This miniproject aimed to analyse the use of dedicated self-study workbooks in the teaching of elementary abstract proof to second-year undergraduates. The project’s educational research dimension evaluated the extent to which the workbooks affect a single body of students’ attitudes and progress over the course of the module.
The project members investigated whether such workbooks help students learn to read, to understand and to write proofs of their own. It was also important to consider whether the preparation and use of such workbooks are an effective use of teaching resources.
In the first phase of the project in 2006/07 Brown, as lecturer, with input from Dunning, designed three self-study workbooks focusing on topics in abstract algebra. The intentions of the workbooks were informed by earlier examples used in the teaching of analysis, Alcock’s expertise in maths educational theory, the literature on learning group theory and the authors’ impressions from their own experiences of teaching.
Two twelve-page workbooks were designed to support the second-year algebra module, ‘Groups, Rings and Fields’, taught at the University of Kent. This course is typically taken by around 25 mathematics students, most of whom go on to study algebra topics in the third year of their course. A further, third, workbook was prepared for the third-year algebra module ‘Polynomials of Several Variables’.
The three workbooks covered Lagrange’s theorem, Factorisation in integral domains, and Cardinals and the size of infinite sets. All workbooks contain a brief but comprehensive review of relevant material, some of which may have already been lectured. Rather than writing out proofs, the workbooks describe the strategy for proofs and leave a series of blank boxes in which the students can fill in the mathematics for themselves.
One aim of the workbook design was that amid the large lecture notes, students would become familiar and comfortable with a few small ‘islands’ of difficult material and become accustomed to working with a balance of abstract ideas and concrete calculations. Additionally, it was hoped that certain key points, such as definitions and theorems, would be reinforced and, if necessary, corrected. The workbooks were first used in 2006/07, replacing in each case a standard piece of assessed coursework.
During the first year of the workbooks’ use the students completed a feedback form seeking qualitative free-text feedback. The responses gave an insight into what students think about their learning and study skills, as well as some concrete feedback that led to small modifications of the workbooks.