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Modelling cycle from Blum & Leiβ (2007). According to Borromeo-Ferri's (2006) cognitive point of view, the situation model is a mental representation of reality.
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Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof...
Contexts in source publication
Context 1
... For example, the book explains that cartographer Edward Wright first explained Mercator's cartographic projection, providing an elegant Euclidean proof of the geometry involved. It is a typical modelling approach between, on the one hand, a cartographic representation system (the situation model, in the meaning of Blum, & Leiß, 2007; see also Fig. 3 for details) and its mathematization in geometry on the other, the proof remaining attached to a discursive genesis activity within the mathematical model. But a diametrically opposed approach is also evoked in the collective book, which we readily assimilate to a cycle of «antimodelling» that starts from the mathematical model to be ...
Context 2
... search for valid solutions, the design factors associated with the problem, the plausible creative processes in some kind of iterative problematic and an innovative line of reasoning behind the constraints, factors and the design choices made. This activity can be seen as a problematic-modelling dynamic between reality and the mathematical world (Fig. 3) which allows, under certain conditions, to get some results and enrich the understanding of reality (Fig. 5). According to the authors: «this problem and most of its variants, are not yet completely solved as far as mathematical research is concerned» (p. 61), which presupposes from the outset that it will be necessary to be inventive. ...
Context 3
... 61), which presupposes from the outset that it will be necessary to be inventive. The algorithmic approach would first be used for programming a mathematical work by gradually approaching a reference situation through successive problem solving or judiciously considered cases. Figure 5. When we attack a complex real problem, the modelling cycle (Fig. 3) is characterized by some back and forth processes that we wish to converge (infundibuliform path), at least until we obtain a stable problematic. At this tipping point, the cycle can continue for results (cycle tightening), or the real situation can be rethought by considering new constraints (cycle widening). The situation model is ...
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Citations
The theory of Mathematical Working Spaces (MWS) is introduced in this chapter. Presenting epistemological and cognitive aspectsCognitivecognitive aspects, we see how the theory of MWS aims to provide tools—theoretical and methodological—for the specific study of mathematical work in which students and teachers effectively engage during mathematics lessons. Some of the main key constructs of the theory are introduced: the notion of mathematical work in relation to Mathematical Working Spaces; the semiotic, instrumentalInstrument and discursive geneses associated with MWS diagramsMathematical Working Space, MWSMWS diagram; the different levels of MWS associated with reference, suitable and personal work, etc. We then demonstrate how these different tools enable the description, characterization and formation of mathematical work. Finally, emphasis is placed on the originality of this theory in the field of mathematics education theories.
When considering the mathematics competences referential (Niss & Højgaard, 2019), whether from Québec Education Program, the Standards of the National Council of Teachers of Mathematics or some large international studies such as the Programme for International StudentStudent Assessment, reasoning, and mathematical proofs are always at the forefront. Although the notion of proof is limited, in elementary school, to mathematical reasoning and conviction, and in post-secondary education, to demonstration and written communication, the evolution of the treatment reserved for deductive reasoning in secondary school has the appearance, depending on the region, of an eternally shilly-shallying between valorization and marginalization.
