Examining mathematical discourse to understand in-service teachers’ mathematical activities

Margot Berger

doi:10.4102/pythagoras.v34i1.197

Abstract

In this article I use Sfard’s theory of commognition to examine the surprising activities of a pair of in-service mathematics teachers in South Africa as they engaged in a particular mathematical task which allowed for, but did not prescribe, the use of GeoGebra. The (pre-calculus) task required students to examine a function at an undefined point and to decide whether a vertical asymptote is associated with this point or not. Using the different characteristics of mathematical discourse, I argue that the words that students use really matter and show how a change in one participant’s use of the term ‘vertical asymptote’ constituted and reflected her learning. I also show how the other participant used imitation in a ritualised routine to get through the task. Furthermore I demonstrate how digital immigrants may resist the use of technology as the generator of legitimate mathematical objects.

Highlights

In this article I examine the activities of a pair of in-service mathematics teachers as they engage in a particular mathematical task which allows for, but does not prescribe, the use of GeoGebra
I use this theory to show how the use of computers as a tool in mathematical learning may require an explicit rewriting of the rules of what counts as mathematical activity
I have used the analytic constructs to explain the reluctance of students to rely on computer-generated mathematical objects as visual mediators

Summary

Introduction

In this article I examine the activities of a pair of in-service mathematics teachers (the students) as they engage in a particular mathematical task which allows for, but does not prescribe, the use of GeoGebra. The students (i.e. the teachers) engage in the task in a surprising way: their use of GeoGebra is very limited and they use various mathematical terms such as ‘asymptote’ and ‘undefined’ very loosely. Cognition is an intrapersonal expression whereas communication is an interpersonal expression of a phenomenon The theory draws both on Vygotsky and Wittgenstein and assumes a view of mathematics and of mathematics learning which resonates strongly with my own experiences of teaching and learning mathematics. How the teachers deal with the affordances and limitations of graphing software, in this case, GeoGebra This has important implications for their use of technology as a teaching tool. It was intended as a tool of amplification rather than of cognitive reorganisation (Pea, 1993)

Background

11. E: asymptote

Conclusion