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In German speaking countries, educational thinking and theorizing on mathematics teaching and learning originated with the establishment of compulsory education for all children and the creation of a school system. Though first efforts go back to the 18th century it does make sense to start this survey with the beginning of the 19th century, with the implication that educational research on mathematics has a history of about two hundred years in German speaking countries. During the 19th century a more and more sophisticated system of publication (journals and books) on mathematics education emerged, the education of mathematics teachers had become more professional and teacher training had developed into one of the main obligations of university teaching. However, didactics of mathematics as an academic discipline is a comparably new achievement. Its establishment began approximately fifty years ago, predominately by creating professorships and opportunities of graduation at universities. After a phase of broad discussion on the identity of the discipline (e.g., in a special issue of ZDM edited by Steiner, 1974), the community of didactics of mathematics steadily expanded, diversified and developed fruitful connections to other neighboring disciplines. This overview intends to outline this development with respect to intuitions, key ideas, research strategies and the connection between research and practice. Selected topics are presented in the following chapters in more detail.

In the development of didactics of mathematics as a professional field in Germany, subject-related approaches play an important role. Their goal was to develop approaches to represent mathematical concepts and knowledge in a way that corresponded to the cognitive abilities of the students without disturbing the mathematical substance. In the 1980s, views upon the nature of learning as well as objects and methods of research in mathematics education changed and the perspective was widened and opened towards new directions. This shift of view issued new challenges to subject-related considerations that are enhanced by the recent discussions about professional mathematical knowledge for teaching.

Considering mathematics education as a DESIGN SCIENCE has strong roots in Germany. E. Ch. Wittmann in particular contributed to the establishment of this approach. From a DESIGN SCIENCE perspective, substantial learning environments play a crucial role. They comprise mathematical tasks which are connected in an operative way, indicative of a specific epistemological structure. In such substantial learning environments, students are actively immersed in learning mathematics, and the learning environments allow for the pursuit of individual and differentiated learning processes. In this chapter, we first address the scope of Design Science and pay attention to characteristics of the learning environments and how teaching experiments can be conducted. We then focus on key ideas and their role as a design principle. In the next section, we provide a comprehensive example of designing a learning environment. Lastly, we shift our attention to the Design Research approach, which complements designing substantial learning environments by empirically studying the initiated learning processes to gain evidence for both theoretical considerations and design principles.

Mathematical modelling plays a prominent role in German mathematics education. The significance of modelling problems in school and in teacher education has increased over the last decades, accompanied by various research projects. In addition, there has been a vivid discussion on the implementation of modelling in schools. This chapter gives on overview on the current state of mathematical modelling in German speaking countries. After a short summary of the development of the past years, a widespread conceptualisation of modelling competence as well as its description within the German Educational standards is presented. Ways of implementing mathematical modelling in classrooms as well as in everyday lessons but also via modelling projects are described and an example of one of these problems worked on by students of grade 9 over two years is given. Furthermore, different modelling cycles are shown and their aims and usage in different circumstances are outlined. In addition, research questions currently being discussed are addressed and an example for a quantitative research project is given.

Section of this chapter is written by Hans Niels Jahnke on the basis of his presentation at ICME 13. Michael Fried was invited to react to this presentation at ICME 13 and elaborated his reaction as Sect. of this chapter. Although the authors are only responsible for their respective parts, the parts belong together and are therefore published here as a joint chapter. The first part analyzes the role of mathematics within the ideas on education of the neo-humanist movement. It refers to the period of around 1800–1850 and concentrates on the thinking of W. von Humboldt and the two catchwords of ‘anti-utilitarianism” and ‘self-directed activity’. From this general educational attitude resulted a certain preference for pure mathematics which had to be balanced against the daily needs of shopkeepers and workmen. A compromise on this issue was developed and implemented in the 1820s. Nevertheless, a strong emphasis on theoretical thinking, understanding and pure science remained for a long time the main stream of educational thinking In the eyes of the neo-humanists this was not a denial of the demands of practical life, but the best way to meet them. In his reaction entitled “Bildung, Paideia, and some undergraduate programs manifesting them,” Michael N. Fried discusses how notions similar to that of Bildung are enshrined in the idea of paideia and the classical concept of the liberal arts. He shows that such ideas also work in modern times in the English speaking world by hinting at examples of prominent colleges in North America.

The first part of this chapter has been written by Rolf Biehler on the basis of his presentation at ICME 13. Mogens Niss was invited to react to this presentation at ICME 13 and elaborated his reaction as the second part of this chapter. Although the authors are only responsible for their respective sections, they both belong together and are therefore published here as a joint chapter. The first part gives a sketch of the discussion on ‘Allgemeinbildung’ (general education for all) and mathematical literacy in Germany from the late 1960s to today. In the 1970s, educational goals for Allgemeinbildung were condensed in different visions, for example, a ‘scientifically educated human being’, a ‘reflected citizen’, an ‘emancipated individual being able to critique society’, and a person ‘well educated for the needs of the economic system’. In the early 1990s, a book by H. W. Heymann on Allgemeinbildung and mathematics education initiated a controversial discussion, which will be critically examined and related to other conceptions. Due to bad results in TIMSS (Third International Mathematics and Science Study) and PISA (Programme for International Student Assessment) starting in the late 1990s, a new discussion on educational goals in mathematics arose and made PISA’s conception of mathematical literacy popular in Germany. However, the idea of mathematical literacy was modified and extended by the German debate, some traits of which can be traced back to Humboldt and the 19th century. In his reaction “Allgemeinbildung, mathematical competencies and mathematical literacy: Conflict or compatibility?” Mogens Niss relates the German discussions to the international development on competence orientation, featuring the KOM project (Competencies and Mathematical Learning), including the various conceptualisations in the PISA frameworks.

How far has the didactics of mathematics developed as a scientific discipline? This question was discussed intensively in Germany during the 1980s, with both affirmative and critical reference to Kuhn and Masterman. In , Hans-Georg Steiner inaugurated a series of international conferences on ‘Theories of Mathematics Education’ (TME), pursuing a scientific program that aimed at founding and developing didactics of mathematics as a scientific discipline. Today, a more bottom-up meta-theoretical approach, the networking of theories, has emerged which has roots in the early days of discussing the developmental of mathematics education as a scientific discipline. This article presents an overview of this thread of development and a brief description of the TME program. Two theories from German-speaking countries are outlined and networked in the analysis of an empirical example that shows their complementary nature traced back to the TME program.

The specific aspects of Classroom Studies, as a focus within the German Speaking Traditions in Mathematics Education Research, rest on the fundamental sociological orientation on mathematics lessons. Initiated by the works of Heinrich Bauersfeld, the first sociological perspective unfolds its power of description by reconstructing social processes regarding the negotiation of meaning and the social constitution of shared knowledge through collective argumentation in the daily practice of mathematic lessons. A second sociological perspective aims at the reconstruction of the conditions and the structure surrounding the construction of performance and success in mathematics lessons.

One of the main goals of research in (mathematics) education is the generation of knowledge on processes of teaching and learning. The approaches of many research projects in German-speaking countries that contributed to achieving this goal during recent decades are diverse. Many of these projects are characterized by narrowly focusing on distinctive phenomena within learning and teaching mathematics, by taking a multi-step approach that develops theory in a series of consecutive studies (often one area of interest is pursued over many years) and by a mixed-method research strategy that integrates different methodological practices. This chapter provides exemplary insight into these kinds of research in mathematics education in German-speaking countries over the last few decades. After a brief glimpse into the beginnings, we deliver four examples that illustrate the features of these kinds of research and also describe the perspectives of researchers by way of short interview excerpts.

Large-scale studies assess mathematical competence in large samples. They often compare mathematical competence between groups of individuals within or between countries. Although large-scale research is part of empirical educational research more generally, it is also linked to more genuine mathematics education research traditions, because sophisticated methods allow for empirical verifications of theoretical models of mathematical competence, and because results from large-scale assessments have influenced mathematics education practices. This chapter provides an overview of large-scale research in mathematics education in German speaking countries over the last decades. After a brief review of historical developments of large-scale assessments in Germany, we focus on the development of competence models in Germany and Austria. At the end of this chapter, we reflect on recent developments and discuss issues of large-scale assessments more generally, including an international perspective.