Skip to main content

Advertisement

SpringerLink
Log in

History, Applications, and Philosophy in Mathematics Education: HAPh—A Use of Primary Sources

  • Published:
Science & Education Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Examples of recent collective volumes are: Blum et al. (2007), Kaiser et al. (2011).

  2. Philosophy in general, on the other hand, is often used within theoretical constructs and frameworks in mathematics education.

  3. The few I am familiar with are: Carson and Rowlands (2007), Chassapis (2010), Daniel et al. (2000), De La Garza et al. (2000), Iversen (2009), Kennedy (2007), Kjeldsen and Blomhøj (2009), Prediger (2010), Rowlands and Davies (2006).

  4. For further discussion of this framework, see also Tzanakis and Thomaidis (2012).

  5. For a brief discussion of these, see also Jankvist and Kjeldsen (2011).

  6. For a discussion of how such solidness may be brought about and evaluated through the construct of anchoring, see Jankvist (2009c, d, 2010, 2011a) and Jankvist and Kjeldsen (2011).

  7. For an example of a research study, where students were asked to make up their minds about the question of discovery or invention of mathematics, see Jankvist (2009d, forthcoming). Yet an example may be found in Rowlands and Davies (2006).

  8. These observations are based on studies reported in Jankvist (2009c, d, 2010, 2011a).

  9. 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.

  10. 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.

  11. 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.

  12. Although not included in this HAPh-module, a relevant source on Hilbert’s views is that of Corry (2004).

  13. 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).

  14. For a modern textbook presentation of Dijkstra’s algorithms, see e.g. Rosen (2003) or Jankvist (2011c).

  15. 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.

  16. For further examples of essay-assignments from this HAPh-module as well as examples of mathematical tasks included in the material, see Jankvist (2011b, 2012a).

  17. For examples of students’ reactions to this HAPh-module, see Jankvist (2012b).

  18. 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.

  19. 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).

  20. 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).

  21. 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.

  22. From a history point of view, such selection of sources of course also significantly reduces the problems associated with Whiggism.

References

  • Barbin, E. (1997). Histoire des Mathématiques: Pourquoi? Comment? Bulletin AMQ, 37(1), 20–25.

    Google Scholar 

  • Barnett, J. H. (2011a). Applications of Boolean algebra: Claude Shannon and circuit design. http://www.cs.nmsu.edu/historical-projects/projects.php.

  • Barnett, J. H. (2011b). Origins of Boolean algebra in logic of classes: George Boole, John Venn and C. S. Peirce. http://www.cs.nmsu.edu/historical-projects/projects.php.

  • Barnett, J. H. (n.d.). Early writings on graph theory—Euler circuits and the Königsberg bridge problem. http://www.math.nmsu.edu/hist_projects.

  • Barnett, J. H., Lodder, J., Pengelley, D., Pivkina, I., & Ranjan, D. (2011). Designing student projects for teaching and learning discrete mathematics and computer science via primary historical sources. In V. Katz, & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education (MAA Notes 78, pp. 187–200). Washington: The Mathematical Association of America.

  • Biggs, N. L., Lloyd, E. K., & Wilson, R. J. (1976). Graph theory 1736–1936. Oxford: Clarendon Press.

    Google Scholar 

  • Blum, W., Galbraith, P. L., Henn, H.-W. & Niss, M. (Eds.). (2007). Modelling and applications in mathematics education. The 14th ICMI Study (New ICMI Studies series 10). New York: Springer.

  • Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—state, trends, and issues in mathematics instruction. Educational Studies in Mathematics, 22, 37–68.

    Article  Google Scholar 

  • Boole, G. (1854). An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities. London: Walton and Maberly.

    Google Scholar 

  • Butterfield, H. (1931/1951). The Whig interpretation of history. New York: Charles Scribner‘s Sons.

  • Carson, R. N., & Rowlands, S. (2007). Teaching the conceptual revolutions in geometry. Science & Education, 16, 921–954.

    Article  Google Scholar 

  • Chassapis, D. (2010). Integrating the philosophy of mathematics in teacher training. In K. Francois & J. P. Van Bendegem (Eds.), Philosophical dimensions in mathematics education (pp. 61–80). New York: Springer.

    Google Scholar 

  • Corry, L. (2004). Hilbert and the axiomatization of physics (1898–1918): From “Grundlagen der Geometrie” to “Grundlagen der Physik” (ARCHIMEDES: New Studies in the History and Philosophy of Science and Technology, Vol. 10). Dordrecht: Kluwer.

  • Daniel, M. F., Lafortune, L., Pallascio, R., & Sykes, P. (2000). A primary school curriculum to foster thinking about mathematics. Encyclopaedia of Philosophy of Education.

  • De La Garza, M. T., Slade-Hamilton, C., & Daniel, M. F. (2000). Philosophy of mathematics in the classroom: Aspects of a tri-national study. Analytic Teaching, 20(2), 88–104.

    Google Scholar 

  • Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1, 269–271.

    Article  Google Scholar 

  • Euler, L. (1736). Solutio prolematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 8, 128–140.

    Google Scholar 

  • Fleischner, H. (1990). Eulerian graphs and related topics. Amsterdam: Elsevier.

    Google Scholar 

  • Freudenthal, H. (1991). Revisiting mathematics education—China lectures. Dordrecht: Kluwer.

    Google Scholar 

  • Fried, M. (2001). Can history of mathematics and mathematics education coexist? Science & Education, 10(4), 391–408.

    Article  Google Scholar 

  • Fried, M. N. (2007). Didactics and history of mathematics: Knowledge and selfknowledge. Educational Studies in Mathematics, 66, 203–223.

    Article  Google Scholar 

  • Fried, M. N. (2010). History of mathematics: Problems and prospects. In E. Barbin, M. Kronfellner, & C. Tzanakis (Eds.), History and epistemology in mathematics education proceedings of the 6th European Summer University (pp. 13–26). Vienna: Holzhausen Publishing Ltd.

    Google Scholar 

  • Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87(2), 81–90.

    Article  Google Scholar 

  • Heeffer, A. (2011). Historical objections against the number line. Science & Education, 20(9), 863–880.

    Article  Google Scholar 

  • Hersh, R. (1997). What is mathematics really?. Oxford: Oxford University Press.

    Google Scholar 

  • Hilbert, D. (1900). Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematike-Congress zu Paris 1900. Gött. Nachr. 1900, 253–297. Göttingen: Vandenhoeck & Ruprecht.

  • Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society, 8, 437–479. Reprinted in: Bulletin (New Series) of the American Mathematical Society, 37(4), 407–436, Article electronically published on June 26, 2000.

  • Iversen, S. M. (2009). Modeling interdisciplinary activities involving mathematics and philosophy. In B. Sriraman, V. Freiman, & N. Lirette-Pitre (Eds.), Interdisciplinarity, creativity, and learning—mathematics with literature, paradoxes, history, technology, and modeling (pp. 147–164). Charlotte: Information Age Publishing.

    Google Scholar 

  • Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., et al. (2000). The use of original sources in the mathematics classroom. In: J. Fauvel, & J. van Maanen (Eds.), History in mathematics education, The ICMI Study (pp. 291–328). Dordrecht: Kluwer.

  • Jankvist, U. T. (2009a). A categorization of the ‘whys’ and ‘hows’ of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235–261.

    Article  Google Scholar 

  • Jankvist, U. T. (2009b). History of modern applied mathematics in mathematics education. For the Learning of Mathematics, 29(1), 8–13.

    Google Scholar 

  • Jankvist, U. T. (2009c). On empirical research in the field of using history in mathematics education. ReLIME, 12(1), 67–101.

    Google Scholar 

  • Jankvist, U. T. (2009d). Using history as a ‘goal’ in mathematics education. Ph.D. thesis, IMFUFA, Roskilde University, Roskilde. Tekster fra IMFUFA, no. 464, 361 pp.

  • Jankvist, U. T. (2010). An emprical study of using history as a ‘goal’. Educational Studies in Mathematics, 74(1), 53–74.

    Article  Google Scholar 

  • Jankvist, U. T. (2011a). Anchoring students’ meta-perspective discussions of history in mathematics. Journal of Research in Mathematics Education, 42(4), 346–385.

    Google Scholar 

  • Jankvist, U. T. (2011b). Designing teaching modules on the history, application, and philosophy of mathematics. In CERME 7, Proceedings of the 7th congress of the European Society for Research in Mathematics Education (WG12) in Poland (10 pp).

  • Jankvist, U. T. (2011c). Historisk fremkomst og moderne anvendelse af grafteori—et matematikfilosofisk undervisningsforløb til gymnasiet. Tekster fra IMFUFA, no. 486, 76 pp.

  • Jankvist, U. T. (2011d). Historisk fremkomst og moderne anvendelse af Boolsk algebra—et matematikfilosofisk undervisningsforløb til gymnasiet. Tekster fra IMFUFA, no. 487, 80 pp.

  • Jankvist, U. T. (2012a). History, application, and philosophy of mathematics in mathematics education: Accessing and assessing students’ overview and judgment. Regular lecture at ICME-12 in Seoul, Korea.

  • Jankvist, U. T. (2012b). A historical teaching module on “the unreasonable effectiveness of mathematics”—the case of Boolean algebra and Shannon circuits. Paper presented at HPM2012 in Daejeon, Korea.

  • Jankvist, U. T. (forthcoming). Changing students’ images of mathematics as a discipline (under review).

  • Jankvist, U. T., & Kjeldsen, T. H. (2011). New avenues for history in mathematics education—mathematical competencies and anchoring. Science & Education, 20(9), 831–862.

    Article  Google Scholar 

  • Kaiser, G., Blum, W., Ferri, R. B., & Stillman, G. (2011). Trends in teaching and learning of mathematical modelling: ICTMA14 (International Perspectives on the Teaching and Learning of Mathematical Modelling). Dordrecht: Springer.

    Book  Google Scholar 

  • Kennedy, N. S. (2007). From philosophical to mathematical inquiry in the classroom. Childhood & Philosophy, 3(6), 1–18.

    Google Scholar 

  • Kjeldsen, T. H. (2011). Uses of history in mathematics education: Development of learning strategies and historical awareness. In CERME 7, Proceedings of the 7th Congress of the European Society for Research in Mathematics Education (WG12) in Poland, 10 pp.

  • Kjeldsen, T. H., & Blomhøj, M. (2009). Integrating history and philosophy in mathematics education at university level through problem-oriented project work. ZDM Mathematics Education, Zentralblatt für Didaktik der Mathematik, 41, 87–104.

    Article  Google Scholar 

  • Kjeldsen, T. H., & Blomhøj, M. (in press). Beyond motivation: history as a method for learning meta-discursive rules in mathematics. Educational Studies in Mathematics. Published Online First (September 23, 2011).

  • Niss, M. (2009). Perspectives on the balance between application and modelling and ‘pure’ mathematics in the teaching and learning of mathematics. In M. Menghini, F. Furinghetti, L. Giacardi, & F. Arzarello (Eds.), The first century of the international commission on mathematical instruction (1908–2008)—reflecting and shaping the world of mathematics education (pp. 69–84). Roma: Istituto della Enciclopedia Italiana fondata da Giovanni Treccani.

    Google Scholar 

  • Niss, M., & Højgaard, T. (Eds.). (2011). Competencies and mathematical learning—ideas and inspiration for the development of mathematics teaching and learning in Denmark. English Edition, October 2011. IMFUFA tekst no. 485. Roskilde: Roskilde University (published in Danish in 2002).

  • Panagiotou, E. N. (2011). Using history to teach mathematics: The case of logarithms. Science & Education, 20(1), 1–35.

    Article  Google Scholar 

  • Pengelley, D. (2011). Teaching with primary historical sources: Should it go mainstream? Can it? In V. Katz & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education (MAA Notes 78, pp. 1–8). Washington: The Mathematical Association of America.

  • Prediger, S. (2010). Philosophical reflections in mathematics classrooms. In K. Francois & J. P. Van Bendegem (Eds.), Philosophical dimensions in mathematics education (pp. 43–58). New York: Springer.

    Google Scholar 

  • Rosen, K. H. (2003). Discrete mathematics and its applications (5th ed.). New York: McGraw Hill.

    Google Scholar 

  • Rowlands, S., & Davies, A. (2006). Mathematics masterclass: Is mathematics invented or discovered? Mathematics in School, 35(2), 2–6. Also reprinted in Mathematics in Schools, 2001, 40(2), 30–34.

  • Shannon, C. E. (1938a). A symbolic analysis of relay and switching circuits. Master’s thesis. Massachusetts Institute of Technology, Cambridge.

  • Shannon, C. E. (1938b). A symbolic analysis of relay and switching circuits. American Institute of Electrical Engineers Transactions, 57, 713–723.

    Article  Google Scholar 

  • Sloane, N. J. A., & Wyner, A. D. (Eds.). (1993). Claude Elwood Shannon: Collected papers. New York: IEEE Press.

    Google Scholar 

  • Tzanakis, C., & Thomaidis, Y. (2012). Classifying the arguments and methodological schemes for integrating history in mathematics education. In B. Sriraman (Ed.), Crossroads in the history of mathematics and mathematics education (The Montana Mathematics Enthusiast Monographs in Mathematics Education, Vol. 12, pp. 247–295). Charlotte: IPA.

  • UVM (Undervisningsministeriet). (2008). ‘Vejledning: Matematik A, Matematik B, Matematik C’. Bilag 35, 36, 37. English translation of title: Ministerial order of 2008—Mathematics levels A, B, and C.

  • Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Science, Systems and Models, Roskilde University, Roskilde, Denmark

    Uffe Thomas Jankvist

Authors
  1. Uffe Thomas Jankvist
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uffe Thomas Jankvist.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11191-012-9470-8

Keywords

Search

Navigation