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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.
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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
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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
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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
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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).
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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
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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
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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).
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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
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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
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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).
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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
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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.
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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
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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
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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.
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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.
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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.
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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
• • •
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