Prospective Teachers’ Interactive Visualization and Affect in Mathematical Problem-Solving

Abstract

Abstract: Research on technology-assisted teaching and learning has identified several families of factors that contribute to the effective integration of such tools. Focusing on one such family, affective factors, this article reports on a qualitative study of 30 prospective secondary school mathematics teachers designed to acquire insight into the affect associated with the visualization of geometric loci using GeoGebra. Affect as a representational system was the approach adopted to gain insight into how the use of dynamic geometry applications impacted students' affective pathways. The data suggests that affect is related to motivation through goals and self-concept. Basic instrumental knowledge and the application of modeling to generate interactive images, along with the use of analogical visualization, played a role in local affect and prospective teachers' use of visualization.Keywords: problem-solving strategies, visual thinking, interactive learning, drawing, diagrams, teacher training, visual representations, reasoning, GeoGebra.1. Experimental conditions and research questions addressedAt present, the predominant lines of research on problem-solving aim to identify underlying assumptions and critical issues, and raise questions about the acquisition of problem-solving strategies, metacognition, and beliefs and dispositions associated with problem-solvers' affect and development (Schoenfeld, 1992; Lester and Kehle, 2003). Problem-solving expertise is assumed to evolve multi-dimensionally (mathematically, metacognitively, affectively) and involve the holistic co-development of content, problem-solving strategies, higher-order thinking and affect, all to varying degrees (English & Sriraman, 2010). This expertise must, however, be set in a specific context. Future research should therefore address the question of how prospective teachers' expertise can be holistically developed.The research described here was conducted with a group of 30 Spanish mathematics undergraduates. These future teachers took courses in advanced mathematics in differential and Riemannian geometry, but worked very little with the classical geometry they would later be teaching. They were accustomed to solving mathematical problems with specific software, mainly in areas such as symbolic calculation or dynamic geometry, but were not necessarily prepared to use these tools as future teachers. Research on teaching in technological contexts (Tapan, 2006) has shown that students are un- or ill-acquainted with mathematics teaching, i.e., they are unaware of how to convey mathematical notions in classroom environments and find it difficult to use software in learning situations. Hence the need to specifically include the classroom use of software in teacher training.This paper addresses certain understudied areas in problem-solving such as visualization and affect, with a view to developing discipline awareness and integrating crucial elements for mathematics education in teacher training. As defined by Mason (1998), teachers' professional development is regarded here as development of attention and awareness. The teacher's role is to create conditions in which students' attention shifts to events and facts of which they were previously unaware. Viewed in those terms, teaching itself can be seen as a path for personal development.The main aim of this essay is to explain that in a dynamic geometry environment, visualization is related to the viewer's affective state. The construction and use of imagery of any kind in mathematical problem-solving constitute a research challenge because of the difficulty of identifying these processes in the individual. The visual imagery used in mathematics is often personal in nature, related not only to conceptual knowledge and belief systems, but laden with affect (Goldin, 2000; Gomez-Chacon, 2000b; Presmeg, 1997). These very personal aspects are what may enhance or constrain mathematical problem-solving (Aspinwall, Shaw, and Presmeg, 1997; Presmeg, 1997), however, and as such should be analyzed, especially in technological contexts. …