Number sense in a secondary school context

Current mathematics education research in New Zealand is directed at primary school or senior secondary school, in the areas of numeracy, technology innovations, student achievement and teacher professional development. This study investigated junior secondary mathematics lessons as contexts for distributed mathematical activity. Two Year 9 classes and one Year 10 class from the same school constituted case studies; two case studies were lower band and one was middle band. All three teachers continued with their normal mathematics programmes. Using an interpretivist methodology, participant observation was the main method used to collect data of sequential mathematics lessons over five weeks. The data was organised around three domains of classroom activity: classroom social norms, mathematical tools, and interaction. The data analysis focussed on aspects of these domains that afforded and constrained the negotiation of mathematical meaning. In each classroom context, social norms associated with lesson events were found to afford student participation. Student participation was afforded by interactional spaces; that is, regular opportunities for students to volunteer mathematical contributions and to adopt different roles in joint activity. Interactional spaces provided students with greater access to more diverse mathematical resources. Students activated interactional spaces in different ways with questions, suggestions, explanations, errors and, in one classroom, a discourse of primary school mathematics. Using a sociocultural perspective, school mathematical artefacts were conceptualised as cognitive mathematical tools. Cognitive tools were identified as representations, thinking devices and analogous models. Tools that were both representations and thinking devices afforded further resources for students, such as mathematical information, cues and other signifiers. The only technological tool in the case study classrooms was the scientific calculator. The roles of the calculator were as a recording device, calculating machine and exploratory object. In these roles, the calculator afforded further resources for students. The findings in the interaction domain challenge the assumption that traditional classroom mathematics programmes have limited discursive opportunities. Students initiated more focussed negotiations in both whole class and localised situations. In each case study, students and the teacher actively contributed mathematical resources. For these lower and middle ability students, access to mathematical resources was a significant affordance for further mathematical activity. In each classroom context, activity was found to be distributed over social norms, tools and interaction. Number sense activity was distributed according to the adapted 'environment' metaphor (Greeno, 1991 ). Students negotiated mathematical meanings by activating and using cognitive tools and discursive resources in the number sense 'environment'