Roles of Metaphor in the Growth of Mathematical Understanding

Abstract

The purpose of this qualitative study was to investigate roles of metaphor in the growth of mathematical understanding in the area of proportionality. To this end, the task of designing software that would help other people learn about proportions was offered to six children ages 13 to 16 during individual interviews.

The process of software design helped to access metaphors students developed for thinking about proportionality. A conceptual framework for the study was based on enactivist perspective. Data analysis was based on the study of sources and targets of metaphors, as represented in learners' actions. The Pirie-Kieren Model for the Growth of Mathematical Understanding, in combination with a model for proportional reasoning development that was created for the study, were used to map learners' knowing actions.

Microworlds created by learners supported metaphoric systems, which in turn helped to coordinate the process of knowledge development in proportionality domain. The model used for mapping this process included the notions of equivalence class, relation and invariance. Findings indicate that each of these notions may be developed, during local and context-specific growth of understanding, through actions in additive, multiplicative and qualitative analogy worlds.

Metaphors serve as a tool for coordination of collecting in these worlds. Moreover, metaphor is a way new understanding grows out of old knowing. In metaphoric structures, sources fade away, and formalized targets become independent entities. This extended process of metaphorising is supported by open-ended, extended tasks.