The Intimate Interplay Between Experimentation and Deduction: Some Classroom Implications

We understand experimental mathematics as the systematic experimental investigation of concrete examples of a mathematical structure in the search for conjectures about its properties. Experiments might be done by pencil-and-paper work, building physical models, and, of course, by using available computer programs for doing time-consuming calculations, geometric constructions, and other types of visualizations. The guiding intention of the chapter is the study of the interplay of experimentation and deduction (proof) in the context of the teaching of mathematics at lower and upper secondary schools as well as the education of teachers of mathematics. The didactical intentions of the chapter make it necessary to take a special philosophical perspective, which might be characterized as a combination of Lakatosian and non-Lakatosian viewpoints (Sect. 2). In Sect. 3 we shortly discuss three historical examples (Euler, Fourier series, Chinese Remainder Theorem) to give a more realistic image of the interplay between experimentation and deduction in contrast to the somewhat artificial character which didactical examples necessarily involve. In the following Sects. 4, 5, 6, 7, and 8, we discuss a few classroom examples of mathematical experimentation. Section 4 is on checking, guessing, and finding a proof, and in Sect. 5 we put a particular emphasis on finding and handling counterexamples. These two sections can be characterized as the classical domain of Lakatosian ideas transferred to the classroom. Section 6 explores the potential of viewing elementary Euclidean geometry from the point of view of statics. This approach has had a long tradition in mathematics since ancient times and represents a transition to discussing experimentation with hypotheses in Sect. 7 on modelling. The latter is definitely a non-Lakatosian viewpoint. Lastly, in Sect. 8 it is concluded that the interplay between experimentation and deduction needs to be incorporated in primary education and extended throughout for the long-term development of sound mathematical thinking and understanding of students.