Advertisement

Mathematics Education Research Journal

, Volume 27, Issue 4, pp 401–421 | Cite as

Introducing linear functions: an alternative statistical approach

  • Caroline Nolan
  • Sandra HerbertEmail author
Original Article

Abstract

The introduction of linear functions is the turning point where many students decide if mathematics is useful or not. This means the role of parameters and variables in linear functions could be considered to be ‘threshold concepts’. There is recognition that linear functions can be taught in context through the exploration of linear modelling examples, but this has its limitations. Currently, statistical data is easily attainable, and graphics or computer algebra system (CAS) calculators are common in many classrooms. The use of this technology provides ease of access to different representations of linear functions as well as the ability to fit a least-squares line for real-life data. This means these calculators could support a possible alternative approach to the introduction of linear functions. This study compares the results of an end-of-topic test for two classes of Australian middle secondary students at a regional school to determine if such an alternative approach is feasible. In this study, test questions were grouped by concept and subjected to concept by concept analysis of the means of test results of the two classes. This analysis revealed that the students following the alternative approach demonstrated greater competence with non-standard questions.

Keywords

Linear functions Secondary school Statistical data Multiple representations Technology 

References

  1. Asp, G., Dowsey, J., Stacey, K., & Tynan, D. (1995). Graphic algebra. Carlton: Curriculum Corporation.Google Scholar
  2. Australian Bureau of Statistics. (2013). “Census At School”, from http://www.abs.gov.au/censusatschool
  3. Australian Curriculum, Assessment and Reporting Authority. (2012). “Australian Curriculum.” from http://www.australiancurriculum.edu.au
  4. Bardini, C., Pierce, R., & Stacey, K. (2004). Teaching linear functions in context with graphics calculators: students’ responses and the impact of the approach on their use of algebraic symbols. International Journal of Science and Mathematics Education, 2(3), 353–376.CrossRefGoogle Scholar
  5. Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters and indeterminate objects or how to go insane in mathematics. In Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 129–136).Google Scholar
  6. Bardini, C., & Stacey, K. (2006). Students’ conceptions of m and c: how to tune a linear function. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 113–120).Google Scholar
  7. Chick, H. L., & Pierce, R. (2012). Teaching for statistical literacy: utilising affordances in real-world data. International Journal of Science and Mathematics Education, 10(2), 339–362.CrossRefGoogle Scholar
  8. Craig, M. (2009). “Evaluating the professional development materials provided by the Australian Bureau of Statistics for the Census At School Program”, from http://web.cs.wpi.edu/∼rek/Projects/ABS_D09.pdf
  9. Ely, R., & Adams, A. E. (2012). Unknown, placeholder, or variable: what is x? Mathematics Education Research Journal, 24(1), 19–38.CrossRefGoogle Scholar
  10. Galbraith, P., Stillman, G., Brown, J., & Edwards, I. (2005). Facilitating mathematical modelling competencies in the middle secondary school. London: International Conference on the Teaching of Mathematical Modelling and Applications.Google Scholar
  11. Garner, S., & Leigh-Lancaster, D. (2003). A teacher-researcher perspective on CAS in senior secondary mathematics. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics education research: innovation, networking, opportunity (pp. 372–379). Geelong: Proceedings of the 26th annual conference of the Mathematics Education Research Group of Australasia.Google Scholar
  12. Gattuso, L. (2006). “Statistics and mathematics. Is it possible to create fruitful links”. Paper presented at the Proceedings of the Seventh International Conference on Teaching Statistics. CD ROM. Salvador (Bahia): International Association for Statistical Education and International Statistical Institute.Google Scholar
  13. Geiger, V., Faragher, R., & Goos, M. (2010). CAS-enabled technologies as ‘agents provocateurs’ in teaching and learning mathematical modelling in secondary school classrooms. Mathematics Education Research Journal, 22(2), 48–68.CrossRefGoogle Scholar
  14. Goos, M., Renshaw, P., & Galbraith, P. (1997). Resisting interaction and collaboration in secondary mathematics classrooms. Brisbane: Proceedings of the Annual Conference of the Australian Association for Research in Education.Google Scholar
  15. Goos, M., Geiger, V., & Dole, S. (2013). Designing rich numeracy tasks. Paper presented at the Task Design in Mathematics Education (pp. 589–597). Oxford: Proceedings of ICMI Study 22.Google Scholar
  16. Greenwood, D., Woodley, S., Vaughan, J., Franca, F., & Goodman, J. (2011). Essential mathematics for the Australian Curriculum: year 9. Melbourne: Cambridge University Press.Google Scholar
  17. Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65–97). New York: Macmillan.Google Scholar
  18. Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter? Mathematics Education Research Journal, 23(4), 455–477.CrossRefGoogle Scholar
  19. Kendal, M., & Stacey, K. (2003). Tracing learning of three representations with the differentiation competency framework. Mathematics Education Research Journal, 15(1), 22–41.CrossRefGoogle Scholar
  20. Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding it up: helping children learn Mathematics, from http://www.nap.edu/catalog.php?record_id=9822
  21. Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: tasks, learning and teaching. Review of Educational Research, 60, 1–64.CrossRefGoogle Scholar
  22. Meyer, J. H., & Land, R. (2005). Threshold concepts and troublesome knowledge (2): epistemological considerations and a conceptual framework for teaching and learning. Higher Education, 49(3), 373–388.CrossRefGoogle Scholar
  23. Mitchelmore, M., & White, P. (2004). Abstraction in mathematics and mathematics learning. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 329–336).Google Scholar
  24. Nathan, M. J., & Kim, S. (2007). Pattern generalization with graphs and words: a cross-sectional and longitudinal analysis of middle school students’ representational fluency. Mathematical Thinking and Learning, 9(3), 193–219.CrossRefGoogle Scholar
  25. O’Keefe, J. (1997). Teacher to teacher: the human scatterplot. Mathematics Teaching in the Middle School, 3(3), 208–209.Google Scholar
  26. Perkins, D., & Blythe, T. (1994). Putting understanding up front. Educational Leadership, 51, 4–4.Google Scholar
  27. Pettersson, K. & Scheja, M. (2012). “Prospective mathematics teachers’ development of understanding of the threshold concept of a function”, http://dx.doi.org/ 10.1007/s10734-009-9244-7
  28. Pierce, R. (2005). Linear functions and a triple influence of teaching on the development of student’s algebraic expectations (Vol. 4, pp. 81–88). Melbourne: Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education.Google Scholar
  29. Pierce, R., Stacey, K., & Ball, L. (2004). Mathematics from still and moving images. In B. Tadich, S. Tobias, C. Brew, B. Beatty, & P. Sullivan (Eds.), Towards excellence in mathematics (pp. 386–395). Melbourne: Monash University.Google Scholar
  30. Roschelle, J., Vahey, P. D., Tatar, D. J., Kaput, J., & Hegedus, S. (2003). Five key considerations for networking in a handheld-based mathematics classroom. Honolulu: Proceedings of the 2003 Joint Meeting of PME and PMENA.Google Scholar
  31. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.Google Scholar
  32. Stacey, K. (2010). “Mathematics teaching and learning to reach beyond the basics” Make it count: what research tells us about effective teaching and learning of mathematics? (pp. 17–19). Melbourne: ACER Research Conference: Teaching Mathematics.Google Scholar
  33. Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modelling in the secondary classroom. In J. Watson & K. Beswick (Eds.), Mathematics: essential research, essential practice (Vol. 2, pp. 688–697). Hobart: Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia.Google Scholar
  34. Tukey, J. W. (1977). “Exploratory data analysis”. Reading, Ma, 231, 32Google Scholar
  35. Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107.CrossRefGoogle Scholar

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2015

Authors and Affiliations

  1. 1.Damascus CollegeCanadianAustralia
  2. 2.Deakin UniversityWarrnamboolAustralia

Personalised recommendations