This miniproject aimed to create a unique DVD resource for mathematicians who, as part of their professional development, want to understand more about how students think as they try to construct mathematical proofs. This resource addressed the difficult transition from calculation to proof usually encountered at the second-year undergraduate level and above. It is suitable for use both in professional development workshops and by interested individuals.
The DVD contains material from two research interviews, in each of which a student attempts to answer proof tasks about upper bounds of sets of real numbers and about functions and maxima. The video content is specially annotated so that half of the screen shows the student as they work on the problem and the other half is white and has subtitles appearing in blue at the bottom and a printed version of what they are writing on the page at the top. This allows the viewer to follow the student’s thinking without being distracted by having to switch between document types. The content is also broken up into short excerpts, each of which is followed by a screen with suggestive questions to facilitate reflection on the material and/or discussion between colleagues.
Watching the content highlights a number of major causes of student difficulty in learning to construct proofs, including:
1. possessing standards of justification different from those held by mathematicians;
2. lack of appreciation for the role of definitions in formal mathematics;
3. errors in logical reasoning, particularly when this involves quantifiers and/or “if…then” statements;
4. restricted knowledge of examples of important mathematical concepts; and,
5. lack of proof strategies and/or difficulty in applying these effectively.
A further aim of the project was to allow viewers to examine and develop their own conceptions and theories of learning by discussing these with others, thereby creating, at least on a temporary basis, what Jaworski might call a community of inquiry [1]. It is worth noting that this opportunity is often lacking in the UK academic experience; in most universities, a single course is taught by a single lecturer, so there is little in the way of natural occasion for in-depth discussion of actual student responses to particular tasks or types of instruction.
One thing this video material does is to provide common experience to a group which can be discussed there and then.
This DVD may be requested from Lara Alcock, Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU; l.j.alcock@lboro.ac.uk
[1] Jaworski, B. (2003) ‘Research practice into/influencing mathematics teaching and learning development: Towards a theoretical framework based on co-learning partnerships’, Educational Studies in mathematics, 54, 249-282.