Abstract
The article first investigates the basis for designing teaching activities dealing with aspects of history, applications, and philosophy of mathematics in unison by discussing and analyzing the different ‘whys’ and ‘hows’ of including these three dimensions in mathematics education. Based on the observation that a use of history, applications, and philosophy as a ‘goal’ is best realized through a modules approach, the article goes on to discuss how to actually design such teaching modules. It is argued that a use of primary original sources through a so-called guided reading along with a use of student essay assignments, which are suitable for bringing out relevant meta-issues of mathematics, is a sensible way of realizing a design encompassing the three dimensions. Two concrete teaching modules on aspects of the history, applications, and philosophy of mathematics—HAPh-modules—are outlined and the mathematical cases of these, graph theory and Boolean algebra, are described. Excerpts of student groups’ essays from actual implementations of these modules are displayed as illustrative examples of the possible effect such HAPh-modules may have on students’ development of an awareness regarding history, applications, and philosophy in relation to mathematics as a (scientific) discipline.
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Notes
Philosophy in general, on the other hand, is often used within theoretical constructs and frameworks in mathematics education.
For further discussion of this framework, see also Tzanakis and Thomaidis (2012).
For a brief discussion of these, see also Jankvist and Kjeldsen (2011).
Chapter 9 in the ICMI Study on History in Mathematics Education; the chapter is written by Jahnke, Arcavi, Barbin, Bekken, Furinghetti, El Idrissi, da Silva and Weeks.
Links are: http://www.math.nmsu.edu/hist_projects/ and http://www.cs.nmsu.edu/historical-projects/ (retrieved on April 15, 2012). In particular the projects by Janet Heine Barnett (n.d., 2011a, b) have served as a source of inspiration for the work discussed in this paper.
The illustrative examples stem from implementations of the two HAPh-modules in a Danish upper secondary class. Danish upper secondary school is 3 years, where students in first year are of age 15–16 years. The class under consideration followed the mathematics-science direction, meaning that they study mathematics through all 3 years of upper secondary school.
Although not included in this HAPh-module, a relevant source on Hilbert’s views is that of Corry (2004).
The proof for the third part of Euler’s result is ascribed to Carl Hierholzer (published posthumous in 1873). For a discussion of the students’ work with these proofs, see Jankvist (2011b).
The problem of finding minimum spanning trees had on several occasions been solved before though: by Prim in 1957; by Kruskal in 1956; by Jarník in 1930; and by Borůkva in 1926.
For examples of students’ reactions to this HAPh-module, see Jankvist (2012b).
As part of the data collection, I video recorded one particular group of students (Group 7 out of seven) while they worked on the mathematical tasks and essay-assignments of the modules, allowing me an insight into this group’s discussions prior to putting their answers down on paper.
In fact, the research study involved a following of this class of students for a two-year period, during which they were given three questionnaires, 1 year apart, and interviewed afterwards in order to evaluate possible developments of their awareness, including beliefs/views/images (Jankvist 2009d), in relation to the KOM-project’s three types of overview and judgment. For a preliminary analysis, see Jankvist (2012a).
For a discussion of Danish upper secondary school teachers’ attitudes toward history, application, and philosophy (or the three types of OJ) in teaching, see Jankvist (2009d).
In Danish upper secondary school students have a national final written exam and a local final oral exam. Activities such as the HAPh-modules have the possibility of being part of the oral exam, if the teacher decides so.
From a history point of view, such selection of sources of course also significantly reduces the problems associated with Whiggism.
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Jankvist, U.T. History, Applications, and Philosophy in Mathematics Education: HAPh—A Use of Primary Sources. Sci & Educ 22, 635–656 (2013). https://doi.org/10.1007/s11191-012-9470-8
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DOI: https://doi.org/10.1007/s11191-012-9470-8