Teachers’ evolving mathematical understandings
What is this research about?
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 pre-service 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
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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
teacher-candidates, and these understandings of models and connections are the focus
of our mathematical work with pre-service 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.
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What did the researchers do?
The study involved 580 pre-service 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 (Kajander, 2010) 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 (Kajander, 2010). 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 Kajander & Mason (2007) and Kajander
(2010).
What did the researchers find?
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
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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.
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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.
How can you use this research?
Our findings suggest that pre-service 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.
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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.
Kajander. A. (2010). Mathematics teacher preparation in an era of curriculum change:
The development of mathematics for teaching. Canadian Journal of Education.
33(1), 228-255.
Kajander. A. (2010). Teachers constructing concepts of mathematics for teaching and
learning: “It’s like the roots beneath the surface, not a bigger garden”. Canadian
Journal of Science, Mathematics and Technology Education. 10(2), 87-102.
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.
Kajander, A. & Mason, R. (2007). Examining teacher growth in professional learning
groups for in-service teachers of mathematics. Canadian Journal of Mathematics,
Science and Technology Education, 7(4), pp. 417-438.
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.
About this summary
This research summary was developed from Ann Kajander’s study:
Kajander. A. (2010). Teachers constructing concepts of mathematics for teaching
and learning: “It’s like the roots beneath the surface, not a bigger garden”.
Canadian Journal of Science, Mathematics and Technology Education. 10(2),
87-102.
This research summary reflects findings from this study only and is not necessarily
representative of the broader body of literature on this subject. Please consult the
original document for complete details about this research. In case of any
disagreement, the original document should be understood as authoritative.
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Other related articles
Kajander, A, & Jarvis, D. (2009). Report of the working group on mathematics for
teaching. Canadian Mathematics Education Forum. Vancouver, BC.
http://math.ca/Events/CMEF2009/reports/wg2-report.pdf
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.
Key words
Math, Mathematics, Preservice teachers, transitions, teacher efficacy, Professional
development (PD)
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