Developing Prospective Elementary Teachers' Flexibility in the Domain of Proportional Reasoning

Abstract

Flexibility in the use of mathematics procedures consists of the ability to employ multiple solution methods across a set of problems, solve the same problem using multiple methods, and choose strategically from among methods so as to reduce computational demands. The purpose of this study was to characterize prospective elementary teachers' (n = 148) flexibility in the domain of proportional reasoning before formal instruction and to test the effects of two versions of an intervention that engaged prospective teachers in comparing different solutions to proportion problems. Results indicate that (a) participants exhibited limited flexibility before formal instruction, (b) the intervention led to significant gains in participants' flexibility that were retained six months after instruction, and (c) varying the source of the problem solutions that participants compared had no discernible effects on their flexibility. Implications for mathematics teacher preparation and for research on flexibility development are provided.

An earlier version of this article was presented at the 2005 Annual Meeting of the American Educational Research Association in Montreal, Canada. The research reported in this paper was supported in part by a grant from the National Science Foundation (Grant 0083429 to the Mid-Atlantic Center for Teaching and Learning Mathematics). The views expressed in the article are those of the authors and not necessarily those of the Foundation. We gratefully acknowledge James Hiebert, Yuichi Handa, and James Beyers for their assistance and feedback.

Notes

¹Of the 154 prospective teachers enrolled in the course, 148 gave consent for their written work to be analyzed. Prospective teachers who were randomly selected and who consented to participate in the interviews earned nominal course credit. Alternative assignments for equivalent course credit were provided for the remaining prospective teachers who either did not consent or were not chosen to participate in the interviews.

²For example, in Problem A, 9 and 12 share a common factor of 3, enabling one to easily scale down the given ratio of 12 inches: 9 miles to 4 inches: 3 miles, which can then easily be scaled up to the desired ratio of 16 inches: 12 miles.