## Abstract

In this paper, we argue that history might have a profound role to play for learning mathematics by providing a self-evident (if not indispensable) strategy for revealing meta-discursive rules in mathematics and turning them into explicit objects of reflection for students. Our argument is based on Sfard’s theory of *Thinking as Communicating*, combined with ideas from historiography of mathematics regarding a multiple perspective approach to the history of practices of mathematics. We analyse two project reports from a cohort of history of mathematics projects performed by students at Roskilde University. These project reports constitute the experiential and empirical basis for our claims. The project reports are analysed with respect to students’ reflections about meta-discursive rules to illustrate how and in what sense history can be used in mathematics education to facilitate the development of students’ meta-discursive rules of mathematical discourse.

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## Notes

For literature on using history in mathematics education, see these references, for example (Arcavi & Bruckheimer, 2000; Bruckheimer & Arcavi, 2000; Fauvel, 1991a, b; Fauvel & van Maanen, 2000; Furinghetti, 2004; Jankvist, 2009a; Jankvist & Kjeldsen, 2011; Katz, 2000; Katz & Michalowicz, 2004; Kjeldsen, 2010, 2011a, b). For special issues on history in mathematics education journals, see

*For The Learning of Mathematics*(vol. 11(2), 1991; vol. 17(1), 1997) as well as*Educational Studies in Mathematics*(vol. 66(2), 2007).See also Fried (2007) for further discussions about the difficulties of integrating history of mathematics into mathematics teaching and learning.

The term whig history was introduced by the British historian Herbert Butterfield in the 1930s. It is explained in Section 3 of the present paper.

For a history of such statements and their role in mathematics, see Lützen (2009).

See e.g. Leo Corry’s introduction (Corry, 2004) as well as the rest of the papers published in

*Science in Context*, 17(1/2), 2004. See also (Daston, 1988; Dahan-Dalmedico, 1996; Epple, 1999; Kjeldsen, Pedersen & Sonne-Hansen, 2004; Kjeldsen 2006) to name just a few where also further references can be found. In history of science, there has been a trend to approach the development of the experimental sciences from such a perspective emphasizing the local nature of research practices; see Buchwald and Franklin (2005, p. 1). For methodological discussions of this approach in history of mathematics, see e.g. Epple (2004, pp. 131–164).This is inspired by the Danish historian Bernard Eric Jensen. We have borrowed the term

*multiple-perspective approach*, which is a direct translation from Danish of his term “flerperspektivisk tilgang”, from his work (Jensen, 2003).For further details on the project-organised study programmes at Roskilde University, see Niss (2001). For a presentation and discussion of how history and philosophy of mathematics and science is integrated in the 2-year introductory science and mathematics programme, also through problem-oriented project work, see Kjeldsen and Blomhøj (2009).

The exact formulation of the problem that guides the students’ project work is a process that often goes on throughout the entire semester where the problem formulation becomes more and more qualified, as the students become more and more familiar and knowledgeable within the scope of their project work.

All the quotes from the students’ report have been translated into English by the authors of the present paper.

The “contract” with the students was that they should provide an agenda for the weekly meeting with their supervisor a few days in advance. As a supervisor, I always try to press the students to deliver something in writing for each meeting accompanied by a reading direction such as: you only need to skim part such and such; in section such and such, we need feedback on the issue of such and such; we consider part such and such to be almost finished, so please read it carefully, etc.

Often students and their supervisor will discuss the problem formulation throughout the entire semester focusing on whether the problem can actually guide the students’ work, whether it is in accordance with what the students are actually investigating and so on. As has been explained (Blomhøj & Kjeldsen, 2009; Kjeldsen & Blomhøj, 2009), the students do not always manage to formulate good research questions that are consistent with what they have researched in their project. In the present case, the problem formulation is way too broad to actually guide the students’ project work. This does not mean that the project work has not been successful, but it indicates that the students’ formulation of their problem is not a precise articulation of what they actually searched for.

It is important to be aware that this was a time when the concept of a function was not clearly defined. Even though there are some explicit, general definitions of a function around in the eighteenth century, there was a gap between the way mathematicians

*used*functions and how they*defined*them (Lützen, 2002).See e.g. (Lützen, 2002).

In Jankvist and Kjeldsen (2011), the lack of impact of history in mathematics education research is discussed and new avenues for history in mathematics educations are proposed together with explanations of theoretical constructs that can be used as guidelines for practice. In Kjeldsen (2011b), a theoretical framework is outlined that can be used to analyse specific implementations that have been realized and as a tool to orient the design of future implementations of history of mathematics in mathematics education.

This suggestion was outlined in at the workshop

*Does history have a significant role to play for the learning of mathematics? Multiple perspective approach to history, and the learning of meta level rules of mathematical discourse*at the The 6th. European Summer University on the History and Epistemology in Mathematics Education (ESU 6) in Vienna, July 19–23, 2010, and interested readers are referred to the forthcoming proceedings of the meeting, see Kjeldsen (2011c). A matrix organised design for using history in mathematics education to elucidate meta-rules of past and present mathematics, to have students reflect upon those, to develop students’ mathematical competence, and general educational skills of independence and autonomy is being tried out in a Danish upper secondary class at the moment. Results from the study will be published in forthcoming papers.

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Kjeldsen, T.H., Blomhøj, M. Beyond motivation: history as a method for learning meta-discursive rules in mathematics.
*Educ Stud Math* **80**, 327–349 (2012). https://doi.org/10.1007/s10649-011-9352-z

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DOI: https://doi.org/10.1007/s10649-011-9352-z