Is there room for conjectures in mathematics? The role of dynamic geometry environments

Ioannis Rizos 1 * , Nikolaos Gkrekas 1
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1 Department of Mathematics, University of Thessaly, Lamia, GREECE
* Corresponding Author
EUR J SCI MATH ED, Volume 11, Issue 4, pp. 589-598. https://doi.org/10.30935/scimath/13204
Published Online: 14 April 2023, Published: 01 October 2023
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ABSTRACT

Proof, as a central and integral part of mathematics, is an essential component of mathematical education and is considered as the basic procedure for revealing the truth of mathematical propositions and for teaching productive reasoning as part of human civilization. Is there therefore room for conjectures in mathematics? In this paper after discussing at a theoretical level the concepts of proof and conjecture, both in a paper-and-pencil environment and in a dynamic geometry environment (DGE) as well as how school practice affects them, we fully explain a task involving various mathematical disciplines, which we tackle using elementary mathematics, in a mathematics education context. On the occasion of the Greek educational system we refer to some parameters of the teaching of geometry in school and we propose an activity, within a DGE, that could enable students to be guided in the formulation and exploration of conjectures. Finally, we discuss the teaching implications of this activity and make some suggestions.
Keywords: conjecture, proof, geometric transformations, dynamic geometry environments, mathematics education

CITATION

Rizos, I., & Gkrekas, N. (2023). Is there room for conjectures in mathematics? The role of dynamic geometry environments. European Journal of Science and Mathematics Education, 11(4), 589-598. https://doi.org/10.30935/scimath/13204

REFERENCES

  • Abboud, M., & Rogalski, J. (2021). Open dynamic situations of classroom use of digital technologies: Investigating teachers’ interventions. Canadian Journal of Science, Mathematics and Technology Education, 21, 424-440. https://doi.org/10.1007/s42330-021-00151-9
  • Abramovich, S., Grinshpan, A. Z., & Milligan, D. L. (2019). Teaching mathematics through concept motivation and action learning. Education Research International, 2019, 3745406. https://doi.org/10.1155/2019/3745406
  • Albano, G., & Dello Iacono, U. (2019). A scaffolding toolkit to foster argumentation and proofs in mathematics: Some case studies. International Journal of Educational Technology in Higher Education, 16, 4. https://doi.org/10.1186/s41239-019-0134-5
  • Arcavi, A., & Hadas, N. (2000). Computer mediated learning: An example of an approach. International Journal of Computers for Mathematical Learning, 5, 25-45. https://doi.org/10.1023/A:1009841817245
  • Arzarello, F., Ferrara, F., & Robutti, O. (2012). Mathematical modelling with technology: The role of dynamic representations. Teaching Mathematics and its Applications: An International Journal of the IMA, 31(1), 20-30. https://doi.org/10.1093/teamat/hrr027
  • Astawa, W. P., Budayasa, K., & Juniati, D. (2018). The process of student cognition in constructing mathematical conjecture. Journal on Mathematics Education, 9(1), 15-26. https://doi.org/10.22342/jme.9.1.4278.15-26
  • Baccaglini-Frank, A. (2019). Dragging, instrumented abduction and evidence, in processes of conjecture generation in a dynamic geometry environment. ZDM-Mathematics Education, 51, 779-791. https://doi.org/10.1007/s11858-019-01046-8
  • Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15, 225-253. https://doi.org/10.1007/s10758-010-9169-3
  • Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof, 7, 8.
  • Bruner, J. S. (1961). The act of discovery. Harvard Educational Review, 31, 21-32.
  • Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377. https://doi.org/10.1016/j.dr.2014.10.001
  • Dockendorff, M., & Solar, H. (2018). ICT integration in mathematics initial teacher training and its impact on visualization: The case of GeoGebra. International Journal of Mathematical Education in Science and Technology, 49(1), 66-84. https://doi.org/10.1080/0020739X.2017.1341060
  • Furinghetti, F., & Paola, D. (2003). To produce conjectures and to prove them within a dynamic geometry environment: A case study. In Proceedings of the 27th Conference of the PME (pp. 397-404).
  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
  • Hanna, G. (2020). Mathematical proof, argumentation, and reasoning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 561-566). Springer. https://doi.org/10.1007/978-3-030-15789-0_102
  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96, 1-16. https://doi.org/10.1007/s10649-017-9761-8
  • Komatsu, K., & Jones, K. (2020). Interplay between paper-and-pencil activity and dynamic-geometry-environment use during generalization and proving. Digital Experiences in Mathematics Education, 6, 123-143. https://doi.org/10.1007/s40751-020-00067-3
  • Leung, A., Baccaglini-Frank, A., & Mariotti, M. A. (2013). Discernment of invariants in dynamic geometry environments. Educational Studies in Mathematics, 84, 439-460. https://doi.org/10.1007/s10649-013-9492-4
  • Liburd, K. K. D., & Jen, H. Y. (2021). Investigating the effectiveness of using a technological approach on students’ achievement in mathematics–Case study of a high school in a Caribbean Country. Sustainability, 13, 5586. https://doi.org/10.3390/su13105586
  • Mariotti, M. A. (2014). Transforming Images in DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), Transformation–A fundamental idea of mathematics education (pp. 155-172). Springer. https://doi.org/10.1007/978-1-4614-3489-4_8
  • McCulloch, A. W., Hollebrands, K., Lee, H., Harisson, T., & Mutlu, A. (2018). Factors that influence secondary mathematics teachers’ integration of technology in mathematics lessons. Computers & Education, 123, 26-40. https://doi.org/10.1016/j.compedu.2018.04.008
  • Mulenga, E. M., & Marban, J. M. (2020). Is COVID-19 the gateway for digital learning in mathematics education? Contemporary Educational Technology, 12(2), ep269. https://doi.org/10.30935/cedtech/7949
  • NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
  • Oxman, V., Stupel, M. (2022). Dynamic investigation of surprising situations with conservation of areas described by functions with parameters. Resonance, 27, 1661-1672. https://doi.org/10.1007/s12045-022-1456-5
  • Patronis, T., & Thomaidis, Y. (1997). On the arithmetization of school geometry in the setting of modern axiomatics. Science & Education, 6, 273-290. https://doi.org/10.1023/A:1008603828653
  • Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587-597. https://doi.org/10.1007/s10648-015-9302-x
  • Rizos, I., & Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686. https://doi.org/10.29333/iejme/11977
  • Rizos, I., & Gkrekas, N. (2022). Teaching and learning sciences within the Covid-19 pandemic era in a Greek University Department. U. Porto Journal of Engineering, 8(1), 73-83. https://doi.org/10.24840/2183-6493_008.001_0008
  • Rizos, I., Kolokotronis, G., & Papanikolaou, A. M. (2023). Investigating the effectiveness of Flipped Classroom model in a Mathematics Education course in Greece. Journal of Mathematics and Science Teacher, 3(1), em021. https://doi.org/10.29333/mathsciteacher/12608
  • Rizos, I., Patronis, T., & Papadopoulou, A. (2021). Difficulties in basic arithmetic and geometry as related to school algebra and the current effect of ‘demathematization’. For the Learning of Mathematics, 41(1), 37-39. https://flm-journal.org/Articles/3FFD6631426EFEAA965B1E1A8EA7E5.pdf
  • Santos-Tiago, M., Camacho-Machín, M., & Olvera-Martínez, C. (2018). High school teachers’ use of a dynamic geometry system to formulate conjectures and to transit from empirical to geometric and algebraic arguments in problem-solving approaches. In N. Amado, S. Carreira, & K. Jones (Eds.), Broadening the scope of research on mathematical problem solving. Springer. https://doi.org/10.1007/978-3-319-99861-9_4
  • Segal, R., Stupel, M., & Oxman, V. (2015). Dynamic investigation of loci with surprising outcomes and their mathematical explanations. International Journal of Mathematical Education in Science and Technology, 47(3), 443-462. https://doi.org/10.1080/0020739X.2015.1075613
  • Sherman, M. (2014). The role of technology in supporting students’ mathematical thinking: Extending the metaphors of amplifier and reorganizer. Contemporary Issues in Technology and Teacher Education, 14(3), 220-246.
  • Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13, 169-181. https://doi.org/10.1080/14926156.2013.784828
  • Tran, T., Nguyen, N. G., Bui, M. D., & Phan, A. H. (2014). Discovery learning with the help of the GeoGebra dynamic software. International Journal of Learning, Teaching and Educational Research, 7(1), 44-57.
  • Vasquez, D. (2015). Enhancing student achievement using GeoGebra in a technology rich environment [Doctoral dissertation, California State Polytechnic University].
  • Velichová, D. (2021). The role of visualization in mathematics. In Proceedings of the 20th SEFI Special Interest Group in Mathematics (pp. 63-68).
  • Wares, A. (2018). Dynamic geometry as a context for exploring conjectures. International Journal of Mathematical Education in Science and Technology, 49(1), 153-159. https://doi.org/10.1080/0020739X.2017.1366559
  • Winter, E., Costello, A., O’Brien, M., & Hickey, G. (2021). Teachers’ use of technology and the impact of COVID-19. Irish Educational Studies, 40(2), 235-246. https://doi.org/10.1080/03323315.2021.1916559
  • Yates, A., Starkey, L., Egerton, B., & Flueggen, F. (2021). High school students’ experience of online learning during COVID-19: The influence of technology and pedagogy. Technology, Pedagogy and Education, 30(1), 59-73. https://doi.org/10.1080/1475939X.2020.1854337
  • Yilmaz Akkurt, Y., & Durmus, S. (2022). Tracing proof schemes: Some patterns and new perspectives. Journal of Research and Advances in Mathematics Education, 7(1), 1-16. https://doi.org/10.23917/jramathedu.v7i1.15740
  • Zhao, Y., & Liu, G. (2019). How do teachers face educational changes in artificial intelligence era. In Proceedings of the 2018 International Workshop on Education Reform and Social Sciences (pp. 47-50). Atlantis Press. https://doi.org/10.2991/erss-18.2019.9