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

## 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

- Alcock, L., & Simpson, A. (2002). Definitions: Dealing with categories mathematically.
*For the Learning of Mathematics*,*22*(2), 28–34.Google Scholar - 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 - 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 - Briggs, W., Cochran, L., & Gillett, B. (2015).
*Calculus: Early transcendentals*(2nd ed.). Boston: Pearson.Google Scholar - 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 - 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 - Cuoco, A., & Curcio, F. (Eds.). (2001).
*The roles of representation in school mathematics: 2001 NCTM yearbook*. Reston: NCTM.Google Scholar - 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 - Hershkowitz, R. (1989). Visualization in geometry -- two sides of the coin.
*Focus on Learning Problems in Mathematics*,*11*(1–2), 61–76.Google Scholar - Jones, S. R. (2013). Understanding the integral: Students' symbolic forms.
*The Journal of Mathematical Behavior*,*32*(2), 122–141.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Lakoff, G. (1987).
*Women, fire, and dangerous things: What categories reveal about the mind*. Chicago: University of Chicago Press.CrossRefGoogle Scholar - 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 - 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 - Presmeg, N. C. (1986). Visualisation in high school mathematics.
*For the Learning of Mathematics*,*6*(3), 42–46.Google Scholar - Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics.
*Educational Studies in Mathematics*,*23*(6), 595–610.CrossRefGoogle Scholar - Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.). (1993).
*Integrating research on the graphical representation of functions*. Mahwah: Lawrence Erlbaum Associates.Google Scholar - 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 - Rosch, E. H., & Mervis, C. B. (1975). Family resemblance: Studies in the internal structure of categories.
*Cognitive Psychology*,*7*, 573–605.CrossRefGoogle Scholar - 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 - 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 - 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 - Stewart, J. (2015).
*Calculus: Early transcendentals*(8th ed.). Boston: Cengage Learning.Google Scholar - 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 - 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 - Thomas, G. B., Weir, M. D., & Hass, J. (2009).
*Thomas' calculus*(12th ed.). Boston: Pearson.Google Scholar - 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 - 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 - Zill, D. G., & Wright, W. S. (2011).
*Calculus: Early transcendentals*(4th ed.). Sudbury: Jones and Bartlett Publishers.Google Scholar - Zimmermann, W., & Cunningham, S. (Eds.). (1991).
*Visualization in teaching and learning mathematics*. Washington DC: Mathematical Association of America.Google Scholar