Skip to main content

Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis

  • 81 Accesses

Part of the Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (ACSHPM)

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-95201-3_11
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   139.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-95201-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   179.99
Price excludes VAT (USA)
Hardcover Book
USD   179.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Notes

  1. 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. 2.

    This example comes from the Preface of Middlemiss (1952).

  3. 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

  • Berkeley G (1951) The Analyst: or A Discourse Addressed to an Infidel Mathematician. Wherein it is examined whether the object, principles, and inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than religious Mysteries and points of Faith. First printed in 1734. In Luce A, Jessop T, editors, The Works of George Berkeley Bishop of Cloyne, Vol. 4, pp. 53–102. Nelson, London Full text, edited by David R. Wilkins, available on line at https://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf.

  • Dedekind R (1965) Stetigkeit und Irrationale Zahlen Friedr. Vieweg & Sohn, Braunschweig, 7th edition. 1st edition, 1872.

    Google Scholar 

  • Devlin K (2005) Plenary address Delivered at a meeting on June 3, 2005 of the Canadian Mathematical Society.

    Google Scholar 

  • Grabiner JV (1997) Was Newton’s Calculus a Dead End? the Continental Influence of Maclaurin’s Treatice of Fluxions. American Mathematical Monthly 104:393–410. Preliminary version presented as an invited address under the title “Was Newton’s Calculus a Dead End? A New Look at the Calculus of Colin Maclaurin” to the Sixteenth Annual Meeting of the Canadian Society for the History and Philosophy of Mathematics, University of Victoria (British Columbia), May 31–June 1, 1990.

    Google Scholar 

  • Heath TL (1912) The Works of Archimedes Cambridge University Press Reprinted by Dover (no date given).

    Google Scholar 

  • Heath TL (1926) The Thirteen Books of Euclid’s Elements Cambridge University Press, second edition Three volumes. Reprinted by Dover, 1956. The text, along with a Java applet to manipulate the diagrams can be found on line at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.

  • Knorr W (1975) The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry Reidel, Dordrecht and Boston and London.

    Google Scholar 

  • Middlemiss RR (1952) College Algebra McGraw-Hill.

    Google Scholar 

  • Nardi E (2014) Reflections on visualization in mathematics and in mathematics education In Fried M, Dreyfus T, editors, Mathematics and Mathematics Education: Searching for Common Ground, Advances in Mathematics Education. Springer.

    Google Scholar 

  • Seldin J (1990) Reasoning in elementary mathematics In Berggren T, editor, Canadian Society for History and Philosophy of Mathematics. Proceedings of the Fifteenth Annual Meeting, Quebec City, Quebec, May 29–May 30, 1989, pp. 151–174 Available on line from http://www.cs.uleth.ca/~seldin under “Publications”.

  • Seldin J (1991) From exhaustion to modern limit theory In Abeles FF, Katz VJ, Thomas RS, editors, Canadian Society for History and Philosophy of Mathematics. Proceedings of the Sixteenth Annual Meeting, Victoria, British Columbia, May 31–June 1, 1990, pp. 120–136 Available on line from http://www.cs.uleth.ca/~seldin under “Publications”.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jonathan P. Seldin .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

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

Download citation