# Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach

## Abstract

This article is concerned with cognitive aspects of students’ struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors, and roots in the domain of reals and in the domain of complex numbers. The article introduces a polysemous approach for structuring students’ concept images in these situations. Post-exchanges from an online forum and excerpts from an interview were analyzed for illustrating the potential of the approach for indicating possible sources of students’ misconceptions and meta-ways of thinking that might make students aware of their mistakes.

## Keywords

Concept image Conceptual change Cross-curricular concepts Epistemological obstacles Polysemy## Notes

### Acknowledgements

I am grateful to Arthur Bakker and to anonymous reviewers for their thorough criticism and insightful suggestions. I wish to thank Sze Looi Chin for proofreading.

## References

- Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education.
*Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education*,*6*, 1–32.CrossRefGoogle Scholar - Austin, J. L., & Howson, A. G. (1979). Language and mathematical education.
*Educational Studies in Mathematics*,*10*(2), 161–197.CrossRefGoogle Scholar - Bingolbali, E., & Monaghan, J. (2008). Concept image revisited.
*Educational Studies in Mathematics*,*68*(1), 19–35.CrossRefGoogle Scholar - Biza, I., & Zachariades, T. (2010). First year mathematics undergraduates’ settled images of tangent line.
*The Journal of Mathematical Behavior*,*29*(4), 218–229.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three.
*Journal for Research in Mathematics Education, (15)*3, 179–202.Google Scholar - Chi, M. T. H. (1992). Conceptual change within and across ontological categories: Examples from learning and discovery in science. In R. Giere (Ed.),
*Cognitive models of science: Minnesota studies in the philosophy of science*(pp. 129–186). Minneapolis, MN: University of Minnesota Press.Google Scholar - Dreyfus, T. (1999). Why Johnny can’t prove.
*Educational Studies in Mathematics*,*38*(1), 85–109.CrossRefGoogle Scholar - Durkin, K., & Shire, B. (1991). Lexical ambiguity in mathematical contexts. In K. Durkin & B. Shire (Eds.),
*Language in mathematical education: Research and Practice*(pp. 71–84). Milton Keynes: Open University Press.Google Scholar - Fischbein, E. (1987).
*Intuition in science and mathematics*. Dordrecht: Reidel.Google Scholar - Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic.
*Journal for Research in Mathematics Education, 25*(2), 116–140.Google Scholar - Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts.
*Educational Studies in Mathematics, 40*(1), 71–90.Google Scholar - Hazzan, O., & Zazkis, R. (2005). Reducing abstraction: The case of school mathematics.
*Educational Studies in Mathematics, 58*(1), 101–119.Google Scholar - IES & NSF. (2013).
*A report from Institute of Education Sciences*. Washington: US Department of Education and the National Science Foundation.Google Scholar - Kleiner, I. (1988). Thinking the unthinkable: The story of complex numbers (with a moral).
*Mathematics Teacher, 81*(7), 583–592.Google Scholar - Kline, M. (1973).
*Why Johnny can’t add: The failure of the New Math*. New York: St. Martin’s Press.Google Scholar - Kontorovich, I. (2016a). √9=? The answer depends on your lecturer.
*Research in Mathematics Education, 18*(3), 284–299.Google Scholar - Kontorovich, I. (2016b).
*Exploring students’ interactions in an asynchronous forum that accompanied a course in linear algebra*. Paper presented at 13th International Congress on Mathematics Education, Hamburg, Germany. from https://www.researchgate.net/publication/306280993_Exploring_students%27_interactions_in_an_online_forum_that_accompanied_a_course_in_linear_algebra?ev=prf_pub. Retrieved 24 Feb 2017 - Kontorovich, I. (2016c). Exploring the image of the square root concept among university students. In C. Csíkos, A. Rausch, & J. Szitányi (Eds.),
*Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 99–106). Szeged: PME.Google Scholar - Kontorovich, I. (2016d). Students’ confusions with reciprocal and inverse functions.
*International Journal of Mathematical Education in Science and Technology, 48*(2), 278–284.CrossRefGoogle Scholar - Kontorovich, I., & Zazkis, R. (2016). Turn vs. shape: Teachers cope with incompatible perspectives on angle.
*Educational Studies in Mathematics, 93*(2), 223–243.Google Scholar - Kontorovich, I., & Zazkis, R. (2017). Mathematical conventions: Revisiting arbitrary and necessary.
*For the Learning of Mathematics, 37*(1), 15–20.Google Scholar - Kuhn, T. S. (1970).
*The structure of scientific revolutions*. Chicago: University of Chicago Press.Google Scholar - Mamolo, A. (2010). Polysemy of symbols: Signs of ambiguity.
*The Mathematics Enthusiast, 7*(2), 247–262.Google Scholar - Mason, J. (1989). Mathematical abstraction seen as a delicate shift in attention.
*For the Learning of Mathematics*,*9*(2), 2–8.Google Scholar - Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limón & L. Mason (Eds.),
*Reconsidering conceptual Change. Issues in theory and practice*(pp. 233–257). Dordrecht: Kluwer Academic Publishers.Google Scholar - Moreno, L. E., & Waldegg, G. (1991). The conceptual evolution of actual mathematical infinity.
*Educational Studies in Mathematics, 22*(3), 211–231.CrossRefGoogle Scholar - Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom.
*International Journal of Educational Research*,*51–52*, 10–27.CrossRefGoogle Scholar - Panaoura, A., Michael-Chrysanthou, P., Gagatsis, A., Elia, I., & Philippou, A. (2016). A structural model related to the understanding of the concept of function: definition and problem solving.
*International Journal of Science and Mathematics Education, 15*(4), 1–18.Google Scholar - Peirce, C. S. (1955).
*Philosophical writings of Peirce*(J. Buchler, Ed.). New York: Dover.Google Scholar - Piaget, J. (1985).
*The equilibration of cognitive structures: The central problem of intellectual development*. Chicago: University of Chicago Press.Google Scholar - Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: how can we characterise it and how can we represent it?
*Educational Studies in Mathematics*,*26*(2–3), 165–190.CrossRefGoogle Scholar - Sandoval, I., & Possani, E. (2016). An analysis of different representations for vectors and planes in ℝ
^{3}.*Educational Studies in Mathematics, 92*(1), 109–127.Google Scholar - Sapir, E. (1970).
*Culture, language and personality: Selected essays*. Berkeley: University of California Press.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.Google Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, development of discourses, and mathematizing*. New York, NY: Cambridge University Press.Google Scholar - Shire, B., & Durkin, K. (1989). Junior school children’s responses to conflict between the spatial and numerical meanings of ‘up’and ‘down’.
*Educational Psychology*,*9*(2), 141–147.CrossRefGoogle Scholar - Sierpińska, A. (1987). Humanities students and epistemological obstacles related to limits.
*Educational Studies in Mathematics*,*18*(4), 371–397.CrossRefGoogle Scholar - Smith, C. P., King, B., & Hoyte, J. (2014). Learning angles through movement: Critical actions for developing understanding in an embodied activity.
*Journal of Mathematical Behavior, 36*, 95–108.Google Scholar - Svennevig, J. (2001). Abduction as a methodological approach to the study of spoken interaction.
*Norskrift*,*103*, 1–22.Google Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*(2), 151–169.Google Scholar - Tirosh, D. (1999). Finite and infinite sets: Definitions and intuitions.
*International Journal of Mathematics in Science and Technology, 30*(3), 341–349.Google Scholar - Tirosh, D., & Even, R. (1997). To define or not to define: The case of \( {\left(-8\right)}^{\frac{1}{3}} \).
*Educational Studies in Mathematics, 33*(3), 321–330.Google Scholar - Tirosh, D., & Stavy, R. (1999). Intuitive rules and comparison tasks.
*Mathematical Thinking and Learning*,*1*(3), 179–194.CrossRefGoogle Scholar - Tsamir, P. (2005). Enhancing prospective teachers’ knowledge of learners’ intuitive conceptions: The case of 'same A-same B'.
*Journal of Mathematics Teacher Education, 8*(6), 469–497.Google Scholar - Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach.
*Learning and Instruction, 14*(5), 453–467.Google Scholar - Van Dooren, W., Lehtinen, E., & Verschaffel, L. (2015). Unraveling the gap between natural and rational numbers.
*Learning and Instruction, 37*, 1–4.Google Scholar - Vosniadou, S. (2014). Examining conceptual development from a conceptual change point of view: The framework approach.
*European Journal of Developmental Psychology, 11*(6), 645–661.Google Scholar - Vosniadou, S., & Skopeliti, I. (2014). Conceptual change from the framework theory side of the fence.
*Science and Education, 23*(7), 1427–1445.Google Scholar - Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching.
*Learning and Instruction, 14*(5), 445–451.Google Scholar - Williams, S. R. (1991). Models of limit held by college calculus students.
*Journal for Research in Mathematics Education*,*22*(3), 219–236.CrossRefGoogle Scholar - Zazkis, R. (1998). Divisors and quotients: Acknowledging polysemy.
*For the Learning of Mathematics, 18*(3), 27–30.Google Scholar - Zazkis, R. (2000). Factors, divisors and multiples: Exploring the web of students’ connections.
*Research in Collegiate Mathematics Education, 4*, 210–238. Google Scholar