Studying teachers' use of metaphors in the context of directednumbers

Abstract

People use metaphors to describe or understand one thing in terms of another. The central idea of this thesis is that metaphors can be used to teach mathematics, particularly abstract topics such as directed numbers. Using directed numbers as a context, this study develops a framework and a coding scheme that can be used as a tool for analysing the use of metaphors in the teaching of mathematics.

The part of the theoretical framework of the coding scheme is based on the work of Lakoff and Johnson (1980) and Lakoff and Nunez (2000). In those studies, the authors classify metaphors used for teaching mathematics into one of three categories: ontological, orientational and structural metaphors. By considering the source domain of metaphors, they can be classified into either grounding or linking metaphors. Similarly, the target domain of the metaphors can be categorized by the intended learning outcomes and by the functions of the metaphors.

One of the primary contributions of this thesis is the development of a coding scheme that is specifically designed to analyze the use of metaphors in mathematics lessons. The scheme was then used and validated through the analysis of mathematics lessons taught by two teachers with contrasting academic backgrounds and teaching experiences. Three lessons taught by each teacher on the topic of directed numbers at Secondary One level were recorded and analysed. The metaphors used by each teacher were identified, coded and analyzed in order to determine how metaphors can be extended and transformed into other metaphors.. Finally, this thesis compared how the two teachers differed in their use of metaphors, particularly in terms of the selection, sequencing and organization of the metaphors used. This can be indicative the level of conceptual learning that is made available for students in their classes.

The research questions: 1. What kinds of metaphors did the teachers use to introduce and explain the concepts and computational processes of directed numbers? 2. What functions did these metaphors serve? 3. What is the developmental path of these metaphors within and across the lessons? 4. What were the differences in the selecting, sequencing, and organization of the metaphor used by the two teachers?

Findings This thesis designed and tested an original coding scheme. The findings revealed that the two teachers had used many kinds of metaphors in their lessons. They were used for classifying different kinds of numbers, constructing concepts, and explaining the properties and computational processes of directed numbers. Most of the metaphors found in this study were used to provide a cognitive function that facilitates the introduction of new mathematical concepts and helps the students make sense of the operational processes; only a few metaphors served a memorable function.

When comparing the use of metaphors by the two teachers, we can analyze their teaching philosophies. Teacher 1’s use of metaphors demonstrated a linear development path from a simple to a more advance perspective, whereas Teacher 2’s use of metaphors revealed more comprehensive, sophisticated and multi-layered perspective.

Significance of the study This study provides insights into the meaning and implication of using metaphors in teaching mathematical concepts. At research level, this study extends the existing work of Lakoff and develops an analytical tool specifically designed to understand the pedagogical values of using metaphors to teach abstract mathematical concepts such as directed numbers. At pedagogical level, the metaphor coding scheme can act as an initial foray into how metaphors can be used in and for teaching. Moreover, the Metaphor-Concept Development Chart developed in this study is a practical tool that can help teachers to analyze and improve their own use metaphors, thereby furthering their professional development and teaching effectivenss.