The three worlds and two sides of mathematics and a visual construction for a continuous nowhere differentiable function
Keywords:mathematical thinking, advanced mathematics, the three worlds of mathematics, secondary-tertiary transition, nowhere differentiable functions
A rigorous and axiomatic-deductive approach is emphasized in teaching mathematics at university-level. Therefore, the secondary-tertiary transition includes a major change in mathematical thinking. One viewpoint to examine such elements of mathematical thinking is David Tall’s framework of the three worlds of mathematics. Tall’s framework describes the aspects and the development of mathematical thinking from early childhood to university-level mathematics. In this theoretical article, we further elaborate Tall’s framework. First, we present a division between the subjective-social and objective sides of mathematics. Then, we combine Tall’s distinction to ours and present a framework of six dimensions of mathematics. We demonstrate this framework by discussion on the definition of continuity and by presenting a visual construction of a nowhere differentiable function and analyzing the way in which this construction is communicated visually. In this connection, we discuss the importance to distinguish the subjective-social from the objective side of mathematics. We argue that the framework presented in this paper can be useful in developing mathematics teaching at all levels and can be applied in educational research to analyze mathematical communication in authentic situations.
Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In J. Kaput, A. Schoenfeld & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education (Vol. 6, pp. 1–32). Washington D.C.: American Mathematical Society and Mathematical Association of America.
Bingolbali, E., & Monaghan, J. (2008). Concept image revisited. Educational studies in Mathematics, 68(1), 19–35.
Bruner, J. S. (1967). Toward a theory of instruction. Cambridge, Mass.: Belknap Press of Harvard University Press.
Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.
Chin, K. E. (2013). Making sense of mathematics: Supportive and problematic conceptions with special reference to trigonometry. (Doctoral dissertation). The University of Warwick.
Clark, M. & Lovric, M. (2009). Understanding secondary–tertiary transition in mathematics. International Journal of Mathematical Education in Science and Technology, 40(6), 755–776.
Di Martino, P., & Gregorio, F. (2019). The mathematical crisis in secondary–tertiary transition. International Journal of Science and Mathematics Education, 17(4), 825–843.
Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89–114.
Dubinsky, E., & McDonald, M. A. (2002). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 275–282). Dordrecht: Kluwer.
Education Committee of the EMS. (2013). Why is university mathematics difficult for students? Solid findings about the secondary-tertiary transition. Newsletter of the European Mathematical Society, 90, 46–48.
Fan, L., & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: A conceptual model with focus on cognitive development. ZDM Mathematics Education, 46(3), 481–492.
Fischbein, E. (1994). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Strässer & B. Winkelmann (Eds.). Didactics of mathematics as a scientific discipline (pp. 231–245). Dordrecht: Kluwer.
Goos, M., & Kaya, S. (2020). Understanding and promoting students’ mathematical thinking: a review of research published in ESM. Educational Studies in Mathematics, 103(1), 7–25.
Gray, E., & Tall, D. (1991). Duality, ambiguity and flexibility in successful mathematical thinking. In F. Furinghetti (Ed.), Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (PME), 2 (pp. 72–79). Assisi, Italy: PME.
Haapasalo, L. (2003). The conflict between conceptual and procedural knowledge: Should we need to understand in order to be able to do, or vice versa. In L. Haapasalo & K. Sormunen (Eds.), Proceedings on the IXX Symposium of the Finnish Mathematics and Science Education Research Association, 86 (pp. 1–20). University of Joensuu: Bulletins of the Faculty of Education.
Hannula, J. (2018). The gap between school mathematics and university mathematics: prospective mathematics teachers’ conceptions and mathematical thinking. Nordic studies in mathematics education 23(1), 67–90.
Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, N. J.: Erlbaum.
Hodges, W. (1985). Building Models by Games. Cambridge: London Mathematical Society Student Texts.
Hähkiöniemi, M. (2006). The role of representations in learning the derivative. (Doctoral dissertation). University of Jyväskylä.
Joutsenlahti, J. (2005). Lukiolaisen tehtäväorientoituneen matemaattisen ajattelun piirteitä: 1990-luvun pitkän matematiikan opiskelijoiden matemaattisen osaamisen ja uskomusten ilmentämänä. [Characteristics of task-oriented Mathematical thinking among students in upper-secondary school]. (Doctoral dissertation). Acta Universitatis Tamperensis 1061, University of Tampere.
Oikkonen, J. (2004). Mathematics between its two faces. In L. Jalonen, T. Keranto and K. Kaila (Eds.). Matemaattisten aineiden opettajan taitotieto – haaste vai mahdollisuus (pp. 23–30). University of Oulu, Finland.
Oikkonen, J. (2008) Good results in teaching beginning math students in Helsinki, ICMI bulletin. 62, p. 74–80.
Oikkonen, J. (2009). Ideas and results in teaching beginning maths students. International Journal of Mathematical Education in Science and Technology, 40:1,127–138.
Resnick, L. B. (1995). Inventing arithmetic: Making children's intuition work in school. In C. A. Nelson (Ed.), Basic and applied perspectives on learning, cognition, and development (pp. 75–101). Lawrence Erlbaum Associates, Inc.
Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22, 1–36.
Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20–26.
Sternberg, R. J. (1996). What is mathematical thinking? In R. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 303–318). Mahwah: Lawrence Erlbaum Associates.
Tall, D. (1982). The blancmange function, continuous everywhere but differentiable nowhere. Mathematical Gazette, 66, 11–22.
Tall, D. (1991). Advanced mathematical thinking. Dordrecht: Kluwer Academic Publishers.
Tall, D. (2004). Thinking through three worlds of mathematics. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 281–288). Bergen University College.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 2008, 20(2), 5–24.
Tall, D. (2013). How humans learn to think mathematically - Exploring the three worlds of mathematics. Cambridge University Press.
Tall, D. & Di Giacomo, S. (2000) Cosa vediamo nei disegni geometrici? (il caso della funzione blancmange), Progetto Alice 1(2), 321–336). [English version: What do we "see" in geometric pictures? (the case of the blancmange function)].
Tall, D. and Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.
Thim, J. (2003). Continuous nowhere differentiable functions. (Master’s thesis). Luleå University of Technology.
Viholainen, A. (2008). Finnish mathematics teacher student’s informal and formal arguing skills in the case of derivative. Nordic Studies in Mathematics Education, 13(2), 71–92.
Weierstrass, K. (1872). Uber continuirliche Functionen eines reellen Arguments, die fur keinen Werth des letzteren einen bestimmten Differentailqutienten besitzen, Akademievortrag. Math. Werke, 71–74.
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