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Technology, Knowledge and Learning

, Volume 22, Issue 1, pp 37–64 | Cite as

Delving into the Nature of Problem Solving Processes in a Dynamic Geometry Environment: Different Technological Effects on Cognitive Processing

  • Ana Kuzle
Original Research
First Online:

Abstract

Students regularly struggle with mathematical tasks, particularly those concerning non-routine problems in geometry. Although educators would like for their learners to transfer their knowledge to non-routine and real-life situations, students run into a number of difficulties. The goal of this exploratory study was to analyze three participants’ problem solving processes in a dynamic geometry software (DGS), and therefore, gain insights about how DGS was used to support solving non-routine geometry problems. Here I viewed the DGS as a cognitive tool that can enhance and reorganize the problem solving process. The three participants were in different phases of their educational career in mathematics and/or mathematics education (bachelor, master, and doctoral student). Only one problem—the Land Boundary Problem—from the TIMSS video study will be discussed here. In this problem, the participants had to straighten a bent fence between two farmers’ land so that each farmer would keep the same amount of land. All three participants solved the problem, but used the same computer-based problem-solving tool differently. While a DGS allowed and supported some participants to discover new methods of thinking, and unanticipated ways of using it, it also inhibited the problem solving processes through development of tool-dependency by some. Its different use was dependent on the presence of managerial decisions, ability to manage different resources, and problem solving experience. Based on these findings, I make recommendations for technology-embedded problem solving with an emphasis on the importance of appropriate tool use in educational settings and offer some teaching methods that may be worthwhile for research.

Keywords

Cognitive tool Dynamic geometry software Mathematical problem solving Metacognition Non-routine geometry problems Teacher education 
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References

  1. 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. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd annual conference of the international group for the psychology of mathematics education (Vol. 2, pp. 32–39). Stellenbosch.Google Scholar
  2. Bereiter, C., & Scardamalia, M. (1987). The psychology of written composition. Hillsdale, NJ: Erlbaum.Google Scholar
  3. Berg, B. L. (2007). Qualitative research methods for the social sciences (6th ed.). Boston, MA: Pearson.Google Scholar
  4. Bransford, J. D., Brown, A. L., & Cocking, R. R. (2004). How people learn. Brain, mind, experience, and school. Washington, DC: National Academy Press.Google Scholar
  5. Brown, A. L. (1987). Metacognition, executive control, self regulation and other more mysterious mechanisms. In F. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation and understanding (pp. 65–116). Hillsdale, NJ: Erlbaum.Google Scholar
  6. Carlson, M. P. (1999). The mathematical behavior of six successful mathematics graduate students: Influences leading to mathematical success. Educational Studies in Mathematics, 40(3), 237–258.CrossRefGoogle Scholar
  7. Carlson, M. P., & Bloom, I. (2005). The cycle nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.CrossRefGoogle Scholar
  8. Dörfler, W. (1991). Der Computer als kognitives Werkzeug und kognitives Medium [Computer as a cognitive tool and a cognitive medium]. In W. Dörfler, W. Peschek, E. Schneider, & K. Wegenkittl (Eds.), Computer - Mensch - Mathematik (pp. 51–76). Wien, Stuttgart: Hölder-Pichler-Tempsky; B.G.Teubner.Google Scholar
  9. Ericsson, K. A., & Simon, H. A. (1993). Protocol analysis: Verbal reports as data (Rev. ed.). Cambridge, MA: MIT Press.Google Scholar
  10. Fey, J. T., Hollenbeck, R. M., & Wray, J. A. (2010). Technology and the mathematics curriculum. In B. J. Reys, R. E. Reys, & R. Rubenstein (Eds.), Mathematics curriculum: Issues, trends, and future directions (pp. 41–49). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  11. Flavell, J. H. (1976). Metacognition aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231–236). Hillsdale, NJ: Erlbaum.Google Scholar
  12. Flavell, J. H. (1981). Cognitive monitoring. In W. P. Dickson (Ed.), Children’s oral communication skills (pp. 35–60). New York, NY: Academic Press.Google Scholar
  13. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16, 163–176.CrossRefGoogle Scholar
  14. Goldhammer, F., Naumann, J., Stelter, A., Toth, K., Rölke, H., & Kleme, E. (2014). The time on task effect in reading and problem solving is moderated by task difficulty and skill: Insights from a computer-based large-scale assessment. Journal of Educational Psychology, 106(3), 608–626.CrossRefGoogle Scholar
  15. Haug, R. (2012). Problemlösen lernen mit digitalen Medien. Förderung grundlegender Problemlösetechniken durch den Einsatz dynamischer Werkzeuge [Problem solving learning with digital media. Promoting the fundamental problem-solving techniques through the use of dynamic tools]. Wiesbaden: Vieweg + Teubner Verlag.Google Scholar
  16. Herrera, M., Preiss, R., & Riera, G. (2008, July). Intellectual amplification and other effects “with,” “of” and “through” technology in teaching and learning mathematics. Paper presented at the Meeting of the International Congress of Mathematics Instruction. Monterrey, Mexico. Retrieved February 13, 2014, from dg.icme11.org/document/get/76.
  17. Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164–192.Google Scholar
  18. Hölzl, R. (2001). Using dynamic geometry software to add contrast to geometric situations—A case study. International Journal of Computers for Mathematical Learning, 6, 63–86.CrossRefGoogle Scholar
  19. Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop, M. A. Clemens, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 323–349). Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  20. Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Research in Mathematics Education, 8(3), 163–180.CrossRefGoogle Scholar
  21. Kim, B., & Reeves, T. C. (2007). Reframing research on learning with technology: In search of the meaning of cognitive tools. Instructional Science, 35(3), 207–256.CrossRefGoogle Scholar
  22. Kultusministerkonferenz. (2003). Bildungsstandards im Fach Mathematik für den mittleren Schulabschluss [Educational standards for the middle school]. Bonn: KMK.Google Scholar
  23. Kuzle, A. (2011). Preservice teachers’ patterns of metacognitive behavior during mathematics problem solving in a dynamic geometry environment. Doctoral dissertation. University of Georgia–Athens, GA.Google Scholar
  24. Kuzle, A. (2013). Patterns of metacognitive behavior during mathematics problem-solving in a dynamic geometry environment. International Electronic Journal of Mathematics Education, 8(1), 20–40.Google Scholar
  25. Kuzle, A. (2015). Nature of metacognition in a dynamic geometry environment. LUMAT – Research and Practice in Math, Science and Technology Education, 3(5), 627–646.Google Scholar
  26. Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1), 151–161.CrossRefGoogle Scholar
  27. Laborde, C., Kynigos, C., Hollebrands, K., & Sträßer, R. (2006). Teaching and learning geometry with technology. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 275–304). Rotterdam: Sense.Google Scholar
  28. Lajoie, S. P. (Ed.). (2000). Computers as cognitive tools: No more walls (Vol. 2). Mahwah, NJ: Erlbaum.Google Scholar
  29. Lajoie, S. P. (2005). Cognitive tools for the mind: The promises of technology: Cognitive amplifiers or bionic prosthetics? In R. J. Sternberg & D. Preiss (Eds.), Intelligence and technology: Impact of tools on the nature and development of human skills (pp. 87–102). Mahwah, NJ: Erlbaum.Google Scholar
  30. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  31. National Council of Teachers of Mathematics. (2005). Technology-supported mathematics learning environments. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  32. Olive, J., & Makar, K. (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain. The 17th ICMI Study (pp. 133–177). New York, NY: Springer.Google Scholar
  33. Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage.Google Scholar
  34. Pea, R. D. (1985). Beyond amplification: Using the computer to recognize mental functioning. Educational Psychologist, 20(4), 167–182.CrossRefGoogle Scholar
  35. Pólya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press. (Original work published 1945).Google Scholar
  36. Proske, A., Damnik, G., & Körndle, H. (2011). Learners-as-Designers: Wissensräume mit kognitiven Werkzeugen aktiv nutzen und konstruieren [Learners-as-designers: Active use and construction of knowledge spaces with cognitive tools]. In T. Köhler & J. Neumann (Eds.), Wissensgemeinschaften. Digitale Medien - Öffnung und Offenheit in Forschung und Lehre (pp. 198–208). Münster: Waxmann.Google Scholar
  37. Salomon, G., & Perkins, D. (2005). Do technologies make us smarter? Intellectual amplification with, of, and through technology. In R. J. Sternberg & D. D. Preiss (Eds.), Intelligence and technology: The impact of tools on the nature and development of human abilities (pp. 71–86). Mahwah, NJ: Erlbaum.Google Scholar
  38. Salomon, G., Perkins, D. N., & Globerson, T. (1991). Partners in cognition: Extending human intelligence with intelligent technologies. Educational Researcher, 20(3), 2–9.CrossRefGoogle Scholar
  39. Schoenfeld, A. H. (1981). Episodes and executive decisions in mathematical problem solving (ERIC Documentation Reproduction Service No. ED201505). Paper presented at the Annual meeting of the American educational research association, Los Angeles, CA.Google Scholar
  40. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.Google Scholar
  41. Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Erlbaum.Google Scholar
  42. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York, NY: Macmillan.Google Scholar
  43. Tall, D. (1989). Concepts images, generic organizers, computers and curriculum change. For the Learning of Mathematics, 9(3), 37–42.Google Scholar
  44. Wilson, J., & Clarke, D. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25–48.CrossRefGoogle Scholar
  45. Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 57–78). New York, NY: Macmillan.Google Scholar
  46. Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. (2007). Research on technology in mathematics education: A perspective of constructs. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1169–1207). Charlotte, NC: Information Age.Google Scholar
  47. Zimmerman, B. J. (2002). Becoming a self-regulated learner: An overview. Theory Into Practice, 41(2), 64–70.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University of PotsdamPotsdamGermany

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