Developing mathematical knowledge for teaching teachers: potentials of history of mathematics in teacher educator training


What mathematical knowledge is required for teaching has been researched by many, with Ball et al. (J Teach Educ 59(5):389–407, 2008) and their practice-based theory of mathematical knowledge for teaching (MKT) primary among them. However, what is required in terms of mathematical knowledge for teacher educator training has been researched much less. The present study builds on an emerging framework on mathematical knowledge for teaching teachers (MKTT) by Zopf (Mathematical knowledge for teaching teachers: the mathematical work of and knowledge entailed by teacher education, University of Michigan, Ann Arbor, 2010) augmented with more recent developments in the field. The empirical basis for the study stems from an implementation of a short course on history in mathematics education at the Danish School of Education, and in particular on two case studies involving three teacher educator students participating in that course. Based on analyses of their developments in terms of MKTT, we point to the potential of using history in both teacher and teacher educator training, and in particular historical source material. As part of the analyses, we also consider the teacher educator students’ development of disciplinary knowledge of mathematics, especially in relation to developing and making use of knowledge of the epistemology of mathematics and mathematical work. We assert that the cases we present here address Zopf’s call for case studies from different teacher education contexts as well as investigations of novice teacher educators, and further contribute to the developing theory of MKTT.

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

Fig. 1
Fig. 2


  1. 1.

    HPM refers to the International Study Group on the Relations between the History and Pedagogy of Mathematics, which is affiliated with the International Commission on Mathematical Instruction (ICMI), and concerns the role history of mathematics in mathematics education.

  2. 2.

    Other studies of history of mathematics in teacher education include: Charalambous et al. (2009), Furinghetti (1997, 2007), Huntley and Flores (2010), and Wang et al. (2018).

  3. 3.

    The course described in this paper was previously analyzed from the point of view of developing teacher competencies (Niss and Højgaard 2011). This analysis is available in Jankvist (2015). An initial discussion of the potential in relation to MKT and MKTT can be found in Jankvist et al. (2016).

  4. 4.

    Alternatively, the educators may hold a master’s degree in mathematics. Note that teacher educators in Denmark are not required to possess a PhD.

  5. 5.

    In 2016 the three other short courses were: “The Relevance Problem of Mathematics Education” taught by Tomas Højgaard; “Mathematics for All” by Lena Lindenskov; and “Technology in Mathematics Education” by guest professors Morten Blomhøj and Morten Misfeldt.

  6. 6.

    As supplemental literature for this lesson, students were encouraged to examine Jankvist and Kjeldsen (2011).

  7. 7.

    Supplementary literature for session 3 included Glaubitz (2011) and Jankvist (2014).

  8. 8.

    Supplementary literature for session 4 included Smestad et al. (2014).

  9. 9.

    In her early work, Ball (1988) also included PCK, and in particular the ability to represent mathematics in different ways in order for other to understand as part of knowledge of mathematics.


  1. Andersen, K., Andersen, I., Gram, K., Holth, K., Jakobsen, I. T., & Mejlbo, L. (1986). Kilder og kommentarer til ligningernes historie. Vejle: Forlaget Trip.

    Google Scholar 

  2. Arcavi, A., Bruckheimer, B., & Ben-Zvi, R. (1982). Maybe a mathematics teacher can profit from the study of the history of mathematics. For the Learning of Mathematics,3(1), 30–37.

    Google Scholar 

  3. Artzt, A., Sultan, A., Curcio, F., & Gurl, T. (2012). A capstone mathematics course for prospective secondary mathematics teachers. Journal of Mathematics Teacher Education,15(3), 251–262.

    Article  Google Scholar 

  4. Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, East Lansing.

  5. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education,59(5), 389–407.

    Article  Google Scholar 

  6. 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.

    Article  Google Scholar 

  7. Castro Superfine, A., & Li, W. (2014). Exploring the mathematical knowledge needed for teaching teachers. Journal of Teacher Education,65(4), 303–314.

    Article  Google Scholar 

  8. Charalambous, C. Y., Panaoura, A., & Philippou, G. (2009). Using the history of mathematics to induce changes in preservice teachers’ beliefs and attitudes: Insights from evaluating a teacher education program. Educational Studies in Mathematics,71(2), 161–180.

    Article  Google Scholar 

  9. 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.

    Article  Google Scholar 

  10. Dirichlet, J. P. G. L. (1837). Über die darstellung ganz willkürlicher funktionen durch sinus- und cosinus reihen. Repertorium der Physik,I, 152–174.

    Google Scholar 

  11. Euler, L. (1748/1988). Introductio in Analysin Infinitorum [Introduction to analysis of the infinite book I] (J. D. Blanton, Trans.). New York: Springer.

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

    Article  Google Scholar 

  13. Furinghetti, F. (1997). History of mathematics, mathematics education, school practice: Case studies linking different domains. For the Learning of Mathematics,17(1), 55–61.

    Google Scholar 

  14. Furinghetti, F. (2007). Teacher education through the history of mathematics. Educational Studies in Mathematics,66(2), 131–143.

    Article  Google Scholar 

  15. Glaubitz, M. R. (2011). The use of original sources in the classroom: Empirical research findings. In E. Barbin, M. Kronfellner, & C. Tzanakis (Eds.), History and epistemology in mathematics education: Proceedings of the 6th European Summer University (pp. 351–362). Vienna: Holzhausen.

    Google Scholar 

  16. Hoover, M., Mosvold, R., Ball, D. L., & Lai, Y. (2016). Making progress on mathematical knowledge for teaching. The Mathematics Enthusiast,13(1–2), 3–34.

    Google Scholar 

  17. Huntley, M. A., & Flores, A. (2010). A history of mathematics course to develop prospective secondary mathematics teachers’ knowledge for teaching. PRIMUS,20(7), 603–616.

    Article  Google Scholar 

  18. 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.

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. Jankvist, U. T. (2013). History, applications, and philosophy in mathematics education: HAPh—A use of primary sources. Science & Education,22(3), 635–656.

    Article  Google Scholar 

  21. Jankvist, U. T. (2014). On the use of primary sources in the teaching and learning of mathematics. In M. R. Matthews (Ed.), International handbook of research in history, philosophy and science teaching (pp. 873–908). Dordrecht: Springer.

    Google Scholar 

  22. Jankvist, U. T. (2015). Teaching history in mathematics education to future mathematics teacher educators. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (pp. 1825–1831). Prague, Czech Republic: Charles University in Prague, Faculty of Education and ERME.

  23. Jankvist, U. T., & Iversen, S. M. (2014). ‘Whys’ and ‘hows’ of using philosophy in mathematics education. Science & Education,23(1), 205–222.

    Article  Google Scholar 

  24. 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 

  25. Jankvist, U. T., & Misfeldt, M. (2015). CAS-induced difficulties in learning mathematics. For the Learning of Mathematics,35(1), 15–20.

    Google Scholar 

  26. Jankvist, U. T., Mosvold, R., & Clark, K. (2016). Mathematical knowledge for teaching teachers: The case of history in mathematics education. In L. Radford, F. Furinghetti, & T. Hausberger (Eds.), International Study Group on the Relations between the History and Pedagogy of Mathematics–Proceedings of the 2016 ICME Satellite Meeting (pp. 441–452). Montpellier, France: IREM de Montpellie.

  27. Jankvist, U. T., Mosvold, R., Fauskanger, J., & Jakobsen, A. (2015). Analysing the use of history of mathematics through MKT. International Journal of Mathematical Education in Science and Technology,46(4), 495–507.

    Article  Google Scholar 

  28. Kim, Y. (2013). Teaching mathematical knowledge for teaching: Curriculum and challenges. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.

  29. Kjeldsen, T. H., & Blomhøj, M. (2012). Beyond motivation: History as a method for learning metadiscursive rules in mathematics. Educational Studies in Mathematics,80(3), 327–349.

    Article  Google Scholar 

  30. Kjeldsen, T. H., & Petersen, P. H. (2014). Bridging history of the concept of function with learning mathematics: Students’ meta-discursive rules, concept formation and historical awareness. Science & Education,23(1), 29–45.

    Article  Google Scholar 

  31. Masingila, J. O., Olanoff, D., & Kimani, P. M. (2017). Mathematical knowledge for teaching teachers: Knowledge used and developed by mathematics teacher educators in learning to teach via problem solving. Journal of Mathematics Teacher Education.

    Article  Google Scholar 

  32. Mirabelle, (2016). Mini-project & appendices, Didactics of Mathematics course. Copenhagen: Danish School of Education, Aarhus University.

    Google Scholar 

  33. Mosvold, R., Jakobsen, A., & Jankvist, U. T. (2014). How mathematical knowledge for teaching may profit from the study of history of mathematics. Science & Education,23(1), 47–60.

    Article  Google Scholar 

  34. Nickalls, R. W. D. (2006). Viète, Descartes and the cubic equation. The Mathematical Gazette,90(518), 203–208.

    Article  Google Scholar 

  35. Niss, M., & Højgaard, T. (Eds.). (2011). Competencies and mathematical learningIdeas and inspiration for the development of mathematics teaching and learning in Denmark. IMFUFA tekst nr. 485/2011, Roskilde University.

  36. Petersen, P. H. (2011). Potentielle vindinger ved inddragelse af matematikhistorie i matematikundervisningen. Master thesis in mathematics, Roskilde University, Roskilde, Denmark. Retrieved from Accessed 7 Jan 2019.

  37. Rose, & Sibyl, (2016). Mini-project & appendices, Didactics of Mathematics course. Copenhagen: Danish School of Education, Aarhus University.

    Google Scholar 

  38. Schou, J., Jess, K, Hansen, H. C., & Skott, J. (2013). Matematik for lærerstuderende - Tal, algebra og funktioner 4.-10. Klassetrin. Frederiksberg: Samfundslitteratur.

  39. Schwab, J. J. (1978). Science, curriculum, and liberal education. Chicago: The University of Chicago Press.

    Google Scholar 

  40. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University Press.

    Google Scholar 

  41. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher,15(2), 4–14.

    Article  Google Scholar 

  42. Smestad, B., Jankvist, U. T., & Clark, K. (2014). Teachers’ mathematical knowledge for teaching in relation to the inclusion of history of mathematics in teaching. Nordic Studies in Mathematics Education,19(3–4), 169–183. (Thematic issue edited by J. Fauskanger & R. Mosvold.).

    Google Scholar 

  43. Tall, D., & Vinner, S. (1981). Concept image and concept definition with particular reference to limits and continuity. Educational Studies in Mathematics,12(2), 151–169.

    Article  Google Scholar 

  44. VIA UC. (2013). Studieordningen for Læreruddannelsen i Aarhus. Modulbeskrivelser. Matematik 4.-10. Klassetrin. Matematikmodul 4: Modellering og undervisningsdifferentiering. Aarhus: VIA University College.

  45. Viète, F. (1646). Opera Mathematica, edited with notes by Frans von Schooten (Leyden, 1646). A facsimile reprint has been issued (Hildesheim: Georg Olms Verlag, 1970).

  46. Waldeana, H. N., & Abraham, S. T. (2014). The effect of an historical perspective on prospective teachers’ beliefs in learning mathematics. Journal of Mathematics Teacher Education,17(4), 303–330.

    Article  Google Scholar 

  47. Wang, K., Wang, X., Li, Y., & Rugh, M. S. (2018). A framework for integrating the history of mathematics into teaching in Shanghai. Educational Studies in Mathematics.

    Article  Google Scholar 

  48. Zopf, D. (2010). Mathematical knowledge for teaching teachers: The mathematical work of and knowledge entailed by teacher education. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.

Download references

Author information



Corresponding author

Correspondence to Kathleen Michelle Clark.

Additional information

Publisher's Note

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

Appendix: Center text from Fig. 1 (partial translation)

Appendix: Center text from Fig. 1 (partial translation)

  • VII.10 Excerpt from Viète’s theory of equations

  •                      Third sentence in another way

  • If, when B is greater than half of D:

  • If A cube − B square thrice into A is equated to B square into D,

  • but

  • B square [thrice] into E − E cube will be equated to B square into D,

  • and there are two right angled triangles with equal hypotenuse B, such that the acute angle subtended by the perpendicular of the first, is triple the acute angle subtended by the perpendicular of the second, while double the base of the first, is D, making the double of the base of the second be A

  • [Let]              1C − 300N be equated to 432;

  • or even let

  •                      300N − 1C be equated to 432.

  • There are two right triangles, of which the common hypotenuse is 10 [radius B] … Moreover double the base of the first is \(\frac{432}{100}\) [chord D] and … the base becomes 2 \(\frac{16}{100}\) … when it is said that 300N − 1C is equated to 432, 9 + R57 [9 +  \(\sqrt {57}\)] or 9 − R57 [9 −  \(\sqrt {57}\)] will be made to be 1 N [chord A]. (adapted from Nickalls 2006, pp. 204–205)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jankvist, U.T., Clark, K.M. & Mosvold, R. Developing mathematical knowledge for teaching teachers: potentials of history of mathematics in teacher educator training. J Math Teacher Educ 23, 311–332 (2020).

Download citation


  • Mathematical knowledge for teaching
  • Mathematical knowledge for teaching teachers
  • Teacher educator students
  • History of mathematics
  • Primary historical sources