Educational Studies in Mathematics

, Volume 91, Issue 3, pp 375–393 | Cite as

Making implicit metalevel rules of the discourse on function explicit topics of reflection in the classroom to foster student learning



Despite the existence of extensive literature on functions, fewer studies used sociocultural views to explore the development of student learning about the concept. This study uses a discursive lens to examine whether an instructional approach that specifically attends to particular metalevel rules in the mathematical discourse on functions supports students’ learning of the concept in a postsecondary mathematics classroom. The findings suggest that such instruction has the potential to foster learning as indicated by the changes in the ways students talked about functions, and their awareness and modifications of the assumptions shaping their thinking about functions.


Functions Teaching experiment Mathematical discourse Metadiscursive rules Student learning Postsecondary education 



I would like to thank the participants and the graduate students David Kamin, Heather Trahan-Martins, Robert Nanna, and James Fanning for their valuable contributions during various stages of this work.

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  1. Bar-Tikva, J. (2009). Old meta-discursive rules die hard. In F. Lin, F. Hsieh, G. Hanna, & M. deVilliers (Eds.), Proceedings of the ICMI Study 19: proof and proving in mathematics education (Vol. 1, pp. 89–93). Taipei: ICMI.Google Scholar
  2. Carlson, M. P. (1998). A cross-sectional investigation of the function concept. In Schoenfeld, A. H., Kaput, J., Dubinsky, E. (Eds.), Research in Collegiate Mathematics Education. III (pp. 114–162), CBMS Issues in Mathematics Education Vol. 7. Providence, Rhode Island: American Mathematical Society.Google Scholar
  3. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: a framework and a study. Journal for Research in Mathematics Education, 33(5), 352–367.CrossRefGoogle Scholar
  4. Dubinsky, E. (1991). In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  5. Eisenberg, T. (1991). Functions and associated learning difficulties. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 140–152). Dordrecht: Kluwer.Google Scholar
  6. Furinghetti, F., & Radford, L. (2008). Contrasts and oblique connections between historical conceptual developments and classroom learning in mathematics. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 626–655). New York: Routledge.Google Scholar
  7. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26, 115–141.Google Scholar
  8. Güçler, B. (2011). Historical junctures in the development of discourse on limits. In B. Ubuz (Ed.), Proceedings of the thirty-fifth annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 465–472). Turkey: Ankara.Google Scholar
  9. Güçler, B. (2013). Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics, 82(3), 439–453.CrossRefGoogle Scholar
  10. Güçler, B. (2014). The role of symbols in mathematical communication: the case of the limit notation. Research in Mathematics Education, 16(3), 251–268.CrossRefGoogle Scholar
  11. Kjeldsen, T. H., & Blomhøj, M. (2012). Beyond motivation: history as a method for learning meta-discursive rules in mathematics. Educational Studies in Mathematics, 80(3), 327–349.Google Scholar
  12. Kjeldsen, T. H., & Petersen, P. H. (2014). Bridging history of the concept of function with learning of mathematics: Students’ meta-discursive rules, concept formation and historical awareness. Science and Education, 23(1), 29–45.CrossRefGoogle Scholar
  13. Kleiner, I. (1989). Evolution of the function concept: a brief survey. College Mathematics Journal, 20, 282–300.Google Scholar
  14. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  15. Monk, G. S. (1994). Students’ understanding of functions in calculus courses. Humanistic Mathematics Network Journal, 9, 21–27.Google Scholar
  16. Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom–the case of function. International Journal of Educational Research, 51, 10–27.CrossRefGoogle Scholar
  17. Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014). Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of calculus. Research in Mathematics Education, 16(2), 182–198.Google Scholar
  18. Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understandings of function. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 27–42). Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  19. Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. The Mathematics Teacher, 81(6), 420–427.Google Scholar
  20. Schubring, G. (2011). Conceptions for relating the evolution of mathematical concepts to mathematics learning—epistemology, history, and semiotics interacting. Educational Studies in Mathematics, 77(1), 79–104.CrossRefGoogle Scholar
  21. 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
  22. Sfard, A. (1992). Operational origin of mathematical objects and the quandary of reification – the case of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: aspects of epistemology and pedagogy (pp. 59–84). Washington, DC: Mathematical Association of America.Google Scholar
  23. Sfard, A. (2001). There is more to discourse than meets the ears: looking at thinking as communicating to learn more about mathematical learning. Educational Studies in Mathematics, 46(1/3), 13–57.Google Scholar
  24. Sfard, A. (2008). Thinking as communicating: human development, The growth of discourses and mathematizing. New York: Cambridge University Press.CrossRefGoogle Scholar
  25. Sfard, A. (2014). University mathematics as a discourse—why, how, and what for? Research in Mathematics Education, 16(2), 199–203.CrossRefGoogle Scholar
  26. Sierpinska, A. (1992). Theoretical perspectives for development of the function concept. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes (Vol. 25, pp. 23–58). Washington DC: Mathematical Association of America.Google Scholar
  27. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 267–307). Hillsdale: Erlbaum.Google Scholar
  28. Tall, D. (1996). Functions and calculus. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 289–325). Dordrecht: Kluwer Academic Publishers.Google Scholar
  29. Viirman, O. (2013). The functions of function discourse—university mathematics teaching from a commognitive standpoint. International Journal of Mathematics Education in Science and Technology, 45(4), 512–527.CrossRefGoogle Scholar
  30. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 356–366.Google Scholar
  31. Weber, E., & Thompson, P. T. (2014). Students’ images of two-variable functions and their graphs. Educational Studies in Mathematics, 87(1), 67–85.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Kaput Center for Research and Innovation in STEM EducationUniversity of Massachusetts DartmouthFairhavenUSA

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