Exercises in mathematical imagining: setting out a teaching instrument that evokes imaginings and utilises visualisations in secondary school mathematics

Abstract

This paper sets out a teaching instrument which could be referred to as ‘mental imagery exercises in mathematics’ or ‘exercises in mathematical visualisation’. Originally developed in the context of the teaching principles of the German-speaking countries, and then based on theories of the German mathematics education literature, the paper will reframe this task design and conclude that the term ‘exercises in mathemat- ical imagining’ is most appropriate. The fact that imagining—i.e. mentally forming and manipulating images—is personal means that this approach enables students to experience mathematics as commencing in their own minds. Teachers are also simultaneously able to gain insight into their students’ thought processes through this approach. In contrast with mental arithmetic, this task design does not focus on training a particular ability. Rather it promotes a specific heuristic strategy, within which students first conceive of and imagine a mathematical topic, and then construct and explore visualisations in order to understand their implications. This paper should be read as a case study of the researcher’s day-to-day teaching. It makes explicit a practice that has for many years produced positive results in terms of students developing mathematical understanding.