Denis Tanguay

Researcher-Designed Resources and Their Adaptation Within Classroom Teaching Practice: Shaping Both the Implicit and the Explicit

Little is known about the ways in which teachers take on research-based resources and adapt them to their own needs. This study focused on the manner in which three teachers who were participating in a research project on the learning of algebra with Computer Algebra System (CAS) technology spontaneously adapted the resources that were designed specifically by the researchers for use in the project. Analysis of the classroom-based observations of teaching practice showed that shaping occurred with respect to all three key features (the mathematics, the students, and the technology) of our researcher-designed resources, whether our intentions with respect to those features were explicitly stated or implicitly suggested. The study also found that embarking with the same set of task-sequences, and sharing the same goal of participating in a research project aimed at developing the technical and theoretical knowledge of algebra students within a CAS-supported environment, can lead to quite different uses of the resources. The teachers brought into the study their own beliefs, knowledge, and customary ways of interacting with their students. The results highlight the differential role that the same resources can play vis-a-vis the dialectical processes of ‘documentational genesis’ whereby resources are viewed as both shaping and being shaped by teaching practice.

Analyse des problemes de geometrie et apprentissage de la preuve au secondaire

RésuméLe présent article rend compte de l’élaboration d’une grille d’analyse des problèmes de géométrie, et de sa mise à l’épreuve par la classification des problèmes et exercices de géométrie synthétique dans une collection de manuels du secondaire parmi les plus utilisées au Québec. Le cadre conceptuel sur lequel s’appuie cette élaboration s’inspire principalement des travaux de Balacheff (1987), Barbin (1988), Brousseau (1998), Hanna (1995) et Rouche (1989), et débouche sur une typologie des preuves de géométrie. La classification des problèmes à partir de cette grille et l’analyse qui en découle nous a permis de conclure sur les aspects de l’apprentissage de la preuve que nous évaluons comme mal « gérés » dans la collection: transition non suffisamment graduelle du sensible au formel (peu de problèmes qui sollicitent une validation hybride, niveau de formalisation trop longtemps stationnaire), prépondérance des applications directes et des déductions locales sur les séquences déductives, intérêt et mode de présentation des résultats qui ne favorisent pas une « attitude de preuve ».Executive SummaryIn this research project, I have attempted to understand how the notion of proof develops during the secondary school student’s learning process. From this perspective, I have first examined official texts in order to identify not only the objectives of the Québec Ministry of Education (MEQ) yearly curriculum (1993–1996) dealing with the learning of proof but also how such curricula propose to accomplishing these objectives. According to these curricula, the learning of proof occurs primarily through the study of geometry, in a more general context ofproblem solving. The next step was to understand, on a second level, how this learning has been transferred into textbooks. In that connection, it was necessary to be able to account for learning progression, in terms of continuity-discontinuity. As the objectives of the curriculum emphasize problem solving, this phase of my analysis required a close analysis of geometry problems. With the objective of developing an analytical grid to be applied to synthetic geometry problems in relation to the type of proof they require, I have attempted to synthesize the reflections on proof contained in the work of Balacheff (1987), Barbin (1988), Brousseau (1998), Hanna (1995), and Rouche (1989), on the basis of what I call ‘schemas of bipolarization.’ Using the schémas suggested by these authors, I then built both a typology of proof and, on the basis of this typology, an analytical grid proper. Both the typology and the grid were developed from the perspective of sources of validation that students are capable of drawing on. The grid broke down all of the problems and exercises of synthetic geometry into seven categories: the direct application (nothing to be validated); the spontaneously seeing the general in one particular case and empirical induction (source of validation: the ‘tangible’ or the ‘perceptible’); mental experience and the empirico-deductive argument (dual sources of validation: reasoned argumentation based on the perceptible); and, finally, the local deduction and the deductive chain or linkage (source of validation: logico-deductive reasoning). I tested out this grid by classifying problems and exercises of synthetic geometric contained in a collection of textbooks among those most widely used in Québec for all the secondary school grades (Secondary I to V, students ages 12 to 17). Based on an analysis of the results, I developed a series of conclusions about aspects of the learning of proof that I view as being poorly handled in these textbooks, including an insufficiently gradual transition from the perceptible to the formal (very few problems that draw on a hybrid validation; over-long stationary formalization; a break during Secondary V); the predominance of direct applications and local deductions over deductive sequences; and a focus and mode of presentation of results that do not foster a ‘proof-oriented attitude.’ This analysis also gave me an opportunity to examine various possible interpretations of what the MEQ curriculum has termed îlot déductif (or ‘local axiomatic’) and the role that has been planned for the geometry of transformations.

Working on Proofs as Contributing to Conceptualization—The Case of IR Completeness

In this chapter, we propose a mathematical and epistemological study about two classical constructions of the real number system, by Dedekind (cuts ) and Cantor (Cauchy sequences ), and the associated proofs of its completeness . In addition, we present two contrasting constructions leaning on decimal expansions. Our analysis points out that Dedekind’s construction fosters a conceptualization of the real numbers leaning strongly on the total ordering of ℚ and ℝ, while putting aside the metrical aspects. By contrast, the more intricate construction through Cauchy sequences calls on complex objects, but yields to a better understanding of the topological relationship between rational and real numbers. We argue that suitable considerations of decimal expansions and of approximation issues enable to connect and complement those two approaches. These analyses highlight the dialectical interplay between syntax and semantics and the crucial role of the definitions of objects at play in proof and proving . The general didactical issue pertains to the potential contribution of analyzing proofs as a means for deepening the understanding of the related objects and of their ensuing conceptualization. We hypothesize that doing so with Dedekind’s cuts, Cauchy sequences and decimal expansions open paths towards improving the conceptualization of the real numbers, by taking into account the triad discreteness/density /continuity .