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Educational Studies in Mathematics

, Volume 97, Issue 3, pp 215–234 | Cite as

Prototype images in mathematics education: the case of the graphical representation of the definite integral

Article

Abstract

Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally.

Keywords

Calculus Definite integral Graphical representation Prototype Imagery 

References

  1. Alcock, L., & Simpson, A. (2002). Definitions: Dealing with categories mathematically. For the Learning of Mathematics, 22(2), 28–34.Google Scholar
  2. Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery: Graphical connections between a function and its derivative. Educational Studies in Mathematics, 33(3), 301–317.CrossRefGoogle Scholar
  3. Bezuidenhout, J., & Olivier, A. (2000). Students' conceptions of the integral. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th conference of the International Group for the Psychology of mathematics education. IGPME: Hiroshima.Google Scholar
  4. Briggs, W., Cochran, L., & Gillett, B. (2015). Calculus: Early transcendentals (2nd ed.). Boston: Pearson.Google Scholar
  5. Carlson, M. P. (1999). The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success. Educational Studies in Mathematics, 40(3), 237–258.CrossRefGoogle Scholar
  6. Clements, D. H., Sarama, J., & DiBiase, A. M. (Eds.). (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  7. Cuoco, A., & Curcio, F. (Eds.). (2001). The roles of representation in school mathematics: 2001 NCTM yearbook. Reston: NCTM.Google Scholar
  8. Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students' understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176.CrossRefGoogle Scholar
  9. Hershkowitz, R. (1989). Visualization in geometry -- two sides of the coin. Focus on Learning Problems in Mathematics, 11(1–2), 61–76.Google Scholar
  10. Jones, S. R. (2013). Understanding the integral: Students' symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141.CrossRefGoogle Scholar
  11. Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38, 9–28.CrossRefGoogle Scholar
  12. Jones, S. R. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann-sum based conceptions in students' explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736.CrossRefGoogle Scholar
  13. Jones, S. R., & Dorko, A. (2015). Students' understandings of multivariate integrals and how they may be generalized from single integral conceptions. The Journal of Mathematical Behavior, 40(B), 154–170.CrossRefGoogle Scholar
  14. Jones, S. R., Lim, Y., & Chandler, K. R. (2017). Teaching integration: How certain instructional moves may undermine the potential conceptual value of the Riemann sum and the Riemann integral. International Journal of Science and Mathematics Education, 15(6), 1075–1095.CrossRefGoogle Scholar
  15. Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the mind. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  16. Lakoff, G. (1999). Cognitive models and prototype theory. In E. Margolis & S. Laurence (Eds.), Concepts: Core readings (pp. 391–421). Cambridge: MIT Press.Google Scholar
  17. Mullis, I. V. S., Martin, M. O., Robitaille, D. F., & Foy, P. (2009). TIMSS advanced 2008 international report: Findings from IEA's study of achievement in advanced mathematics and physics in the final year of secondary school. Chestnut Hill: TIMSS & PIRLS International Study Center, Boston College.Google Scholar
  18. Presmeg, N. C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 42–46.Google Scholar
  19. Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23(6), 595–610.CrossRefGoogle Scholar
  20. Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.). (1993). Integrating research on the graphical representation of functions. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  21. Rosch, E. H. (1973). Natural categories. Cognitive Psychology, 4, 328–350.CrossRefGoogle Scholar
  22. Rosch, E. H. (1978/1999). Principles of categorization. In E. Margolis & S. Laurence (Eds.), Concepts: Core readings (pp. 189–206). Cambridge: MIT Press.Google Scholar
  23. Rosch, E. H., & Mervis, C. B. (1975). Family resemblance: Studies in the internal structure of categories. Cognitive Psychology, 7, 573–605.CrossRefGoogle Scholar
  24. Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362–389.CrossRefGoogle Scholar
  25. Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33(1), 230–245.CrossRefGoogle Scholar
  26. Simmons, C., & Oehrtman, M. (2017). Beyond the product structure for definite integrals Proceedings of the 20th special interest group of the Mathematical Association of America on research in undergraduate mathematics education. San Diego: SIGMAA on RUME.Google Scholar
  27. Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Boston: Cengage Learning.Google Scholar
  28. Tall, D. O. (1991). Intuition and rigor: The role of visualization in the calculus. In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 105–120). Washington, DC: Mathematical Association of America.Google Scholar
  29. Tall, D. O., & Bakar, M. (1992). Students' mental prototypes for functions and graphs. International Journal of Mathematics Education in Science and Technology, 23(1), 39–50.CrossRefGoogle Scholar
  30. Thomas, G. B., Weir, M. D., & Hass, J. (2009). Thomas' calculus (12th ed.). Boston: Pearson.Google Scholar
  31. Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–459.Google Scholar
  32. Yerushalmy, M., & Shternberg, B. (2001). Charting a visual course to the concept of function. In A. Cuoco & F. Curcio (Eds.), The roles of representations in school mathematics: 2001 NCTM yearbook (pp. 251–268). Reston: NCTM.Google Scholar
  33. Zill, D. G., & Wright, W. S. (2011). Calculus: Early transcendentals (4th ed.). Sudbury: Jones and Bartlett Publishers.Google Scholar
  34. Zimmermann, W., & Cunningham, S. (Eds.). (1991). Visualization in teaching and learning mathematics. Washington DC: Mathematical Association of America.Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Brigham Young UniversityProvoUSA

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