the test did not include

multiple choice questions. Furthermore, questions that would assess TCs’ pedagogical

skills were not included, i.e., the tests did not ask TCs how they would explain algorithms,

describe and make connections between concepts, or examine misconceptions in students’

work. Rather, this study focused on participants’ MCK, namely numerical operation skills,

which required TCs to demonstrate basic math knowledge.

The assessment was reviewed by several math and non-math teacher education

instructors for feedback on each of the items. The feedback was generally positive and all

the instructors felt that the questions were reasonable for TCs to complete. Many also

noted that this math test would help TCs better understand their own basic knowledge

and the aggregated results would help set a positive direction for the Master of Teaching

(MT) program.

TCs were informed approximately two months ahead of time that they would be

completing the basic math assessment a week prior to classes. It was also clearly

communicated that this pre-test was not at all high stakes, that is, the results were not

factored into their marks and passing the test was not a requirement to complete their MT

degree. Rather, the pre-test was designed as a diagnostic assessment to determine

strengths and areas for improvement in one’s own basic math proficiency. Candidates

were offered a practice test to help them prepare for the math assessment. The post-test

took place after the first year of the MT program, allowing TCs to review their progress

and identify areas of MCK that still required further focus and learning.

TCs completed the pre- and post-tests in their cohort groups of about 25 at a time. There

was one supervisor per cohort who administered the assessments. Supervisors

encouraged participants to show their work and write out as much of their thinking on the

test paper. Participants had up to 90 minutes to complete the questions, and the majority

of them finished within an hour. The pre- and post-tests were assessed and returned to

TCs with feedback on how to improve their basic math knowledge. Furthermore, data

results were aggregated to determine areas of strengths, needs, and next steps for

developing MCK.

The semi-structured individual interviews gave the researchers an in-depth

understanding of TCs’ experiences of their math development. In this study, interviews

859

who considered themselves competent in math, as well as those who felt that math was a

struggle. Four TCs with low confidence in math and four TCs with high confidence in math

were invited to participate. These qualitative data focused on exploring TCs’ feelings

toward the math test and observations of their year one and two math classes. The

combination of the quantitative and qualitative data allowed for the researchers to

identify common errors made by TCs on the math tests, as well as identify attitudes and

beliefs toward their math learning and teaching in the program.

Instrument Development and Administration

The math pre- and post-tests were collaboratively developed by a committee of math

teacher education instructors. The committee met several times over a six month period

to discuss and create questions. The goal for the pre-test was to determine the entry

points of teacher candidates’ (TCs’) numeracy operation skills. The goal of the post-test

was to determine gains in math content knowledge (MCK) that TCs achieved. Committee

members carefully based test questions on the Number Sense and Numeration strand in

The Ontario Curriculum, Grades 1-8: Mathematics (2005), mostly at grades 5 and 6, with

approximately 10% of the questions at the grade 7 level focused on integers. The tests

comprised of several questions in the following numeracy areas: addition, subtraction,

multiplication, division, fractions, percent, decimals, ratio, order of operations, and

integers. Calculators were not permitted and the format of the test did not include

multiple choice questions. Furthermore, questions that would assess TCs’ pedagogical

skills were not included, i.e., the tests did not ask TCs how they would explain algorithms,

describe and make connections between concepts, or examine misconceptions in students’

work. Rather, this study focused on participants’ MCK, namely numerical operation skills,

which required TCs to demonstrate basic math knowledge.

The assessment was reviewed by several math and non-math teacher education

instructors for feedback on each of the items. The feedback was generally positive and all

the instructors felt that the questions were reasonable for TCs to complete. Many also

noted that this math test would help TCs better understand their own basic knowledge

and the aggregated results would help set a positive direction for the Master of Teaching

(MT) program.

TCs were informed approximately two months ahead of time that they would be

completing the basic math assessment a week prior to classes. It was also clearly

communicated that this pre-test was not at all high stakes, that is, the results were not

factored into their marks and passing the test was not a requirement to complete their MT

degree. Rather, the pre-test was designed as a diagnostic assessment to determine

strengths and areas for improvement in one’s own basic math proficiency. Candidates

were offered a practice test to help them prepare for the math assessment. The post-test

took place after the first year of the MT program, allowing TCs to review their progress

and identify areas of MCK that still required further focus and learning.

TCs completed the pre- and post-tests in their cohort groups of about 25 at a time. There

was one supervisor per cohort who administered the assessments. Supervisors

encouraged participants to show their work and write out as much of their thinking on the

test paper. Participants had up to 90 minutes to complete the questions, and the majority

of them finished within an hour. The pre- and post-tests were assessed and returned to

TCs with feedback on how to improve their basic math knowledge. Furthermore, data

results were aggregated to determine areas of strengths, needs, and next steps for

developing MCK.

The semi-structured individual interviews gave the researchers an in-depth

understanding of TCs’ experiences of their math development. In this study, interviews

859

10.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

were conducted with eight TCs (four with low confidence in math; four with high

confidence in math) and included the following questions: 1) Has your content knowledge

of math changed during the two courses in this program? 2) What was the most valuable

experience during the program in your development of math content knowledge? 3) What

was the most challenging experience during the program in your development of math

content knowledge? 4) How has your confidence as a math teacher changed during this

program? 5) Have your conceptions of how to teach math changed during this program?

and 6) How could the program be improved? The interviewer promoted a conversational

atmosphere by occasionally checking for understanding of the answers offered to

questions (Yin, 2009). For each question, interviewees were asked to offer details and/or

examples. Each interview was completed within a 30-45 minute time period. The

interviews were audio-recorded and transcriptions were completed and analyzed by the

researchers after the submission of final grades of any researcher associated with an

Two researchers were used in the coding process of each of the interviews to promote and

confirm inter-reliability (Kurasaki, 2000). Based on the theoretical underpinnings

embedded in this study, an initial set of codes was developed (Creswell, 2009, Denzin &

Lincoln, 2000), e.g., procedural knowledge, confidence to teach math. Further codes were

also added during the coding phase, e.g., mindset. The researchers engaged in an iterative

process whereby interviews were analyzed to ultimately identify themes and create

meaning (Sandelowski, 2001). This process provided additional opportunities for the

researchers to reflect and analyze upon the transcripts and comments. The coded data

was further reviewed to develop categories across the interviews, thereby grouping

content in terms of parallel and divergent concepts (Patton, 2002). Broad themes were

then established to frame the consistencies of ideas within and throughout the categories

(Baxter, 1991; Polit & Hungler, 1999).

Data analysis for this study included overall pre- and post-test results of teacher

candidates’ (TCs’) numeracy operation skills. Researchers identified low scoring items on

the pre-test and assessed any statistical significances between pre- and post-test results.

In conjunction with the quantitative results, transcripts of interviews were examined

through content analysis and coding techniques. Both quantitative and qualitative results

generated a comprehensive understanding of TCs’ math content knowledge (MCK) and

their experiences as math learners and teachers. This study’s results revealed three major

themes: 1) MCK tests support reflective practice; 2) instructors of and activities within

math courses are vital; and 3) practicum placements in math are essential for math

knowledge for teaching (MKT). Furthermore, these themes highlight specific areas of

need in TCs’ MCK and MKT, as they relate to this study’s theoretical methodology and

prior research.

Overall Results of Pre- and Post-Test

The pre- and post-tests comprised of 61 and 66 data points respectively. Committee

members added five additional questions to the post-test for future pre- and post-tests.

These additional questions were not part of the calculations in this study to support

comparability of the results of the pre- and post-tests. The 61 comparable questions

covered areas in addition, subtraction, multiplication, division, fractions, percent,

decimals, ratio, order of operations, and integers (see Table 1 for itemization of

assessment questions).

860

were conducted with eight TCs (four with low confidence in math; four with high

confidence in math) and included the following questions: 1) Has your content knowledge

of math changed during the two courses in this program? 2) What was the most valuable

experience during the program in your development of math content knowledge? 3) What

was the most challenging experience during the program in your development of math

content knowledge? 4) How has your confidence as a math teacher changed during this

program? 5) Have your conceptions of how to teach math changed during this program?

and 6) How could the program be improved? The interviewer promoted a conversational

atmosphere by occasionally checking for understanding of the answers offered to

questions (Yin, 2009). For each question, interviewees were asked to offer details and/or

examples. Each interview was completed within a 30-45 minute time period. The

interviews were audio-recorded and transcriptions were completed and analyzed by the

researchers after the submission of final grades of any researcher associated with an

Two researchers were used in the coding process of each of the interviews to promote and

confirm inter-reliability (Kurasaki, 2000). Based on the theoretical underpinnings

embedded in this study, an initial set of codes was developed (Creswell, 2009, Denzin &

Lincoln, 2000), e.g., procedural knowledge, confidence to teach math. Further codes were

also added during the coding phase, e.g., mindset. The researchers engaged in an iterative

process whereby interviews were analyzed to ultimately identify themes and create

meaning (Sandelowski, 2001). This process provided additional opportunities for the

researchers to reflect and analyze upon the transcripts and comments. The coded data

was further reviewed to develop categories across the interviews, thereby grouping

content in terms of parallel and divergent concepts (Patton, 2002). Broad themes were

then established to frame the consistencies of ideas within and throughout the categories

(Baxter, 1991; Polit & Hungler, 1999).

Data analysis for this study included overall pre- and post-test results of teacher

candidates’ (TCs’) numeracy operation skills. Researchers identified low scoring items on

the pre-test and assessed any statistical significances between pre- and post-test results.

In conjunction with the quantitative results, transcripts of interviews were examined

through content analysis and coding techniques. Both quantitative and qualitative results

generated a comprehensive understanding of TCs’ math content knowledge (MCK) and

their experiences as math learners and teachers. This study’s results revealed three major

themes: 1) MCK tests support reflective practice; 2) instructors of and activities within

math courses are vital; and 3) practicum placements in math are essential for math

knowledge for teaching (MKT). Furthermore, these themes highlight specific areas of

need in TCs’ MCK and MKT, as they relate to this study’s theoretical methodology and

prior research.

Overall Results of Pre- and Post-Test

The pre- and post-tests comprised of 61 and 66 data points respectively. Committee

members added five additional questions to the post-test for future pre- and post-tests.

These additional questions were not part of the calculations in this study to support

comparability of the results of the pre- and post-tests. The 61 comparable questions

covered areas in addition, subtraction, multiplication, division, fractions, percent,

decimals, ratio, order of operations, and integers (see Table 1 for itemization of

assessment questions).

860

11.
Learning to be a Math Teacher / Reid & Reid

Table 1. Itemization of Assessment Questions

Itemization of Assessment Questions

Numeracy Concept Test Points

Addition 3

Subtraction 4

Multiplication 5

Division 4

Fractions 13

Percent 2

Combination of Fractions, Percent and Decimals 16

Ratio 3

Order of Operations 5

Integers 6

Test Total 61

Note. The basic math content knowledge pre- and post-tests assessed numeracy skills in the areas of addition,

subtraction, multiplication, division, fractions, percent, ratio, decimals, order of operations, and integers.

There was a total of 61 data points that were scored in this pre- and post-test. This chart reveals the

breakdown of the test questions and number of data points for each area.

In general, junior/intermediate (J/I) teacher candidates (TCs) scored slightly higher on

several questions when compared to primary/junior (P/J) scores (pre- and post-test). For

each test item, an unpaired t-test was calculated between the J/I and P/J scores in order to

determine any statistical difference between the two groups. The results indicated

statistical significance between the J/I and P/J groups for two pre-test and three post-test

questions (i.e., p < 0.05). However, it was expected that 5% of these test questions would

yield false positives based on significant differences generated because of random

variation. Five percent is approximately three questions out of the 61 items. For this

reason, it would be difficult to make a case that these questions were somehow indicative

of a fundamental difference between P/J and J/I participants in terms of their abilities to

solve these specific types of questions.

The overall mean score of 151 participants for the pre- and post-tests was 81.54% and

84.62% respectively. This indicated an increase that is considered extremely statically

significant (i.e., p < 0.0001). Overall, there were several questions that TCs improved

significantly over the year. Although the improved post-test scores demonstrated

enhanced math content knowledge (MCK), there were numeracy skills that continued to

challenge many of the participants, for example, division of four-digit by two-digit

numbers, order of operations, and word problems involving percentages. It is not

surprising that not all areas improved on the post-test. During interviews with TC

participants, some identified that they were math anxious and reminisced about dropping

math as soon as possible in high school … “And then when math became an option, like

enough … I’m done with math!” Another TC noted that she had not focused on math in

some time and the math test helped identify gaps in her understanding … “I went to the

math test and I realized that there are a lot of principles that I haven’t studied for or

reviewed ... there were lots of challenges throughout.” Overall, most of the TCs commented

that the pre-test identified math areas where their content knowledge was weak, thus

allowing them to prioritize their own math learning.

Low Scoring Numeracy Skills

To focus on test items that presented difficulty to teacher candidates (TCs), the

researchers of this study analyzed items where less than two-thirds of the participants

861

Table 1. Itemization of Assessment Questions

Itemization of Assessment Questions

Numeracy Concept Test Points

Addition 3

Subtraction 4

Multiplication 5

Division 4

Fractions 13

Percent 2

Combination of Fractions, Percent and Decimals 16

Ratio 3

Order of Operations 5

Integers 6

Test Total 61

Note. The basic math content knowledge pre- and post-tests assessed numeracy skills in the areas of addition,

subtraction, multiplication, division, fractions, percent, ratio, decimals, order of operations, and integers.

There was a total of 61 data points that were scored in this pre- and post-test. This chart reveals the

breakdown of the test questions and number of data points for each area.

In general, junior/intermediate (J/I) teacher candidates (TCs) scored slightly higher on

several questions when compared to primary/junior (P/J) scores (pre- and post-test). For

each test item, an unpaired t-test was calculated between the J/I and P/J scores in order to

determine any statistical difference between the two groups. The results indicated

statistical significance between the J/I and P/J groups for two pre-test and three post-test

questions (i.e., p < 0.05). However, it was expected that 5% of these test questions would

yield false positives based on significant differences generated because of random

variation. Five percent is approximately three questions out of the 61 items. For this

reason, it would be difficult to make a case that these questions were somehow indicative

of a fundamental difference between P/J and J/I participants in terms of their abilities to

solve these specific types of questions.

The overall mean score of 151 participants for the pre- and post-tests was 81.54% and

84.62% respectively. This indicated an increase that is considered extremely statically

significant (i.e., p < 0.0001). Overall, there were several questions that TCs improved

significantly over the year. Although the improved post-test scores demonstrated

enhanced math content knowledge (MCK), there were numeracy skills that continued to

challenge many of the participants, for example, division of four-digit by two-digit

numbers, order of operations, and word problems involving percentages. It is not

surprising that not all areas improved on the post-test. During interviews with TC

participants, some identified that they were math anxious and reminisced about dropping

math as soon as possible in high school … “And then when math became an option, like

enough … I’m done with math!” Another TC noted that she had not focused on math in

some time and the math test helped identify gaps in her understanding … “I went to the

math test and I realized that there are a lot of principles that I haven’t studied for or

reviewed ... there were lots of challenges throughout.” Overall, most of the TCs commented

that the pre-test identified math areas where their content knowledge was weak, thus

allowing them to prioritize their own math learning.

Low Scoring Numeracy Skills

To focus on test items that presented difficulty to teacher candidates (TCs), the

researchers of this study analyzed items where less than two-thirds of the participants

861

12.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

answered correctly (see Table 2). For 10 of the 61 pre-test items, participants struggled to

answer the items correctly. More specifically, the percentage of participants that

answered the items correctly ranged from 33.55% - 63.65%. When the 10 pre- and post-

test items where compared through a paired t-test, six items indicated an increase that

was considered extremely significant (items 1-6), three items did not indicate a

statistically significant increase (items 7-9), and one item did not indicate a statistically

significant decrease (item 10).

Table 2. Low Scoring Numeracy Skills

Low Scoring Numeracy Skills (< Two-Thirds of TCs answered correctly)

Numeracy Skill Sample Items & Answers Pre-Test Post-Test Paired

(Diff) t-test

Divide a 4-digit number by a 2- 1246 ÷ 45 = 27.69 37.09% 57.62% t = 3.7170

digit number to the hundredth (+20.53%) p < 0.001

Represent in a fraction. How much chocolate is left? 60.26% 80.13% t = 4.3140

Represent in a fraction. (+19.87%) p < 0.0001

Convert a percent number into Convert 167% into a fraction. 58.94% 78.15% t = 4.6030

a fraction. (+19.21%) p < 0.0001

Divide a 4-digit number by a 2- 9768 ÷ 38 = 257.05 33.77% 51.66% t = 3.7256

digit number to the hundredth (+17.89%) p < 0.001

Divide a 4-digit number by a 2- 7727 ÷ 25 = 309 R-2 54.30% 70.20% t = 3.4460

digit number using whole (+15.90%) p < 0.001

Divide a 4-digit number by a 2- 2487 ÷ 16 = 155 R-7 63.58% 77.48% t = 3.4845

digit number using whole (+13.90%) p < 0.001

Multiply a decimal number by 8.273 x 15 = 124.095 59.60% 66.23% t = 1.3910

a whole number. (+6.63%) p > 0.1

Calculate the percentage of a 55% of 110 = 60.5 62.91% 65.56% t = 0.6017

whole number. (+2.65%) p > 0.5

Use order of operations to 13 – 6 + 8 = 15 55.63% 56.95% t = 0.2878

solve. (+1.32%) p > 0.5

Solve problem using numeracy Kiana has read 120 pages of her 55.63% 47.68% t = 1.5565

skills in percent, fractions, book – 120 pages equals to 40% (-7.95%) p > 0.1

and/or decimals. of the entire book. How many

pages does she have left to

read?

Kiana has 180 pages left to read.

Note. This table displays questions from the basic math content pre-test where less than two-thirds

of the participants were able to answer these items correctly. Results are reported in order of the

highest to lowest gains on the post-test scores. Each test item is connected to a numeracy skill. A

paired t-test was performed comparing the pre- and post-test results for the 10 items.

862

answered correctly (see Table 2). For 10 of the 61 pre-test items, participants struggled to

answer the items correctly. More specifically, the percentage of participants that

answered the items correctly ranged from 33.55% - 63.65%. When the 10 pre- and post-

test items where compared through a paired t-test, six items indicated an increase that

was considered extremely significant (items 1-6), three items did not indicate a

statistically significant increase (items 7-9), and one item did not indicate a statistically

significant decrease (item 10).

Table 2. Low Scoring Numeracy Skills

Low Scoring Numeracy Skills (< Two-Thirds of TCs answered correctly)

Numeracy Skill Sample Items & Answers Pre-Test Post-Test Paired

(Diff) t-test

Divide a 4-digit number by a 2- 1246 ÷ 45 = 27.69 37.09% 57.62% t = 3.7170

digit number to the hundredth (+20.53%) p < 0.001

Represent in a fraction. How much chocolate is left? 60.26% 80.13% t = 4.3140

Represent in a fraction. (+19.87%) p < 0.0001

Convert a percent number into Convert 167% into a fraction. 58.94% 78.15% t = 4.6030

a fraction. (+19.21%) p < 0.0001

Divide a 4-digit number by a 2- 9768 ÷ 38 = 257.05 33.77% 51.66% t = 3.7256

digit number to the hundredth (+17.89%) p < 0.001

Divide a 4-digit number by a 2- 7727 ÷ 25 = 309 R-2 54.30% 70.20% t = 3.4460

digit number using whole (+15.90%) p < 0.001

Divide a 4-digit number by a 2- 2487 ÷ 16 = 155 R-7 63.58% 77.48% t = 3.4845

digit number using whole (+13.90%) p < 0.001

Multiply a decimal number by 8.273 x 15 = 124.095 59.60% 66.23% t = 1.3910

a whole number. (+6.63%) p > 0.1

Calculate the percentage of a 55% of 110 = 60.5 62.91% 65.56% t = 0.6017

whole number. (+2.65%) p > 0.5

Use order of operations to 13 – 6 + 8 = 15 55.63% 56.95% t = 0.2878

solve. (+1.32%) p > 0.5

Solve problem using numeracy Kiana has read 120 pages of her 55.63% 47.68% t = 1.5565

skills in percent, fractions, book – 120 pages equals to 40% (-7.95%) p > 0.1

and/or decimals. of the entire book. How many

pages does she have left to

read?

Kiana has 180 pages left to read.

Note. This table displays questions from the basic math content pre-test where less than two-thirds

of the participants were able to answer these items correctly. Results are reported in order of the

highest to lowest gains on the post-test scores. Each test item is connected to a numeracy skill. A

paired t-test was performed comparing the pre- and post-test results for the 10 items.

862

13.
Learning to be a Math Teacher / Reid & Reid

When comparing the pre- and post-test results, as shown in Table 2, six items indicated an

increase that was considered extremely significant. Dividing 4-digit numbers by 2-digit

numbers to the hundredth decimal (item 2 and 4) proved to be the most difficult items –

with only 37.09% of participants responding accurately for item 2 and 33.77% answering

correctly for item 4 on the pre-test. While each item indicated a significant gain in the

post-test, many TCs were still not able to answer the items correctly, i.e., 42.38% and

48.34% respectively. Division with whole remainders (items 5 and 6) was also

challenging, with an initial mean of 54.30% for item 5 and 63.58% for item 6. Each item

showed a significant gain with more than 70% of TCs answering correctly on the post-test

for both items. At the start of the program, converting a percent number into a fraction

(item 3) had an average of 58.94%. For the post-test, 78.15% of TCs were able to convert

a percent number into a fraction, a significant gain. One item involved the representation

of chocolate bars as fractions (item 1); this item scored at 60.26% on the pre-test. A

significant gain was shown with an increase of 19.78% on the post-test.

When examining items 7-10 in Table 2, the difference between pre- and post-test items

were not considered to be statistically significant. Calculating a number sentence using

order of operations resulted with 55.63% achievement (item 9), with a minimal gain of

1.32%. Similarly, when multiplying a decimal number by a whole number (item 7) and

calculating the percentage of a whole number (item 8), the items only demonstrated gains

of 6.63% and 2.65% respectively. Lastly, solving a word problem that required knowledge

of percent, fractions and/or decimals (item 10) scored 47.68% on the post-test, a decrease

of 7.95%.

Although the overall results of the TCs had increased on the post-test, the researchers

further reviewed TCs who did not demonstrate proficiency upon commencing the

program in order to track their progress. Specifically, the group of TCs who did not

perform at or above 70% on the pre-test were further analyzed. For this group, almost

three-quarters of the TCs’ post-test scores remained under 70%. Furthermore, the

researchers reviewed the remaining TCs who scored below 70% on the post-test and

found that they had scored less than or equal to 75% on the pre-test.

It is worth noting that the MT math classes were focused on developing math knowledge

for teaching, lesson planning, assessment, and constructivist approaches to teaching math.

Although math content knowledge (MCK) was not an intentional focus of these classes,

numeracy operation skills were reinforced during math activities and problem solving

tasks. Even though not specifically asked during the interviews, many TCs shared their

own experiences as a math learner in middle and high school. It was acknowledged by

most of the interviewees, that they relied on memorized procedures to answer the pre-test

questions. These types of questions in a traditional math classroom would typically be

taught through rote learning methods. For example, division would be taught via

algorithm dependency without conceptually understanding why and how the algorithms

work. In all cases, a traditional style of past learning experiences in middle and high

school was described by TCs … “I would say [the math teacher] was stringent, it was like

here’s a product, here’s a solution, do it. No critical thinking, no application, no flexibility

Math Content Knowledge (MCK) Tests Support Reflective Practice

The pre- and post-tests focused on the numeracy operation skills, primarily for grades 5

and 6. Regardless of the math confidence noted by the interviewees, all teacher

candidates (TCs) described how their performance on the test shed light on their

competencies. For those with confidence in math, it reinforced that they had a foundation

of content knowledge required to teach math ... “what also helped to give me confidence was

863

When comparing the pre- and post-test results, as shown in Table 2, six items indicated an

increase that was considered extremely significant. Dividing 4-digit numbers by 2-digit

numbers to the hundredth decimal (item 2 and 4) proved to be the most difficult items –

with only 37.09% of participants responding accurately for item 2 and 33.77% answering

correctly for item 4 on the pre-test. While each item indicated a significant gain in the

post-test, many TCs were still not able to answer the items correctly, i.e., 42.38% and

48.34% respectively. Division with whole remainders (items 5 and 6) was also

challenging, with an initial mean of 54.30% for item 5 and 63.58% for item 6. Each item

showed a significant gain with more than 70% of TCs answering correctly on the post-test

for both items. At the start of the program, converting a percent number into a fraction

(item 3) had an average of 58.94%. For the post-test, 78.15% of TCs were able to convert

a percent number into a fraction, a significant gain. One item involved the representation

of chocolate bars as fractions (item 1); this item scored at 60.26% on the pre-test. A

significant gain was shown with an increase of 19.78% on the post-test.

When examining items 7-10 in Table 2, the difference between pre- and post-test items

were not considered to be statistically significant. Calculating a number sentence using

order of operations resulted with 55.63% achievement (item 9), with a minimal gain of

1.32%. Similarly, when multiplying a decimal number by a whole number (item 7) and

calculating the percentage of a whole number (item 8), the items only demonstrated gains

of 6.63% and 2.65% respectively. Lastly, solving a word problem that required knowledge

of percent, fractions and/or decimals (item 10) scored 47.68% on the post-test, a decrease

of 7.95%.

Although the overall results of the TCs had increased on the post-test, the researchers

further reviewed TCs who did not demonstrate proficiency upon commencing the

program in order to track their progress. Specifically, the group of TCs who did not

perform at or above 70% on the pre-test were further analyzed. For this group, almost

three-quarters of the TCs’ post-test scores remained under 70%. Furthermore, the

researchers reviewed the remaining TCs who scored below 70% on the post-test and

found that they had scored less than or equal to 75% on the pre-test.

It is worth noting that the MT math classes were focused on developing math knowledge

for teaching, lesson planning, assessment, and constructivist approaches to teaching math.

Although math content knowledge (MCK) was not an intentional focus of these classes,

numeracy operation skills were reinforced during math activities and problem solving

tasks. Even though not specifically asked during the interviews, many TCs shared their

own experiences as a math learner in middle and high school. It was acknowledged by

most of the interviewees, that they relied on memorized procedures to answer the pre-test

questions. These types of questions in a traditional math classroom would typically be

taught through rote learning methods. For example, division would be taught via

algorithm dependency without conceptually understanding why and how the algorithms

work. In all cases, a traditional style of past learning experiences in middle and high

school was described by TCs … “I would say [the math teacher] was stringent, it was like

here’s a product, here’s a solution, do it. No critical thinking, no application, no flexibility

Math Content Knowledge (MCK) Tests Support Reflective Practice

The pre- and post-tests focused on the numeracy operation skills, primarily for grades 5

and 6. Regardless of the math confidence noted by the interviewees, all teacher

candidates (TCs) described how their performance on the test shed light on their

competencies. For those with confidence in math, it reinforced that they had a foundation

of content knowledge required to teach math ... “what also helped to give me confidence was

863

14.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

the math test … because it turned out I was very good and I still remembered everything and

I understood the math, so that was also very good for my confidence.” For other TCs, the test

encouraged them to set goals for future personal learning … “[The test] hit on all the things

that you're going to need to know in order to teach [math] and so I think it was a great

opportunity because if you had a problem, you would have been able to see where it was.”

TCs’ personal learning was further facilitated by the math instructors in the MT program.

After the pre-tests, some classes collectively reviewed different ways in which the same

answers could be calculated … “we went through [the test] afterwards and it really helped

… we talked about how there are several different ways to solve a problem, even when it

wasn't the traditional method.” This concept of deconstructing and comparing various

methods to solve a problem was consistently reinforced throughout the MT math classes.

During the interviews, several TCs suggested changes to the test in the future, since not all

of their peers thought that completing the test was a priority; 30 TCs did not take part in

the pre- and/or post test. TCs explained how busy their schedules were due to course

work, assignments, and preparation for practicums, therefore it was easy to opt out of the

test. With this in mind, a few interviewees felt the need to make the pre- and post-test

mandatory … “I wondered if the math test could be mandated.” Another recommendation

included covering math curriculum higher than grade 6 … “in second year you could offer a

different test … if it was grade 7, 8, 9 and 10 … knowing that this test will include the higher

grades that we might be teaching, then it would interest others.” This is a relevant proposal

because TCs who are Junior/Intermediate (J/I) qualified, would likely be required to teach

math at the grade 7 and 8 level, and may even find themselves having to teach grades 9

and 10 math. Perhaps more importantly, several participants discussed additional

enticements to take part in the pre- and post-test. For example, the provision of

supplementary math training in the form of university sponsored tutoring was mentioned

to support TCs who were not confident in math.

Instructors of and Activities within Math Courses are Vital

When asked about experiences in the Master of Teaching (MT) math classes, all

interviewees provided positive responses with examples of how their conceptions of math

and/or teaching of math were supported. A common element across all interviews

included the importance of the MT instructors … “the instructors who are teaching math,

they actually love math” … “it was really amazing, she does have passion towards

mathematics.” These instructors were able to create safe environments to explore the

misconceptions that many teacher candidates (TCs) still had about learning and teaching

math; moving beyond the algorithms that TCs were previously overly reliant on.

Reflection on their learning as a TC, as well as past learning as a math student, was viewed

as an essential part of developing as teachers … “In this class, we started to learn different

concepts and I thought, why didn’t I learn this way” … “learning about the pedagogy has

allowed me to relate differently to my students and really dive into understanding the issues

in math.” The MT math classes offered the TCs contemporary models of instruction to

compare with their own experiences as math learners, thereby challenging their

conceptions of learning and teaching math.

All TCs discussed their changing perceptions and beliefs about teaching math based on

their interactions in the MT math classes … “I guess without having taken [instructor’s]

course, I wouldn’t have thought about math instruction differently than the way I was

taught.” TCs transformed how they viewed the curriculum by prioritizing students’ math

needs and figuring out how to make math accessible for all students. The course

experiences challenged TCs’ mindsets developed over time in traditional math classes. For

example, this involved spending time on the process of solving problems instead of only

the answer. Finding multiple ways to arrive at a solution was encouraged, as well as

864

the math test … because it turned out I was very good and I still remembered everything and

I understood the math, so that was also very good for my confidence.” For other TCs, the test

encouraged them to set goals for future personal learning … “[The test] hit on all the things

that you're going to need to know in order to teach [math] and so I think it was a great

opportunity because if you had a problem, you would have been able to see where it was.”

TCs’ personal learning was further facilitated by the math instructors in the MT program.

After the pre-tests, some classes collectively reviewed different ways in which the same

answers could be calculated … “we went through [the test] afterwards and it really helped

… we talked about how there are several different ways to solve a problem, even when it

wasn't the traditional method.” This concept of deconstructing and comparing various

methods to solve a problem was consistently reinforced throughout the MT math classes.

During the interviews, several TCs suggested changes to the test in the future, since not all

of their peers thought that completing the test was a priority; 30 TCs did not take part in

the pre- and/or post test. TCs explained how busy their schedules were due to course

work, assignments, and preparation for practicums, therefore it was easy to opt out of the

test. With this in mind, a few interviewees felt the need to make the pre- and post-test

mandatory … “I wondered if the math test could be mandated.” Another recommendation

included covering math curriculum higher than grade 6 … “in second year you could offer a

different test … if it was grade 7, 8, 9 and 10 … knowing that this test will include the higher

grades that we might be teaching, then it would interest others.” This is a relevant proposal

because TCs who are Junior/Intermediate (J/I) qualified, would likely be required to teach

math at the grade 7 and 8 level, and may even find themselves having to teach grades 9

and 10 math. Perhaps more importantly, several participants discussed additional

enticements to take part in the pre- and post-test. For example, the provision of

supplementary math training in the form of university sponsored tutoring was mentioned

to support TCs who were not confident in math.

Instructors of and Activities within Math Courses are Vital

When asked about experiences in the Master of Teaching (MT) math classes, all

interviewees provided positive responses with examples of how their conceptions of math

and/or teaching of math were supported. A common element across all interviews

included the importance of the MT instructors … “the instructors who are teaching math,

they actually love math” … “it was really amazing, she does have passion towards

mathematics.” These instructors were able to create safe environments to explore the

misconceptions that many teacher candidates (TCs) still had about learning and teaching

math; moving beyond the algorithms that TCs were previously overly reliant on.

Reflection on their learning as a TC, as well as past learning as a math student, was viewed

as an essential part of developing as teachers … “In this class, we started to learn different

concepts and I thought, why didn’t I learn this way” … “learning about the pedagogy has

allowed me to relate differently to my students and really dive into understanding the issues

in math.” The MT math classes offered the TCs contemporary models of instruction to

compare with their own experiences as math learners, thereby challenging their

conceptions of learning and teaching math.

All TCs discussed their changing perceptions and beliefs about teaching math based on

their interactions in the MT math classes … “I guess without having taken [instructor’s]

course, I wouldn’t have thought about math instruction differently than the way I was

taught.” TCs transformed how they viewed the curriculum by prioritizing students’ math

needs and figuring out how to make math accessible for all students. The course

experiences challenged TCs’ mindsets developed over time in traditional math classes. For

example, this involved spending time on the process of solving problems instead of only

the answer. Finding multiple ways to arrive at a solution was encouraged, as well as

864

15.
Learning to be a Math Teacher / Reid & Reid

reflecting on the math skills needed to formulate an accurate answer. For many, these

were unfamiliar practices that contradicted the traditional methods of teaching math. “We

basically looked at mathematics as having multiple ways to get the answers and creating our

own questions which was completely foreign to me … it allowed us to explore ourselves as

teachers in a new light.” This was further exemplified by MT instructors who emphasized

the importance of reasoning and justifications of math solutions … “we were taught to

embrace students explaining their thinking and showing their different strategies … it’s more

about understanding how they got there.”

TCs challenged themselves as math students and teachers, through various hands-on

opportunities in the MT math courses. Examples of these experiences included working

with manipulatives, probability and simulation activities, and deconstructing common

algorithms. These experiential activities reinforced different ways of teaching and

learning math concepts. TCs often stated they were so engaged in some of the activities

that they didn’t realize that multiple math skills were being addressed until they reflected

on and discussed the specific math skills involved. This led to shifts in mindsets which

changed conceptions of the composition of a math class … “before, I didn’t necessarily think

about doing the hands-on problem-solving activities, it’s absolutely made me more likely to

do these activities.” Through such course experiences, TCs challenged their own beliefs

about how learning in a math classroom occurs. This was an important step towards

reform math instruction, however, TCs required further opportunities to actualize these

new beliefs and hands-on activities in a real classroom. Hence, practicum placements

were of utmost importance for TCs to develop their confidence as math teachers.

Practicum Placements in Math are Essential for Math Knowledge for Teaching (MKT)

During the two-year Master of Teaching (MT) program, teacher candidates (TCs) complete

four practicum blocks, two each year. The practicum block is a full-time experience within

a classroom for four weeks. The TC is matched with an associate teacher, who is an

Ontario certified teacher recommended by a school leader. An important part of the

placement involves observation of the associate teacher, students, and classroom

program. Through the partnership developed between the associate teacher and the TC,

the associate teacher incrementally releases responsibility onto the TC to take on

planning, teaching, and assessing for the class.

Although TCs were not specifically asked about their practicum placements, the

importance of teaching math during practicum was discussed by all interviewees. In most

cases, TCs shared about practicum placements when asked about their confidence as a

math teacher and/or the development of their math content knowledge (MCK). Two TCs

had not taught math during their practicums at that point in time, with one more

practicum left. These TCs strongly expressed the lack of math teaching as a deficit to their

development. One TC described her first practicum with an associate teacher who felt that

sharing the math teaching would … “cause confusion” … for her students. Fortunately, this

TC felt that her associate teacher modelled an effective math program … “I was able to

observe … she had routines already established for the kids … I learned techniques of how to

deal with math.” Although the TC wanted to teach math during this practicum, at least she

had a positive role model. For the majority of the interviewees, it was stressed that math

was a critical part of their placement and professional learning … “I was really lucky

because I actually taught math quite a few times during my practicums … that was a huge

help, combined with the [MT math] courses, actually trying it out.” This appreciation for

teaching math in the practicum was globally shared by those with high and low self-

proclaimed levels of confidence in math.

865

reflecting on the math skills needed to formulate an accurate answer. For many, these

were unfamiliar practices that contradicted the traditional methods of teaching math. “We

basically looked at mathematics as having multiple ways to get the answers and creating our

own questions which was completely foreign to me … it allowed us to explore ourselves as

teachers in a new light.” This was further exemplified by MT instructors who emphasized

the importance of reasoning and justifications of math solutions … “we were taught to

embrace students explaining their thinking and showing their different strategies … it’s more

about understanding how they got there.”

TCs challenged themselves as math students and teachers, through various hands-on

opportunities in the MT math courses. Examples of these experiences included working

with manipulatives, probability and simulation activities, and deconstructing common

algorithms. These experiential activities reinforced different ways of teaching and

learning math concepts. TCs often stated they were so engaged in some of the activities

that they didn’t realize that multiple math skills were being addressed until they reflected

on and discussed the specific math skills involved. This led to shifts in mindsets which

changed conceptions of the composition of a math class … “before, I didn’t necessarily think

about doing the hands-on problem-solving activities, it’s absolutely made me more likely to

do these activities.” Through such course experiences, TCs challenged their own beliefs

about how learning in a math classroom occurs. This was an important step towards

reform math instruction, however, TCs required further opportunities to actualize these

new beliefs and hands-on activities in a real classroom. Hence, practicum placements

were of utmost importance for TCs to develop their confidence as math teachers.

Practicum Placements in Math are Essential for Math Knowledge for Teaching (MKT)

During the two-year Master of Teaching (MT) program, teacher candidates (TCs) complete

four practicum blocks, two each year. The practicum block is a full-time experience within

a classroom for four weeks. The TC is matched with an associate teacher, who is an

Ontario certified teacher recommended by a school leader. An important part of the

placement involves observation of the associate teacher, students, and classroom

program. Through the partnership developed between the associate teacher and the TC,

the associate teacher incrementally releases responsibility onto the TC to take on

planning, teaching, and assessing for the class.

Although TCs were not specifically asked about their practicum placements, the

importance of teaching math during practicum was discussed by all interviewees. In most

cases, TCs shared about practicum placements when asked about their confidence as a

math teacher and/or the development of their math content knowledge (MCK). Two TCs

had not taught math during their practicums at that point in time, with one more

practicum left. These TCs strongly expressed the lack of math teaching as a deficit to their

development. One TC described her first practicum with an associate teacher who felt that

sharing the math teaching would … “cause confusion” … for her students. Fortunately, this

TC felt that her associate teacher modelled an effective math program … “I was able to

observe … she had routines already established for the kids … I learned techniques of how to

deal with math.” Although the TC wanted to teach math during this practicum, at least she

had a positive role model. For the majority of the interviewees, it was stressed that math

was a critical part of their placement and professional learning … “I was really lucky

because I actually taught math quite a few times during my practicums … that was a huge

help, combined with the [MT math] courses, actually trying it out.” This appreciation for

teaching math in the practicum was globally shared by those with high and low self-

proclaimed levels of confidence in math.

865

16.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

A bi-product of teaching math during practicum described by TCs was the interplay

between MCK and knowledge for teaching math … “there was a difference between being

very good at math and being able to teach it.” Although the practicum was identified as

helpful when discussing their confidence as a math teacher or their conceptions of how to

teach math had changed, a few also discussed how teaching math induced stress for

themselves or their peers. When some TCs found out that they were teaching math, they

shared their angst with others, wondering about resources and feeling a general sense of

panic. Fortunately, for almost all TC participants who taught math during their

practicum(s), confidence in teaching math improved … “when I think about teaching my

first lesson versus my last, I already felt more confident and I knew what worked” and “you

don’t get comfortable knowing when you need to adjust unless you have actually tried it.”

Many TCs discussed the nuances of getting to know the learners in the classroom and

attempting to meet their needs … “it’s one of the only ways where you can figure out where

kids have the gaps in their understanding of math and where they’re going to have

Discussion and Implications

After analyzing pre- and post-tests and the semi-structured interviews, researchers

discovered that TCs gained numeracy operation skills and content knowledge over the

year, as well as increased confidence in teaching math. The significant findings from this

study were further analyzed by the researchers to consider implications such as possible

revisions to the pre- and post-tests, as well as enhancements to the MT program. The

recommendations are presented in the following sections: 1) establish minimum

standards; 2) raise the stakes of the post-test; 3) interplay of procedural and conceptual

knowledge; and 4) coherence between math courses and practicum.

Establish Minimum Standards

Although gains in math content knowledge (MCK) were observed in the post-tests, and

interview participants described changes in their math knowledge for teaching (MKT),

there were some areas in their content knowledge that did not improve. Specifically,

almost three-quarters of the teacher candidates (TCs) who scored less than 70% on the

pre-test, remained below 70% on the post-test. In addition, a few of the TCs who scored

75% or below on the pre-test also struggled in their MCK development, scoring below

70% on the post-test. These findings raise the question about what is reasonably expected

of TCs to know and understand in basic math prior to their teacher education training. In

Ontario, grade 6 students who perform below level three, that is below 70%, on the

provincial math assessment do not meet the provincial standard (EQAO, 2016a). With this

standard in mind, the researchers assert the need for minimal math content knowledge

standards for TCs. Without basic MCK, TCs will likely struggle to engage their students in

achieving math concepts, and further develop their own MKT. As noted in the research,

teachers with inadequate content knowledge find it difficult to explain math concepts,

provide models, and make connections to support understanding (Ponte & Chapman,

2008; Thames & Ball, 2010). For these reasons, the researchers highly recommend that

TCs who score 75% or below on the pre-test receive supplementary math support beyond

the existing offerings in the Master of Teaching (MT) math classes. Currently, TCs receive

recommended online resources for low-scoring areas on their test. The researchers

suggest that further math support through face-to-face or online tutoring would greatly

benefit those TCs who performed poorly. Although this would come at an expense to the

program, the researchers strongly feel this investment would be worthwhile due to the

impact TCs could have, each potentially teaching math to hundreds of students in the

years following graduation.

866

A bi-product of teaching math during practicum described by TCs was the interplay

between MCK and knowledge for teaching math … “there was a difference between being

very good at math and being able to teach it.” Although the practicum was identified as

helpful when discussing their confidence as a math teacher or their conceptions of how to

teach math had changed, a few also discussed how teaching math induced stress for

themselves or their peers. When some TCs found out that they were teaching math, they

shared their angst with others, wondering about resources and feeling a general sense of

panic. Fortunately, for almost all TC participants who taught math during their

practicum(s), confidence in teaching math improved … “when I think about teaching my

first lesson versus my last, I already felt more confident and I knew what worked” and “you

don’t get comfortable knowing when you need to adjust unless you have actually tried it.”

Many TCs discussed the nuances of getting to know the learners in the classroom and

attempting to meet their needs … “it’s one of the only ways where you can figure out where

kids have the gaps in their understanding of math and where they’re going to have

Discussion and Implications

After analyzing pre- and post-tests and the semi-structured interviews, researchers

discovered that TCs gained numeracy operation skills and content knowledge over the

year, as well as increased confidence in teaching math. The significant findings from this

study were further analyzed by the researchers to consider implications such as possible

revisions to the pre- and post-tests, as well as enhancements to the MT program. The

recommendations are presented in the following sections: 1) establish minimum

standards; 2) raise the stakes of the post-test; 3) interplay of procedural and conceptual

knowledge; and 4) coherence between math courses and practicum.

Establish Minimum Standards

Although gains in math content knowledge (MCK) were observed in the post-tests, and

interview participants described changes in their math knowledge for teaching (MKT),

there were some areas in their content knowledge that did not improve. Specifically,

almost three-quarters of the teacher candidates (TCs) who scored less than 70% on the

pre-test, remained below 70% on the post-test. In addition, a few of the TCs who scored

75% or below on the pre-test also struggled in their MCK development, scoring below

70% on the post-test. These findings raise the question about what is reasonably expected

of TCs to know and understand in basic math prior to their teacher education training. In

Ontario, grade 6 students who perform below level three, that is below 70%, on the

provincial math assessment do not meet the provincial standard (EQAO, 2016a). With this

standard in mind, the researchers assert the need for minimal math content knowledge

standards for TCs. Without basic MCK, TCs will likely struggle to engage their students in

achieving math concepts, and further develop their own MKT. As noted in the research,

teachers with inadequate content knowledge find it difficult to explain math concepts,

provide models, and make connections to support understanding (Ponte & Chapman,

2008; Thames & Ball, 2010). For these reasons, the researchers highly recommend that

TCs who score 75% or below on the pre-test receive supplementary math support beyond

the existing offerings in the Master of Teaching (MT) math classes. Currently, TCs receive

recommended online resources for low-scoring areas on their test. The researchers

suggest that further math support through face-to-face or online tutoring would greatly

benefit those TCs who performed poorly. Although this would come at an expense to the

program, the researchers strongly feel this investment would be worthwhile due to the

impact TCs could have, each potentially teaching math to hundreds of students in the

years following graduation.

866

17.
Learning to be a Math Teacher / Reid & Reid

Raise the Stakes of the Post-Test

Similar to the research of CBMS (2012) and Thames and Ball (2010), all teacher

candidates (TCs) described the value of deeply understanding the math content in order to

develop appropriate lessons for their students, that is, possessing the specialized math

knowledge to make sound pedagogical decisions. Importantly, a few TCs suggested the

need for the tests, or at least the post-test, to be more difficult, by adding questions

specifically designed for grade 7 and 8. They felt that this would not only elevate the

status of the tests, but also encourage TCs to further develop the MCK necessary to be

effective teachers. This is a critical concept that promotes the importance of math

knowledge required to teach. Studies have shown the deep knowledge base teachers

require in order to teach math effectively, even in the primary grades (Ball et al., 2005).

Based on the findings from this study, the researchers recommend that the Master of

Teaching (MT) program: 1) increase the difficulty of the pre- and post-tests (e.g.,

consisting of questions from grades 5 – 8; 35% grade 5, 35% grade 6, and 15% grade 7,

and 15% grade 8), and 2) have the post-test score count toward a small percentage of the

TCs’ final math grade, for example, 10% of the final math mark. The researchers feel this

will positively increase the stakes of the test and propel TCs to invest in their MCK by

studying and seeking help where needed. With increased MCK, the researchers also have

confidence that TCs will be better positioned to further develop math knowledge for

teaching during MT math classes and practicums.

Interplay of Procedural and Conceptual Knowledge

The findings from the basic content knowledge test illustrated some of the challenges

teacher candidates (TCs) faced with specific numeracy operation skills. It was evident that

many participants still struggled with questions that required procedural knowledge

involving several steps such as multiplication of decimal numbers and solving word

problems involving percent, fractions, and/or decimals. The results from the present

study align with other research indicating the critical relationship between procedural and

conceptual knowledge (Ambrose, 2004; Heibert, 2013). TCs with a strong math

background described understanding the math concepts that were investigated in their

Master of Teaching (MT) math classes, as well as feeling confident to teach math. Other

TCs who found math to be more challenging, revealed how they were dependent on the

memorization of the steps and therefore their deep understanding of concepts was

compromised. TCs’ overreliance on procedural knowledge was also found by researchers

such as Tirosh (2000).

A major part of the MT math classes engaged TCs in activities that modelled math

approaches to promote conceptual understandings of math that go beyond the

memorization and application of algorithms. TCs gained both procedural and conceptual

knowledge of math that led to the development of math knowledge for teaching during

their practicums. This finding aligns with research about the importance of building the

conceptual knowledge of TCs as this leads to shifts in pedagogical beliefs and practices in

the classroom (Ball, 1990a; Kajander, 2007, 2010). Most importantly, the TCs described

that the math lessons they planned for their students were different from how they

themselves experienced math in school. Instead of traditional math practices, the TCs

carefully planned math activities with multiple entry points for their students and ensured

that their students were able to discuss and reflect on their problem-solving strategies.

Due to these findings, the researchers suggest that the MT math courses continue to

investigate opportunities to challenge TCs’ perceived notions of teaching math, providing

multiple opportunities to build content and procedural knowledge through rich, hands on

learning opportunities.

867

Raise the Stakes of the Post-Test

Similar to the research of CBMS (2012) and Thames and Ball (2010), all teacher

candidates (TCs) described the value of deeply understanding the math content in order to

develop appropriate lessons for their students, that is, possessing the specialized math

knowledge to make sound pedagogical decisions. Importantly, a few TCs suggested the

need for the tests, or at least the post-test, to be more difficult, by adding questions

specifically designed for grade 7 and 8. They felt that this would not only elevate the

status of the tests, but also encourage TCs to further develop the MCK necessary to be

effective teachers. This is a critical concept that promotes the importance of math

knowledge required to teach. Studies have shown the deep knowledge base teachers

require in order to teach math effectively, even in the primary grades (Ball et al., 2005).

Based on the findings from this study, the researchers recommend that the Master of

Teaching (MT) program: 1) increase the difficulty of the pre- and post-tests (e.g.,

consisting of questions from grades 5 – 8; 35% grade 5, 35% grade 6, and 15% grade 7,

and 15% grade 8), and 2) have the post-test score count toward a small percentage of the

TCs’ final math grade, for example, 10% of the final math mark. The researchers feel this

will positively increase the stakes of the test and propel TCs to invest in their MCK by

studying and seeking help where needed. With increased MCK, the researchers also have

confidence that TCs will be better positioned to further develop math knowledge for

teaching during MT math classes and practicums.

Interplay of Procedural and Conceptual Knowledge

The findings from the basic content knowledge test illustrated some of the challenges

teacher candidates (TCs) faced with specific numeracy operation skills. It was evident that

many participants still struggled with questions that required procedural knowledge

involving several steps such as multiplication of decimal numbers and solving word

problems involving percent, fractions, and/or decimals. The results from the present

study align with other research indicating the critical relationship between procedural and

conceptual knowledge (Ambrose, 2004; Heibert, 2013). TCs with a strong math

background described understanding the math concepts that were investigated in their

Master of Teaching (MT) math classes, as well as feeling confident to teach math. Other

TCs who found math to be more challenging, revealed how they were dependent on the

memorization of the steps and therefore their deep understanding of concepts was

compromised. TCs’ overreliance on procedural knowledge was also found by researchers

such as Tirosh (2000).

A major part of the MT math classes engaged TCs in activities that modelled math

approaches to promote conceptual understandings of math that go beyond the

memorization and application of algorithms. TCs gained both procedural and conceptual

knowledge of math that led to the development of math knowledge for teaching during

their practicums. This finding aligns with research about the importance of building the

conceptual knowledge of TCs as this leads to shifts in pedagogical beliefs and practices in

the classroom (Ball, 1990a; Kajander, 2007, 2010). Most importantly, the TCs described

that the math lessons they planned for their students were different from how they

themselves experienced math in school. Instead of traditional math practices, the TCs

carefully planned math activities with multiple entry points for their students and ensured

that their students were able to discuss and reflect on their problem-solving strategies.

Due to these findings, the researchers suggest that the MT math courses continue to

investigate opportunities to challenge TCs’ perceived notions of teaching math, providing

multiple opportunities to build content and procedural knowledge through rich, hands on

learning opportunities.

867

18.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

Coherence between Math Courses and Practicum

The math courses in the Master of Teaching (MT) program were noted by all interviewees

as being vital to their math learning as students and math teachers. Nevertheless, these

courses could not reproduce an actual classroom with real students for teacher candidates

(TCs) to interact with and teach. These experiences could only be gained through

practicums that involved teaching math to students. However, one cannot assume that

teaching math automatically improves math content knowledge (MCK), math knowledge

for teaching (MKT), and/or confidence in teaching math. Unfortunately, two TCs

described practicum placements that were not aligned to the pedagogical outcomes in the

MT program. These narratives suggest the need for coherence between the MT math

courses and practice teaching. In a few practicum placements, TCs felt pressure to teach

math through a prescribed and transmissive method, that is, memorization of procedures

and assigning math text book pages and worksheets. This type of instruction aligns with

the evidence found by CBMS (2012) and Ma (1999) where math teachers struggled with

explaining the deeper conceptual knowledge of math, only feeling comfortable to teach

through algorithms and procedures. In an ideal world, all practicum experiences would

embody exemplary math environments for TCs to work within. However, as noted in the

research, teachers have varying levels of math proficiency and in many cases, the

knowledge base may be lacking (Ball, Hill & Bass, 2005; Cai & Wang, 2010; Vistro-Yu,

2013). Based on these findings, the researchers assert that teacher education programs

and school boards should involve supporting associate teachers to nurture risk-taking by

allowing TCs to dialogue, question, and trial math teaching strategies with intentional

reflection (Reid, 2013). Through this process, associate teachers and TCs engage in

relevant discourse and work together to improve their math teaching skills and thereby

positively impact students’ math achievement.

How much basic math content knowledge TCs ought to know prior to entering their

teacher education program is an area that has not been widely examined. This current

research offers important knowledge about the math content knowledge (MCK) of TCs, as

well as the major impact that math courses and practicums have during their teacher

education experiences. Regardless of the basic numeracy skill levels that TCs possessed

upon entering the program, their Master of Teaching (MT) math classes and practicums

contributed to both MCK and their beliefs about how to teach math. Nevertheless, the

researchers of this study posit that those TCs with low MCK, based on the pre-test results,

would have benefited from additional support, e.g., tutoring. This recommendation is

derived from the fact that some items in the post-test revealed no significant

improvements and several items still posed a challenge for TCs. Additional support in the

form of small group or one-on-one tutoring would enhance TCs’ MCK, especially for those

who experienced difficulty learning the curriculum content required for their practicum

teaching. Other ways in which teacher education programs can raise the importance of

MCK include: minimum standards on entrance or exit exams and the inclusion of the post-

test scores as a percentage of final math grades. The researchers of this study believe that

all teacher education programs should support compulsory minimum math knowledge

requirements. This focus on the foundational MCK skills of TCs is critical for the successful

development of math knowledge for teaching (MKT) capacities. Ultimately, this will raise

the significance of teaching math for understanding and increase the abilities for all math

educators to support effective math environments for students to flourish as

• • •

868

Coherence between Math Courses and Practicum

The math courses in the Master of Teaching (MT) program were noted by all interviewees

as being vital to their math learning as students and math teachers. Nevertheless, these

courses could not reproduce an actual classroom with real students for teacher candidates

(TCs) to interact with and teach. These experiences could only be gained through

practicums that involved teaching math to students. However, one cannot assume that

teaching math automatically improves math content knowledge (MCK), math knowledge

for teaching (MKT), and/or confidence in teaching math. Unfortunately, two TCs

described practicum placements that were not aligned to the pedagogical outcomes in the

MT program. These narratives suggest the need for coherence between the MT math

courses and practice teaching. In a few practicum placements, TCs felt pressure to teach

math through a prescribed and transmissive method, that is, memorization of procedures

and assigning math text book pages and worksheets. This type of instruction aligns with

the evidence found by CBMS (2012) and Ma (1999) where math teachers struggled with

explaining the deeper conceptual knowledge of math, only feeling comfortable to teach

through algorithms and procedures. In an ideal world, all practicum experiences would

embody exemplary math environments for TCs to work within. However, as noted in the

research, teachers have varying levels of math proficiency and in many cases, the

knowledge base may be lacking (Ball, Hill & Bass, 2005; Cai & Wang, 2010; Vistro-Yu,

2013). Based on these findings, the researchers assert that teacher education programs

and school boards should involve supporting associate teachers to nurture risk-taking by

allowing TCs to dialogue, question, and trial math teaching strategies with intentional

reflection (Reid, 2013). Through this process, associate teachers and TCs engage in

relevant discourse and work together to improve their math teaching skills and thereby

positively impact students’ math achievement.

How much basic math content knowledge TCs ought to know prior to entering their

teacher education program is an area that has not been widely examined. This current

research offers important knowledge about the math content knowledge (MCK) of TCs, as

well as the major impact that math courses and practicums have during their teacher

education experiences. Regardless of the basic numeracy skill levels that TCs possessed

upon entering the program, their Master of Teaching (MT) math classes and practicums

contributed to both MCK and their beliefs about how to teach math. Nevertheless, the

researchers of this study posit that those TCs with low MCK, based on the pre-test results,

would have benefited from additional support, e.g., tutoring. This recommendation is

derived from the fact that some items in the post-test revealed no significant

improvements and several items still posed a challenge for TCs. Additional support in the

form of small group or one-on-one tutoring would enhance TCs’ MCK, especially for those

who experienced difficulty learning the curriculum content required for their practicum

teaching. Other ways in which teacher education programs can raise the importance of

MCK include: minimum standards on entrance or exit exams and the inclusion of the post-

test scores as a percentage of final math grades. The researchers of this study believe that

all teacher education programs should support compulsory minimum math knowledge

requirements. This focus on the foundational MCK skills of TCs is critical for the successful

development of math knowledge for teaching (MKT) capacities. Ultimately, this will raise

the significance of teaching math for understanding and increase the abilities for all math

educators to support effective math environments for students to flourish as

• • •

868

19.
Learning to be a Math Teacher / Reid & Reid

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Alsup, J., (2004). A comparison of constructivist and traditional instruction in mathematics.

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Alsup, J., & Sprigler, M. (2003). A comparison of traditional and reform mathematics curricula in an

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Ambrose, R. (2004). Initiating change in prospective elementary school teachers’ orientations to

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Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved

problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of

research on teaching (4th ed.) (pp. 433-456). New York, NY: Macmillan.

Ball, D. L., Sleep, L., Boerst, T. A., & Bass, H. (2009). Combining the development of practice and the

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doi:10.1086/596996

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Bartell, T. G., Webel, C., Bowen, B., & Dyson, N. (2013). Prospective teacher learning: recognizing

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strategies and underlying counting schemes. International Journal of Science and

Mathematics Education, 9(1), 1-24. doi:10.1007/s10763-010-9202-y

Cai, J., & Wang, T. (2010). Conceptions of effective mathematics teaching within a cultural context:

Perspectives of teachers from China and the United States. Journal of Mathematics Teacher

Education. 13(3), 265-287. doi: 10.1007/s10857-009-9132-1

Conference Board of the Mathematical Sciences. (2012). Issues in mathematics education: Vol. 17.

The mathematical education of teachers II. Providence, RI: American Mathematical Society.

Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches

(3rd ed.). Thousand Oaks, CA: Sage.

Creswell, J. W., & Clark, V. L. (2007). Designing and conducting mixed methods research. Thousand

Oaks, CA: Sage.

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secondary teachers. Journal of Mathematics Teacher Education 3(1), 47-76.

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Mathematics Education, 30(1), 3-19. doi:10.2307/749627

Hiebert, J. (2013). Conceptual and procedural knowledge: The case of mathematics. Routledge.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Stigler, J. W. (2003).

Understanding and improving mathematics teaching: Highlights from the TIMSS 1999

Video Study. Phi Delta Kappan, 84(10), 768-775.

Hiebert, J., Stigler, J. W., Jacobs, J. K., Givvin, K. B., Garnier, H., Smith, M., … Gallimore, R. (2005).

Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS

1999 video study. Educational Evaluation and Policy Analysis, 27(2), 111-132.

doi:10.3102/01623737027002111

Hill, H. & Ball, D. L. (2004). Learning mathematics for teaching: results from California’s

mathematics professional development institutes. Journal for Research in Mathematics

Education, 35(5), 330-351. doi:10.2307/30034819

Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of mathematical knowledge for

teaching. Phi Delta Kappan, 91(2), 68-71. doi:10.1177/003172170909100215

Hill, H., Rowan, B., & Ball, D. (2005). Effects of teachers’ mathematical knowledge for teaching on

student achievement. American Education Research Journal, 42(2), 371-406.

doi:10.3102/00028312042002371

Kagan, D. (1992). Professional growth among preservice and beginning teachers. Review of

Educational Research, 62(2), 129-168. doi:10.3102/00346543062002129

Kajander, A. (2007). Unpacking mathematics for teaching: A study of preservice elementary

teachers’ evolving mathematical understandings and beliefs. Journal of Teaching and

Learning, 5(1), 33-54.

Kajander, A. (2010). Elementary Mathematics Teacher Preparation in an Era of Reform: The

Development and Assessment of Mathematics for Teaching. Canadian Journal of Education,

33(1), 228-255.

870

Darling-Hammond, L., & Youngs, P. (2002). Defining “highly qualified teachers”: What does

“scientifically-based research” actually tell us? Educational Researcher, 31(9), 13-25.

Denzin, N., & Lincoln, Y. (2000). Handbook of qualitative research (2nd ed.). Thousand Oaks, CA:

Sage.

Education Quality and Accountability Office. (2016a). Highlights of the provincial results:

Assessments of reading, writing and mathematics, primary division (grades 1-3) and junior

division (grades 4-6): English-language students, 2015-2016. Toronto, ON: Education

Quality and Accountability Office.

Education Quality and Accountability Office. (2016b). Programme for international student

assessment (PISA), 2015: Highlights of Ontario student results. Toronto, ON: Education

Quality and Accountability Office.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual

knowledge falls through the cracks: Complexities of learning to teach mathematics for

understanding . Journal for Research in Mathematics Education, 24(1), 8-40.

doi:10.2307/749384

Frykholm, J.A. (1999). The impact of reform: Challenges for mathematics teacher preparation.

Journal of Mathematics Teacher Education, 2(1), 79-105. doi:10.1023/A:1009904604728

Grover, B. W., & Connor, J. (2000). Characteristics of the college geometry course for preservice

secondary teachers. Journal of Mathematics Teacher Education 3(1), 47-76.

doi:10.1023/A:1009921628065

Hiebert, J. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of

research on mathematics teaching and learning (pp. 65-97). New York, NY: Macmillan.

Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for Research in

Mathematics Education, 30(1), 3-19. doi:10.2307/749627

Hiebert, J. (2013). Conceptual and procedural knowledge: The case of mathematics. Routledge.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Stigler, J. W. (2003).

Understanding and improving mathematics teaching: Highlights from the TIMSS 1999

Video Study. Phi Delta Kappan, 84(10), 768-775.

Hiebert, J., Stigler, J. W., Jacobs, J. K., Givvin, K. B., Garnier, H., Smith, M., … Gallimore, R. (2005).

Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS

1999 video study. Educational Evaluation and Policy Analysis, 27(2), 111-132.

doi:10.3102/01623737027002111

Hill, H. & Ball, D. L. (2004). Learning mathematics for teaching: results from California’s

mathematics professional development institutes. Journal for Research in Mathematics

Education, 35(5), 330-351. doi:10.2307/30034819

Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of mathematical knowledge for

teaching. Phi Delta Kappan, 91(2), 68-71. doi:10.1177/003172170909100215

Hill, H., Rowan, B., & Ball, D. (2005). Effects of teachers’ mathematical knowledge for teaching on

student achievement. American Education Research Journal, 42(2), 371-406.

doi:10.3102/00028312042002371

Kagan, D. (1992). Professional growth among preservice and beginning teachers. Review of

Educational Research, 62(2), 129-168. doi:10.3102/00346543062002129

Kajander, A. (2007). Unpacking mathematics for teaching: A study of preservice elementary

teachers’ evolving mathematical understandings and beliefs. Journal of Teaching and

Learning, 5(1), 33-54.

Kajander, A. (2010). Elementary Mathematics Teacher Preparation in an Era of Reform: The

Development and Assessment of Mathematics for Teaching. Canadian Journal of Education,

33(1), 228-255.

870

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Learning to be a Math Teacher / Reid & Reid

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic: 2nd Grade (2nd ed.). New York, NY:

Teachers College Press.

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interview data. Field Methods, 12(3), 179-194. doi:10.1177/1525822X0001200301

Lo, J.-J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division. Journal of

Mathematics Teacher Education, 15(6), 481-500. doi:10.1007/s10857-012-9221-4

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' Understanding of

fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum

Associates.

McCormick, R. (1997). Conceptual and procedural knowledge. International Journal of Technology

and Design Education, 7(1-2), 141-159.

McDougall, D., Ross J., & Jaafar, S. (2006). PRIME ten dimensions of mathematical education:

Research study. Toronto, ON: Nelson.

Morris, A., Hiebert, J., & Spitzer, S. (2009). Mathematical knowledge for teaching in planning and

evaluating instruction: What can preservice teachers learn? Journal for Research in

Mathematics Education 40(5). 491-529.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards

for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school

mathematics. Reston, VA: Author.

Ontario Ministry of Education. (2005). The Ontario Curriculum, Grades 1-8: Mathematics. Toronto,

ON: Ontario Ministry of Education.

Ontario Ministry of Education. (2014). Achieving Excellence: A Renewed Vision for Education in

Ontario. Toronto, ON: Ontario Ministry of Education.

Parekh, G. (2014) ‘Social citizenship and disability: Identity, belonging, and the structural

organization of education’. PhD Thesis, York University, Toronto.

Philipp, R., Ambrose, R., Lamb, L., Sowder, J., Schappelle, B., Sowder, L., . . . Chauvot, J. (2007). Effects

of early field experiences on the mathematical content knowledge and beliefs of

prospective elementary school teachers: An experimental study. Journal for Research in

Mathematics Education, 38(5), 438-476.

Polit, D. F., & Hungler, B. P. (1999). Nursing research: Principles and methods (6th ed.). New York,

NY: J.B. Lippincott Company.

Ponte, J. P. & Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development.

In L. D. English (Ed.) Handbook of international research in mathematics education:

Directions for the 21st century (2nd Edition, pp. 225 - 263). New York: Routledge.

Reid, M. (2013). Mathematics Content Knowledge, Mathematics Teacher Efficacy, and Pedagogy: An

Examination of the Three Constructs in Mathematics Preservice Elementary Education

(doctoral thesis). University of Calgary, Calgary, Canada.

Rittle-Johnson, B. & Koedinger, K. (2002). Comparing instructional strategies for integrating

conceptual and procedural knowledge. In Mewborn, D.S., Sztajin, P., White, D.Y., Wiegel,

H.G., Bryant, R.L. & Nooney, K. (Eds.) Proceedings of the 24th annual meeting of the North

American Chapters of the International Group for the Psychology of Mathematics Education

(pp. 969-978). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and

Environmental Education.

Rowan, B., Chiang, F., & Miller, R. J. (1997). Using research on employees’ performance to study the

effects of teachers on students’ achievement. Sociology of Education, 70(4), 256–284.

doi:10.2307/2673267

871

Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic: 2nd Grade (2nd ed.). New York, NY:

Teachers College Press.

Kurasaki, K. S. (2000). Intercoder reliability for validating conclusions drawn from open-ended

interview data. Field Methods, 12(3), 179-194. doi:10.1177/1525822X0001200301

Lo, J.-J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division. Journal of

Mathematics Teacher Education, 15(6), 481-500. doi:10.1007/s10857-012-9221-4

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' Understanding of

fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum

Associates.

McCormick, R. (1997). Conceptual and procedural knowledge. International Journal of Technology

and Design Education, 7(1-2), 141-159.

McDougall, D., Ross J., & Jaafar, S. (2006). PRIME ten dimensions of mathematical education:

Research study. Toronto, ON: Nelson.

Morris, A., Hiebert, J., & Spitzer, S. (2009). Mathematical knowledge for teaching in planning and

evaluating instruction: What can preservice teachers learn? Journal for Research in

Mathematics Education 40(5). 491-529.

National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards

for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school

mathematics. Reston, VA: Author.

Ontario Ministry of Education. (2005). The Ontario Curriculum, Grades 1-8: Mathematics. Toronto,

ON: Ontario Ministry of Education.

Ontario Ministry of Education. (2014). Achieving Excellence: A Renewed Vision for Education in

Ontario. Toronto, ON: Ontario Ministry of Education.

Parekh, G. (2014) ‘Social citizenship and disability: Identity, belonging, and the structural

organization of education’. PhD Thesis, York University, Toronto.

Philipp, R., Ambrose, R., Lamb, L., Sowder, J., Schappelle, B., Sowder, L., . . . Chauvot, J. (2007). Effects

of early field experiences on the mathematical content knowledge and beliefs of

prospective elementary school teachers: An experimental study. Journal for Research in

Mathematics Education, 38(5), 438-476.

Polit, D. F., & Hungler, B. P. (1999). Nursing research: Principles and methods (6th ed.). New York,

NY: J.B. Lippincott Company.

Ponte, J. P. & Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development.

In L. D. English (Ed.) Handbook of international research in mathematics education:

Directions for the 21st century (2nd Edition, pp. 225 - 263). New York: Routledge.

Reid, M. (2013). Mathematics Content Knowledge, Mathematics Teacher Efficacy, and Pedagogy: An

Examination of the Three Constructs in Mathematics Preservice Elementary Education

(doctoral thesis). University of Calgary, Calgary, Canada.

Rittle-Johnson, B. & Koedinger, K. (2002). Comparing instructional strategies for integrating

conceptual and procedural knowledge. In Mewborn, D.S., Sztajin, P., White, D.Y., Wiegel,

H.G., Bryant, R.L. & Nooney, K. (Eds.) Proceedings of the 24th annual meeting of the North

American Chapters of the International Group for the Psychology of Mathematics Education

(pp. 969-978). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and

Environmental Education.

Rowan, B., Chiang, F., & Miller, R. J. (1997). Using research on employees’ performance to study the

effects of teachers on students’ achievement. Sociology of Education, 70(4), 256–284.

doi:10.2307/2673267

871

22.
International Electronic Journal of Elementary Education Vol.9, Issue 4, 851-872, June 2017

Sandelowski, M. (2001). Real qualitative researchers do not count: The use of numbers in

qualitative research. Research in Nursing & Health, 24(3), 230-240. doi:10.1002/nur.1025

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4-14. doi:10.3102/0013189X015002004

Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational

Review, 57(1), 1-22. doi:10.17763/haer.57.1.j463w79r56455411

Sowder, J. T. (2007). The mathematical education and development of teachers. In F. K. Lester (Ed.),

Second handbook of research on mathematics teaching and learning (pp. 157 – 224). Reston,

VA: National Council of Teachers of Mathematics. doi:10.1234/12345678

Stigler, J.W., & Hiebert, J. (1997). Understanding and improving classroom mathematics instruction:

An overview of the TIMSS video study. Phi Delta Kappan, 79(1), 14-21.

Tabachnik, R. B., & Zeichner, K. M. (1984). The impact of students teaching experience on the

development of teacher perspectives. Journal of Research in Teacher Education, 35(6), 28-

36. doi:10.1177/002248718403500608

Tella, A. (2008). Teacher variables as predictors of academic achievement of primary school pupils

mathematics. International Electronic Journal of Elementary Education, 1(1), 16-33.

Thames, M., & Ball, D. L. (2010). What math knowledge does teaching require? Teaching Children

Mathematics, 17(4), 220-229.

Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case

of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.

Vistro-Yu, C. P. (2013). Cross-national studies on the teaching and learning of mathematics: Where

do we go from here? ZDM Mathematics Education, 45(1), 145-151. doi:10.1007/s11858-

013-0488-4

Yackel, E., Underwood, D., & Elias, N. (2007). Mathematical tasks designed to foster a

reconceptualized view of early arithmetic. Journal of Mathematics Teacher Education, 10(4-

6), 351-367.DOI 10.1007/s10857-007-9044-x

Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Thousand Oaks, CA: Sage.

872

Sandelowski, M. (2001). Real qualitative researchers do not count: The use of numbers in

qualitative research. Research in Nursing & Health, 24(3), 230-240. doi:10.1002/nur.1025

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4-14. doi:10.3102/0013189X015002004

Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational

Review, 57(1), 1-22. doi:10.17763/haer.57.1.j463w79r56455411

Sowder, J. T. (2007). The mathematical education and development of teachers. In F. K. Lester (Ed.),

Second handbook of research on mathematics teaching and learning (pp. 157 – 224). Reston,

VA: National Council of Teachers of Mathematics. doi:10.1234/12345678

Stigler, J.W., & Hiebert, J. (1997). Understanding and improving classroom mathematics instruction:

An overview of the TIMSS video study. Phi Delta Kappan, 79(1), 14-21.

Tabachnik, R. B., & Zeichner, K. M. (1984). The impact of students teaching experience on the

development of teacher perspectives. Journal of Research in Teacher Education, 35(6), 28-

36. doi:10.1177/002248718403500608

Tella, A. (2008). Teacher variables as predictors of academic achievement of primary school pupils

mathematics. International Electronic Journal of Elementary Education, 1(1), 16-33.

Thames, M., & Ball, D. L. (2010). What math knowledge does teaching require? Teaching Children

Mathematics, 17(4), 220-229.

Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case

of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.

Vistro-Yu, C. P. (2013). Cross-national studies on the teaching and learning of mathematics: Where

do we go from here? ZDM Mathematics Education, 45(1), 145-151. doi:10.1007/s11858-

013-0488-4

Yackel, E., Underwood, D., & Elias, N. (2007). Mathematical tasks designed to foster a

reconceptualized view of early arithmetic. Journal of Mathematics Teacher Education, 10(4-

6), 351-367.DOI 10.1007/s10857-007-9044-x

Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Thousand Oaks, CA: Sage.

872