Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Can slope be negative in 3-space? Studying concept image of slope through collective definition construction


Developing deep conceptual understanding of what Ma (1999) calls fundamental mathematics is a well-accepted goal of teacher education. This paper presents a microanalysis of an intriguing episode within a course designed to encourage such understanding. An adaptation of Krummheuer’s (1995) elaboration of Toulmin’s (1958/2003) diagrams is used to examine video recordings and transcripts of a group of graduate students in secondary mathematics education grappling with the idea of a three-dimensional line having negative slope. The graduate students’ understandings of slope are examined using an expansion of Stump’s (1999, 2001b) categories of conceptions of slope. The episode ends in an interesting impasse, in which the graduate students agree to pursue the idea no further, purposely ignoring the question of negative slope, despite the clear intention of the task. The analysis explores the argumentation, factors of the learning environment, and conceptions of slope that may have contributed to this impasse.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    Note that defining is an activity in which meaning is assigned to a concept. When done collaboratively, the process often involves argumentation, the negotiation and discussion between collaborators with certain, preexisting ideas as they attempt to reach mutually acceptable conclusions.

  2. 2.

    Pseudonyms are used.

  3. 3.

    References to color indicate work that was done in the analysis process.


  1. Azcarate, C. (1992). Estudio de los esquemas conceptuales y de los perfiles de unos alumnos de segundo de BUP en relación con el concepto de pendiente de una recta [A study of the conceptual schemes and the profiles of some BUP students in relation to the concept of the slope of a line]. Epsilon, 24, 9–22.

  2. Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth: Heinemann.

  3. Burbules, N. C. (1993). Dialogue in teaching: Theory and practice. New York: Teacher College.

  4. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.

  5. de Villiers, M. (1998). To teach definitions in geometry or teach to define? In A. Olivier & K. Newstead (Eds.), Proceedings of the twenty-second international conference for the Psychology of Mathematics Education (Vol. 2, pp. 248–255). Stellenbosch: Program Committee.

  6. Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527–548.

  7. Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66, 3–21.

  8. Jewitt, C. (2008). Multimodality and literacy in school classrooms. Review of Research in Education, 32(1), 241–267.

  9. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: interaction in classroom cultures (pp. 229–269). Hillsdale: Lawrence Erlbaum Associates.

  10. Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. Journal of Mathematical Behavior, 26, 60–82.

  11. Leikin, R., & Winicki-Landman, G. (2001). Defining as a vehicle for professional development of secondary school mathematics teachers. Mathematics Education Research Journal, 3, 62–73.

  12. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah: Erlbaum.

  13. McClain, K. (2002). Teacher’s and students’ understanding: The role of tools and inscriptions in supporting effective communication. Journal of the Learning Sciences, 11(2&3), 217–249.

  14. McGee, D., Moore-Russo, D., Lomen, D., Ebersole, D., & Marin-Quintero, M. (2008). Physical manipulatives for visualizing multivariable concepts and how they can reform mathematics courses such as algebra, precalculus and calculus. Paper presented at the Joint Meetings of the American Mathematical Society and the Mathematical Association of America, San Diego, CA.

  15. Pirie, S., & Schwarzenberger, R. (1988). Mathematical discussion and mathematical understanding. Educational Studies in Mathematics, 19(4), 459–470.

  16. Rasslan, S., & Vinner, S. (1995). In L. Meira & D. Carraher (Eds.), Proceedings of the 19 th international conference for the Psychology of Mathematics Education (Vol. 2, pp. 264–271). Recife: PME.

  17. Stump, S. (1997). Secondary mathematics teachers’ knowledge of the concept of slope. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL. (ERIC Document Reproduction Service No. ED 408193).

  18. Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144.

  19. Stump, S. (2001a). Developing preservice teachers’ pedagogical content knowledge of slope. Journal of Mathematical Behavior, 20, 207–227.

  20. Stump, S. (2001b). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2), 81–89.

  21. 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.

  22. Toulmin, S. E. (2003). The uses of argument (updated ed.). New York: Cambridge University Press. First published in 1958.

  23. Valsiner, J. (1994). Culture and human development: A co-constructivist perspective. In P. van Geert, L. P. Mos, & W. J. Baker (Eds.), Annals of theoretical psychology (Vol. 10, pp. 247–298). New York: Plenum.

  24. Vidakovic, D., & Martin, W. O. (2004). Small-group searches for mathematical proofs and individual reconstructions of mathematical concepts. Journal of Mathematical Behavior, 23, 465–492.

  25. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 346–366.

  26. Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 163–201). Hillsdale: Lawrence Erlbaum Associates.

  27. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer.

  28. Walter, J. G., & Gerson, H. (2007). Teachers’ personal agency: Making sense of slope through additive structures. Educational Studies in Mathematics, 65, 205–233.

  29. Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21, 423–440.

  30. Zaslavsky, O., Sela, H., & Leron, U. (2002). Being sloppy about slope: The effect of changing the scale. Educational Studies in Mathematics, 49, 119–140.

  31. Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.

Download references

Author information

Correspondence to Deborah Moore-Russo.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Moore-Russo, D., Conner, A. & Rugg, K.I. Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educ Stud Math 76, 3–21 (2011). https://doi.org/10.1007/s10649-010-9277-y

Download citation


  • Secondary mathematics teachers
  • Slope
  • Teacher education
  • Concept image
  • Argumentation