Foo Him Ho

The problem-solving approach in the teaching of number theory

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.

Exponential function and its derivative revisited

Most of the available proofs for rely on results involving either power series, uniform convergence or a round-about definition of the natural logarithm function ln(x) by the definite integral , and are thus not readily accessible by high school teachers and students. Even instructors of calculus courses avoid showing the complete proof to their undergraduate students because a direct and elementary approach is missing. This short article fills in this gap by supplying a simple proof of the aforementioned basic calculus fact.

An elementary proof of the identity

This article gives an elementary proof of the famous identity Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Eulers and Nevilles identities, as well as the identity .

The Impact of Online Video Suite on the Singapore Pre-service Teachers’ Buying-In to Innovative Teaching of Factorisation via AlgeCards

A group of pre-service teachers at one institute of education was assigned to record their reflections as they viewed an online video suite which recorded (1) a secondary mathematics teacher employing a Concrete-Pictorial-Abstract (CPA)-based approach to teaching quadratic expansion and factorisation to a group of low-ability secondary 2 (14+) mathematics students; (2) students using concrete manipulative, AlgeCards, and pictorial representations, Rectangle Diagram, to perform quadratic expansion and factorisation; and (3) teachers, including a Head of Department (HOD), sharing their teaching experience in the use of the teaching package based on the abovementioned CPA approach. This chapter studied the impact of the online video suite on the pre-service teachers’ willingness to buy-in to innovative teaching of factorisation via AlgeCards through inspecting the written responses of these teachers to preset Milestone Tasks placed at different points along the progression of the video watching. The term ‘buy-in’ refers to the evidential shift of teacher’s belief in the applicability of the lesson innovation. We identified the salient features of the video suite which were responsible for bringing about pre-service teachers’ buying-in to the use of AlgeCards in teaching factorisation. Based on these findings, this chapter makes some recommendations on the design of teacher preparation method course using video technology.