Visual Organisers for Formal Mathematics

Formal mathematics involves definitions and deductions in a manner which is quite different from the mental processes of school mathematics. Formal definitions of function, limit, continuity, differentiation and integration (both Riemann and Lebesgue) involve possibilities that often conflict with the students’ previous experience, leading to confusion and alienation. Examples given to ,motivate - definitions invariably have specific properties that do not follow logically from the definition itself. For instance, examples of sequences are usually given by formulae so that the sequence ,gets closer and closer - to the limit, without actually reaching it. Consequently, many students believe that this is an essential property of the limit concept. Functions are nearly always given by formulae whose graphs look ,smooth - so that students have difficulty imagining anything different. When discontinuities are exemplified by drawing a graph, the picture is often represented as a number of curved pieces with a ,jump - at the point under consideration. The result is a widespread belief that a typical function is given by a formula and is continuous except at occasional isolated points. In this way the student builds up a personal concept image of the concepts at variance with the theory.