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

- 663 Downloads
- 11 Citations

## 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.

## Keywords

Mathematics Education Boolean Algebra Minimum Span Tree Teaching Module Original Text## 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle 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.Google Scholar - Carson, R. N., & Rowlands, S. (2007). Teaching the conceptual revolutions in geometry.
*Science & Education,**16*, 921–954.CrossRefGoogle 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.Google Scholar - Daniel, M. F., Lafortune, L., Pallascio, R., & Sykes, P. (2000). A primary school curriculum to foster thinking about mathematics.
*Encyclopaedia of Philosophy of Education.*Google Scholar - 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.CrossRefGoogle 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.CrossRefGoogle Scholar - Fried, M. N. (2007). Didactics and history of mathematics: Knowledge and selfknowledge.
*Educational Studies in Mathematics,**66*, 203–223.CrossRefGoogle 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.CrossRefGoogle Scholar - Heeffer, A. (2011). Historical objections against the number line.
*Science & Education,**20*(9), 863–880.CrossRefGoogle 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle 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.Google Scholar - Jankvist, U. T. (2010). An emprical study of using history as a ‘goal’.
*Educational Studies in Mathematics,**74*(1), 53–74.CrossRefGoogle 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).Google Scholar - Jankvist, U. T. (2011c). Historisk fremkomst og moderne anvendelse af grafteori—et matematikfilosofisk undervisningsforløb til gymnasiet.
*Tekster fra IMFUFA*, no. 486, 76 pp.Google Scholar - Jankvist, U. T. (2011d). Historisk fremkomst og moderne anvendelse af Boolsk algebra—et matematikfilosofisk undervisningsforløb til gymnasiet.
*Tekster fra IMFUFA*, no. 487, 80 pp.Google Scholar - 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.Google Scholar - 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.Google Scholar - Jankvist, U. T. (forthcoming).
*Changing students’ images of mathematics as a discipline*(under review).Google Scholar - Jankvist, U. T., & Kjeldsen, T. H. (2011). New avenues for history in mathematics education—mathematical competencies and anchoring.
*Science & Education,**20*(9), 831–862.CrossRefGoogle 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.CrossRefGoogle 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.Google Scholar - 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.CrossRefGoogle 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).Google Scholar - 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).Google Scholar - Panagiotou, E. N. (2011). Using history to teach mathematics: The case of logarithms.
*Science & Education,**20*(1), 1–35.CrossRefGoogle 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.Google Scholar - 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.Google Scholar - Shannon, C. E. (1938a).
*A symbolic analysis of relay and switching circuits*. Master’s thesis. Massachusetts Institute of Technology, Cambridge.Google Scholar - Shannon, C. E. (1938b). A symbolic analysis of relay and switching circuits.
*American Institute of Electrical Engineers Transactions,**57*, 713–723.CrossRefGoogle 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.Google Scholar - 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.Google Scholar
- Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences.
*Communications in Pure and Applied Mathematics,**13*(1), 1–14.CrossRefGoogle Scholar