The miniproject team wrote a short book entitled ‘Ideas from Mathematics Education: An Introduction for Mathematicians’ designed to be an accessible introduction to three main ideas that can help mathematicians to better understand their students’ thinking and to design instruction accordingly. The book addresses the need for resources that allow mathematicians to derive useful pedagogical content knowledge from mathematics education research literature, without having to immerse themselves in its technicalities.
The book is intended to be particularly useful for new lecturers enrolled on PGCHE programmes, but would also be of interest for experienced lecturers looking for further insight into their students’ thinking.
The guide focuses upon what students need to learn in order to succeed in undergraduate mathematics. Specifically, what they need to learn in order to make sense of their early encounters with abstract definitions and proofs, which are seen as key to making a successful transition to university-level mathematical thinking.
In brief, the main points of the three chapters within the guide are:
1. Definitions: Students are often unaware of the status of formal definitions within mathematical theory and attempt to work with concepts informally instead;
2. Mathematical objects: Many mathematical constructs can be understood as processes, but need to be understood as objects in order for higher level reasoning to make sense; and,
3. Two reasoning strategies: Two sensible ways of approaching a proof-related task are distinguished; each demands a range of skills and some individuals may have preferences for one or the other.
The discussion of each of these issues is based on research involving close observations of students’ learning of undergraduate mathematics. A number of quotations from students are included to illustrate their thinking.
The guide is very much about theories of learning mathematics, as opposed to general theories of learning or generic advice on ‘good practice’ in teaching. It is believed that this is important because mathematical content is central to a lecturer’s work in constructing lectures, notes, problem sheets etc., and because understanding students’ likely interpretations of this content is therefore directly relevant to this day-to-day design work.
The ideas described in the guide should prove helpful in planning mathematics teaching and in responding to students’ individual questions, because they allow the lecturer to think systematically about underlying difficulties that manifest themselves in a variety of misunderstandings and errors.
It is not suggested that there is a unique ‘best way to teach’, but many teaching strategies can be thought of in terms of how they address the issues raised.