mathematics Article How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics John Mason 1,2 1 Department of Mathematics and Statistics, Open University, Milton Keynes MK7 6AA, UK;
[email protected] 2 Department of Education, University of Oxford, 15 Norham Gardens, Oxford OX2 6PY, UK Abstract: The theme of ‘undoing a doing’ is applied to the ubiquitous action of exchange, showing how exchange pervades a school’s mathematics curriculum. It is possible that many obstacles encountered in school mathematics arise from an impoverished sense of exchange, for learners and possibly for teachers. The approach is phenomenological, in that the reader is urged to undertake the tasks themselves, so that the pedagogical and mathematical comments, and elaborations, may connect directly to immediate experience. Keywords: doing and undoing; exchange; substitution 1. Introduction Arithmetic is seen here as the study of actions (usually by numbers) on numbers, whereas calculation is an epiphenomenon: useful as a skill, but only to facilitate recognition Citation: Mason, J. How the Theme of relationships between numbers. Particular relationships may, upon articulation and of ‘Doing and Undoing’ Applied to analysis (generalisation), turn out to be properties that hold more generally. This way of the Action of Exchange Reveals thinking sets the scene for the pervasive mathematical theme of inverse, also known as Overlooked Core Ideas in School doing and undoing (Gardiner [1,2]; Mason [3]; SMP [4]). Exploiting this theme brings to Mathematics. Mathematics 2021, 9, the surface core ideas, such as exchange, which, although appearing sporadically in most 1530.