Bibliography 2: Preservice Teachers' Conceptions

Preservice Teachers' Conceptions of Mathematics
and How to Teach It

by Annie and John Selden

June 5, 1997

This is a supplement to the Research Sampler column of the same name.

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 univalence.
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: Macmillan.

A good overview of the research on teachers' beliefs and conceptions of mathematics teaching and learning (up to about 1989).
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, 25(4), 346-370.

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 classes.
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 being transitive.