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This paper intends to describe a phenomenon observed when children aged 6 and 7 years old, who had not yet received formal instruction about division, tried to solve a problem of distribution by means of drawings. The phenomenon called "frames of meaning" attempts to explain why some children were successful in solving problems and others were not. We illustrate this hypothesis empirically through a case study. This hypothesis is a powerful theoretical and methodological tool for understanding the underlying mathematical thinking and arithmetic problem-solving abilities of children this age. However, it is necessary to continue to expand our analysis and the application of this framework. This would lead to reconsider conventional educational and evaluative practices. In this article we try to point the way by which these changes could be made. [ABSTRACT FROM AUTHOR]

The goal of this study is to characterize the processes involved in mathematical proof, in the context of geometry, from a cognitive perspective. In particular, the focus of this study is the characterization of the interaction between reasoning processes and verification procedures that secondary students use when solving geometry problems in a pencil-and-paper environment. The results show that the different ways of using alternative verification procedures to validate a proposition are related to the different outcomes of the reasoning in problems that require a proof. [ABSTRACT FROM AUTHOR]

This paper reports the presence or absence of a systemic thinking in students, when they solve the linear extension problem, which consists of determining a linear transformation through the images of the vectors of a basis. The problem is formulated geometrically, making use of the tools of the Cabri-géomètre II software. The difficulties that students present when face this problem, probably are due to that they do not carry out the adequate connections among the concepts involved. This phenomenon can be studied from the point of view of the theoretical approximation the theoretical thinking versus the practical thinking (Sierpinska, 2000). One of the characteristics of the theoretical thinking is that tries to be focused in the establishment and study of the relations between the concepts and its characterization inside a system that also contains other concepts (Sierpinska, et al. 2002). [ABSTRACT FROM AUTHOR]

In this paper, we show that the hidden curriculum approach is relevant for math teaching at different levels of the French education system, from senior high school to tertiary education to first year of math secondary teacher training. Considering that the existence of learning necessities non explicitly cared about by any institution is a phenomenon that is impossible to avoid, we propose several ways to manage with the difficulties that creates this curriculum ignored by the education system. [ABSTRACT FROM AUTHOR]

Causing a cognitive conflict can be one way to make students aware of the inadequacy of the concepts that they possess and the methods that they use in solving a problem. Apart from discussions in a cooperative group and the use of mathematical games, the mathematical situations themselves have to possess certain characteristics to be able to provoke a conflict and furthermore facilitate the confrontation between the previous knowledge and new knowledge to be acquired. Moreover teachers need to have direct experiences at their level with this kind of activities, before they can employ them in their classes. In this paper we report the results of a study in which a cryptography activity was applied with a group of university and high school teachers, where the concepts of sets and binary operations came into play. [ABSTRACT FROM AUTHOR]