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Groups to the Rescue: Responding to Situations of Contingency

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Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers

Part of the book series: Research in Mathematics Education ((RME))

Abstract

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.

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References

  • Ameis, J. A. (2011). The truth about PEDMAS. Mathematics Teaching in the Middle School, 16(7), 414–420.

    Google Scholar 

  • 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 

  • Childs, L. N. (2009). A concrete introduction to higher algebra (3rd ed.). New York, NY: Springer.

    Book  Google Scholar 

  • Devlin, K. J. (1981). Sets, functions and logic: Basic concepts of university mathematics. New York, NY: Springer.

    Book  Google Scholar 

  • Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305.

    Article  Google Scholar 

  • Dummit, D. S., & Foote, R. M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: John Wiley and Sons.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Glidden, P. L. (2008). Prospective elementary teachers’ understanding of order of operations. School Science and Mathematics, 108(4), 130–136.

    Article  Google Scholar 

  • Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Hadar, N., & Hadass, R. (1981). Between associativity and commutativity. International Journal of Mathematical Education in Science and Technology, 12(5), 535–539.

    Article  Google Scholar 

  • 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 

  • Koichu, B. (2008). If not, what yes? International Journal of Mathematical Education in Science and Technology, 39(4), 443–454.

    Article  Google Scholar 

  • Kontorovich, I., & Zazkis, R. (2017). Mathematical conventions: Revisiting arbitrary and necessary. For the Learning of Mathematics, 37(1), 29–34.

    Google Scholar 

  • Larsen, S. (2010). Struggling to disentangle the associative and commutative properties. For the Learning of Mathematics, 30(1), 37–42.

    Google Scholar 

  • Mason, J. (2002). Researching your own practice: The discipline of noticing. London, UK: Routledge Falmer.

    Google Scholar 

  • Musser, G. L., Burger, W. F., & Peterson, B. E. (2010). Mathematics for elementary teachers: A contemporary approach (9th ed.). Hoboken, NJ: Wiley.

    Google Scholar 

  • Polya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined). New York, NY: Wiley.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Stewart, J. (2008). Calculus: Early transcendentals (6th ed.). Belmont, CA: Thomson Learning.

    Google Scholar 

  • Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of Mathematics. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Wasserman, N. H. (2018). Knowledge of nonlocal mathematics for teaching. Journal of Mathematical Behavior, 49, 116–128.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Zazkis, R. (2017). Order of operations: On conventions, mnemonics and knowledge-in-use. For the Learning of Mathematics, 37(3), 18–20.

    Google Scholar 

  • Zazkis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208.

    Article  Google Scholar 

  • Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (−1): Prospective secondary mathematics teachers explain. Journal of Mathematical Behavior, 43, 98–110.

    Article  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.

    Article  Google Scholar 

  • Zazkis, R., & Mamolo, A. (2011). Reconceptualising knowledge at the mathematical horizon. For the Learning of Mathematics, 31(2), 8–13.

    Google Scholar 

  • Zazkis, R., & Rouleau, A. (2018). Order of operations: On convention and met-before acronyms. Educational Studies in Mathematics, 97(2), 143–162.

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Education, Simon Fraser University, Burnaby, BC, Canada

    Rina Zazkis & Ofer Marmur

Authors
  1. Rina Zazkis
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  2. Ofer Marmur
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Corresponding author

Correspondence to Rina Zazkis .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Science and Technology, Teachers College, Columbia University, New York, NY, USA

    Nicholas H. Wasserman

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Zazkis, R., Marmur, O. (2018). Groups to the Rescue: Responding to Situations of Contingency. In: Wasserman, N. (eds) Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99214-3_17

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  • DOI: https://doi.org/10.1007/978-3-319-99214-3_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99213-6

  • Online ISBN: 978-3-319-99214-3

  • eBook Packages: EducationEducation (R0)

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