Preservice Teachers' Conceptions of Mathematics
How to Teach It
by Annie and John Selden
June 5, 1997
This is a supplement to the Research Sampler column of the
Return to the Bibliography or
the Research Sampler.
- Borko, H., Eisenhart, M., Brown, C. A., Underhill R.
G., Jones, D. & Agard P. C. (1992). Learning to teach
hard mathematics: Do novice teachers and their
instructors give up too easily?, Journal for Research
in Mathematics Education, 24(1), 23(3), 194-222.
Besides describing how one preservice teacher's
(Ms. Daniels') beliefs about teaching for understanding
could not be realized due to her lack of both subject
matter and pedagogical content knowledge (of the
division-of-fractions algorithm), this article describes
preservice teachers' unrealistic expectations that from
their methods course they could learn (by rote)
ready-to-use explanations (i.e., concrete
representations) for all algorithms they might be
required to teach one day.
- 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.
Explores messages, and unresolved tensions, about
teaching for procedural and conceptual knowledge that
one preservice teacher (Ms. Daniels) got over the course
of a year, from her methods course and the teachers and
administrations at four placement schools.
- Even, R. (1993). Subject-matter knowledge and
pedagogical content knowledge: Prospective secondary
teachers and the function concept, Journal for Research
in Mathematics Education, 24(2), 94-116.
Describes preservice secondary teachers' limited conceptions
of functions as formulas, as having "nice graphs," etc.,
with little or no appreciation for the "arbitrariness" of
the modern definition of function or the concept of
- Fennema, E. & Franke, M. L. (1992). Teachers' knowledge
and its impact. In D. A. Grouws, NCTM Handbook of
Research on Mathematics Teaching and Learning, (pp.
147-164). New York: Macmillan.
A good overview of the research on teachers' knowledge of
mathematics, mathematical representations, and students'
cognitions (up to about 1991).
- Gregg, J. (1995). The tensions and contradictions of
the school mathematics tradition, Journal for Research in
Mathematics Education, 26(5), 442-466.
An ethnographic study of one beginning high school
mathematics teacher's attempts to cope when she comes up
against students whose previous experience of mathematics is
as procedures, conventions, and rules. She eventually
constitutes her classes as procedurally as possible (e.g.,
because her geometry students dislike and have difficulty
with proofs, her job becomes planning them -- she only asks
students to name the procedures and theorems she's used).
Gregg sees such proceduralization as maximizing the
appearance of student and teacher competence, while
concurrently blaming students' failure to learn mathematics
on their (supposed) lack of ability.
- Simon, M. A. & Blume, G. W. (1994). Building and
understanding multiplicative relationships: A study of
prospective elementary teachers, Journal for Research
in Mathematics Education, 25(5), 472-494.
First discusses preservice elementary teachers' general
inability to distinguish additive from multiplicative
situations, the inadequacy of the repeated addition
interpretation of multiplication for many problems
(e.g., ratio), and the difference between numerical and
quantitative reasoning (cf., P. Thompson, in The
Development of Multiplicative Reasoning in the Learning of
Mathematics, G. Harel and J. Confrey, Eds., SUNY Press,
1994). Then analyzes the conceptual complexity of relating
a procedural conception of area (take two linear measures,
multiply them, and affix the label "square units") with an
image of a region actually constituted of squares.
- Thompson, A. G. (1992). Teachers' beliefs and
conceptions: A synthesis of the research. In D. A.
Grouws, NCTM Handbook of Research on Mathematics
Teaching and Learning, (pp. 127-146). New York:
A good overview of the research on teachers' beliefs and
conceptions of mathematics teaching and learning (up to about
- Wilson, M. R. (1994). One preservice secondary
teacher's understanding of function: The impact of a
course integrating mathematical content and pedagogy,
Journal for Research in Mathematics Education,
A case-study of 20-year old Molly, who initially saw
functions in terms of computational activities (function
machines, point plotting, vertical line test) and had a
very procedural view of mathematics. Through various
innovative instructional activities described in the
paper, Molly's understanding of functions grew, but her
approach to teaching them did not. She saw such activities
as merely providing breaks in the monotony of lecture
- Zazkis, R. & Campbell, S. (1996). Prime decomposition:
Understanding uniqueness, The Journal of Mathematical
Behavior, 15(2), 207-218.
A study of preservice teachers' (mis-)understandings of prime
decomposition, along with some suggestions for why they have
difficulties, e.g., the concept of divisibility is
constructed prior to the concept of indivisibility. Also,
many of these students believe prime decomposition means
decomposition into small primes.
- Zazkis, R. & Gunn, C. (1997). Sets, subsets, and the
empty set: Students' constructions and mathematical
conventions, Journal of Computers in Mathematics and
Science Teaching, 16(1), 131-169.
A study of preservice teachers' (mis-)conceptions about sets
after that topic had been covered in class, along with some
suggestions about why students have difficulties, e.g.,
that "is an element of" and "belongs to" are viewed as
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the Research Sampler column.