The three worlds and two sides of mathematics and a visual construction for a continuous nowhere differentiable function

Juha Oikkonen; Jani Hannula

doi:10.31129/LUMAT.10.2.1693

Authors

Juha Oikkonen Department of Mathematics and Statistics, University of Helsinki, Finland
Jani Hannula Viikki teacher training school, University of Helsinki, Finland https://orcid.org/0000-0002-6067-4686

DOI:

https://doi.org/10.31129/LUMAT.10.2.1693

Keywords:

mathematical thinking, advanced mathematics, the three worlds of mathematics, secondary-tertiary transition, nowhere differentiable functions

Abstract

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.

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