Teaching Mathematics and its Applications - Advance Access

A proof of the converse of the Pythagorean proposition

Scimone, A. — 2009-05-31

The article presents a demonstration of the converse of the Pythagorean Theorem based on the reductio ad absurdum. This is necessary to overcome the discrepancy, noticed by pupils, between the Euclidean purpose to demonstrate that the given triangle is right-angled and the auxiliary figure by which the given triangle is drawn as if it were already a right-angled one. To the eyes of students this does not make the Euclidean reasoning clear.

The immortal ant and the expanding balloon

Griffiths, M. — 2009-05-19

In this article we consider, via a specific modelling example, the educational benefits to be gained from running mathematical activities with our sixth-form and undergraduate students that, in modern parlance, might be termed ‘rich tasks’. The idea for this modelling activity arose while the author was reading a popular-science book on cosmology (in particular, on the possible shapes of the universe). Light travelling around the universe was likened to an ant crawling around a balloon. A statement in the book regarding the ant's progress around the balloon did not entirely ring true with the author, and his subsequent investigations led to the activity described here. We explore several scenarios associated with the model in order both to pre-empt possible paths taken by the students and to be able to provide some guidance when necessary. Suggestions are given as to how the activity may be extended, and then, after highlighting the numerous educational benefits, we consider the potential pitfalls and difficulties associated with the delivery of tasks such as these.

Teaching mathematics understandings for transfer

Jones, J. L., Jones, K. A., Vermette, P. J. — 2009-05-15

Promoting student understanding for transfer is an illusive hallmark of effective mathematics instruction. While much research has shown the necessity of promoting understanding for transfer, less attention has been paid to actual pedagogical strategies that can be used to promote transfer of mathematical ideas. Using Fogarty et al. (1992, How to Teach for Transfer, The mindful school, Palatine, Illinois: Skylight Publishing) ‘ten tools for teaching for transfer’ as a model, the authors of this piece provide numerous suggestions designed specifically for promoting transfer of concepts in secondary mathematics.

The Bolyai lamp, a new math manipulative modelling the hyperbolic plane

Gyorfi, Z. — 2007-11-23

The author, Zoltan Györfi, has requested that this article is withdrawn from publication.