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Mathematics Education Research Journal

, Volume 30, Issue 4, pp 499–523 | Cite as

Investigating Years 7 to 12 students’ knowledge of linear relationships through different contexts and representations

  • Karina J. Wilkie
  • Michal Ayalon
Original Article
First Online:

Abstract

A foundational component of developing algebraic thinking for meaningful calculus learning is the idea of “function” that focuses on the relationship between varying quantities. Students have demonstrated widespread difficulties in learning calculus, particularly interpreting and modeling dynamic events, when they have a poor understanding of relationships between variables. Yet, there are differing views on how to develop students’ functional thinking over time. In the Australian curriculum context, linear relationships are introduced to lower secondary students with content that reflects a hybrid of traditional and reform algebra pedagogy. This article discusses an investigation into Australian secondary students’ understanding of linear functional relationships from Years 7 to 12 (approximately 12 to 18 years old; n = 215) in their approaches to three tasks (finding rate of change, pattern generalisation and interpretation of gradient) involving four different representations (table, geometric growing pattern, equation and graph). From the findings, it appears that these students’ knowledge of linear functions remains context-specific rather than becoming connected over time.

Keywords

Algebra Correspondence Covariation Functional thinking Linear functions Secondary mathematics 
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References

  1. Australian Curriculum Assessment and Reporting Authority. (2014). The Australian curriculum: mathematics. Retrieved from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10.
  2. Ayalon, M., & Wilkie, K. J. (under review). Exploring relationships among task design, prior experiences, and students’ responses to linear functions tasks.Google Scholar
  3. Ayalon, M., Watson, A., & Lerman, S. (2015). Functions represented as linear sequential data: relationships between presentation and student responses. Educational Studies in Mathematics, 90(3), 321–339.CrossRefGoogle Scholar
  4. Ayalon, M., Watson, A., & Lerman, S. (2016). Progression towards functions: students’ performance on three tasks about variables from grades 7 to 12. International Journal of Science and Mathematics Education, 14(6), 1153–1173.CrossRefGoogle Scholar
  5. Bardini, C. & Pierce, R. (2014). Overcoming algebraic misconceptions that inhibit students’ progress in mathematical sciences. Retrieved from Office for Learning and Teaching website: http://www.olt.gov.au/resource-overcoming-algebraic-misconceptions.
  6. Blanton, M., & Kaput, J. (2004). Elementary grades students' capacity for functional thinking. In M. Høines & A. Fuglestad (Eds.), Proceedings of the 28th annual meeting of International Group for the Psychology of Mathematics Education (pp. 135–142): IGPME.Google Scholar
  7. Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Murphy Gardiner, A. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17(1), 34–63.CrossRefGoogle Scholar
  8. 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), 352–378.CrossRefGoogle Scholar
  9. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Charlotte, NC: Information Age Publishing.Google Scholar
  10. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 135–164.CrossRefGoogle Scholar
  11. 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
  12. De Beer, H., Gravemeijer, K., & van Eijck, M. (2015). Discrete and continuous reasoning about change in primary school classrooms. ZDM Mathematics Education, 47, 981–996.CrossRefGoogle Scholar
  13. English, L. D., & Warren, E. (1998). Introducing the variable through pattern exploration. The Mathematics Teacher, 91(2), 166–170.Google Scholar
  14. Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. Curcio (Eds.), The roles of representation in school mathematics (pp. 1–23). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  15. Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten—grade 2. Reston, VA: The National Council of Teachers of Mathematics.Google Scholar
  16. Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81(1), 85–101.CrossRefGoogle Scholar
  17. Hershkowitz, R., Arcavi, A., & Bruckheimer, M. (2001). Reflections on the status and nature of visual reasoning—the case of the matches. International Journal of Mathematical Education in Science and Technology, 32(2), 255–265.CrossRefGoogle Scholar
  18. Jurdak, M. E., & Mouhayar, R. R. (2014). Trends in the development of student level of reasoning in pattern generalisation tasks across grade level. Educational Studies in Mathematics, 85, 75–92.CrossRefGoogle Scholar
  19. Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. L. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Taylor & Francis Group.Google Scholar
  20. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 707–762). Charlotte, NC: National Council of Teachers of Mathematics, Information Age Publishing.Google Scholar
  21. Knuth, E. J. (2000). Student understanding of the Cartesian connection: an exploratory study. Journal for Research in Mathematics Education, 31(4), 500–507.CrossRefGoogle Scholar
  22. Kruteskii, V. (1976). The psychology of mathematical ability in school children. Chicago: University of Chicago Press.Google Scholar
  23. Küchemann, D. (2010). Using patterns generically to see structure. Pedagogies, 5(3), 233–250.CrossRefGoogle Scholar
  24. Lannin, J. K. (2005). Generalisation and justification: the challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258.  https://doi.org/10.1207/s15327833mtl0703_3 CrossRefGoogle Scholar
  25. Leinhardt, G., Zaslavsky, O., & Stein, M. (1990). Functions, graphs and graphing: tasks, learning and teaching. Review of Educational Research, 60(1), 37–42.CrossRefGoogle Scholar
  26. Lesh, R. (1981). Applied mathematical problem solving. Educational Studies in Mathematics, 12(2), 235–264.CrossRefGoogle Scholar
  27. Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: on multiple perspectives and representations of linear functions and connections among them. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.), Integrating research on the graphical representation of functions (pp. 69–100). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  28. Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). What is your theory? What is your rule? Fourth graders build an understanding of function through patterns and generalising problems. In C. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (70th yearbook of the National Council of Teachers of Mathematics) (pp. 155–168). Reston, VA: NCTM.Google Scholar
  29. Nitsch, R., Fredebohm, A., Bruder, R., Kelava, A., Naccarella, D., Leuders, T., & Wirtz, M. (2015). Students’ competencies in working with functions in secondary mathematics education—empirical examination of a competence structure model. International Journal of Science and Mathematics Education, 13(3), 657–682.CrossRefGoogle Scholar
  30. Oehrtman, M., Carlson, M. & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 27–42): Mathematical Association of America.Google Scholar
  31. Orton, J., Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 121–136). London: Redwood Books Ltd.Google Scholar
  32. Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: the multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38(5), 507–530.Google Scholar
  33. Romberg, T. A., Carpenter, T. P., & Fennema, E. (1993). Toward a common research perspective. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.), Integrating research on the graphical representation of functions (pp. 1–9). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  34. Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problem-solving performance. Journal for Research in Mathematics Education, 10, 173–187.CrossRefGoogle Scholar
  35. 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.CrossRefGoogle Scholar
  36. Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. L. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133–160). New York, NY: Taylor & Francis Group.Google Scholar
  37. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164.CrossRefGoogle Scholar
  38. Sutherland, R. (2002). A comparative study of algebra curricula: London: Qualifications and Curriculum Authority. Google Scholar
  39. Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ pre-instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.CrossRefGoogle Scholar
  40. 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
  41. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education (Vol. 1, pp. 33–57). Laramie, WY: University of Wyoming.Google Scholar
  42. 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, VA: National Council of Teachers of Mathematics.Google Scholar
  43. Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30, 124–147.CrossRefGoogle Scholar
  44. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12: NCTM 1988 yearbook (pp. 8–19). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  45. Van Dooren, W., De Bock, D., & Verschaffel, L. (2012). How students understand aspects of linearity: Searching for obstacles in representational flexibility. In T. Y. Tso (Ed.), Proceedings of the 36th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 179–186). Taipei: PME.Google Scholar
  46. Warren, E., & Cooper, T. (2008). Generalising the pattern rule for visual growth patterns: actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171–185.CrossRefGoogle Scholar
  47. Wilkie, K. J. (2016). Students’ use of variables and multiple representations in generalising functional relationships prior to secondary school. Educational Studies in Mathematics, 93(3), 333–361.CrossRefGoogle Scholar
  48. 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

© Mathematics Education Research Group of Australasia, Inc. 2018

Authors and Affiliations

  1. 1.Monash UniversityMelbourneAustralia
  2. 2.University of HaifaHaifaIsrael

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