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A Model for Task Design with Focus on Exploration, Explanation, and Generalization in a Dynamic Geometry Environment

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Abstract

The increasing availability of new technologies in schools provides new possibilities for the integration of technology in mathematics education. However, research has shown that there is a need for new kinds of task that utilize the affordances provided by new technology. Numerous studies have demonstrated that dynamic geometry environments provide opportunities for students to engage in mathematical activities such as exploration, conjecturing, explanation, and generalization. This paper presents a model for design of tasks that promote these kinds of mathematical activity, especially tasks that foster students to make generalizations. This model has been primarily developed to suit the use of dynamic environments in tackling geometrical locus problems. The model was initially constructed in the light of previous literature. This initial model was used to design a concrete example of such a task situation which was tested in action through a case study with two doctoral students. Findings from this case study were used to guide revision of the initial model.

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Notes

  1. A task-situation consists of a sequence of tasks. The term “task-situation” is adopted from Kieran and Saldanha (2008).

  2. Arzarello et al. (2002) use the notion “semi-dragable point” when they refer to a point that belongs to an object.

  3. The text within square brackets should be adapted to the circumstances under consideration.

  4. This formulation suits a common type of geometrical locus problems.

  5. The sum of the distances from a point on the ellipse to the two foci is independent of the choice of point, i.e. the sum is constant.

  6. The foci are F and a point A on the line through F and F′, where FA/AF′ = FM/MP.

  7. A homothety is a transformation determined by a certain point Q that carries each point P into a point M on the straight line PQ in accordance with the rule QM = kQP, where k is a constant number, not equal to zero. The homothetic image of a figure F is the figure formed by the set of all points M homothetic to the points which constitutes the figure F. A homothety is a special case of a similarity.

References

  • Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129.

    Article  Google Scholar 

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in cabri environments. ZDM the International Journal on Mathematics Education, 34(3), 66–72.

    Google Scholar 

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

    Article  Google Scholar 

  • Chazan, D. (1990a). Quasi-empirical views of mathematics and mathematics teaching. Interchange, 21(1), 14–23.

    Article  Google Scholar 

  • Chazan, D. (1990b). Students’ microcomputer-aided exploration in geometry. Mathematics Teacher, 83(8), 628–635.

    Google Scholar 

  • De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Doorman, M., Drijvers, P., Dekker, T., van den Heuvel-Panhuizen, M., de Lange, J., & Wijers, M. (2007). Problem solving as a challenge for mathematics education in the Netherlands. ZDM the International Journal on Mathematics Education, 39(5), 405–418.

    Article  Google Scholar 

  • Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–234.

    Article  Google Scholar 

  • Edwards, L. D. (1997). Exploring the territory before proof: Student‘s generalizations in a computer microworld for transformation geometry. International Journal of Computers for Mathematical Learning, 2(3), 187–215.

    Article  Google Scholar 

  • Furinghetti, F., & Paola, D. (2003). To produce conjectures and to prove them within a dynamic geometry environment: A case study. Proceedings of the Twenty Seventh Annual Conference of the International Group for the Psychology of Mathematics Education, 2, 404.

    Google Scholar 

  • Guven, B. (2008). Using dynamic geometry software to gain insight into a proof. International Journal of Computers for Mathematical Learning, 13(3), 251–262.

    Article  Google Scholar 

  • Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1), 127–150.

    Article  Google Scholar 

  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5–23.

    Article  Google Scholar 

  • Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 877–908). Dordrecht: Kluwer.

    Google Scholar 

  • Healy, L., & Hoyles, C. (2002). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235–256.

    Article  Google Scholar 

  • Hitt, F., & Kieran, C. (2009). Constructing knowledge via a peer interaction in a CAS environment with tasks designed from a task—technique—theory perspective. International Journal of Computers for Mathematical Learning, 14(2), 121–152.

    Article  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(1), 63–86.

    Article  Google Scholar 

  • Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 121–128). Dordrecht: Kluwer.

    Google Scholar 

  • Kieran, C., & Saldanha, L. (2008). Designing tasks for the codevelopment of conceptual and technical knowledge in CAS activity: An example from factoring. In G. W. Blume & K. M. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 2, cases and perspectives (pp. 393–414). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Laborde, C. (2002). Integration of technology in the design of geometry tasks with cabri-geometry. International Journal of Computers for Mathematical Learning, 6(3), 283–317.

    Article  Google Scholar 

  • Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM the International Journal on Mathematics Education, 43(3), 325–336.

    Article  Google Scholar 

  • Leung, A., & Lopez-Real, F. (2002). Theorem justification and acquisition in dynamic geometry: A case of proof by contradiction. International Journal of Computers for Mathematical Learning, 7(2), 145–165.

    Article  Google Scholar 

  • Lin, F. L., Yang, K. L., Lee, K. H., Tabach, M., & Stylianides, G. (2012). Principles of task design for conjecturing and proving. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education. the 19th ICMI study (pp. 305–325). Berlin: Springer.

    Google Scholar 

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

    Article  Google Scholar 

  • Marrades, R., & Gutierrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1), 87–125.

    Article  Google Scholar 

  • Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Dorchester: Pearson Education.

    Google Scholar 

  • Mogetta, C., Olivero, F., & Jones, K. (1999). Providing the motivation to prove in a dynamic geometry environment. Proceedings of the British Society for Research into Learning Mathematics, 19(2), 96.

    Google Scholar 

  • Olivero, F., & Robutti, O. (2007). Measuring in dynamic geometry environments as a tool for conjecturing and proving. International Journal of Computers for Mathematical Learning, 12(2), 135–156.

    Article  Google Scholar 

  • Öner, D. (2008). Supporting students’ participation in authentic proof activities in computer supported collaborative learning (CSCL) environments. International Journal of Computer Supported Collaborative Learning, 3(3), 343–359.

    Article  Google Scholar 

  • Pierce, R., & Ball, L. (2009). Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes. Educational Studies in Mathematics, 71(3), 299–317.

    Article  Google Scholar 

  • Pólya, G. (1945). How to solve it. Princton: Princeton University Press.

    Google Scholar 

  • Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319–325.

    Article  Google Scholar 

  • Ruthven, K. (2009). Towards a naturalistic conceptualisation of technology integration in classroom practice: The example of school mathematics. Education & Didactique, 3(1), 131–159.

    Article  Google Scholar 

  • Ruthven, K., Hennessy, S., & Deaney, R. (2008). Constructions of dynamic geometry: A study of the interpretative flexibility of educational software in classroom practice. Computers & Education, 51(1), 297–317.

    Article  Google Scholar 

  • Santos-Trigo, M., & Espinosa-Perez, H. (2002). Searching and exploring properties of geometric configurations using dynamic software. International Journal of Mathematical Education in Science and Technology, 33(1), 37–50.

    Article  Google Scholar 

  • Sinclair, M. P. (2003). Some implications of the results of a case study for the design of pre-constructed, dynamic geometry sketches and accompanying materials. Educational Studies in Mathematics, 52(3), 289–317.

    Article  Google Scholar 

  • Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332.

    Article  Google Scholar 

  • Weber, K. (2005). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. The Journal of Mathematical Behavior, 24(3–4), 351–360.

    Article  Google Scholar 

  • Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(2), 209–234.

    Article  Google Scholar 

  • Yerushalmy, M. (1993). Generalization in geometry. In J. L. Schwartz, M. J. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of? (pp. 57–84). USA: Lawrence Erlbaum.

    Google Scholar 

  • Zehavi, N., & Mann, G. (2011). Development process of a praxeology for supporting the teaching of proofs in a CAS environment based on teachers’ experience in a professional development course. Technology Knowledge and Learning, 16(2), 153–181.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Karlstad University, 651 88, Karlstad, Sweden

    Maria Fahlgren & Mats Brunström

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  1. Maria Fahlgren
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  2. Mats Brunström
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Correspondence to Maria Fahlgren.

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Fahlgren, M., Brunström, M. A Model for Task Design with Focus on Exploration, Explanation, and Generalization in a Dynamic Geometry Environment. Tech Know Learn 19, 287–315 (2014). https://doi.org/10.1007/s10758-014-9213-9

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  • DOI: https://doi.org/10.1007/s10758-014-9213-9

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