Complementary to existingnormativemodels, in this paper we suggest adescriptivephase 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, adescriptivemodel seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such adescriptivemodel 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? Ourdescriptivemodel 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.

The aim of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. For this purpose, 36 task-based interviews with pre-service primary and secondary mathematics teachers working in pairs who were given two structured problem-posing situations were conducted. Through an inductive-deductive category development, five types of activities (situation analysis, variation, generation, problem-solving, evaluation) were identified. These activities were coded in so-called episodes, allowing time-covering analyses of the observed processes. Recurring transitions between these episodes were observed, through which a descriptive phase model was derived. In addition, coding of the developed episode types was validated for its interrater agreement.

In research on mathematical problem posing, a broad spectrum of different situations is used to induce the activity of posing problems. This review aims at characterizing these so-called problem-posing situations by conducting three consecutive analyses: (1) By analyzing the openness of potential problem-posing situations, the concept of “mathematical posing” is concretized. (2) The problem-posing situations are assigned to the categories free, semi-structured, and structured by Stoyanova and Ellerton to illustrate the distribution of situations used in research. (3) Finally, the initial problems of the structured problem-posing situations are analyzed with regard to whether they are routine or non-routine problems. These analyses are conducted on 271 potential problem-posing situations from 241 systematically gathered articles on problem posing. The purpose of this review is to provide a framework for the identification of differences between problem-posing situations.

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.

This article aims to develop a framework for the characterisation of problem-posing activities. The framework links three theoretical constructs from research on problem posing, problem solving, and psychology: (1) problem posing as an activity of generating new or reformulating given problems, (2) emerging tasks on the spectrum between routine and non-routine problems, and (3) metacognitive behaviour in problem-posing processes. These dimensions are first conceptualised theoretically. Afterward, the application of these conceptualised dimensions is demonstrated qualitatively using empirical studies on problem posing. Finally, the framework is applied to characterise problem-posing activities within systematically gathered articles from high-ranked journals on mathematics education to identify focal points and under-represented activities in research on problem posing.