Making sense of negative numbers through metaphorical reasoning. (English)

Bergsten, Christer (ed.) et al., Perspectives on mathematical knowledge. Proceedings of MADIF 6, the 6th Swedish mathematics education research seminar, Stockholm, Sweden, January 29‒30, 2008. Linköping: Svensk Förening för MatematikDidaktisk Forskning (ISBN 91-973934-5-2/pbk). Skrifter vrån Svensk Förening för MatematikDidaktisk Forskning (SMDF) 6, 30-40 (2009).

Summary: The concept of negative numbers is such an abstract concept that it has been argued that it can only be understood through symbolic reasoning. However, others argue that all mathematical concepts are understood through metaphors. Previous research has identified three important aspects of understanding negative numbers: direction and multitude, proficiency in arithmetic operations, and the meaning of the minus sign. This study further explores the theory of conceptual metaphors and metaphorical reasoning by investigating the use of models and metaphorical reasoning when dealing with negative numbers. The data consists of test results from 99 students in the teacher training program and follow-up group interviews. The results show that some students’ difficulties seem to be a consequence of their use of metaphorical reasoning using a metaphor that is insufficient. However, metaphorical reasoning also seems to be helpful for students who are aware of the limitations of the metaphor. Enlightened use of metaphorical reasoning, i.e., being aware of the potentials and constraints of models and metaphors, could therefore be described as a fourth important aspect of understanding negative numbers.