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

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## 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## References

- 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 - Bereiter, C., & Scardamalia, M. (1987).
*The psychology of written composition*. Hillsdale, NJ: Erlbaum.Google Scholar - Berg, B. L. (2007).
*Qualitative research methods for the social sciences*(6th ed.). Boston, MA: Pearson.Google Scholar - 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 - 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 - 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 - 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 - 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 - Ericsson, K. A., & Simon, H. A. (1993).
*Protocol analysis: Verbal reports as data (Rev. ed.)*. Cambridge, MA: MIT Press.Google Scholar - 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 - 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 - 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 - Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance.
*Journal for Research in Mathematics Education,**16*, 163–176.CrossRefGoogle Scholar - 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 - 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 - 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. - 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 - 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 - 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 - Kantowski, M. G. (1977). Processes involved in mathematical problem solving.
*Journal for Research in Mathematics Education,**8*(3), 163–180.CrossRefGoogle Scholar - 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 - Kultusministerkonferenz. (2003).
*Bildungsstandards im Fach Mathematik für den mittleren Schulabschluss [Educational standards for the middle school]*. Bonn: KMK.Google Scholar - 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 - 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 - 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 - 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 - 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 - Lajoie, S. P. (Ed.). (2000).
*Computers as cognitive tools: No more walls*(Vol. 2). Mahwah, NJ: Erlbaum.Google Scholar - 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 - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - National Council of Teachers of Mathematics. (2005).
*Technology-supported mathematics learning environments*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - 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 - Patton, M. Q. (2002).
*Qualitative research and evaluation methods*. Thousand Oaks, CA: Sage.Google Scholar - Pea, R. D. (1985). Beyond amplification: Using the computer to recognize mental functioning.
*Educational Psychologist,**20*(4), 167–182.CrossRefGoogle Scholar - Pólya, G. (1973).
*How to solve it: A new aspect of mathematical method*. Princeton, NJ: Princeton University Press. (Original work published 1945).Google Scholar - 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 - 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 - 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 - 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 - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. Orlando, FL: Academic Press.Google Scholar - 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 - 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 - Tall, D. (1989). Concepts images, generic organizers, computers and curriculum change.
*For the Learning of Mathematics,**9*(3), 37–42.Google Scholar - Wilson, J., & Clarke, D. (2004). Towards the modelling of mathematical metacognition.
*Mathematics Education Research Journal,**16*(2), 25–48.CrossRefGoogle Scholar - 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 - 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 - Zimmerman, B. J. (2002). Becoming a self-regulated learner: An overview.
*Theory Into Practice,**41*(2), 64–70.CrossRefGoogle Scholar