Abstract
This paper discusses ideas for a different approach to teaching the foundations of mathematical analysis. The main idea is to avoid the use of what Keith Devlin in 2005 called “formal definitions,” which are definitions that nobody can understand without working with them. For students without mathematical maturity, these definitions can be difficult to understand and use. This paper discusses an approach that uses the history of mathematics to first develop fundamental concepts and only introduces formal definitions after the concepts are understood. The audience for this approach is third-year undergraduate students. Several examples are provided in the paper.
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Notes
- 1.
For the importance of visualization in the learning of mathematics, see Nardi (2014) as well as the publications of eric.ed.gov on the Role of Visualization in the Teaching and Learning of Mathematics and especially Mathematical Analysis as for example in the article of Miguel de Guzman in https://files.eric.ed.gov/fulltext/ED472047.pdf.
- 2.
This example comes from the Preface of Middlemiss (1952).
- 3.
Given the lack of a proper universal set in axiomatic set theory, we cannot formally define cardinal numbers as equivalence classes, but we can indicate how the idea behind this is approximately that of such equivalent classes to be covered.
References
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Kamareddine, F., Seldin, J.P. (2022). Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-95201-3_11
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