The impact of teachers' knowledge on the connection between technology supported exploration and mathematical proof

Helena Rocha 1 *
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1 CICS.NOVA, Faculdade de Ciências e Tecnologia - Universidade NOVA de Lisboa, PORTUGAL
* Corresponding Author
EUR J SCI MATH ED, Volume 11, Issue 4, pp. 635-649.
Published Online: 15 May 2023, Published: 01 October 2023
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Technology is recognized for its potential to implement exploration tasks. The ease and speed with which it becomes possible to observe many cases of a situation, allows the development of conjectures and brings conviction about their veracity. Mathematical proof, assumed as the essence of Mathematics, tends to appear to the students as something dispensable. Based on KTMT – Knowledge for Teaching Mathematics with Technology model, this study intends to understand the impact of the teachers’ knowledge on mathematical proof in a context of technology integration. The study adopts a qualitative and interpretative methodology, based on case study, analyzing the practice of one teacher. The conclusions emphasize the relevance of the teacher’s MTK – Mathematics and Technology Knowledge, and TLTK – Teaching and Learning and Technology Knowledge. The teacher's MTK guides her decisions, leading her to focus on helping students understand the meaning of conjecture and proof, valuing, at the same time, the relevance of algebraic manipulations. However, the teacher’s TLTK guides her practice, where the knowledge about the students is determinant. The study provides evidence about the difficulty of articulating proof and technology, but it also clarifies the relevance of this articulation and of how the teacher’s KTMT can impact the teacher’s decisions.


Rocha, H. (2023). The impact of teachers' knowledge on the connection between technology supported exploration and mathematical proof. European Journal of Science and Mathematics Education, 11(4), 635-649.


  • Aristidou, M. (2020). Is mathematical logic really necessary in teaching mathematical proofs? Athens Journal of Education, 7, 99-122.
  • Biza, I., Nardi, E., & Zachariades, T. (2010). Teachers’ views on the role of visualization and didactical intentions regarding proof. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Cerme 6 - Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 261-270). INRP.
  • Blanton, M., & Stylianou, D. (2014). Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs. Journal of Mathematical Behavior, 34, 76-98.
  • Bleiler-Baxter, S., & Pair, J. (2017). Engaging students in roles of proof. Journal of Mathematical Behavior, 47, 16-34.
  • Cabassut, R., Conner, A., Ísçimen, F., Furinghetti, F., Jahnke, H., & Morselli, F. (2012). Concepts of proof – in research and teaching. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 169-190). Springer.
  • Davis, P., & Hersh, R. (1983). A experiência matemática [The math experience]. Gradiva.
  • Dawkins, P., & Weber, K. (2017). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics, 95, 123-142.
  • De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
  • De Villiers, M. (1999). Rethinking proof with Sketchpad. Key Curriculum Press.
  • De Villiers, M. (2012). An illustration of the explanatory and discovery functions of proof. Pythagoras, 33(3), 1-8. v33i3.193
  • De Villiers, M. (2020). Proof as a means of discovery. International Journal of Mathematical Education in Science and Technology, 51(3), 451-455.
  • Goos, M., & Bennison, A. (2008). Surveying the technology landscape: teachers’ use of technology in secondary mathematics classrooms. Mathematics Education Research Journal, 20(3), 102-130.
  • Hanna, G. (2001). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
  • Hanna, G. (2014). The width of a proof. PNA, 9(1), 29-39.
  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31, 396-428.
  • Hsieh, F., Horng, W., & Shy, H. (2012). From exploration to proof production. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 279-304). Springer.
  • Knuth, E. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.
  • Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. Journal of Mathematical Behavior, 29, 1-10.
  • Mejía-Ramos, J. (2005). Aspects of proof in mathematics research. In D. Hewitt (Ed.), Proceedings of the British Society for Research into Learning Mathematics (pp. 61-66). Open University.
  • Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108, 1017-1054.
  • Miyakawa, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94, 223-239.
  • Reid, D. (2011). Understanding proof and transforming teaching. In Proceedings of 33rd annual meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (pp. 15-30). PME-NA.
  • Rocha, H. (2015). O formalismo matemático num contexto de utilização da tecnologia [Mathematical formalism in a context of technology use]. In Atas do XXVI Seminário de Investigação em Educação Matemática [Proceedings of the XXVI Seminar on Research in Mathematics Education] (pp. 22-35). APM.
  • Rocha, H. (2019). Mathematical proof: From mathematics to school mathematics. Philosophical Transactions of the Royal Society A, 377(2140), 1-12.
  • Rocha, H. (2020a). Graphical representation of functions using technology: a window to teacher knowledge. Teaching Mathematics and its Applications, 39(2), 105-126.
  • Rocha, H. (2020b). Using tasks to develop pre-service teachers’ knowledge for teaching mathematics with digital technology. ZDM, 52(7), 1381-1396.
  • Schoenfeld, A. (2009). The soul of mathematics. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: a K-16 perspective (pp. xii–xvi). Taylor & Francis Group.
  • Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
  • Smith, J. (2006). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. Journal of Mathematical Behavior, 25, 73-90.
  • Steele, M., & Rogers, K. (2012). Relationships between mathematical knowledge for teaching and teaching practice: The case of proof. Journal of Mathematics Teacher Education, 15(2), 159-180.
  • Stylianides, A., & Ball, D. (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, 307-332.
  • Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315-351). Sense Publishers.
  • Stylianides, G., Stylianides, A., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237-266). National Council of Teachers of Mathematics.
  • Tabach, M., & Trgalová, J. (2019). The knowledge and skills that mathematics teachers need for ICT integration: The issue of standards. In G. Aldon & J. Trgalová (Eds.), Technology in mathematics teaching (pp. 183-203). Springer.
  • Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng,Y. (2012). Cognitive development of proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 13-49). Springer.
  • Yin, R. K. (2017). Case study research and applications: Design and methods. Sage.