The Definition of Mathematics: Philosophical and Pedagogical Aspects

Abstract

There is a strange fact that many works written with the purpose to explain what is mathematics, somehow avoid the issue. This paper is aimed at filling this gap. After discussing various descriptions of mathematics as they appear in literature, it is suggested that mathematics is an essentially linguistic activity characterized by association of words with precise meanings. Educational implications of this idea are considered in the light of (1) a strong tendency of most humans to the fuzzy way of thought as described by the dual-process theory developed by researchers of human reasoning; (2) the fact that the main mathematics-related skill needed in our computerized, informationsaturated society is a proper mind-set rather than knowledge of some facts or techniques.The proposed definition of mathematics can enrich debates on its nature, in particular, connecting natural human psychological characteristics with philosophical questions of the origin of mathematics.

Factors influencing the transition to university service mathematics: part 2 a qualitative study

“…This definition could have many educational implications for society in general, i.e. the general public's perception of mathematics education, as it is believed that mathematics education cannot be built around some specific subject matter if it does not want to lose its relevance (Khait, 2005). In that paper, there is a call for change from such a topic-oriented mathematics education.…”

Designing a curriculum based on four purposes: let mathematics speak for itself

“…This is one of the reasons why some refer to mathematics as an essentially linguistic activity and many call mathematics a language. While one may argue that too much emphasis on learning mathematical abstraction undermines practising wider skills contributing to employability, we find it informative to refer to an argument in Khait (2005). Teaching mathematics as an essentially linguistic activity operating with abstract objects assigned precise meanings, as opposed to teaching algorithmic procedures, helps create a mind-set essential for living in the computerised and industrialised society of today (Khait, 2005).…”

“…that provide the basis that for defining aspects of what mathematics should be taught and how it should be taught (see, e.g. Khait, 2005;Voskoglou, 2018).…”

“…However, it seems to us that such a statement allows different interpretations. While for a mathematical "outsider" it may seem that learning and applying mathematics in a context as opposed to learning abstract mathematics is better linked to its interdisciplinary and cross-curriculum aspects, mathematical "insiders" may agree with Timothy Gowers who emphasised that it is precisely the abstract nature of mathematics that makes mathematics a very powerful tool of cooperation (for pedagogical implications of defining and interpreting mathematics by mathematical 'insiders' and 'outsiders', see, for example Khait, 2005). We therefor start with discussing what abstraction means in mathematics.…”

Investigating the changes in teachers' pedagogical practices through the use of the Mathematics Reasoning Heuristic (MRH) approach

“…Mathematics, by its nature, is perceived differently by different communities. For example, for mathematicians, mathematics is a process of creation and exploration which they are constantly studying to expand their repertoire of mathematical knowledge by engaging in mathematical activities (i.e., proof, refutation, counterargument) (Khait, 2005); whereas, for ordinary people or students, or even for some mathematics teachers, mathematics is "a collection of preestablished facts, rules, and techniques essentially having to do with numbers and (at best) geometric shapes" (Borasi, 1992, p. 158). When math teachers see school mathematics as nothing but a set of rules, their belief shapes their "instructional decisions about curriculum choices, teaching strategies, and classroom organization and management" (p. 157).…”