Personal Geometrical Working Space: a Didactic and Statistical Approach
In this paper, we study answers that pre-service teachers gave in an exercise of Geometry. Our purpose is to gain a better understanding of what we call the geometrical working space (espace de travail géAoméAtrique). We first conduct a didactical study b
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Summary. In this paper, we study answers that pre-service teachers gave in an exercise of Geometry. Our purpose is to gain a better understanding of what we call the geometrical working space (espace de travail géométrique). We first conduct a didactical study based on the notion of geometrical paradigms that leads to a classification of student’s answers. Then, we use statistical tools to precise the previous analysis and explain students’ evolution during their training. Key words: Geometry, Didactic, Paradigm, Geometrical Working Space, Teachers Training.
1 Presentation of the study Various theoretical tools have been developed to study the teaching of geometry and, in the case of teacher training, two of them have been here preferred: geometrical paradigms and geometrical working spaces (GWS; in French: Espace du Travail Géométrique). Using these tools, our research focused on the following hypothesis, which our work abundantly supports: In education, the sole term geometry evokes several distinct paradigms. By and large, these paradigms reflect the breaks observed between the various academic cycles in the teaching and learning of geometry. In our view, the field of geometry can be mapped out according to three paradigms, only two of which — Geometry I and II — play a part in today’s secondary education. Each paradigm is global and coherent enough to define and structure geometry as a discipline and to set up respective working spaces suitable to solve a wide class of problems. Based on these premises, we built a training device designed to make future teachers aware of these paradigms and of their role as cause of certain misunderstandings in a classroom setting. The construction and evaluation of the device requires a precise analysis of A. Kuzniak: Personal Geometrical Working Space: a Didactic and Statistical Approach, Studies in Computational Intelligence (SCI) 127, 185–202 (2008) www.springerlink.com © Springer-Verlag Berlin Heidelberg 2008
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students’ spontaneous use of paradigms as they solve geometrical problems. This analysis is meant to understand better the geometrical working space of each student. Existing research provided the elements that lead to distinguishing among four groups of students, each corresponding to a specific approach to the study of geometry. In this paper, we wish to examine what specific contribution statistical methods can bring to that research. More precisely, we focus on the two following sets of questions: • The first set bears on the classification resulting from our didactical analysis. Does statistical analysis produce the same outcomes as the initial analysis? Which new elements, if any, emerging from implicative analysis, help better understand the various classes of students and thus predict some of the changes observed during training sessions? • The second series of questions is concerned with automating the process of sorting students according the classes defined above. Indeed, in addition to being demanding, the didactic analysis calls for adv
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