PERCEPTIONS, OPERATIONS AND PROOF IN UNDERGRADUATE MATHEMATICS

Teaching and learning undergraduate mathematics involves the introduction of ways of thinking that at the same time are intended to be more precise and logical, yet which operate in ways that are unlike students’ previous experience. When we think of a vector, in school it is a quantity with magnitude and direction that may be visualized as an arrow, or a symbol with coordinates that can be acted upon by matrices. In university mathematics it is an element in an axiomatic vector space. As I reflected on this situation I realised that these three entirely different ways of thinking apply in general throughout the whole of mathematics [7, 8]. The two ways encountered in school depend on the one hand on our physical perception and action and dynamic thought experiments as we think about relationships, on the other they depend on operations that we learn to perform such as counting and sharing which in turn are symbolised as mathematical concepts such as number and fraction. At university, all this is turned on its head and reformulated in terms of axiomatic systems and formal deduction. Our previous experiences are now to be refined and properties are only valid if they can be proved from the axioms and definitions using mathematical proof. The formal approach gives a huge bonus. No longer do proofs depend on a particular situation: they will hold good in any future situation we may meet provided only that the new context satisfies the specific axioms and definitions. However, the new experience is also accompanied by mental confusion as links, previously connected in perception and action, now require reorganisation as formal deductions, and subtle implicit links from experience may be at variance with the new formal setting. Further analysis of the development of mathematical thinking reveals three quite different forms of thinking and development that I term conceptual embodiment, operational symbolism and axiomatic formalism. These operate in such different ways—not only at a given point in time, but also