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Proportional reasoning is a key topic both at school and in everyday life. However, students are often misled by their preconceptions regarding proportions. Our hypothesis is that these limitations can be mitigated by working on alternative ways of categorizing situations that enable more adequate inferences. Multiple categorization triggers flexibility, which enables reinterpreting a problem statement and adopting a more relevant point of view. The present study aims to show the improvements in proportional reasoning after an intervention focusing on such a multiple categorization. Twenty-eight 4th and 5th grade classes participated in the study during one school year. Schools were classified by the SES of their neighborhood. The experimental group received 12 math lessons focusing on flexibly envisioning a situation involving proportional reasoning from different points of view. At the end of the school year, compared to a control group, the experimental group had better results on the posttest when solving proportion word problems and proposed more diverse solving strategies. The analyses also show that the performance gap linked to the school's SES classification was reduced. This offers promising perspectives regarding multiple categorization as a path to overtake preconceptions and develop cognitive flexibility at school.

We propose to assess conceptual knowledge of mathematical notions by having recourse to isomorphic word problems. We assumed that failing to solve isomorphic problems is an indicator of lack of conceptual knowledge. To reach these conclusions, two experiments were conducted among 4th and 5th grade students. In experiment 1, each student had to solve one isomorphic word problem allowing two solving strategies. Results from experiment 1 showed that semantic context promoted a one-sided perspective, constraining the strategies used. The second experiment questioned if the strategy choices by students reflected a lack of procedural knowledge or difficulty in conceptual knowledge. Students solved several problems with the explicit task of proposing two strategies. Results showed that students struggled to identify the conceptual structure of the problems: they changed their strategy depending on semantic contexts, and did not succeed in adopting two encodings for the same problem. This lack of flexibility does not reflect weak procedural knowledge but difficulties in conceptual knowledge. Once conceptual knowledge is characterized as the ability to get over dependence on semantic context, using word problems as assessment tools for conceptual development opens up educational perspectives.

In problem solving situation, cognitive flexibility appears tobe a major skill. Fostering cognitive flexibility is therefore aspecific stake in mathematics education. This researchintroduces a learning method to develop mathematicalconcepts when solving word arithmetic problems. The studywas conducted with 8 classes (4 th -5 th Grades) from high-priority education schools in the Paris area following thisprotocol: pre-tests, 5 learning sessions for experimental andcontrol groups, post-tests. During learning sessions, studentsstudied arithmetic word problems that can be solved in twodifferent ways: an expansion strategy and a factorization one.The experimental teaching method, based on arecategorization principle, allowed experimental students toimprove more than the control students in ability to use thefactorization strategy even in contexts where it is the lessintuitive and to consider the two successful strategies.Educational entailments of our finding are discussed.

Introduction: Many studies in physical education (PE) have sought to identify and categorize the modes of student interaction in order to gain greater insight into the nature of cooperative activity. More others recent studies have examined how modes of interaction evolve on the basis of the modes of collective activity that they generate. These studies have shown to describe and explain the interactions among individuals and the processes they generate, which then lead to the construction, deconstruction or reconstruction of different interaction modes. Although some studies have sought to describe the dynamics of student interactions, very few have quantified these dynamics. By doing so, however, researchers might gain a new perspective on student interaction modes that inspires new designs for teaching in PE, thereby having professional impact. The present study extends this research by investigating the dynamics of student interaction, with a focus on the emergence of interaction modes during orienteering lessons. For this purpose, the study was conducted within the methodological framework of course of action theory, which is an effective approach for examining activity in natural situations to provide insight into the experience of activity from the actors perspective. Method: The study was conducted in two classes of seventh-grade students (about 12 years old) in which 16 students volunteered and were available for post-action interviews immediately after the lessons under study. These volunteers were placed in eight affinity-based dyads. The teachers planned orienteering lessons at similar levels of difficulty and duration but modified the lessons across a range of contextual features. Two categories of data were collected: (1) data from audiovisual recordings as the students searched for the checkpoints and (2) verbalization data during the post-action interviews with the students. The data were processed in two steps: one qualitative, the other quantitative. The qualitative step consisted of processing the data of the student experiences to characterize their interactions in the three different contexts. In the quantitative step, the data from the first step were graphically represented to depict the interaction dynamics within the student dyads. Results and discussion: The qualitative analysis showed the emergence of three modes of student interaction shared across each learning context: co-construction, confrontation and delegation. The quantitative analysis revealed the percentages of the different modes of interaction and therefore characterized the interaction dynamics. Our results showed that the interaction dynamics within the dyads were both unique and similar in the task contexts in terms of both ratios of change and distribution. Results are discussed across two major points of interest: (1) the observation of the same interaction modes whatever the context yet with quite different dynamics and (2) proposals for PE teacher interventions.

Introduction: One of the aims of physical education (PE) is to develop social skills such as cooperation, teamwork, and mutual helping among students. Cooperation is a broad research topic, implicating several disciplines in the human sciences (e.g. psychology, sociology, linguistics, philosophy). It is also an important topic in various domains of practice like organizational management, ergonomics, sports performance, and PE and sports pedagogy. Studies in sport have shown that cooperation between partners is not automatically produced by the cooperative structure of the learning tasks. In this exploratory study, we focused on the links between cooperation and trust judgments about one's partner. We characterized the processes by which students construct trust judgments in dyadic cooperative interactions during climbing lessons. This study was carried out within the theoretical and methodological frameworks of the 'course of action.' Method: Two middle-school students in 10th-grade participated in this study. They formed a fixed dyad based on personal affinity. During the four climbing lessons under study, the students alternated the roles of climber and belayer. Their behaviors and communications were recorded on video and after each session they participated in self-confrontation interviews. From these data, we then reconstructed their courses of experience and focused particularly on preoccupations, meaningful elements in the situation, and mobilized knowledge. The elements contributing to the construction of the climber's trust judgments about his belayer were identified. Changes in these judgments were characterized in relation to the modes of cooperation between the students. Twenty-seven climbs were analyzed. Results and discussion: The results showed that the climber's trust judgments focused on two dimensions of the belayer's activity: the reliability of both the belay and his advice for success in the task. To build these trust judgments, the climber combined knowledge about his belayer (the partner's climbing skills, his mastery of safety techniques, his familiarity with the climbing equipment, and his typical attitudes in class) and interpretations of the events during belayer--climber cooperation (the attention shown by the partner, his requests for advice from other students, and the teacher's interventions). Moreover, the results showed the relationship between the development of each student's trust judgments about his partner and the positive and negative dynamics of cooperation between the students. Three typical connections were identified between the climber's trust judgments about his partner and the climber's involvement in cooperation. The results are discussed on the basis of two points: (a) the dynamic and composite character of the construction of trust judgments and (b) the construction of trust judgments as a condition for genuine cooperation between students. The students in the situation of climber displayed typical processes to build a trust judgment about their partner, and this trust judgment appears to be an important element in promoting and regulating cooperative interactions between students. Conclusions: We propose pedagogical perspectives for teachers with regard to understanding trust judgments in climbing and student cooperation. We also suggest new research perspectives with the objective of fully elucidating the dimensions of trust involved in cooperation.

After the conquests of Alexander the Great (336-323 BCE) and the imposition of a Greco-Macedonian domination over the major part of the territories that spread to today’s Pakistan, populations of Greek origin came to install themselves in the new cities created by Alexander and his successors, the Seleucid kings. The objective of this article is to show how the tools elaborated in the framework of studies on diasporas can help us to understand better the communities that these populations formed in Iran and in central Asia, the most eastern regions of Greek space and the least well documented because available sources are rare and very dispersed. A recent article by Stéphane Dufoix that explores the notion of diaspora and defines its possible significations furnishes the bases of a new discussion.