Teachers’ Evolving Mathematical Understandings
This work was funded by the NSERC University of Manitoba CRYSTAL grant Understanding the Dynamics
of Risk and Protective Factors in Promoting Success in Science and Mathematics Education
Project Summary
A deep conceptual understanding of elementary mathematics as appropriate for teaching is increasingly
thought to be an important aspect of elementary teacher capacity. This six year study explored over 500
preservice teachers' initial mathematical understandings and how these understandings developed during a
mathematics methods course for upper elementary (junior-intermediate) teacher candidates in a mid-sized
Ontario faculty of education. The methods course has been more recently supplemented by a newly designed
optional course in mathematics for teaching. Teacher candidates choosing the optional course were initially
weaker in terms of mathematical understanding than their peers, yet showed stronger mathematical
development after engaging in the extra hours of specialized mathematics coursework the course provides. As
a result of this study, a rigorous final examination in mathematics concepts as required for classroom teaching
now forms a required component of the methods course at our institution, and teacher candidates must now
pass this examination with a grade of at least 60% in order to graduate from the program.
Background and Purpose
Ontario mathematics education initiatives have been influenced by the fundamental changes to the elementary
and secondary mathematics curricula, which are grounded in notions of inquiry and problem-solving. These
documents advocate mathematical learning experiences that include exploration and problem solving rather
than more traditional learning environments typically consisting of applying formulas to routine problems. A
consequence of implementing this vision of mathematics teaching and learning is that teachers themselves
now require a deeper mathematical knowledge, which includes more than knowledge of the given curriculum
(Silverman & Thompson, 2008). Rather, teachers need an understanding which is flexible enough to apply in
different problem-solving situations, and which allows them to support and diagnose student methods, enables
them to understand alternate models and strategies, and helps them pose appropriate questions (Ball, Thames,
& Phelps, 2008). Making explicit links among models and examples and mathematical methods that work in
general also requires a depth of understanding which we find typically not available to our new teachercandidates, and these understandings of models and connections are the focus of our mathematical work with
preservice teachers.
In Canada, recent work has advocated for the importance of increasing the amount of specialized mathematics
coursework for mathematics education. In the Policy Statement issued by the Working Group on Mathematics
for Teaching (Kajander & Jarvis, 2009), a minimum of 100 hours of mathematics course work is
recommended for prospective teachers, with an emphasis on experiences that address the specialized content
knowledge necessary for teaching mathematics. One of the aspects of this specialized content knowledge is a
deep, flexible and connected understanding of school mathematics, termed conceptual knowledge, which is
needed in order to construct classroom-appropriate experiences and models, pose questions, and diagnose
student methods and errors. Conceptual knowledge should ideally underpin and connect with procedural
knowledge, which is an ability to apply general methods and procedures. Alternately, a more traditional model
of mathematics instruction involves the direct teaching-by-transmission of procedures, formulas and rules.
Such procedures may or may not be accompanied by conceptual understanding. The concern is that teachers,
especially elementary teachers, are typically not given enough time and opportunity to develop these deep
mathematical understandings as needed for effective teaching.
Method
The study involved 580 preservice junior intermediate teachers, as well as about 50 classroom teachers, and
used a mixed-methods design. A validated paper and pencil survey called the Perceptions of Mathematics
survey (Removed for Review) was administered to teacher-candidates at the beginning and end of their
mathematics methods course, during five one-year cohorts. Participants included graduates of both the
previous and current Ontario curricula. As well, a subset of the participants were interviewed at the beginning
and end of the course to further explore their growth in knowledge and beliefs about mathematics. Methods
course grades, assignment work, and written examination responses were also examined as potential data
sources. As well, classroom teachers participating in school-board administered in-service were given the
same survey, and six of these classroom teachers were also interviewed as well as observed while teaching.
Participant interviews were transcribed and analyzed, and survey scores were analyzed statistically (Removed
for Review). As well, individual item responses on the survey were coded as to response type. Survey
responses were scored as procedural knowledge (evidence of the ability to compute correct answers to
standard upper elementary mathematics questions) and conceptual knowledge (evidence of understanding of
the underlying concepts, meaning, related models, and reasoning). For further details on the instrument see
Removed for Review and Removed for Review.
Findings
The study consistently showed that, while they were typically able to compute answers to computational
questions drawn from the junior-intermediate mathematics curriculum Expectations, teacher candidates
entering the mathematics methods course struggled with providing any explanation, model, example or
reasoning to support their numeric responses. Mean scores on the conceptual knowledge survey items
consistently hovered in the 10% range at the start of the methods course, while means of procedural
knowledge items were much higher (typically about 70%). In the first two years of the study, no graduates of
the reformed Ontario elementary and secondary mathematics curricula had yet reached the professional year
program, with only a few in the third year of the study. However, by years four and five, a number of
participants had experienced the new curriculum for at least part of their school years. No growth in overall
conceptual knowledge mean scores was observed at the pretest over the five years of the study however.
Significant growth in conceptual understanding was observed in all years of the study by the end of the
methods course. However we remained concerned that conceptual understanding was generally far from what
we considered sufficient for effective classroom teaching. Hence two program changes were instigated in year
three of the study, namely the introduction of an optional course in “Mathematics for Teaching” which
participants could elect to take in parallel with the methods course, and a mandatory final examination in
mathematics concepts as needed for teaching. The examination, which must now be passed with a grade of
60% to graduate from our program, is designed each year by faculty working in conjunction with a panel of
expert classroom teachers, mathematicians, and in-service professionals.
These program changes may have helped to support the growth in development we have noticed in the latter
years of the study. As well, our data show that the participants who chose to take the extra optional
“Mathematics for Teaching” course started out significantly weaker in terms of understanding, but caught up
to their peers by the end of the program. Figure 1 illustrates conceptual knowledge findings for participants
who chose to take the extra course, compared with their peers who did not, during years four and five of the
study.
10
9
8
7
6
5
4
3
2
1
0
6.07
5.97
Did not take MT
course
Took MT course
1.27
0.67
preCK
postCK
Figure 1. Conceptual knowledge (CK) scores for the pre and post test of the two years of data. The maximum scores for the both the
pretest and post-test are 10. Mean scores are given for each of the two groups: those who did not take the Math for Teaching (MT)
course and those who did.
Practical Implications
Our findings suggest that preservice teachers benefit measurably from extra opportunities to learn
mathematics as required for classroom teaching. Further, our data suggest that without such opportunities,
teachers may not be prepared to support problem-based learning as described in the Ontario curriculum, given
that such learning requires teachers to understand concepts, models, reasoning, and alternate strategies. As
well teachers must construct problem-based lessons, help students address misconceptions, and pose good
questions. All these abilities require strong conceptual understandings on the part of teachers, and our study
suggests that these understandings are not typically developed prior to entering teacher education. Further, we
argue that significantly increased opportunities for teachers to develop mathematical understanding as needed
for effective teaching need to be offered more generally to support the enactment of the inquiry-based vision
of the Ontario mathematics curriculum.
References
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
Journal of Teacher Education, 59(5), 389-407.
Removed for Review
Removed for Review
Kajander, A., & Jarvis, D. (2009). Report of the working group of elementary mathematics for teaching.
Canadian Mathematics Education Forum. Simon Fraser University, Vancouver, B.C.
Removed for Review
Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical
knowledge for teaching. Journal of Mathematics Teacher Education, 11, 499-511.