Visualising number : a study of children's developing sense of number in the computational medium of Boxer
The aim of this study is to investigate children's developing sense of number in the computational medium of Boxer. Boxer's combination of graphical and symbolic elements afforded the opportunity for children to visualise numbers in an operational way while simultaneously offering insights into how this operational approach mediated their thinking. There were three inter-related aspects to the study, with visualisation being the common feature of all three. • (i) How does the visual structure of Boxer influence students' (aged 9 - 11) ability to program? • (ii) What interpretations do students place on a number system extended beyond the natural numbers and how did they choose to represent these? • (iii) How can the learning environment of Boxer be exploited as a context for developing students' sense of number? Following an exploratory study, pedagogical models for investigating issues (i) and (iii) were developed. For issue (i), Boxer was exploited as a means of introducing itself. This, in turn, meant documenting the issues involved in a process of iterative design. For issue (iii), a model of learning was developed which proposed that the children should construct their own microworlds. Following an off-computer investigation of issue (ii), the model was refined to that of children constructing operational computational objects, and the research aim broadened to include an investigation of how these objects mediated their expression of number. This part of the research consisted of a longitudinal study lasting two years. It entailed case studies with four pairs of children, while the rest of the class learned Boxer independently. None of the children had previous Boxer or Logo experience. The research setting was a normal classroom in an inner-London primary school. Data for issues (i) and (iii) was obtained by means of video recordings and annotated print-outs, while data for issue (ii) was obtained by written records and audio recordings. Evidence from the research suggests that students' programming is significantly more structured in Boxer relative to Logo, and this structure is directly related to the visual nature of Boxer. Moreover, data from the number studies suggests that this visual structure was also instrumental in providing students with the means to connect number processes with number concepts, thus enabling them to engage with number ideas which might otherwise have been beyond their reach.