Teaching and learning fractions has traditionally been one of the most problematic areas in primary school mathematics. Several studies have suggested that one of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (i.e., part-whole, ratio, operator, quotient, and measure). Kieren was the first to establish that the concept of fractions is not a single construct, but consists of several interrelated subconstructs. Later on, in the early 1980s, Behr et al. built on Kieren's conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence, and problem solving. In the present study we used this theoretical model as a reference point to investigate students' constructions of the different subconstructs of fractions. In particular, using structural equation modeling techniques to analyze data of 646 fifth and sixth graders' performance on fractions, we examined the associations among the different subconstructs of fractions as well as the extent to which these subconstructs explain students' performance on fraction operations and fraction equivalence. To a great extent, the data provided support to the associations included in the model, although, they also suggested some additional associations between the notions of the model. We discuss these findings taking into consideration the context in which the study was conducted and we provide implications for the teaching of fractions and suggestions for further research.

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This article focuses on the construction, description and testing of a theoretical model of problem posing. We operationalize procesess that are frequently described in problem solving and problem posing literature in order to generate a model. We name these processes editing quantitative information, their meanings or relationships, selecting quantitative information, comprehending and organizing quantitative information by giving it meaning or creating relations between provided information, and translating quantitative information from one form to another. The validity and the applicability of the model is empirically tested using five problem-posing tests with 143 6 th grade students in Cyprus. The analysis shows that three different categories of students can be identified. Category 1 students are able to respond only to the comprehension tasks. Category 2 students are able to respond to both the comprehension and translation tasks, while Category 3 students are able to respond to all types of tasks. The results of the study also show that students are more successful in first posing problems that involve comprehending processes, then translation processes and finally editing and selecting processes. Kurzreferat:. Gegenstand des Artikels ist die Konstruktion, Beschreibung und das Testen eines theoretischen Modells für das Problemstellen. Die eigentlich hinlänglich bekannten Prozesse, die in der Literatur über Problemlösen und Problemstellen beschrieben werden, sind Ausgangspunkt für eine Operationalisierung. Die Autoren unterscheiden die folgenden Prozesse: Editieren quantitativer Informationen, das Zuweisen von Bedeutungen oder Beziehungen, das (bewusste) Auswählen von quantitativen Informationen, das Verstehen und Organisieren quantitativer Informationen (durch inhaltliche Zuordnung von Bedeutung oder Kontextherstellung) und das Übersetzen von Informationen in andere Kontexte. Die Validität und die Brauchbarkeit des Modells werden anhand von fünf Tests des Problemstellens bei 143 Schülern (Klasse 6) in Zypern getestet. Die Analyse zeigt, dass drei unterschiedliche Kategorien von Schülern identifiziert werden können. Bei Gruppe 1 handelt es sich um Schüler, die lediglich auf die Verstehensaufgabe reagieren, während sich Gruppe 2 aus Schülern zusammensetzt, die sowohl den Kontext erfassen als auch eine Übersetzung vornehmen. Schüler aus Gruppe 3 reagieren auf alle Typen der Aufgabe. Die Ergebnisse der Studie belegen überdies, dass Schüler bei erstmaligem Problemstellen erfolgreicher mit Kontexten umgehen, bei denen es um Verstehensprozesse geht, als dass sie Übersetzungsprozesse oder schließlich Auswahlprozesse umsetzen können. ZDM-Classifikation: C30, D50

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The recent development of powerful new technologies such as dynamic geometry software (DGS) with drag capability has made possible the continuous variation of geometric configurations and allows one to quickly and easily investigate whether particular conjectures are true or not. Because of the inductive nature of the DGS, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical concern. In this article we discuss the implications of the development of this new software for the teaching of proof and making proof meaningful to students. We describe how three prospective primary school teachers explored problems in geometry and how their constructions and conjectures led them "see" proofs in DGS.