This miniproject aimed to support the MSOR community’s teaching of proof to pure mathematicians through: research in this area, producing a comprehensive guide, and promoting the guide to the MSOR community.
Work towards the selection of the themes the guide touched on consisted largely of drawing on relevant literature and on the previous studies made by the project holders in this area.
With regard to the former, the project holders searched the mathematics education research literature in order to identify tried-and-tested recommendations regarding the teaching of Proof to mathematics undergraduates.
With regard to the latter, these studies included the recently completed ‘Engaging mathematicians as educational co-researchers’.
The guide, ‘How to prove it: a brief guide for teaching proof to year 1 mathematics undergraduates’ consists of five sections, each highlighting a significant aspect of teaching and learning Proof:
1. conceptualising formal mathematical reasoning and the necessity for Proof;
2. the role of examples in Proof: the tension between the general and the particular and proof-by-example;
3. the role of examples in Proof: Proof by Counterexample;
4. Proof by Mathematical Induction: the step from n to n+1; and,
5. Proof by Contradiction: spotting the contradiction.
Each section is further split in three sub-sections:
1. Data sample;
2. Some mathematics education research findings; and,
3. Transforming theory into pedagogical practice.
For the data sample sub-section, examples from the data that was collected in the course of the project holders’ previous studies and that illustrates the theme. This usually starts with examples of student written responses to a mathematical problem that required the use of a particular aspect of Proof that the theme of the section aims to explore. It continues with a brief reference to the comments made on the student responses by the mathematicians interviewed in that study and concludes with a listing of issues that the reader is invited to consider in the light of these examples.
Following consultations with mathematicians at University of East Anglia, University of Birmingham, The Open University, The University of Swansea, Heriot Watt University Edinburgh, and a working session at the Mathematical Association Conference 2003, analyses were made to help identify cross-episode patterns in attitudes, beliefs and practices. Of particular interest was to the collaborative nature of the research: the vibrancy of the groups’ views and the enthusiasm with which the members of the group engaged in the conversation helped the content of the conversations escalate beyond the remit of the pre-determined themes. Indeed the participants, by constantly re-shaping the focus of the discussion, were determining the actual content of the data and eventual focus of the research. They were thus becoming co-researchers.