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At the K-12 level one assumes that mathematically gifted students identified by out-of-level testing are also creative in their work. In professional mathematics, “creative” mathematicians constitute a very small subset within the field. At this level, mathematical giftedness does not necessarily imply mathematical creativity but the converse is certainly true. In the domain of mathematics, are the words creativity and giftedness synonyms? In this article, the constructs of mathematical creativity and mathematical giftedness are developed via a synthesis and analysis of the general literature on creativity and giftedness. The notions of creativity and giftedness at the K-12 and professional levels are compared and contrasted to develop principles and models that theoretically “maximize” the compatibility of these constructs. The relevance of these models is discussed with practical considerations for the classroom. The paper also significantly extends ideas presented by Usiskin (2000).

Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians' creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity .

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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