Variations on a theme: introducing new representations of fraction into two classrooms

Abstract

Two teachers in different schools participated in a research project looking at the use of technology based representations of mathematical objects. Each used the same software, incorporating a novel representation of fraction as a dynamic functional relationship between values on two number lines. They planned together, discussing the characteristics of the software, the educational goals and modes of use as well as sharing resources and ideas about student tasks. In practice, the lessons each taught were very different and the ways in which students made use of the software also differed substantially. Influences on the nature of teachers’ incorporation of new elements into their pedagogic practice are discussed, including consideration of explicit and implicit theoretical frameworks and of institutional and cultural contexts. CONTEXT OF THE STUDY The study reported here was carried out as part of the work of the European Research Team TELMA (Technology Enhanced Learning in Mathematics), part of the Kaleidoscope Network of Excellence 1 . TELMA has brought together six research teams from four countries, working in the area of digital technologies and mathematics education. After initial discussion, sharing and synthesising of the work of the teams, a small-scale methodological experiment was planned and carried out with the aim of making theoretical frameworks and the role of institutional and cultural contexts in forms of use of technology more explicit. In this so-called “crossexperimentation”, each team designed and conducted an experiment using a technological tool designed by a team from another country to support learning in the area of fractions. The methodology used for this collaborative work is discussed in Artigue et al. (2007). Some of the outcomes have been reported in the project deliverables 2 and will be published elsewhere in the future. Analysis across national contexts has allowed insights into systemic differences. However, in this paper I will focus only on the conduct and outcomes of the experiment carried out by the London team, based at the Institute of Education. E-SLATE FRACTION-SLIDERS MICROWORLD The tool used was a microworld constructed using E-Slate (http://e-slate.cti.gr/), a toolkit for building exploratory environments devised by the Athens team (NKUAETL). The Fraction-sliders microworld offers two main types of representation of fraction: “sliders” or dynamic number lines and symbols in the form of Logo. These allow a fraction to be represented (a) as a relationship between values shown by positions on two linked number lines (sliders) and (b) as a number entered into a Logo procedure. The numbers entered in the Logo procedure determine the