The aim of this study is to describe and analyse the structure of 3D geometry thinking by identifying different types of reasoning and to examine their relation with spatial ability. To achieve this goal, two tests were administered to students in grades 5 to 9. The results of the study showed that 3D geometry thinking could be described by four distinct types of reasoning which refer to the representation of 3D objects, spatial structuring, conceptualisation of mathematical properties and measurement. The analysis of the study also showed that 3D geometry types of reasoning and spatial abilities should be modelled as different constructs. Finally, it was concluded that students' spatial abilities, which consist of spatial visualisation, spatial orientation and spatial relations factors, are a strong predictive factor of the four types of reasoning in 3D geometry thinking.

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

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.

A theoretical model describing young students’ (Grades 1–3) functional-thinking modes was formulated and validated empirically (n = 345), hypothesizing that young students’ functional-thinking modes consist of recursive patterning, covariational thinking, correspondence-particular, and correspondence-general factors. Data analysis suggested that functional-thinking tasks can be categorized on the basis of the proposed model. Analysis traced three categories of students that represent different functional-thinking profiles. Category 1 students exhibited a recursive-thinking profile. Category 2 students utilized a combination of recursive and contextual strategies and exhibited an emergent covariational and correspondence-particular thinking. Category 3 students approached functional-thinking situations flexibly, using a combination of covariational and correspondence strategies. A structural model showed two parallel paths from recursive patterning to correspondence-general through correspondence-particular or covariational.

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