Fractions are well known to be difficult to learn. Various hypotheses have been proposed in order to explain those difficulties: fractions can denote different concepts; their understanding requires a conceptual reorganization with regard to natural numbers; and using fractions involves the articulation of conceptual knowledge with complex manipulation of procedures. In order to encompass the major aspects of knowledge about fractions, we propose to distinguish between conceptual and procedural knowledge. We designed a test aimed at assessing the main components of fraction knowledge. The test was carried out by fourth-, fifth- and sixth-graders from the French Community of Belgium. The results showed large differences between categories. Pupils seemed to master the part-whole concept, whereas numbers and operations posed problems. Moreover, pupils seemed to apply procedures they do not fully understand. Our results offer further directions to explain why fractions are amongst the most difficult mathematical topics in primary education. This study offers a number of recommendations on how to teach fractions.

ABSTRACT-Fractions constitute a stumbling block in mathematics education. To improve children's understanding of fractions, we designed an intervention based on learningby-doing activities, which focused on the representation of the magnitude of fractions. Participants were 292 Grade 4 and 5 children. Half of the classes received experimental instruction, while the other half pursued their usual lessons. For 10 weeks, they played five different games using cards representing fractions (e.g., Memory and Blackjack). Wooden disks helped them represent and manipulate fractions while playing games. Our results showed an improvement in the conceptual understanding of fractions. The findings confirmed that the usual practice in teaching fractions is largely based on procedural knowledge and provides only minimal opportunities for children to conceptualize the meaning and magnitude of fractional notations. Furthermore, our results demonstrate that a short intervention inducing children to manipulate, compare, and evaluate fractions improves their ability to associate fractional notations with numerical magnitude.

Recherche et perspectives Développement, validation et implantation d'un outil novateur critérié d'évaluation de la progression des compétences des résidents en médecine familiale Development, validation and implementation of an innovative criterion-based tool to assess family medicine residents' progress

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