Inverse function: Pre-service teachers’ techniques and meanings


Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers’ and students’ inverse function meanings. Responding to this call, we characterize 25 pre-service teachers’ inverse function meanings as inferred from our analysis of clinical interviews. After summarizing relevant research, we describe the methodology and theoretical framework we used to interpret the pre-service teachers’ activities. We then present data highlighting the techniques the pre-service teachers used when responding to tasks that involved analytical and graphical representations of functions and inverse functions in both decontextualized and contextualized situations and discuss our inferences of their meanings based on their activities. We conclude with implications for the teaching and learning of inverse function and areas for future research.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    Here we use “variable” to refer to a notational letter or symbol.

  2. 2.

    When creating tasks, we attempted to use notation consistent with what we conjectured the students were introduced through schooling. No students questioned or objected to the notation in any prompt. For instance, students could have questioned using f (x) (i.e., a function’s output value) to name a function (i.e., f).

  3. 3.

    It can be argued all of her techniques consistently involve switching something or performing a transformation. However, we did not infer Alyssa conceived each technique in terms of some underlying invariance.

  4. 4.

    Students used f (x) and y interchangeably. See Thompson (2013) for more on muddled uses of function notation.

  5. 5.

    The other student turned her focus to determining and working with analytically represented functions and did not return to the context.


  1. Bergeron, L., & Alcantara, A. (2015). IB mathematics comparability study: Curriculum & assessment comparison. United Kingdom: UK NARIC. Retrieved from

  2. Blyth, T. S. (2006). Lattices and ordered algebraic structures. London, UK: Springer Science & Business Media.

  3. Breidenback, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23(3), 147–185.

    Google Scholar 

  4. Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). Dordrecht: Kluwer Academic.

  5. Brown, C., & Reynolds, B. (2007). Delineating four conceptions of function: A case of composition and inverse. In T. Lamberg & L. R. Wiest (Eds.), Proceedings of the 29th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education (pp. 190–193). Lake Tahoe, NV: University of Nevada, Reno.

  6. Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Shoenfeld, J. J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education, III: Issues in mathematics education (Vol. 7, pp. 114–162). Providence: American Mathematical Society.

  7. Carlson, M. P., Madison, B., & West, R. D. (2015). A study of students’ readiness to learn calculus. International Journal of Research in Undergraduate Mathematics Education, 1(2), 209–233.

    Article  Google Scholar 

  8. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). Mahwah: Lawrence Erlbaum Associates.

  9. Dugopolski, M. (2009). Fundamentals of precalculus. Boston: Pearson Education.

    Google Scholar 

  10. Engelke, N., Oehrtman, M., & Carlson, M. P. (2005). Composition of functions: Precalculus students' understandings. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education (pp. 570–577). Roanoke, VA: Virginia Tech.

  11. Even, R. (1992). The inverse function: Prospective teachers use of 'undoing'. International Journal of Mathematical Education in Science and Technology, 23(4), 557–562.

    Google Scholar 

  12. Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Mahwah: Lawrence Erlbaum Associates, Inc.

  13. Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its development. Mathematical Thinking and Learning, 7(1), 27–50.

    Article  Google Scholar 

  14. Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. The Journal of Mathematical Behavior, 16(2), 145-165.

    Article  Google Scholar 

  15. Larson, R., Hostetler, R. P., & Edwards, B. H. (2001). Precalculus with limits: A graphing approach (3rd ed.). Boston: Houghton Mifflin Company.

  16. Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64.

    Article  Google Scholar 

  17. Lucus, C. A. (2005). Composition of functions and inverse function of a function: Main ideas as perceived by teachers and preservice teachers (Unpublished doctoral dissertation). Canada: Simon Fraser University, BC.

  18. Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38(1-3), 135–161.

    Article  Google Scholar 

  19. Merriam, S. B., & Tisdell, E. J. (2005). Qualitative research: A guide to design and implementation. San Francisco: John Wiley & Sons.

    Google Scholar 

  20. Moore, K. C. (2014). Signals, symbols, and representational activity. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.), Epistemic algebraic students: Emerging models of students' algebraic knowing (Vol. 4, pp. 211–235). Laramie: University of Wyoming.

  21. Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. R. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157.

    Article  Google Scholar 

  22. Paz, T., & Leron, U. (2009). The slippery road from actions on objects to functions and variables. Journal for Research in Mathematics Education, 40(1), 18-39.

    Google Scholar 

  23. Phillips, N. (2015). Domain, co-domain and causation: A study of Britney’s conception of function. In T. Fukawa-Connelly, N. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the 18th annual Conference on Research in Undergraduate Mathematics Education (pp. 893–895). Pittsburgh: SIGMAA on RUME.

  24. Piaget, J. (1968). Six psycological studies (A. Tenzer, Trans.). New York: Random House.

  25. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah: Erlbaum.

    Google Scholar 

  26. Stewart, J., Redlin, L., & Watson, S. (2012). Precalculus: Mathematics for calculus, sixth edition (6th ed.). Belmont: Brooks/Cole Cegage Learning.

    Google Scholar 

  27. Strauss, A. L., & Corbin, J. M. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks: Sage Publications.

    Google Scholar 

  28. Sullivan, M., & Sullivan III, M. (2007). Precalculus: Concepts through functions, a unit circle approach to trigonometry. Upper Saddle River: Pearson Education.

    Google Scholar 

  29. Swokowski, E. W., & Cole, J. A. (2012). Precalculus: Functions and graphs (12 ed.). Belmont, CA: Cengage Learning.

  30. Thompson, P., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), The compendium for research in mathematics education. Washington, DC: NCTM.

  31. Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sépulveda (Eds.), Proceedings of the annual meeting of the International Group for the Psychology of mathematics education (Vol. 1, pp. 31–49). Morélia: PME.

  32. Thompson, P. W. (2013). Why use f(x) when all we really mean is y? OnCore, The Online Journal of the Arizona Association of Mathematics Teachers, 18-26.

  33. Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 435–461). New York: Taylor & Francis.

    Google Scholar 

  34. Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.), Epistemic algebraic students: Emerging models of students' algebraic knowing (Vol. 4, pp. 1–24). Laramie: University of Wyoming.

  35. Vidakovic, D. (1996). Learning the concept of inverse function. Journal of Computers in Mathematics and Science Teaching, 15(3), 295–318.

    Google Scholar 

  36. Vidakovic, D. (1997). Learning the concept of inverse function in a group versus individual environment. In E. Dubinsky, D. Mathews, & B. E. Reynolds (Eds.), Readings in cooperative learning for undergraduate mathematics (Vol. 44, pp. 175–196). Washington, DC: The Mathematical Association of America.

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

  38. Zazkis, R., & Kontorovich, I. (2016). A curious case of superscript (− 1): Prospective secondary mathematics teachers explain. The Journal of Mathematical Behavior, 43, 98–110.

    Article  Google Scholar 

Download references


This material is based upon work supported by the NSF under Grant No. DRL-1350342. Any opinions, findings, conclusions, or recommendations expressed are those of the authors. We thank Neil Hatfield for directing us to a notation for inverse function and Nick Wasserman for his feedback on an earlier draft.

Author information



Corresponding author

Correspondence to Teo Paoletti.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Paoletti, T., Stevens, I.E., Hobson, N.L.F. et al. Inverse function: Pre-service teachers’ techniques and meanings. Educ Stud Math 97, 93–109 (2018).

Download citation


  • Function
  • Inverse function
  • Pre-service secondary teachers
  • Meanings