Advanced Mathematical Thinking

Abstract

This chapter presents the principal currents and results in recent cognitive research on advanced mathematical thinking processes. The ambiguity in the phrase advanced mathematical thinking processes is intentional: The word advanced can and should refer to either mathematics or processes or both. Advanced mathematical thinking processes thus include on the one hand, thinking about topics in advanced mathematics, which in this context means mathematics beyond Euclidean geometry and intermediate algebra, and on the other hand, advanced processes of mathematical thinking such as abstracting, proving, and reasoning under hypothesis. Although the domain of mathematics subsumed under the term advanced mathematics is vast, only a restricted number of topics are commonly taught in upper secondary and beginning college classes. Far more hours of teaching and learning are spent on analysis than on any other topic in mathematics, and it is therefore not surprising that the majority of research on the teaching and learning of advanced mathematics has been concerned with topics in analysis, among them functions, differentiation and integration, and differential equations. This tendency has been strengthened by the fact that the teaching of analysis poses a large number of nontrivial problems. There do not seem to be clear-cut characteristics that set advanced mathematical concepts apart from those in elementary mathematics. Each advanced concept is, however, based on more elementary concepts and cannot be grasped without a solid and sometimes very specific understanding of these.