Skip to main content
SpringerLink
Log in

What do mathematicians wish to teach teachers about the discipline of mathematics?

  • Published:
Journal of Mathematics Teacher Education Aims and scope Submit manuscript

Abstract

This study investigates what mathematicians, who teach academic mathematics courses to secondary school mathematics teachers, want to teach them about the discipline of mathematics. Data sources included interviews with five research mathematicians. The data analysis revealed that expanding teachers’ knowledge about the discipline of mathematics was one of the interviewees’ main objectives. The mathematicians referred to nine characteristics that can be grouped into three key aspects: (1) essence of mathematics, (2) doing mathematics, and (3) worth of mathematics. The characteristics of the discipline of mathematics generated in our study are in line with ideas and practices widely advocated in the last decades for the teaching and learning of mathematics, and with contemporary philosophical approaches regarding mathematics. The potential of the characteristics to serve as a conceptual framework in studies that focus on teacher knowledge about the discipline of mathematics is discussed.

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

Similar content being viewed by others

References

  • Arnold, V. I. (2014). Mathematical understanding of nature: Essays on amazing physical phenomena and their understanding by mathematicians. AMS.

  • Arnold, V. I. (1998). On teaching mathematics. Russian Mathematical Surveys, 53(1), 229–236.

    Article  Google Scholar 

  • Atiyah, M., & Bourbaki, A. (2007). Secret society of mathematicians. Notices of the AMS, 54(9), 1150–1152.

    Google Scholar 

  • Balaguer, M. (2009). Realism and anti-realism in mathematics. In A. D. Irvine (Ed.), Philosophy of Mathematics (pp. 35–101). Elsveier.

    Chapter  Google Scholar 

  • Baldinger, E. E. (2018). Learning mathematical practices to connect abstract algebra to high school algebra. In N. H. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers (pp. 211–239). Springer International Publishing. https://doi.org/10.1007/978-3-319-99214-3_11

    Chapter  Google Scholar 

  • Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466.

    Article  Google Scholar 

  • Barton, B., Clark, M., & Sheryn, L. (2010). Collective dreaming: A school-university interface. New Zealand Journal of Mathematics, 40, 15–31.

    Google Scholar 

  • Bass, H. (2009). How do you know that you know? Making believe in mathematics. Distinguished University Professor Lecture.

    Google Scholar 

  • Bass, H. (2017). Designing opportunities to learn mathematics theory-building practices. Educational Studies in Mathematics, 95(3), 229–244. https://doi.org/10.1007/s10649-016-9747-y

    Article  Google Scholar 

  • Beswick, K. (2007). Teachers’ beliefs that matter in secondary mathematics classrooms. Educational Studies in Mathematics, 65, 95–120. https://doi.org/10.1007/s10649-006-9035-3

    Article  Google Scholar 

  • Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educational Studies in Mathematics, 79(1), 127–147. https://doi.org/10.1007/s10649-011-9333-2

    Article  Google Scholar 

  • Beswick, K., & Goos, M. (2018). Mathematics teacher educator knowledge: What do we know and where to from here? Journal of Mathematics Teacher Education, 21(5), 417–427. https://doi.org/10.1007/s10857-018-9416-4

    Article  Google Scholar 

  • CBMS - Conference board of the mathematical sciences. (2012). MET II - The mathematical education of teachers II. American Mathematical society and mathematical association of America. https://www.cbmsweb.org/archive/MET2/met2.pdf

  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. SAGE.

  • Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. SAGE.

  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521–544. https://doi.org/10.1007/BF00315943

  • Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioners’ views. ZDM - The International Journal on Mathematics Education, 43(6–7), 941–950. https://doi.org/10.1007/s11858-011-0346-1

  • Even, R., Artstein, Z., & Elbaum-Cohen, A. (2018). The Rothschild-Weizmann master’s program for practicing mathematics teachers. In K-12 Mathematics Education in Israel: Vol. Volume 13 (pp. 235–242). World Scientific. https://doi.org/10.1142/9789813231191_0025

  • Felbrich, A., Kaiser, G., & Schmotz, C. (2012). The cultural dimension of beliefs: An investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries. ZDM - Mathematics Education, 44(3), 355–366. https://doi.org/10.1007/s11858-012-0418-x

    Article  Google Scholar 

  • Fine, K. (2012). Mathematics: Discovery or invention? Think, 11(32), 11–27. https://doi.org/10.1017/S1477175612000188

    Article  Google Scholar 

  • Franklin, J. (2008). Aristotelian realism. In A. D. Irvine (Ed.), Philosophy of mathematics (pp. 103–155). Elsveier.

    Google Scholar 

  • Freudenthal, H. (2002). Revisiting mathematics education: China lectures. Kluwer Academic Publisher.

    Google Scholar 

  • Hadamard, J. (1996). The mathematician’s mind: The psychology of invention in the mathematical field. Princeton University Press.

    Google Scholar 

  • Halmos, P. R. (1981). Applied mathematics is bad mathematics. In L. A. Steen (Ed.), Mathematics tomorrow (pp. 9–20). Springer.

    Chapter  Google Scholar 

  • Hodge, A. M., Gerberry, C. V., Moss, E. R., & Staples, M. E. (2010). Purposes and perceptions: What do university mathematics professors see as their role in the education of secondary mathematics teachers? Primus, 20(8), 646–663. https://doi.org/10.1080/10511970902810733

    Article  Google Scholar 

  • Hoffmann, A., & Even, R. (2019). Contribution of academic mathematics to teacher learning about the essence of mathematics. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the psychology of mathematics education (Vol. 2, pp. 360–367). PME.

  • Hoffmann, A., & Even, R. (2021). What do teachers learn about the discipline of mathematics in academic mathematics courses? In M. Inparshita, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 51–60). PME.

  • Horsten, L. (2019). Philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University.

    Google Scholar 

  • Jackson, A., & Rossi, H. (1996a). Views on high school mathematics education. Notices of the AMS, 43(8), 866–873.

    Google Scholar 

  • Jackson, A., & Rossi, H. (1996b). Views on high school mathematics education. Notices of the AMS, 43(9), 979–986.

    Google Scholar 

  • Jankvist, U. T., Aguilar, M. S., Misfeldt, M., & Koichu, B. (2021). Launching implementation and replication studies in mathematics education (IRME). Implementation and Replication Studies in Mathematics Education, 1(1), 1–19. https://doi.org/10.1163/26670127-01010001

    Article  Google Scholar 

  • Klein, Felix. (2016). Elementary mathematics from a higher standpoint. Springer. https://doi.org/10.1007/978-3-662-49442-4

    Book  Google Scholar 

  • Koichu, B. (2014). Reflections on problem-solving. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & mathematics education: Searching for common ground (pp. 113–135). Springer.

    Chapter  Google Scholar 

  • Lai, Y., & Ahrens, S. (2023). Some mathematicians’ perceived and envisioned instructional relationships in secondary teaching and teaching secondary teachers. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-022-09567-y

    Article  Google Scholar 

  • Lakatos, I. (1976a). Falsification and the methodology of scientific research programmes. In S. G. Harding (Ed.), Can Theories be Refuted? Essays on the Duhem-Quine Thesis (pp. 205–259). Springer. https://doi.org/10.1007/978-94-010-1863-0_14

  • Lakatos, I. (1976b). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.

    Book  Google Scholar 

  • Leikin, R., Zazkis, R., & Meller, M. (2018). Research mathematicians as teacher educators: Focusing on mathematics for secondary mathematics teachers. Journal of Mathematics Teacher Education, 21(5), 451–473. https://doi.org/10.1007/s10857-017-9388-9

    Article  Google Scholar 

  • Lovász, L. (1998). One mathematics (pp. 10–15). The Berliner Intelligencer.

    Google Scholar 

  • Mancosu, P. (Ed.). (2008). Mathematical explanation: Why it matters? In The philosophy of mathematical practice (1st ed.). Oxford University Press.

    Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Pearson.

    Google Scholar 

  • Misfeldt, M., Jankvist, U. T., & Aguilar, M. S. (2016). Teachers’ beliefs about the discipline of mathematics and the use of technology in the classroom. International Electronic Journal of Mathematics Education, 11(2), 395–419. https://doi.org/10.29333/iejme/341

    Article  Google Scholar 

  • Wasserman, N. H. (2018). Part III - Exploring approaches to secondary teacher education: Engaging in abstract algebra in relation to developing disciplinary practices. In Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers (pp. 188–333). Springer.

  • Polya, G. (1962). Mathematical discovery. Wiley.

    Google Scholar 

  • Polya, G. (2014). How to solve it: A new aspect of mathematical method. Princeton University Press.

    Book  Google Scholar 

  • Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of well-taught mathematics courses. Educational Psychologist, 23(2), 145–166.

    Article  Google Scholar 

  • Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. https://doi.org/10.4324/9780203843000

    Book  Google Scholar 

  • Schwab, J. J. (1978). Education and the structure of the disciplines. In Science, Curriculum, and Liberal Education, pp. 229–272. (Original work published 1961).

  • Shamash, J., Barabash, M., & Even, R. (2018). From equations to structures: Modes of relevance of abstract algebra to school mathematics as viewed by teacher educators and teachers. In N. H. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers (pp. 241–262). Springer International Publishing. https://doi.org/10.1007/978-3-319-99214-3_12

    Chapter  Google Scholar 

  • Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41(1), 13–27. https://doi.org/10.1007/s11858-008-0114-z

    Article  Google Scholar 

  • Su, F. (2020). Mathematics for human flourishing. Yale University Press.

    Book  Google Scholar 

  • Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127. https://doi.org/10.1007/BF00305892

    Article  Google Scholar 

  • Warshauer, H. K. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18(4), 375–400. https://doi.org/10.1007/s10857-014-9286-3

    Article  Google Scholar 

  • Weir, A. (2021). Formalism in the philosophy of mathematics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2021). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/entries/formalism-mathematics/.

  • Wigner, E. P. (1990). The unreasonable effectiveness of mathematics in the natural sciences. In R. E. Mickens (Ed.), Mathematics and Science (pp. 291–306). World Scientific. https://doi.org/10.1142/9789814503488_0018

    Chapter  Google Scholar 

  • Wu, H. H. (2011). The mis-education of mathematics teachers. Notices of the AMS, 58(3), 372–384.

    Google Scholar 

  • Wu, H. H. (2016). Teaching school mathematics: Algebra. American Mathematical Society.

    Book  Google Scholar 

  • Yan, X., Marmur, O., & Zazkis, R. (2020). Calculus for teachers: Perspectives and considerations of mathematicians. Canadian Journal of Science, Mathematics and Technology Education, 20, 355–374. https://doi.org/10.1007/s42330-020-00090-x

    Article  Google Scholar 

  • Yang, X., Kaiser, G., König, J., & Blömeke, S. (2020). Relationship between pre-service mathematics teachers’ knowledge, beliefs and instructional practices in China. ZDM - Mathematics Education, 52(2), 281–294. https://doi.org/10.1007/s11858-020-01145-x

    Article  Google Scholar 

  • Yasukawa, K., Skovsmose, O., & Ravn, O. (2012). Shaping and being shaped by mathematics: Examining a technology of rationality. In O. Skovsmose & B. Greer (Eds.), Opening the cage: Critique and politics of mathematics education (pp. 265–286). Sense Publishers.

    Chapter  Google Scholar 

  • Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12(4), 263–281. https://doi.org/10.1080/10986061003786349

    Article  Google Scholar 

  • Ziegler, G. M., & Loos, A. (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. In G. Kaiser (Ed.), Proceedings of the 13th international congress on mathematical education (pp. 63–77). Springer International Publishing. https://doi.org/10.1007/978-3-319-62597-3_5

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    Anna Hoffmann & Ruhama Even

Authors
  1. Anna Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruhama Even
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Hoffmann.

Ethics declarations

Funding

No funding was received for conducting this study.

Data availability

Not applicable.

Code availability

Not applicable.

Conflict of interest

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hoffmann, A., Even, R. What do mathematicians wish to teach teachers about the discipline of mathematics?. J Math Teacher Educ (2023). https://doi.org/10.1007/s10857-023-09577-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10857-023-09577-4

Keywords

Search

Navigation