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Unpacking hidden views: seven ways to treat your formula

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Abstract

Formulas are involved in most parts of the mathematical curriculum in upper secondary education and in everyday mathematics classrooms, but research shows that students have difficulties using formulas adequately. When students are presented with a task, the task activates a conceptual frame in the students, making them perceive formulas in a specific way, thereby affecting their mathematical behaviour. In this paper, building ideal types of patterns of mathematical behaviours is used to conceptualise ‘view on formula’ specified by several specific views. The concept of ‘view on formula’ is applied in an analysis of a classroom episode pointing to reasons for the difficulties students have with handling formulas. When views are either missing or not used in a flexible way this can lead to an unsuccessful handling of formulas. Also, students’ views on formula may indicate what knowledge is missing for solving a task involving formulas. Taken together, this points to the importance of paying attention to views on formulas in the everyday mathematics classroom.

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Notes

  1. By handle we mean to treat in a particular way, that is to deal with, or behave towards (a thing or person), in a certain and particular manner. (see https://dictionary.cambridge.org/).

  2. References for Peirce come from the Collected Papers, CP. The first number denotes the volume and the second the paragraph. That is CP 2.228 is paragraph 228 in volume 2.

  3. Prototypes of the remaining five ideal types can be made accessible on demand by contacting the authors.

  4. The ideal types respective views will be stated with a capital first letter and within single quotation marks.

  5. Although a recipe can be seen as a process of inserting numbers and calculating a result (this being on the arithmetic level), the algebraic formula itself is fixed; hence, the ‘Recipe’ is static.

  6. We thank Simon & Garfunkel for the inspiration to the title of this paper: “50 ways to leave your lover”.

References

  • Artigue, M., Assude, T., Grugeon, B., & Lenfant, A. (2001). Teaching and learning algebra: Approaching complexity through complementary perspectives. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference in Melbourne (vol. 1, pp. 21–32). University of Melbourne. http://hdl.handle.net/11343/35000. Accessed 21 June 2021.

  • Atkin, A. (2013). Peirce’s theory of signs. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/peirce-semiotics/. Accessed 21 June 2021.

  • Arzarello, F., Bazzini, L., & Chiappini, G. (2001). A model for analysing algebraic processes of thinking. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 61–81). Kluwer Academic Press. https://doi.org/10.1007/0-306-47223-6_4

  • Behr, M., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92(1), 13–15.

    Google Scholar 

  • Bikner-Ahsbahs, A. (2015). Empirically grounded building of ideal types. A methodical principle of constructing theory in the interpretative research in mathematics education (Transl.). In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 105–135). Springer. (Original work publ. 2003) https://doi.org/10.1007/978-94-017-9181-6_5

  • Bikner-Ahsbahs, A., Sabena, C., Arzarello, F., & Krause, C. (2014). Semiotic and theoretic control within and across conceptual frames. In C. Nicol, P. Liljedahl, S. Oesterle, & D. Allan, (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 2, pp. 153–160). PME. http://www.igpme.org

  • Blumer, H. (1969). Symbolic interactionism: Perspective and method. University of California Press.

  • Blumer, H. (1980). Mead and Blumer: The convergent methodological perspectives of social behaviorism and symbolic interactionism. American Sociological Review, 45(3), 409–419. https://doi.org/10.2307/2095174

    Article  Google Scholar 

  • Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 99–119). Kluwer Academic Press. https://doi.org/10.1007/0-306-47223-6_6

  • Bosch, M. (2015). Doing research within the anthropological theory of the didactic: The case of school algebra. In Cho S. (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 51–69). Springer. https://doi.org/10.1007/978-3-319-17187-6_4

  • Cobb, P., Yackel, E., & McClain, K. (2000, Eds.). Symbolizing and communicating in mathematics classrooms. Perspectives on discourse, tools, and instructional design. Routledge. Publ. https://doi.org/10.4324/9781410605351

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103–131. https://doi.org/10.1007/s10649-006-0400-z

    Article  Google Scholar 

  • Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236. https://doi.org/10.5951/tcm.6.4.0232

    Article  Google Scholar 

  • Hefendehl-Hebeker, L., & Rezat, S. (2015). Algebra: Leitidee Symbol und Formalisierung. In R. Bruder, L. Hefendehl-Hebeker, B. Schmidt-Thieme, & H.-G. Weigand (Eds.) Handbuch der Mathematikdidaktik. Springer Spektrum. https://doi.org/10.1007/978-3-642-35119-8_5

  • Hattikudur, S., & Alibali, M. W. (2010). Learning about the equal sign: Does comparing with inequality symbols help? Journal of Experimental Child Psychology, 107(1), 15–30. https://doi.org/10.1016/j.jecp.2010.03.004

    Article  Google Scholar 

  • Janvier, C. (1996). Modeling and the initiation into algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 225–236). Kluwer Academic Publisher. https://doi.org/10.1007/978-94-009-1732-3_17

  • Jungwirth, H. (2003). Interpretative Forschung in der Mathematikdidaktik—ein Überblick für Irrgäste, Teilzieher und Standvögel. Zentralblatt für Didaktik der Mathematik, 35(5), 189–200. https://doi.org/10.1007/BF02655743

  • Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/bf00311062

    Article  Google Scholar 

  • Kieran, C. (1992). The learning and teaching of algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). Information Age. https://doi.org/10.1007/BF00311062

  • Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Information Age

  • Kluge, S. (1999). Empirisch begründete Typenbildung. Zur Konstruktion von Typen und Typologien in der qualitativen Sozialforschung. Leske+ Budrich. https://doi.org/10.1007/978-3-322-97436-5

  • Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13(9), 514–519. https://doi.org/10.5951/MTMS.13.9.0514

    Article  Google Scholar 

  • Lemke, J. L. (1998). Teaching all the languages of science: Words, symbols, images, and actions. Paper presented at the Conference on science education (Barcelona). http://academic.brooklyn.cuny.edu/education/jlemke/papers/barcelon.htm. Accessed 21 June 2021.

  • Lins, R. C. (2001). The production of meaning for algebra: A perspective based on a theoretical model of Semantic Fields. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 37–60). Kluwer Academic Press. https://doi.org/10.1007/0-306-47223-6_3

  • McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24(3), 367–385. https://doi.org/10.1207/s1532690xci2403_3

    Article  Google Scholar 

  • Peirce, C. S. (1965). Collected papers of Charles Sanders Peirce. C. Hartshorne, P. Weiss, & A. Burks (Eds). Harvard University Press /Belknap Press

  • Prediger, S. (2010). How to develop mathematics for teaching and for understanding. The case of meanings of the equal sign. Journal of Mathematics Teacher Education, 13(1), 73–93. https://doi.org/10.1007/s10857-009-9119-y

    Article  Google Scholar 

  • Ransdell, J. (1977). Some leading ideas of Peirce’s semiotic. International Association for Semiotic Studies. https://arisbe.sitehost.iu.edu/menu/library/aboutcsp/ransdell/leading.htm. Accessed 18 June 2021.

  • Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education, 24(5/6), 561–590. https://doi.org/10.1007/s11191-015-9749-7

    Article  Google Scholar 

  • Sáenz-Ludlow, A., & Walgamuth, C. (1998). Third graders’ interpretations of equality and the equal symbol. Educational Studies in Mathematics, 35(2), 153–187. https://doi.org/10.1023/A:1003086304201

    Article  Google Scholar 

  • Scheiner, T. (2019). If we want to get ahead, we should transcend dualisms and foster paradigm pluralism. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education, ICME13-Monograph (pp. 511–532). SpringerOpen. https://doi.org/10.1007/978-3-030-15636-7_27

  • Schou, M. H. (2017). The roles of mathematical symbols in teacher instruction. In E. Norén, H. Palmér, & A. Cooke (Eds.), Proceedings for The Eighth Nordic Conference on Mathematics Education. Swedish Society for Research in Mathematics Education

  • Scherr, R. E., & Hammer, D. (2009). Student behavior and epistemological framing: Examples from collaborative active-learning activities in physics. Cognition and Instruction, 27(2), 147–174. https://doi.org/10.1080/07370000902797379

    Article  Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36. https://doi.org/10.1007/BF00302715

    Article  Google Scholar 

  • Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—the case of algebra. Educational Studies in Mathematics, 26(2/3), 191–228. https://doi.org/10.1007/BF01273663

    Article  Google Scholar 

  • Sherin, B. L. (2001). How students understand physics equations. Cognition and Instruction, 19(4), 479–541. https://doi.org/10.1207/S1532690XCI1904_3

    Article  Google Scholar 

  • Siller, H.-S., & Roth, J. (2016). Grundvorstellungen als Basis und Bezugsnorm–das Beispiel Terme. Praxis des Mathematikunterrichts in der Schule, 70, 2–8.

  • Stake, R. E. (1995). The art of case study research. Sage Publisher

  • Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective. Springer Nature. https://doi.org/10.1007/b104944

    Article  Google Scholar 

  • Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. In B. Moses (Ed.), Algebraic thinking, Grades K-12: Readings from NCTM’s school based journals and other publications, (pp. 7–13). National Council of Teachers in Mathematics

  • Weber, M. (1949). “Objectivity” in social science and social policy. In Shils, E. A., & Finch, H. A. (Eds.), The methodology of the social sciences (pp. 49–112), (Transl.). The Free Press. (Original work publ. 1922) https://archive.org/stream/maxweberonmethod00webe#page/n3/mode/2up. Accessed 21 June 2021.

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  1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Marit Hvalsøe Schou

  2. Faculty 3 of Mathematics and Computer Science, Bremen University, Bremen, Germany

    Angelika Bikner-Ahsbahs

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  1. Marit Hvalsøe Schou
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  2. Angelika Bikner-Ahsbahs
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Schou, M.H., Bikner-Ahsbahs, A. Unpacking hidden views: seven ways to treat your formula. Educ Stud Math 109, 639–659 (2022). https://doi.org/10.1007/s10649-021-10092-7

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