Defining moments in the graphing calculator solution of a cubic function task

Authors

  • Jill Brown
  • Gloria Stillman

DOI:

https://doi.org/10.7146/nomad.v11i3.147992

Abstract

A case study investigated cognitive, mathematical, and technological processes undertaken by senior secondary students as they searched for a complete graph of a difficult cubic function using a graphing calculator. Intensive qualitative macroanalysis identified several defining moments in the solution process. Those related to use of scale marks and identification of key function features are presented. Students’ understanding of scale marks varied and this impacted on the efficiency and elegance of their solution. A range of calculator features was used in identifying key feature coordinates. These were not always used successfully or with an understanding of the mathematics underpinning their operation.

References

Anderson, M., Bloom, L., Mueller, U. & Pedler, P. (1999). The impact of the graphics calculator on the assessment of calculus and modeling. International Journal of Mathematical Education in Science and Technology, 30 (4), 489-498. https://doi.org/10.1080/002073999287770

Arnold, S. (1998). Technology and mathematics education: From calculators to multimedia. In N. Ellerton (Ed.), Issues in mathematics education: a contemporary perspective (pp.173-198). Perth, Western Australia: Mathematics, Science and Technology Education Centre, Edith Cowan University.

Binder, M. (1995). A calculator investigation of an interesting polynomial. The Mathematics Teacher, 88 (7), 558-560. https://doi.org/10.5951/MT.88.7.0558

Brown, J. (2002). A complete graph, a graphing calculator environment and student understanding. In C. Vale, J. Roumeliotis & J. Horwood (Eds.), Valuing mathematics in society learning (pp.365-375). Melbourne, Australia: Mathematical Association of Victoria.

Brown, J. (2003). Defining moments in determining a complete graph in a graphing calculator teaching and learning environment. In L. Bragg, C. Campbell, G. Herbert & J. Mousley (Eds.), Proceedings of the 26th annual conference of the Mathematics Education Research Group of Australasia (MERGA), Geelong, Australia (pp.168-175). Sydney, Australia: MERGA.

Cavanagh, M. & Mitchelmore, M. (2000). Student misconceptions in interpreting basic graphic calculator displays. In T. Nakahara & M. Koyama (Eds.), Proceedings of 24th conference of the international group for the Psychology of Mathematics Education (Vol. 2, pp.23-27). Hiroshima, Japan: Program Committee.

Demana, F. & Waits, B. K. (1990). The role of technology in teaching mathematics. Mathematics Teacher, 81 (1), 25-31.

Dick, T. P. (1996). Much more than a toy: graphing calculators in secondary school calculus. In P. Gómez & B. Waits (Eds.), Roles of calculators in the classroom (chap. 5). Retrieved September 9, 2006 from http://ued.uniandes. edu.co/servidor/em/recinf/tg18/Dick/Dick-1.html

Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17 (1), 105-121. https://doi.org/10.1016/S0732-3123(99)80063-7

Fey, J. (2006). Connecting technology and school mathematics [Review of the book The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument]. Journal for Research in Mathematics Education, 37 (4), 348-352.

Goldenberg, E. P. (1987). Believing is seeing: how preconceptions influence the perceptions of graphs. In J. C. Bergeron, N. Herscovics & C. Kieran, (Eds.), Proceedings of 11th international conference for the Psychology of Mathematics Education (Vol. 1, pp.197-203). Montreal, Quebec: Program Committee.

Goldenberg, E. P., Harvey, W., Lewis, P. G., Umiker, R. J., West, J. & Zodhiates, P. (1988). Mathematical, technical, and pedagogical challenges in the graphical representation of functions. (Tech. Rep. No. TR88-4). Cambridge, MA: Harvard University, Educational Technology Centre.

Guin, D., Ruthven, K. & Trouche, L. (Eds.) (2005). The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument. Dordrecht, The Netherlands: Kluwer. https://doi.org/10.1007/b101602

Hiebert, J. & Leferve, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of mathematics (pp.1-27). Hillsdale, NJ: Lawrence Erlbaum.

Huberman, A. M. & Miles, M. B. (2002). The qualitative researcher's companion. Thousand Oaks, CA: Sage. https://doi.org/10.4135/9781412986274

Kooij, H. van der (2001, December). Functional algebra with the use of the graphing calculator. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of 12th International Commission on Mathematics Instruction study conference (Vol. 2, pp.606-612). Melbourne, Australia: University of Melbourne.

Leinhardt, G., Zaslavsky, O. & Stein, M. K. (1990). Functions, graphs and graphing: tasks, learning, and teaching. Review of Educational Research, 60 (1), 1-64. https://doi.org/10.3102/00346543060001001

Lloyd, G. M. & Wilson, M. (1998). Supporting innovation: the impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29 (3), 248-274. https://doi.org/10.2307/749790

Merriam, S. B. (Ed.). (2002). Qualitative research in practice: examples for discussion and analysis. San Francisco, CA: Jossey-Bass.

Mitchelmore, M. & Cavanagh, M. (2000). Student's difficulties in operating a graphics calculator. Mathematics Education Research Journal, 12 (3), 254-268. https://doi.org/10.1007/BF03217088

Ruthven, K. (1995). Pressing on: towards a considered calculator use. In L. Burton & B. Jaworski (Eds.), Technology in mathematics teaching: a bridge between teaching and learning (pp.231-256). London, England: Chartwell- Bratt.

Schoenfeld, A. H. (1992). On paradigms and methods: what do you do when the ones you know don't do what you want them to? Issues in the analysis of data in the form of videotapes. The Journal of the Learning Sciences 2 (2), 179-214. https://doi.org/10.1207/s15327809jls0202_3

Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage.

Steele, D. (1995). The Wesley College technology enriched graphing project (Unpublished masters thesis). University of Melbourne, Victoria, Australia.

Strauss, A. & Corbin, J. (1998). Basics of qualitative research (2nd. ed.). Thousand Oaks, CA: Sage.

Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12 (2), 151-169. https://doi.org/10.1007/BF00305619

Teese, R. (2000). Academic success and social power: examinations and inequality. Melbourne, Australia: Melbourne University Press.

Victorian Board of Studies (VBOS) (1999). Mathematics study design. Melbourne: Australia: Author.

Vinner, S. & T. Dreyfus, (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20 (4), 356-366. https://doi.org/10.2307/749441

Wedege, T. (1999). To know or not to know - mathematics, that is a question of context. Educational Studies in Mathematics, 39 (1-3), 202-227. https://doi.org/10.1023/A:1003871930181

Williams, C. G. (1993). Looking over their shoulders: some difficulties students have with graphing calculators. Mathematics and Computer Education, 27 (3), 198-202.

Zaslavasky, O., Sela, H. & Leron, U. (2002). Being sloppy about slope: the effect of changing the scale. Educational Studies in Mathematics, 49 (1), 119-140. https://doi.org/10.1023/A:1016093305002

Zbiek, R. M. (2003). Using research to influence teaching and learning with computer algebra systems. In J. T. Fey, A. Cuoco, C. Keiran, L. McMullin & R. M. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp.197-216). Reston, VA: National Council of Teachers of Mathematics.

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Published

2006-09-10

How to Cite

Brown, J., & Stillman, G. (2006). Defining moments in the graphing calculator solution of a cubic function task. NOMAD Nordic Studies in Mathematics Education, 11(3), 57–84. https://doi.org/10.7146/nomad.v11i3.147992

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