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Space of fuzziness: avoidance of deterministic decisions in the case of the inverse function

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Abstract

We investigate responses of prospective secondary school teachers to a task related to the inverse function concept. We utilize ideas of fuzzy logic as a theoretical lens in analyzing the participants’ demonstrated avoidance of refuting the existence of an inverse to a quadratic function. The analysis is based on 29 responses to a scripting task, in which participants composed a hypothetical dialog between a teacher and students. The findings demonstrate three categories of fuzzy evaluations of a mathematical statement: (a) false blurred to be true; (b) almost false; and (c) almost true. We offer possible reasons for the prevalence of fuzzy thinking in general, and in discussing the inverse of a function in particular.

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Notes

  1. This analogy extends Vergnaud’s (1996) characterization of concepts-in-action and theorems-in-action.

  2. “Non-contradiction” (a) and “excluded middle” (b) refer to traditional Aristotelian laws of logic: (a) a statement cannot be both true and false; (b) a statement is either true or false.

  3. We call attention to the mathematical convention of considering the domain of a function to be the maximal set of values for which the function is defined, unless explicitly stated otherwise. Accordingly, we regard the domain of the function y = x2 − 4x + 5 as ℝ and the function as non-invertible.

  4. Emphasized here, not in original text.

  5. We interpret the reference to a “one-on-one function” as confusion in terminology, referring to a one-to-one function.

References

  • Bayazit, I., & Gray, E. (2004). Understanding inverse functions: The relationship between teaching practice and student learning. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp. 103-110). Bergen, Norway: PME.

  • Bojadziev, G., & Bojadziev, M. (1997). Fuzzy logic for business, finance, and management. Singapore: World Scientific.

    Book  Google Scholar 

  • Brown, C., & Reynolds, B. (2007). Delineating four conceptions of function: A case of composition and inverse. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 190–193). Stateline (Lake Tahoe), NV: University of Nevada, Reno.

  • Bubp, K. M. (2016). Students’ explicit, unwarranted assumptions in “proofs” of false conjectures. In T. Fukawa-Connelly, N. E. Infante, M. Wawro, & S. Brown (Eds.), Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education (pp. 588–595). Pittsburgh, Pennsylvania.

  • Byers, W. (2010). How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton: Princeton University Press.

    Google Scholar 

  • Durkin, K., & Shire, B. (1991). Primary school children's interpretations of lexical ambiguity in mathematical descriptions. Journal of Research in Reading, 14(1), 46–55.

    Article  Google Scholar 

  • Engelke, N., Oehrtman, M., & Carlson, M. (2005). Composition of functions: Precalculus students' understandings. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1–8). Virginia Tech: PME.

  • Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521–544.

    Article  Google Scholar 

  • Even, R. (1992). The inverse function: Prospective teachers' use of ‘undoing’. International Journal of Mathematical Education in Science and Technology, 23(4), 557–562.

    Google Scholar 

  • Gómez-Chacón, I. M. (2017). Emotions and heuristics: The state of perplexity in mathematics. ZDM Mathematics Education, 49(3), 323–338.

    Article  Google Scholar 

  • Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching, 15(2), 273–289.

    Article  Google Scholar 

  • Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288.

    Article  Google Scholar 

  • Koichu, B., & Zazkis, R. (2013). Decoding a proof of Fermat's little theorem via script writing. Journal of Mathematical Behavior, 32(3), 364–376.

    Article  Google Scholar 

  • Kosko, B. (1993). Fuzzy thinking: The new science of fuzzy logic. New York: Hyperion.

    Google Scholar 

  • Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Lehrer, R., & Franke, M. L. (1992). Applying personal construct psychology to the study of teachers' knowledge of fractions. Journal for Research in Mathematics Education, 23(3), 223–241.

    Article  Google Scholar 

  • Leron, U., & Hazzan, O. (1997). The world according to Johnny: A coping perspective in mathematics education. Educational Studies in Mathematics, 32(3), 265–292.

    Article  Google Scholar 

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

    Google Scholar 

  • Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 27–42). Washington, D.C.: MAA.

    Chapter  Google Scholar 

  • Ouvrier-Buffet, C. (2011). A mathematical experience involving defining processes: In-action definitions and zero-definitions. Educational Studies in Mathematics, 76(2), 165–182.

    Article  Google Scholar 

  • Paoletti, T., Stevens, I. E., Hobson, N. L. F., Moore, K. C., & LaForest, K. R. (2018). Inverse function: Pre-service teachers’ techniques and meanings. Educational Studies in Mathematics, 97, 93–109.

    Article  Google Scholar 

  • Quine, W. V. (1948). On what there is. The Review of Metaphysics, 2(5), 21–38.

    Google Scholar 

  • Saliu, S. (2005). Constrained subjective assessment of student learning. Journal of Science Education and Technology, 14(3), 271–284.

    Article  Google Scholar 

  • Stevenson, A., & Lindberg, C. A. (Eds.). (2010). New Oxford American Dictionary (3rd ed.). New York: Oxford University Press.

    Google Scholar 

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

    Book  Google Scholar 

  • Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 219–239). Mahwah: Erlbaum.

    Google Scholar 

  • Vidakovic, D. (1996). Learning the concept of inverse function. Journal of Computers in Mathematics and Science Teaching, 15(3), 295–318.

    Google Scholar 

  • Vinner, S. (1990). Inconsistencies: Their causes and function in learning mathematics. Focus on Learning Problems in Mathematics, 12(3–4), 85–98.

    Google Scholar 

  • Voskoglou, M. G. (2011). Fuzzy logic and uncertainty in mathematics education. International Journal of Applications of Fuzzy Sets, 1, 45–64.

    Google Scholar 

  • Voskoglou, M. G. (2013). Fuzzy logic as a tool for assessing students’ knowledge and skills. Education Sciences, 3(2), 208–221.

    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 

  • Weber, R. P. (1990). Basic content analysis (2nd ed.). Newbury Park, California: Sage Publications.

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

    Article  Google Scholar 

  • Zazkis, R. (1995). Fuzzy thinking in non-fuzzy situations: Understanding students' perspective. For the Learning of Mathematics, 15(3), 39–41.

    Google Scholar 

  • Zazkis, R. (2000). Using code-switching as a tool for learning mathematical language. For the Learning of Mathematics, 20(3), 38–43.

    Google Scholar 

  • Zazkis, R. (2002). Language of number theory: metaphor and rigor. In S. R. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 83–96). Westport, CT: Ablex Publishing.

  • 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., & Levy, B.-S. (2001). Truth value of mathematical statements: Can fuzzy logic illustrate students' decision making? Focus on Learning Problems in Mathematics, 23(4), 1–26.

    Google Scholar 

  • Zazkis, R., Liljedahl, P., & Sinclair, N. (2009). Lesson plays: Planning teaching vs. teaching planning. For the Learning of Mathematics, 29(1), 40–47.

    Google Scholar 

  • Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: A tool for research and professional development. Dordrecht: Springer.

    Book  Google Scholar 

  • Zazkis, R., & Zazkis, D. (2014). Script writing in the mathematics classroom: Imaginary conversations on the structure of numbers. Research in Mathematics Education, 16(1), 54–70.

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Ofer Marmur & Rina Zazkis

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  1. Ofer Marmur
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  2. Rina Zazkis
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Correspondence to Ofer Marmur.

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Marmur, O., Zazkis, R. Space of fuzziness: avoidance of deterministic decisions in the case of the inverse function. Educ Stud Math 99, 261–275 (2018). https://doi.org/10.1007/s10649-018-9843-2

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