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Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real, not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions: (1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also, successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry environments can be characterised and compared with our model.

Interest in creativity in mathematics education research is increasing, and the field of research is growing. Yet, research on creativity and the notions (we use this wording to accumulate understandings, beliefs, and ideas about the construct) of creativity that are addressed in empirical research are diverse and difficult to organize in an overview, with different theoretical backgrounds and theoretical assumptions underlying them. The aim of this article is therefore to provide a systematic overview of notions of creativity addressed in recent empirical research on mathematical education. We conducted a systematic literature review, guided by the question, What notions of creativity are addressed in current mathematics education research and what theoretical foundations do they rely on? The article gives an overview of the five predominant notions of creativity that were identified in current empirical research in mathematics education from 2006 to 2019. We describe and evaluate these notions and identify trends that will help to structure this diverse field of research.

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