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Students’ Use of Prior Conceptions of Symbols in Finding an Equation for a Horizontal Translation of the Graph of a Function

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Abstract

Research has identified some sources of student difficulty in understanding the horizontal translation of a function. The present study presents and focuses on another source for the difficulties: student conceptions of symbols involved in the horizontal translation of the graph of a function. We explored which student conceptions of the symbols f, x, and y are problematic or supportive in finding the equation of the translated graph when a given graph of y = f(x) is translated −2 units horizontally. For example, student conceptions of x as a general label for any point on the x-axis or as an independent variable of only a given original function emerged as problematic, while a conception of x as an independent variable of a function to describe was supportive. It seemed crucial for students to recruit some background ideas about horizontal translation in order to activate the supportive conceptions of f, x, and y and make them powerful. Our study suggests that horizontal translation of the graph of a function provides an opportunity for students to reflect on their prior conceptions of f, x, and y and restructure the conceptions to be comprehensive.

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Notes

  1. In this study, we use the phrase “horizontal translation of a function or y = f(x)” rather than “horizontal translation of the graph of a function or y = f(x)” for convenience.

  2. Amy used f ′ to represent the translated graph. In this paper, f ′ has nothing to do with the derivative of f.

References

  • Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.

    Google Scholar 

  • Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.

    Article  Google Scholar 

  • Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the Learning of Mathematics, 25(2), 42–47.

    Google Scholar 

  • Ayers, T., Davis, G., Dubinski, E., & Lewin, G. (1988). Computer experiences in learning composition of functions. Journal for Research in Mathematics Education, 19(3), 246–259.

    Article  Google Scholar 

  • Baker, B., Hemenway, C. & Trigueros, M. (2001). On transformations of basic functions. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (Vol. 1, pp. 41–47). University of Melbourne.

  • Borba, M. C., & Confrey, J. (1996). A student’s construction of transformations of functions in a multi representational environment. Educational Studies in Mathematics, 31(3), 319–337.

    Article  Google Scholar 

  • Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–285.

    Article  Google Scholar 

  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative research. Sage.

    Google Scholar 

  • Dilling, F., & Witzke, I. (2020). The use of 3D-printing technology in calculus education: Concept formation processes of the concept of derivative with printed graphs of functions. Digital Experiences in Mathematics Education, 6, 320–339.

    Article  Google Scholar 

  • Edwards, L. D. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70(2), 127–141.

    Article  Google Scholar 

  • Eisenberg, T., & Dreyfus, T. (1994). On understanding how students learn to visualize function transformations. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education (Vol. 1, pp. 45–68). American Mathematical Society.

    Chapter  Google Scholar 

  • Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Basic Books.

    Google Scholar 

  • Fauconnier, G. (2005). Compression and emergent structure. Language and Linguistics, 6(4), 523–538.

    Google Scholar 

  • Faulkenberry, E. D., & Faulkenberry, T. J. (2010). Transforming the way we teach function transformations. Mathematics Teacher, 104(1), 29–33.

    Article  Google Scholar 

  • Gerson, H. & Walter, J. (2008). How blending illuminates understandings of calculus. In Electronic Proceedings for the Eleventh Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics. Retrieved from http://sigmaa.maa.org/rume/crume2008/Proceedings/Gerson%20LONG.pdf

  • Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Aldine Publishing Company.

    Google Scholar 

  • Hall, B., & Giacin, R. (2013). Exploring function transformations using the common core. Mathematics Teacher, 107(2), 132–137.

    Article  Google Scholar 

  • Hazzan, O., & Zazkis, R. (2005). Reducing abstraction: The case of school mathematics. Educational Studies in Mathematics, 58(1), 101–119.

    Article  Google Scholar 

  • Holton, J. A. (2010). The coding process and its challenges. The Grounded Theory Review, 9(1), 21–40.

    Google Scholar 

  • Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22, 55–72.

    Article  Google Scholar 

  • Kinzel, M. (1997). Signifiers and counterparts: Building a framework for analyzing students’ use of symbols. In G. W. Blume & M. K. Heid (Eds.), Yearbook: Teaching and learning mathematics with technology. Pennsylvania Council of Teachers of Mathematics.

  • Lage, A. E., & Gaisman, M. T. (2006). An analysis of students’ ideas about transformations of functions. Advanced Mathematical Thinking, 2, 23–30.

    Google Scholar 

  • Núñez, R. (2005). Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals. Journal of Pragmatics, 37, 1717–1741.

    Article  Google Scholar 

  • Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 27–52). Mathematical Association of America.

    Chapter  Google Scholar 

  • Pimm, D. (1995). Symbols and meanings in school mathematics. Routledge.

    Google Scholar 

  • Scheiner, T. (2016). New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies. Educational Studies in Mathematics, 91(2), 165–183.

    Article  Google Scholar 

  • Stewart, J., Redlin, L., Watson, S. & Panman, P. (2011). College algebra: Concepts and contexts. Cengage Learning.

  • Yanik, H. B. (2011). Prospective middle school teachers’ preconception of geometric translation. Educational Studies in Mathematics, 78(2), 231–260.

    Article  Google Scholar 

  • Yoon, C., Thomas, M. O. J., & Dreyfus, T. (2011). Grounded blends and mathematical gesture spaces: Developing mathematical understandings via gestures. Educational Studies in Mathematics, 78(3), 371–393.

    Article  Google Scholar 

  • Zandieh, M., Roh, K. H., & Knapp, J. (2014). Conceptual blending: Student reasoning when proving “conditional implies conditional” statements. The Journal of Mathematical Behavior, 33, 209–229.

    Article  Google Scholar 

  • Zazkis, R., Liljedahl, P., & Gadowsky, K. (2003). Conceptions of function translation: Obstacles, intuitions, and rerouting. The Journal of Mathematical Behavior, 22(4), 437–450.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics Education, Gyeongin National University of Education, lncheon, Korea

    Jaehoon Yim

  2. Mathematics Department, Eastern Washington University, 316 Kingston, Cheney, WA, 99004, USA

    Hyung Sook Lee

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  1. Jaehoon Yim
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  2. Hyung Sook Lee
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Correspondence to Hyung Sook Lee.

Appendix. Sample questions from the first questionnaire

Appendix. Sample questions from the first questionnaire

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Yim, J., Lee, H.S. Students’ Use of Prior Conceptions of Symbols in Finding an Equation for a Horizontal Translation of the Graph of a Function. Int J of Sci and Math Educ 20, 1699–1717 (2022). https://doi.org/10.1007/s10763-021-10230-w

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  • DOI: https://doi.org/10.1007/s10763-021-10230-w

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