Exploring Advanced Mathematics Courses and Content for Secondary Mathematics Teachers

  • Nicholas H. WassermanEmail author
Part of the Research in Mathematics Education book series (RME)


Over the past century, mathematicians and mathematics educators have explored various ways in which abstract algebra is related to school mathematics. These have included Felix Klein's work as a mathematician—in which his synthesis of the study of geometry through abstract algebraic structures has proved influential on our approach to teaching secondary students geometry even today; his work as a mathematics teacher educator—famous for his observation of the ``double—discontinuity'' that secondary teachers face in their mathematical preparation; the provocative and controversial New Math curricular reforms in the 1960s in the USA, which reorganized and restructured the content of school mathematics to be more in accord with formal set theory and the study of algebraic structures; and various studies about, and investigations of, teachers' (advanced) mathematical knowledge in relation to their practices in the classroom and their student's outcomes. These efforts, and others, have considered the connection between school mathematics and the study of the abstract algebra structures they comprise.


  1. Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures. Paper presented at the 43rd Jahrestagung der Gesellschaft fur Didaktik der Mathematik, Oldenburg.Google Scholar
  2. 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.CrossRefGoogle Scholar
  3. Begle, E. G. (1972). Teacher knowledge and pupil achievement in algebra (NLSMA technical report number 9). Palo Alto, CA: Stanford University, School Mathematics Study Group.Google Scholar
  4. Bremigan, E. G., Bremigan, R. J., & Lorch, J. D. (2011). Mathematics for secondary school teachers. Washington, DC: Mathematical Association of America (MAA).Google Scholar
  5. Cofer, T. (2015). Mathematical explanatory strategies employed by prospective secondary teachers. International Journal of Research in Undergraduate Mathematics Education, 1(1), 63–90.CrossRefGoogle Scholar
  6. Common Core State Standards in Mathematics (CCSSM). (2010). Retrieved from:
  7. Conference Board of Mathematical Sciences (CBMS). (2001). The mathematical education of teachers. Retrieved from:
  8. Conference Board of Mathematical Sciences (CBMS). (2012). The mathematical education of teachers II. Providence, RI: American Mathematical Society and Mathematical Association of America.CrossRefGoogle Scholar
  9. Cuoco, A. (2000). Meta-problems in mathematics. The College Mathematics Journal, 31(5), 373–378.CrossRefGoogle Scholar
  10. Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15(4), 375–402.CrossRefGoogle Scholar
  11. Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305.CrossRefGoogle Scholar
  12. Eisenberg, T. A. (1977). Begle revisited: Teacher knowledge and student achievement in algebra. Journal for Research in Mathematics Education, 8(3), 216–222.CrossRefGoogle Scholar
  13. Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioner’s views. ZDM, 43(6–7), 941–950.CrossRefGoogle Scholar
  14. Goulding, M., Hatch, G., & Rodd, M. (2003). Undergraduate mathematics experience: Its significance in secondary mathematics teacher preparation. Journal of Mathematics Teacher Education, 6(4), 361–393.CrossRefGoogle Scholar
  15. Klein, F. (1932). Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (trans. Hedrick, E. R. & Noble, C. A.). Mineola, NY: Macmillan.Google Scholar
  16. Larsen, S. (2009). Reinventing the concepts of groups and isomorphisms: The case of Jessica and Sandra. Journal of Mathematical Behavior, 28(2–3), 119–137.CrossRefGoogle Scholar
  17. Larsen, S., Johnson, E., & Weber, K. (Eds.). (2013). The teaching abstract algebra for understanding project: Designing and scaling up a curriculum innovation. Journal of Mathematical Behavior (Special Issue), 32(4), 691–790.Google Scholar
  18. Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.CrossRefGoogle Scholar
  19. Moreira, P. C., & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: Some conflicting elements. Journal of Mathematics Teacher Education, 11(1), 23–40.CrossRefGoogle Scholar
  20. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  21. Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 551–579). Reston, VA: National Council of Teachers of Mathematics (NCTM).Google Scholar
  22. Rhoads, K. (2014). High school mathematics teachers’ use of beliefs and knowledge in high quality instruction. Unpublished doctoral dissertation.Google Scholar
  23. Sultan, A., & Artzt, A. F. (2011). The mathematics that every secondary school math teacher needs to know. New York, NY: Routledge.Google Scholar
  24. Ticknor, C. S. (2012). Situated learning in an abstract algebra classroom. Educational Studies in Mathematics, 81(3), 307–323.CrossRefGoogle Scholar
  25. Usiskin, Z., Peressini, A., Marchisotto, E. A., & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. Upper Saddle River, NJ: Pearson.Google Scholar
  26. Wasserman, N. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science Mathematics and Technology Education, 16(1), 28–47.CrossRefGoogle Scholar
  27. Wasserman, N. (2017). Making sense of abstract algebra: Exploring secondary teachers’ understanding of inverse functions in relation to its group structure. Mathematical Thinking and Learning, 19(3), 181–201.CrossRefGoogle Scholar
  28. Wasserman, N. (2018). Nonlocal mathematical knowledge for teaching. Journal of Mathematical Behavior, 49(1), 116–128.CrossRefGoogle Scholar
  29. Wasserman, N., Fukawa-Connelly, T., Villanueva, M., Mejia-Ramos, J. P., & Weber, K. (2017). Making real analysis relevant to secondary teachers: Building up from and stepping down to practice. PRIMUS, 27(6), 559–578.CrossRefGoogle Scholar
  30. Wasserman, N., & Weber, K. (2017). Pedagogical applications from real analysis for secondary mathematics teachers. For the Learning of Mathematics, 37(3), 14–18.Google Scholar
  31. Wasserman, N., Weber, K., & McGuffey, W. (2017). Leveraging real analysis to foster pedagogical practices. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20th annual conference on research in undergraduate mathematics education (RUME) (pp. 1–15). San Diego, CA: RUME.Google Scholar
  32. Wasserman, N., Weber, K., Villanueva, M., & Mejia-Ramos, J. P. (2018). Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis. Journal of Mathematical Behavior, 50(1), 74–89.CrossRefGoogle Scholar
  33. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.CrossRefGoogle Scholar
  34. Weber, K., & Larsen, S. (2008). Teaching and learning group theory. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 137–149). Washington, DC: Mathematical Association of America (MAA).Google Scholar
  35. Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12(4), 263–281.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Technology, Teachers CollegeColumbia UniversityNew YorkUSA

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