## Abstract

This is a theoretical article proposing a way of organizing and structuring the discussion of why and how to use the history of mathematics in the teaching and learning of mathematics, as well as the interrelations between the arguments for using history and the approaches to doing so. The way of going about this is to propose two sets of categories in which to place the arguments for using history (the “whys”) and the different approaches to doing this (the “hows”). The arguments for using history are divided into two categories; history as a tool and history as a goal. The ways of using history are placed into three categories of approaches: the illumination, the modules, and the history-based approaches. This categorization, along with a discussion of the motivation for using history being one concerned with either the inner issues (in-issues) or the metaperspective issues (meta-issues) of mathematics, provides a means of ordering the discussion of “whys” and “hows.”

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## Notes

Ontogenesis is the evolution of a single organism (or individual) from the fertilization of an egg to adulthood and phylogenesis is the evolution of an entire species. The recapitulation argument is also sometimes referred to as the genetic principle; however, the genetic principle is also a principle of education, i.e., an approach, and it will be discussed as such later.

Examples of other papers discussing the possible benefits to be gained by in-service and preservice teachers from knowing and learning about the history of mathematics are Freudenthal (1981), Heiede (1996), and Barabash and Guberman-Glebov (2004). Regarding empirical studies on this matter, the following may be mentioned: Philippou and Christou (1998), Waldegg (2004), Su (2007), Horng (2007), Arcavi and Isoda (2007), Furinghetti (2007), and Goodwin (2007).

A general discussion of gains and losses of classification may be found in the book by Bowker and Star (1999).

Some of the empirical studies already mentioned do this to some degree. In addition to these may be mentioned, among others, McBride and Rollins (1977), Fraser and Koop (1978), and Lit, Siu, and Wong (2001). Further examples of such empirical studies may be found in the recently published proceedings from ESU5, the proceedings from HPM 2008 (published on a CD-ROM), and the forthcoming proceedings from working group 15 at CERME6.

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## Acknowledgements

I would especially like to thank Mogens Niss, who provided me with the basic idea for this article. Also thanks to Tinne Hoff Kjeldsen, Man-Keung Siu, Jan van Maanen, and Abraham Arcavi for fruitful discussions. Thanks as well to the editor and the anonymous ESM reviewers for constructive revision. Thanks to Merete Carlsen and Jørgen Larsen for linguistic and technical assistance respectively, and thanks to Imajean Gray^{†} for proofreading.

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Jankvist, U.T. A categorization of the “whys” and “hows” of using history in mathematics education.
*Educ Stud Math* **71**, 235–261 (2009). https://doi.org/10.1007/s10649-008-9174-9

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DOI: https://doi.org/10.1007/s10649-008-9174-9

### Keywords

- Using history in mathematics education
- Whys and hows
- History as a tool, history as a goal
- Indispensability of arguments
- In-issues and meta-issues
- Illumination, modules, and history-based approaches
- Genetic principle