This ICME-13 Topical Survey reviews the state-of-the-art by first exploring the roots and scope of design science. Second, it presents two examples of current design science projects that focus on substantial learning environments including a student and a teacher perspective. Subsequently, the book elaborates on how empirical research can be conceptualised within design science. Lastly, it explores developments in design science from a national and international perspective, while also discussing current trends in design research. Within the German-language tradition, considering 'mathematics education as a design science' primarily draws on the works of Wittmann. The core of this approach constitutes designing and investigating learning environments that involve substantial mathematics. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.

This ICME-13 Topical Survey reviews the state-of-the-art by first exploring the roots and scope of design science. Second, it presents two examples of current design science projects that focus on substantial learning environments including a student and a teacher perspective. Subsequently, the book elaborates on how empirical research can be conceptualised within design science. Lastly, it explores developments in design science from a national and international perspective, while also discussing current trends in design research. Within the German-language tradition, considering ‘mathematics education as a design science’ primarily draws on the works of Wittmann. The core of this approach constitutes designing and investigating learning environments that involve substantial mathematics.

This ICME-13 Topical Survey reviews the state-of-the-art by first exploring the roots and scope of design science. Second, it presents two examples of current design science projects that focus on substantial learning environments including a student and a teacher perspective. Subsequently, the book elaborates on how empirical research can be conceptualised within design science. Lastly, it explores developments in design science from a national and international perspective, while also discussing current trends in design research. Within the German-language tradition, considering 'mathematics education as a design science' primarily draws on the works of Wittmann. The core of this approach constitutes designing and investigating learning environments that involve substantial mathematics.

This open access book shares revealing insights into the development of mathematics education research in Germany from 1976 (ICME 3 in Karlsruhe) to 2016 (ICME 13 in Hamburg). How did mathematics education research evolve in the course of these four decades? Which ideas and people were most influential, and how did German research interact with the international community? These questions are answered by scholars from a range of fields and in ten thematic sections: (1) a short survey of the development of educational research on mathematics in German speaking countries (2) subject-matter didactics, (3) design science and design research, (4) modelling, (5) mathematics and Bildung 1810 to 1850, (6) Allgemeinbildung, Mathematical Literacy, and Competence Orientation (7) theory traditions, (8) classroom studies, (9) educational research and (10) large-scale studies. During the time span presented here, profound changes took place in German-speaking mathematics education research. Besides the traditional fields of activity like subject-matter didactics or design science, completely new areas also emerged, which are characterized by various empirical approaches and a closer connection to psychology, sociology, epistemology and general education research. Each chapter presents a respective area of mathematics education in Germany and analyzes its relevance for the development of the research community, not only with regard to research findings and methods but also in terms of interaction with the educational system. One of the central aspects in all chapters concerns the constant efforts to find common ground between mathematics and education. In addition, readers can benefit from this analysis by comparing the development shown here with the mathematical education research situation in their own country.

No one disputes how important it is, in today's world, to prepare students to un derstand mathematics as well as to use and communicate mathematics in their future lives. That task is very difficult, however. Refocusing curricula on funda mental concepts, producing new teaching materials, and designing teaching units based on 'mathematicians' common sense' (or on logic) have not resulted in a better understanding of mathematics by more students. The failure of such efforts has raised questions suggesting that what was missing at the outset of these proposals, designs, and productions was a more profound knowledge of the phenomena of learning and teaching mathematics in socially established and culturally, politically, and economically justified institutions - namely, schools. Such knowledge cannot be built by mere juxtaposition of theories in disci plines such as psychology, sociology, and mathematics. Psychological theories focus on the individual learner. Theories of sociology of education look at the general laws of curriculum development, the specifics of pedagogic discourse as opposed to scientific discourse in general, the different possible pedagogic rela tions between the teacher and the taught, and other general problems in the inter face between education and society. Mathematics, aside from its theoretical contents, can be looked at from historical and epistemological points of view, clarifying the genetic development of its concepts, methods, and theories. This view can shed some light on the meaning of mathematical concepts and on the difficulties students have in teaching approaches that disregard the genetic development of these concepts.

This open access book features a selection of articles written by Erich Ch. Wittmann between 1984 to 2019, which shows how the “design science conception” has been continuously developed over a number of decades. The articles not only describe this conception in general terms, but also demonstrate various substantial learning environments that serve as typical examples. In terms of teacher education, the book provides clear information on how to combine (well-understood) mathematics and methods courses to benefit of teachers. The role of mathematics in mathematics education is often explicitly and implicitly reduced to the delivery of subject matter that then has to be selected and made palpable for students using methods imported from psychology, sociology, educational research and related disciplines. While these fields have made significant contributions to mathematics education in recent decades, it cannot be ignored that mathematics itself, if well understood, provides essential knowledge for teaching mathematics beyond the pure delivery of subject matter. For this purpose, mathematics has to be conceived of as an organism that is deeply rooted in elementary operations of the human mind, which can be seamlessly developed to higher and higher levels so that the full richness of problems of various degrees of difficulty, and different means of representation, problem-solving strategies, and forms of proof can be used in ways that are appropriate for the respective level. This view of mathematics is essential for designing learning environments and curricula, for conducting empirical studies on truly mathematical processes and also for implementing the findings of mathematics education in teacher education, where it is crucial to take systemic constraints into account.

This open access book discusses several didactic traditions in mathematics education in countries across Europe, including France, the Netherlands, Italy, Germany, the Czech and Slovakian Republics, and the Scandinavian states. It shows that while they all share common features both in the practice of learning and teaching at school and in research and development, they each have special features due to specific historical and cultural developments. The book also presents interesting historical facts about these didactic traditions, the theories and examples developed in these countries.