Educational Studies in Mathematics

, Volume 97, Issue 3, pp 255–272 | Cite as

Comparison of students’ understanding of functions in classes following English and Israeli national curricula

  • Anne Watson
  • Michal Ayalon
  • Stephen Lerman


This paper arises from a study of how concepts related to understanding functions develop for students across the years of secondary/high school, using small samples from two different curricula systems: England and Israel. We used a survey consisting of function tasks developed in collaboration with teachers from both curriculum systems. We report on 120 higher achieving students, 10 from each of English and Israeli, 12–18 years old. Iterative and comparative analysis identified similarities and differences in students’ responses and we conjecture links between curriculum, enactment, task design, and students’ responses. Towards the end of school, students from both curriculum backgrounds performed similarly on most tasks but approached these by different routes, such as intuitive or formal and with different understandings, including correspondence and covariational approaches to functions.


Functions Curriculum Students’ understanding of functions 


  1. Ainley, J., & Pratt, D. (2005). The significance of task design in mathematics education: Examples from proportional reasoning. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 93–122). Melbourne: PME.Google Scholar
  2. Ayalon, M., Watson, A., & Lerman, S. (2015). Functions represented as linear sequential data: Relationships between presentation and student responses. Educational Studies in Mathematics, 90, 321–339.Google Scholar
  3. Ayalon, M., Watson, A., & Lerman, S. (2016a). Progression towards functions: Students’ performance on three tasks about variables from grades 7 to 12. International Journal of Science and Mathematics Education, 14, 1153–1173.Google Scholar
  4. Ayalon, M., Watson, A., & Lerman, S. (2016b). Reasoning about variables in 11 to 18 year olds: Informal, schooled and formal expression in learning about functions. Mathematics Education Research Journal, 28, 379–404.Google Scholar
  5. Ayalon, M., Watson, A., & Lerman, S. (2017). Students’ conceptualizations of function. Research in Mathematics Education, 19(1), 1–19.CrossRefGoogle Scholar
  6. Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5–23). Berlin: Springer.Google Scholar
  7. Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Research Sampler Series, 9, The Mathematical Association of America Notes Online. Retrieved August 15, 2017, from
  8. Clement, J. (1985). Misconceptions in graphing. In L. Streefland (Ed.), Proceedings of the 9th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 369–375). Noordwijkerhout: PME.Google Scholar
  9. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135–164.CrossRefGoogle Scholar
  10. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.CrossRefGoogle Scholar
  11. Daniels, H. (Ed.). (1993). Charting the agenda: Educational activity after Vygotsky. London: Routledge.Google Scholar
  12. Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students: Linearity smoothness and periodicity. Focus on Learning Problems in Mathematics, 5(3), 119–132.Google Scholar
  13. Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In E. Dubinsky & G. Harel (Eds.), The concept of function. Aspects of epistemology and pedagogy (pp. 85–106). Washington, DC: The Mathematical Association of America.Google Scholar
  14. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39, 111–129.Google Scholar
  15. Janvier, C. (1981). Use of situations in mathematics education. Educational Studies in Mathematics, 12, 113–122.CrossRefGoogle Scholar
  16. Karplus, R. (1978). Intellectual development beyond elementary school IX: Functionality, a survey (Advancing education through science oriented programs, report ID-51). Berkeley: University of California.Google Scholar
  17. Leinhardt, G., Zaslavsky, O., & Stein, M. S. (1990). Functions, graphs and graphing: Tasks, learning, and teaching. Review of Educational Research, 1, 1–64.Google Scholar
  18. Lerman, S. (2001). Accounting for accounts of learning mathematics: Reading the ZPD in videos and transcripts. In D. Clarke (Ed.), Perspectives on practice and meaning in mathematics and science classrooms (pp. 53–74). Dordrecht: Kluwer.Google Scholar
  19. Ministry of Education. (2009). Mathematics curriculum for grades 7–9. Retrieved (in Hebrew) from
  20. Monk, G. S. (1994). Students’ understanding of functions in calculus courses. Humanistic Mathematics Network Journal, 1(9), 7.CrossRefGoogle Scholar
  21. Qualifications and Curriculum Authority [QCA]. (2007). The national curriculum: Statutory requirements for key stages 3 and 4. Retrieved from
  22. Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM – The International Journal on Mathematics Education, 40, 88–96.Google Scholar
  23. Rivera, F. D., & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalisations involving linear figural patterns. ZDM: International Journal in Mathematics Education, 40, 65–82.CrossRefGoogle Scholar
  24. Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Rotterdam: Sense.CrossRefGoogle Scholar
  25. Saldanha, L., & Thompson, P. W. (1998). Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson, K. R. Dawkins, M. Blanton, W. N. Coloumbe, J. Kolb, K. Norwood, & L. Stiff (Eds.), Proceedings of the 20th annual meeting of the Psychology of Mathematics Education North American Chapter (Vol. 1, pp. 298–303). Raleigh: North Carolina State University.Google Scholar
  26. 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, 1–36.Google Scholar
  27. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59–84). Washington, DC: Mathematical Association of America.Google Scholar
  28. Sierpinska, A. (1992). On understanding the notion of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 25–28). Washington, DC: Mathematical Association of America.Google Scholar
  29. Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33, 259–281.CrossRefGoogle Scholar
  30. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20, 147–164.CrossRefGoogle Scholar
  31. Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems. ZDM–International Journal in Mathematics Education, 40, 97–110.CrossRefGoogle Scholar
  32. Swan, M. (1980). The language of functions and graphs. Nottingham: Shell Centre for Mathematical Education, University of Nottingham.Google Scholar
  33. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495–511). New York: Macmillan.Google Scholar
  34. Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229–274.CrossRefGoogle Scholar
  35. Thompson, P. W., & Carlson, M. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). Reston: National Council of Teachers of Mathematics.Google Scholar
  36. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23, 57–86.Google Scholar
  37. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20, 356–366.CrossRefGoogle Scholar
  38. Wilmot, D. B., Schoenfeld, A. H., Wilson, M., Champney, D., & Zahner, W. (2011). Validating a learning progression in mathematical functions for college readiness. Mathematical Thinking and Learning, 13(4), 259–291.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.University of OxfordOxfordEngland
  2. 2.University of HaifaRehovotIsrael
  3. 3.London South Bank UniversityLondonEngland

Personalised recommendations