Groups to the Rescue: Responding to Situations of Contingency

  • Rina ZazkisEmail author
  • Ofer Marmur
Part of the Research in Mathematics Education book series (RME)


We extend the argument on the importance of teachers’ knowledge of advanced mathematics, group theory in particular. We present several examples in which familiarity with properties of groups guided the teacher’s responses in unexpected instructional situations, to which we refer as situations of contingency. We further describe how the situations inspired the design of tasks that capitalize on the applicability of advanced knowledge, and strengthen learners’ connections between different areas of mathematics.


Situations of contingency Group theory Advanced mathematical knowledge 


  1. Ameis, J. A. (2011). The truth about PEDMAS. Mathematics Teaching in the Middle School, 16(7), 414–420.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. Childs, L. N. (2009). A concrete introduction to higher algebra (3rd ed.). New York, NY: Springer.CrossRefGoogle Scholar
  4. Devlin, K. J. (1981). Sets, functions and logic: Basic concepts of university mathematics. New York, NY: Springer.CrossRefGoogle Scholar
  5. 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
  6. Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: John Wiley and Sons.Google Scholar
  7. Even, R. (2011). The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: Practitioners’ views. ZDM Mathematics Education, 43(6–7), 941–950.CrossRefGoogle Scholar
  8. Glidden, P. L. (2008). Prospective elementary teachers’ understanding of order of operations. School Science and Mathematics, 108(4), 130–136.CrossRefGoogle Scholar
  9. Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.Google Scholar
  10. Hadar, N., & Hadass, R. (1981). Between associativity and commutativity. International Journal of Mathematical Education in Science and Technology, 12(5), 535–539.CrossRefGoogle Scholar
  11. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge. Journal for Research in Mathematics Education, 39(4), 372–400.Google Scholar
  12. Koichu, B. (2008). If not, what yes? International Journal of Mathematical Education in Science and Technology, 39(4), 443–454.CrossRefGoogle Scholar
  13. Kontorovich, I., & Zazkis, R. (2017). Mathematical conventions: Revisiting arbitrary and necessary. For the Learning of Mathematics, 37(1), 29–34.Google Scholar
  14. Larsen, S. (2010). Struggling to disentangle the associative and commutative properties. For the Learning of Mathematics, 30(1), 37–42.Google Scholar
  15. Mason, J. (2002). Researching your own practice: The discipline of noticing. London, UK: Routledge Falmer.Google Scholar
  16. Musser, G. L., Burger, W. F., & Peterson, B. E. (2010). Mathematics for elementary teachers: A contemporary approach (9th ed.). Hoboken, NJ: Wiley.Google Scholar
  17. Polya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined). New York, NY: Wiley.Google Scholar
  18. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255–281.CrossRefGoogle Scholar
  19. Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359–371.CrossRefGoogle Scholar
  20. Stewart, J. (2008). Calculus: Early transcendentals (6th ed.). Belmont, CA: Thomson Learning.Google Scholar
  21. Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of Mathematics. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  22. Wasserman, N. H. (2016). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education, 16(1), 28–47.CrossRefGoogle Scholar
  23. Wasserman, N. H. (2017). Making sense of Abstract Algebra: Exploring secondary teachers’ understandings of inverse functions in relation to its group structure. Mathematical Thinking and Learning, 19(3), 181–201.CrossRefGoogle Scholar
  24. Wasserman, N. H. (2018). Knowledge of nonlocal mathematics for teaching. Journal of Mathematical Behavior, 49, 116–128.CrossRefGoogle Scholar
  25. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  26. Weber, K., & Larsen, S. (2008). Teaching and learning group theory. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 139–151). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  27. Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student teachers: The case of binary operation. Journal for Research in Mathematics Education, 27(1), 67–78.CrossRefGoogle Scholar
  28. Zazkis, R. (2017). Order of operations: On conventions, mnemonics and knowledge-in-use. For the Learning of Mathematics, 37(3), 18–20.Google Scholar
  29. Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208.CrossRefGoogle Scholar
  30. Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (−1): Prospective secondary mathematics teachers explain. Journal of Mathematical Behavior, 43, 98–110.CrossRefGoogle Scholar
  31. 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
  32. Zazkis, R., & Mamolo, A. (2011). Reconceptualising knowledge at the mathematical horizon. For the Learning of Mathematics, 31(2), 8–13.Google Scholar
  33. Zazkis, R., & Rouleau, A. (2018). Order of operations: On convention and met-before acronyms. Educational Studies in Mathematics, 97(2), 143–162.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of EducationSimon Fraser UniversityBurnabyCanada

Personalised recommendations