The problem-solving approach in the teaching of number theory

Abstract

Mathematical problem solving is the mainstay of the mathematics curriculum for Singapore schools. In the preparation of prospective mathematics teachers, the authors, who are mathematics teacher educators, deem it important that pre-service mathematics teachers experience non-routine problem solving and acquire an attitude that predisposes them to adopt a Pólya-style approach in learning mathematics. The Practical Worksheet is an instructional scaffold we adopted to help our pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. The Worksheet was initially used in a design experiment aimed at teaching problem solving in a secondary school. In this paper, we describe an application and adaptation of the MProSE (Mathematical Problem Solving for Everyone) design experiment to a university level number theory course for pre-service mathematics teachers. The goal of the enterprise was to help the pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. Our analysis of the pre-service mathematics teachers’ work shows that the MProSE design holds promise for mathematics courses at the tertiary level.

Keywords:

• problem solving
• mathematics teacher education
• number theory

Acknowledgement

This research was partially supported by the National Institute of Education research grant OER 22/12 TTL.

Notes

As Pólya's [6] four-stage model and Schoenfeld's [1] framework of four components are now well-known, we assume the readers are familiar with them and thus are not included here.

We have renamed the fourth stage of Looking Back as Check and Expand.

All names of student teachers used in this article are pseudonyms.

The work of these student teachers as reported in this paragraph and the one before may have been considered by them to be Stages 2 and 3 of Pólya's model. Indeed some of them place these workings under Sections 2 and 3 of the worksheet (see Figure 2). However, we consider these as preliminary attempts prior to the main attack of the stated problem, so we classified them under Stage 1.