An important part of the whole : The role of metaphors in the teaching and learning of mathematics

This study aims to investigate the role of metaphor in the teaching and learning of mathematics. Two different studies help to fulfil this aim. The first study involves a configurative literature review of empirical mathematics education research where two interpretations of the metaphor concept are applied: conceptual metaphor and extraneous metaphor, with the purpose of distinguishing patterns in metaphor concept use. The analysis found explicit, vague, or absent theoretical definitions of the metaphor concept as well as several metaphor examples, and generated three categories of concept use: 1) Conceptual metaphor, 2) Extraneous metaphor, and 3) Potentially conceptual/extraneous. The review stresses the need for a theoretical and empirical distinction between different types of metaphors, to sharpen the analytical tools available to researchers and, by extension, to more clearly highlight the specific function of metaphor in teachers’ classroom work. The second study examines the presence of three conceptual metaphors as the basis for rational numbers as fractions in Swedish textbooks for years 1–3 and which aspects of fractions are distinguished in the student responses required in these books. The results are related to process-object theory (Sfard, 1991), intending to discern the different degrees of reification of the mathematical objects that hypothetically have the opportunity to develop with the help of these metaphors. The results show, in three of the book series, an abundance of labelling exercises related to the metaphor Arithmetic is object construction, mainly with images of geometric shapes. One book series reversely introduces fractions with The measuring stick metaphor and Arithmetic is motion along a path, with representations of number lines in focus. All in all, there is a significant variation between the textbooks, which could have consequences for instruction. This thesis shows the importance of theoretically distinguishing conceptual metaphor from extraneous metaphor, and of understanding their different roles in a teaching and learning context. The thesis also shows how the conceptual metaphors underlying the mathematical concepts describe different aspects of these concepts and that several conceptual metaphors need to be taken into account, together with a consideration of different degrees of reification of the mathematical objects. In addition, the thesis shows that metaphors, as they are understood in Conceptual metaphor theory, point to similar conceptual and discursive processes that can be found in Sfard’s (1991, 2008) more established theories on mathematical conceptualisation