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The learning of mathematics starts early but remains far from any theoretical considerations: pupils’ mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using...
This paper uses the framework of “three worlds of mathematics” (Tall 2004a, b) to chart the development of mathematical thinking from the thought processes of early childhood to the formal structures of set-theoretic definition and formal proof. It sees the development of mathematical thinking building on experiences that the individual has met before, as the child coordinates perceptions and actions...
This paper illustrates two examples of contexts for approaching validation, framed by a research program on semiotic mediation in the mathematics classroom. The question of how to approach theoretical reasoning, a major problem in mathematics education, is especially difficult when physical, truly tangible, artefacts are in play. Whenever a student is given a mathematical task to solve by using some...
Recently, as part of a larger research project, we carried out a long-term teaching experiment in the ninth and tenth grades of a scientific high school aimed at getting students to approach issues of validation and of teaching of mathematical proof. Assuming a Vygotskian perspective, we focused on the social construction of knowledge and on semiotic mediation as accomplished by the teacher through...
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