Problem Solving Trajectories in a Dynamic Mathematics Environment: The Geometer’s Sketchpad

  • Michael J. Bosse Appalachian State University

Abstract

Student problem solving in the context of a dynamic mathematics environment (DME) has previously been investigated primarily through the lens of whether or not the student could complete a problem solving task. Herein, we investigate what trajectories students employ in the realms of mathematics, technology, and problem solving as they attempt to complete tasks and which of these trajectories are more helpful than others. Notably, it was determined that these trajectories are idiosyncratic, nonlinear, and iterative and that, while some trajectories help problem solving, others harm the problem solving process. Among other findings, it was determined that student access to technology may not assist their mathematical problem solving and may at times hinder it even further.

Author Biography

Michael J. Bosse, Appalachian State University

Michael. J. Bossé is the Distinguished Professor of Mathematics Education and MELT Program Director at Appalachian State University, Boone, NC. He teaches undergraduate and graduate courses and is active in providing professional development to teachers in North Carolina and around the nation. His research focuses on learning, cognition, and curriculum in K-16 mathematics.

References

Authors1 (2014).

Authors2 (2016).

Authors2 (2015).

Authors3 (2012).

Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in Cabri and modalities of transition from conjectures to proofs in geometry. Proceedings of the 22th annual conference of the International Group for the Psychology of Mathematics Education (pp. 32-39). South Africa.

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt für Didaktik der Mathematik, 34(3), 66-72.

Baccaglini-Frank, A., & Mariotti, M. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225-253.

Bogdan, R. C., & Biklen, S. K. (2003). Qualitative research for education: An introduction to theories and methods (4 ed.). Boston: Allyn and Bacon.

Authors4 (2014).

Authors5 (2016)

Carpenter, T. P. (1989). Teaching as problem solving. In R. I. Charles and E. A. Silver (Eds), The teaching and assessing of mathematical problem solving (pp.187-202). USA: National Council of Teachers of Mathematics.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2-33.

Creswell, W. J. (2003). Research Design: Qualitative, quantitative and mixed methods approaches (2 ed.). London: Sage Publications.

DePeau, E. A. & Kalder, R. S. (2010). Using dynamic technology to present concepts through multiple representations. Mathematics Teacher, 104(4), 268-273.

de Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. International Journal of Mathematical Education in Science and Technology, 35(5), 703–724.

Dick, T. P., & Hollebrands, K. F. (2011). Introduction to Focus in high school mathematics: Technology to support reasoning and sense making. In T. P. Dick & K. F. Hollebrands (Eds.), Focus in high school mathematics: Technology to support reasoning and sense making (pp. xi - xvii). Reston, VA: National Council of Teachers of Mathematics.

Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131.

Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317-333.

Finzer, W. & Bennett, D. (1995). Technology tips: From drawing to construction with The Geometer’s Sketchpad. Mathematics Teacher, 88(5), 428-431.

Garofalo, J., Drier, H., Harper, S., Timmerman, M.A., & Shockey, T. (2000). Promoting appropriate uses of technology in mathematics teacher preparation. Contemporary Issues in Technology and Teacher Education [Online serial], 1(1). Available: http://www.citejournal.org/vol1/iss1/currentissues/mathematics/article1.htm

Goldenberg, E. P. & Cuoco, A., A. 1998. “What is dynamic geometry?” In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 351-367). Mahwah, NJ: Lawrence Erlbaum Associates.

Gonzalez, G., & Herbst, P. (2009). Students' conceptions of congruency through the use of dynamic geometry software. International Journal of Computers for Mathematical Learning(4), 153-182.

Healy, L., Hölzl. R., Hoyles, C., Noss, R. (1994). Messing up. Micromath, 10, 14-17.

Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Human, P., Murray, H., . . . Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(12), 12-21.

Hollebrands, K. F., & Lee, H. S. (2012). Preparing to teach mathematics with technology: An integrated approach to geometry. Dubuque, IA: Kendall Hunt.

Hollebrands, K. F., McCulloch, A. W., & Chandler, K. (2015, November). High school students’ uses of dragging for examining geometric representations of functions. Poster presented at the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. East Lansing, MI.

Hollebrands, K. & Smith, R. (2009). The impact of dynamic geometry software on secondary students’ learning of geometry: Implications from research. In T. Craine & R. Rubenstein (Eds.), NCTM 2009 yearbook: Understanding geometry for a changing world (pp. 221-232). Reston, VA: NCTM.

Hölzl, R. (1996). How does ‘dragging’ affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1. 169-187.

Hoyles, C., & Noss, R. (1994). Dynamic geometry environments: What’s the point? Mathematics Teacher, 87(9), 716-717.

Isiksal, M., & Askar, P. (2005). The effect of spreadsheet and dynamic geometry software on the achievement and self-efficacy of 7th-grade students. Educational Research, 47(3), 333-350.

Jones, K. (2000). Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44(1/2), 55-85.

Kaput, J. (1987). Towards a theorey of symbol use in mathematics. In C. Janvier (Ed.), Problems or representation in the teaching and learning of mathematics (pp. 159-195). Hillsdale, NJ: Lawrence Erlbaum Associates.

Lee, H. S., & Hollebrands, K. F. (2006). Students’ use of technological features while solving a mathematics problem. Journal of Mathematical Behavior, 25, 252-266.

Lee, H. S., Harper, S., Driskell, S. O., Kersaint, G., & Leatham, K. (2012). Teachers’ statistical problem solving with dynamic technology: Research results across multiple institutions. Contemporary Issues in Technology and Teacher Education [Online serial], 12(3). http://www.citejournal.org/vol12/iss3/mathematics/article1.cfm.

Lester, F. K. Jr., Masingila, J. O., Mau, S. T., Lambdin, D. V., dos Santon, V. M. and Raymond, A. M. (1994). Learning how to teach via problem solving. In Aichele, D. and Coxford, A. (Eds.), Professional Development for Teachers of Mathematics (pp. 152-166). Reston, Virginia: NCTM.

Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. International Journal of Mathematical Education in Science & Technology, 37(6), 665-679.

Miles, M. B. & Huberman, M. N. (1994). Qualitative data analysis: an expanded sourcebook. Thousand Oaks, CA: Sage.
National Council of Teachers of Mathematics [NCTM]. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Authors.

National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Authors.

National Governors Association Center for Best Practices, Council of Chief State School Officers (2010). Common core state standards in mathematics. Washington D.C.: Authors.

Olivero, F., & Robutti, O. (2002). An exploratory study of students' measurement activity in a dynamic geometry environment. In Proceedings of CERME2 (Vol. 1, pp. 215-226).

Pajares, F. (1996). Self-efficacy beliefs and mathematical problem-solving of gifted students. Contemporary Educational Psychology, 25, 325-344.

Pajares, F., & Kranzler, J. (1995). Self-efficacy beliefs and general mental ability in mathematical problem-solving. Contemporary Educational Psychology, 20, 426-443.

Pandiscio, E. A. (1996). Exploring the link between preservice teachers’ conception of proof and the use of dynamic geometry software. School Science and Mathematics, 105(2), 216–221.Ruthven, K., Hennessy, S., & Deaney, R. (2008). Constructions of dynamic geometry: A study of the interpretive flexibility of educational software in classroom practice. Computers & Education 51(1): 297-317.

Schoenfeld, A. (1992). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics. In D. A. Grouws (Ed.) Handbook for research on mathematics teaching and learning (pp. 334-370). New York: Macmillian.

Schoenfeld, A. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.), Mathematical Thinking and Problem Solving (pp. 53-69). Hillsdale, NJ: Lawrence Erlbaum Associates.

Stacey, K., & Groves, S. (1985). Strategies for problem solving. Melbourne, Victoria: VICTRACC.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research. London: Sage Publications Ltd.

Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp.189-236). Hillsdale, N.J: Lawrence Erlbaum.

Wilson, J., Fernandez, M., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.) Research ideas for the classroom: High school mathematics (pp. 57-78). New York: MacMillan.
Published
2018-08-22