Mathematical Problem Posing: Conceptual Considerations and Empirical Investigations for Understanding the Process of Problem Posing

Description

Mathematical problem posing as the substantive formulation of mathematical problems is an activity that lies at the heart of mathematics. In recent years, research in mathematics education has endeavored to gain insights into problem posing―conceptually as well as empirically. In problem-posing research, there has been a focus on analyzing products, that is, the posed problems. Insights into the processes that lead to these products, however, have so far been lacking. Within four journal articles, summarized in this cumulative dissertation, the author attempts to contribute to the understanding of problem-posing processes through conceptual considerations and empirical investigations. The conceptual part consists of a conducted systematic literature review to investigate problem-posing situations and problem-posing activities. The studies in the empirical part deal with the analyses of problem-posing processes of pre-service mathematics teachers from a macroscopic and microscopic perspective. The aim is to develop coherent and meaningful conceptual perspectives for analyzing empirical observations of problem-posing processes. Foreword Danksagung Abstract Contents I Introduction 1 Motivation 2 Objectives 3 Thesis Structure II Theoretical frame 4 Problem posing from a subject-matter didactics perspective 4.1 Illustration with a geometric example 4.2 Conceptions of problem posing – Four mathematical miniatures 4.2.1 Problem posing as generating new problems Mathematical miniature Summarizing empirical research 4.2.2 Problem posing as reformulating a given problem for problem solving Mathematical miniature Summarizing empirical research 4.2.3 Problem posing as reformulating a given problem for investigation Mathematical miniature Summarizing empirical research 4.2.4 Problem posing as constructing tasks for others Mathematical miniature Summarizing empirical research 5 Problem posing and its connection to related constructs in mathematics education 5.1 Modelling 5.2 Proving 5.2.1 Proving through problem posing 5.2.2 Problem posing through proving 5.3 Problem Solving 5.3.1 Problem posing as a subset of problem solving 5.3.2 Problem solving as a subset of problem posing 5.3.3 Problem posing and problem solving as complementary, but different in nature 5.4 Creativity and Giftedness III Journal articles 6 Rethinking problem-posing situations: A review 6.1 Introduction 6.2 Theoretical Background 6.2.1 What is problem posing? 6.2.2 What is posing? 6.2.3 What is a problem? 6.2.4 Openness of problems 6.2.5 Categorisations of problem-posing situations 6.2.6 Research questions 6.3 Design 6.3.1 Data sampling 6.3.2 Data analysis 6.4 Analysis 1: Is it posing? 6.4.1 Discussion of Analysis 1: Mathematical problem posing 6.5 Analysis 2: Is it free, semi-structured, or structured? 6.5.1 Discussion of Analysis 2: Unstructured situations 6.6 Analysis 3: Is it a routine or a non-routine problem? 6.6.1 Discussion of Analysis 3: Routine and non-routine initial problems 6.7 Conclusion 6.7.1 Implications for practice 6.7.2 Implications for research 7 Developing a framework for characterising problem-posing activities: A review 7.1 Introduction 7.2 What is problem posing? 7.2.1 Definition of problem posing 7.2.2 Non-posing activities 7.3 The proposed framework 7.3.1 Dimension 1: Generating and reformulating 7.3.2 Dimension 2: Routine and non-routine problems 7.3.3 Dimension 3: Metacognitive behaviour 7.4 Procedure of the literature review 7.5 Qualitative application of the framework 7.5.1 Dimension 1: Generating and reformulating 7.5.2 Dimension 2: Routine and non-routine problems 7.5.3 Dimension 3: Metacognitive behaviour 7.6 Quantitative application of the framework 7.7 Conclusion and discussion 8 The process of problem posing: Development of a descriptive process model of problem posing 8.1 Introduction 8.2 Theoretical Background 8.2.1 Problem Posing 8.2.2 Process of problem posing – State of research 8.2.3 Research questions 8.3 The study 8.3.1 Data collection 8.3.2 Data evaluation 8.4 Results 8.4.1 Category development of episode types in problem posing 8.4.2 Derivation of a descriptive phase model for problem posing 8.5 Discussion 9 Identifying metacognitive behavior in problem-posing processes. Development of a framework and a proof of concept 9.1 Introduction 9.2 Theoretical Background 9.2.1 Problem Posing 9.2.2 Metacognition 9.2.3 Research on Metacognition in Mathematics Education 9.2.4 Research on Metacognition in Problem Posing 9.3 Research objectives 9.4 Methods 9.4.1 Research design for data collection 9.4.2 Data analysis – Assessment of metacognitive behavior 9.5 Results 9.5.1 Development of a framework for identifying problem-posing-specific aspects of metacognitive behavior 9.5.2 Proof of concept: The cases of Tino & Ulrich and Valerie & Wenke 9.6 Discussion 9.7 Conclusion IV Discussion & Outlook 10 Summary 11 Potential for further research (1,2): (1,3): (1,4): (2,3): (2,4): (3,4): V Listings Bibliography List of Tables List of Figures VI Appendix A Online supplement of journal article 2 A.1 List of all 47 included articles included in the review A.2 Categorisation of all 47 articles within the developed framework B Online supplement of journal article 3