International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education
“Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Can C, Clark KM. “Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics. INT ELECT J MATH ED. 2020;15(3), em0610. https://doi.org/10.29333/iejme/8566
APA 6th edition
In-text citation: (Can & Clark, 2020)
Reference: Can, C., & Clark, K. M. (2020). “Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics. International Electronic Journal of Mathematics Education, 15(3), em0610. https://doi.org/10.29333/iejme/8566
Chicago
In-text citation: (Can and Clark, 2020)
Reference: Can, Cihan, and Kathleen Michelle Clark. "“Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics". International Electronic Journal of Mathematics Education 2020 15 no. 3 (2020): em0610. https://doi.org/10.29333/iejme/8566
Harvard
In-text citation: (Can and Clark, 2020)
Reference: Can, C., and Clark, K. M. (2020). “Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics. International Electronic Journal of Mathematics Education, 15(3), em0610. https://doi.org/10.29333/iejme/8566
MLA
In-text citation: (Can and Clark, 2020)
Reference: Can, Cihan et al. "“Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics". International Electronic Journal of Mathematics Education, vol. 15, no. 3, 2020, em0610. https://doi.org/10.29333/iejme/8566
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Can C, Clark KM. “Because You’re Exploring this Huge Abstract Jungle…”: One Student’s Evolving Conceptions of Axiomatic Structure in Mathematics. INT ELECT J MATH ED. 2020;15(3):em0610. https://doi.org/10.29333/iejme/8566

Abstract

For several decades, literature on the history and pedagogy of mathematics has described how history of mathematics is beneficial for the teaching and learning of mathematics. We investigated the influence of a history and philosophy of mathematics (HPhM) course on students’ progress through the lens of various competencies in mathematics (e.g., mathematical thinking and communicating) as a result of studying mathematical ideas from the perspective of their historical and philosophical development. We present outcomes for one student, whom we call Michael, resulting from his learning experiences in an HPhM course at university. We use the framework from the Competencies and Mathematical Learning project (the Danish KOM project) to analyze the evolution of Michael’s competencies related to axiomatic structure in mathematics. We outline three aspects of axiomatic structure to situate our analysis: Truth, Logic, and Structure. Although our analysis revealed that Michael’s views and knowledge of axiomatic structure demonstrate need for his further development, we assert what he experienced during the HPhM course regarding his mathematical thinking and communication about axiomatic structure is promising support for his future mathematical studies. Finally, we argue that a HPhM course has potential to support students’ progress in advanced mathematics at university.

References

  • Barnett, J. H., Lodder, J., & Pengelley, D. (2014). The pedagogy of primary historical sources in mathematics: Classroom practice meets theoretical frameworks. Science & Education, 23(1), 7-27. https://doi.org/10.1007/s11191-013-9618-1
  • Bernardes, A., & Roque, T. (2018). History of matrices: Commognitive conflicts and reflections on metadiscursive rules. In K. M. Clark, T. H. Kjeldsen, S. Schorcht, & C. Tzanakis (Eds.), Mathematics, education and history: Towards a harmonious partnership. ICME-13 Monographs (pp. 209-227). Cham, Switzerland: Springer International Publishing.
  • Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221-232. https://doi.org/10.1080/00029890.1950.11999523
  • Brown, J. R. (1999). Philosophy of mathematics: An introduction to the world of proofs and pictures. New York, NY: Routledge.
  • Clark, K. M. (2012). History of mathematics: illuminating understanding of school mathematics concepts for prospective mathematics teachers. Educational Studies in Mathematics, 81(1), 67-84. https://doi.org/10.1007/s10649-011-9361-y
  • Corry, L. (2007). Axiomatics between Hilbert and the New Math: Diverging views on mathematical research and their consequences on education. The International Journal for the History of Mathematics Education, 2(2), 21-37.
  • Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston, MA: Birkhäuser.
  • Dunham, W. (1994). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Hoboken, NJ: John Wiley & Sons.
  • Fauvel, J. (1991). Using history in mathematics education. For the Learning of Mathematics, 11(2), 3-6.
  • Furinghetti, F. (2020). Rethinking history and epistemology in mathematics education. International Journal of Mathematical Education in Science and Technology, 51(6), 967-994. https://doi.org/10.1080/0020739X.2019.1565454
  • Hersh, R. (1998). What is mathematics, really? London, UK: Vintage.
  • Hintikka, J. (2011). What is the axiomatic method? Synthese, 183(1), 69-85. https://doi.org/10.1007/s11229-009-9668-8
  • Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71(3), 235-261. https://doi.org/10.1007/s10649-008-9174-9
  • Jankvist, U. T. (2010). An empirical study of using history as a ‘goal.’ Educational Studies in Mathematics, 74(1), 53-74. https://doi.org/10.1007/s10649-009-9227-8
  • Jankvist, U. T. (2011). Anchoring students’ metaperspective discussions of history in mathematics. Journal for Research in Mathematics Education, 42(4), 346-385. https://doi.org/10.5951/jresematheduc.42.4.0346
  • Jankvist, U. T., & Kjeldsen, T. H. (2011). New avenues for history in mathematics education: Mathematical competencies and anchoring. Science & Education, 20(9), 831-862. https://doi.org/10.1007/s11191-010-9315-2
  • Katz, V. J. (2009). History of mathematics: An introduction. Boston, MA: Pearson Education.
  • Kjeldsen, T. H., & Blomhøj, M. (2012). Developing students’ reflections on the function and status of mathematical modeling in different scientific practices: History as a provider of cases. Science & Education, 22(9), 2157-2171. https://doi.org/10.1007/s11191-012-9555-4
  • Kjeldsen, T. H., & Petersen, P. H. (2014). Bridging history of the concept of function with learning of mathematics: Students’ meta-discursive rules, concept formation and historical awareness. Science & Education, 23(1), 29-45. https://doi.org/10.1007/s11191-013-9641-2
  • Mertens, D. M. (2005). Research and evaluation in education and psychology: Integrating diversity with quantitative, qualitative, and mixed methods. Thousand Oaks, CA: SAGE.
  • Mueller, I. (1969). Euclid’s Elements and the axiomatic method. The British Journal for the Philosophy of Science, 20(4), 289-309. https://doi.org/10.1093/bjps/20.4.289
  • National Council of Teachers of Mathematics. (NCTM). (1969). Historical topics for the mathematics classroom. Reston, VA: Author. (31st NCTM Yearbook, reprinted 1989)
  • Niss, M., & Højgaard, T. (Eds.). (2011). Competencies and mathematical learning: Ideas and inspiration for the development of mathematics teaching and learning in Denmark. IMFUFA tekst, Nr. 485-2011. Roskilde, Denmark: Roskilde University.
  • Patton, M. Q. (1990). Qualitative evaluation and research methods. Newbury Park, CA: SAGE.
  • Putnam, H. (1998). What is mathematical truth? In T. Tymoczko (Ed.), New directions in the philosophy of mathematics: An anthology (pp. 49-65). Princeton, NJ: Princeton University Press.
  • Rodin, A. (2014). Formal axiomatic method and the twentieth century mathematics. In Axiomatic method and category theory (pp. 73-97). Cham, Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-00404-4
  • Rossman, G. B., & Rallis, S. F. (2003). Learning in the field: An introduction to qualitative research. Thousand Oaks, CA: SAGE.
  • Speer, N. M., Smith, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior, 29(2), 99-114. https://doi.org/10.1016/j.jmathb.2010.02.001
  • Stake, R. E. (2000). Case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 435-454). Thousand Oaks, CA: Sage Publications.
  • Witzke, I., Clark, K. M., Struve, H., & Stoffels, G. (2018). Addressing the transition from school to university. In K. M. Clark, T. H. Kjeldsen, S. Schorcht, & C. Tzanakis (Eds.), Mathematics, education and history: Towards a harmonious partnership. ICME-13 Monographs (pp. 61-82). Cham, Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-73924-3_4

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