Our regular feature highlighting an article or research paper that will, we hope, have a helpful bearing on your teaching of mathematics. You can find previous features in this series here.
In this issue we share with you a paper first presented at the BSRLM Proceedings in 2006. Celia Hoyles’s (the founding Director of NCETM) and Dietmar Küchemann’s paper Secondary School Pupils’ Approaches to Proof-related Tasks in Geometry gives an account of two pupils’ attempts to solve two GCSE geometry questions involving circle theorems. The paper identifies some of the characteristics of such tasks, some of the pupils’ emerging strategies, and some of the difficulties the pupils encountered, especially with using the givens and extracting information contained in diagrams.
The paper starts by recording how the pupils find (a) angle CBA and (b) angle CDA in the diagram above. Subsequently they work on a different problem:
Here AOB is a diameter, TCS is a tangent, angle ABC = 57˚ and pupils are asked to find (a) angle CAB and (b) angle ACS.
The paper presents a transcript of the pupil/researcher conversations as they work through the problem: we think it’s a conversation that will sound familiar to you, and that you’ll recognise many of the misconceptions and difficulties that the pupils encounter. We hope that this paper will help you steer your pupils’ development of more secure conceptual understanding and procedural fluency, so that they start to tackle circle theorem questions, not least on GCSE exam papers, successfully and confidently. Let us know what you think.