When Is a Proof a Proof?

In his famous talk at ICME 2 (Exeter 1972) the French mathematician R. Thom pointed out that any conception of mathematics teaching necessarily rests on a certain view of mathematics (Thom 1973, 204). As a consequence mathematics education cannot develop without close links to mathematics.

The Alpha and Omega of Teacher Education: Organizing Mathematical Activities

The aim of this paper is to describe an introductory mathematics course for primary student teachers and to explain the philosophy behind it. The paper is structured as follows: It starts with a general plea for placing the mathematical training of any category of students into their professional context. Then the context of primary education in Germany, with its strong emphasis on the principle of learning by discovery, is sketched.

Unfolding the Educational and Practical Resources Inherent in Mathematics for Teaching Mathematics

The objective of this introductory chapter is to explain the common rationale behind the papers of this volume. The structure is as follows. The first section shows that learning environments are a natural way to address teachers in their main role, teaching, and that therefore this approach is promising for improving mathematics teaching in an effective way. The section ends with a teaching model based on Guy Brousseau’s theory of didactical situations.

Structure-Genetic Didactical Analyses—Empirical Research “of the First Kind”

In mathematics education, theories of teaching and learning based on disciplines different from mathematics (“imported” theories) are widely dominating the field. This imbalance greatly reduces the impact of mathematics education both on teacher education and on the teaching practice. In order to return to a balanced situation it is necessary to pay more attention to theories which are based on mathematics. As an example of such a “homegrown” theory, the paper presents the structure-genetic didactical analysis, the research method of mathematics education conceived of as a “design science”.

Collective Teaching Experiments: Organizing a Systemic Cooperation Between Reflective Researchers and Reflective Teachers in Mathematics Education

The success of any substantial innovation in mathematics teaching depends crucially on the ability and readiness of teachers to make sense of this innovation and to transform it effectively and creatively to their context. This refers not only to the design and the implementation of learning environments but also to their empirical foundation. Empirical studies conducted in the usual style are not the only option for supporting the design empirically. Another option consists of uncovering the empirical information that is inherent in mathematics by means of structure-genetic didactical analyses. In this chapter, a third option is proposed as particularly suited to bridge the gap between didactical theories and practice: collective teaching experiments.

Understanding and Organizing Mathematics Education as a Design Science–Origins and New Developments

The objective of this paper is to revisit briefly the conception of mathematics education as a design science as it has been evolving alongside the developmental research in the project Mathe 2000 from 1987 to 2012 to report in some detail on recent developments, as concerns both conceptual and practical issues. The paper is a plea for appreciating and (re-)installing “well-understood mathematics” as the natural foundation for teaching and learning mathematics.

Standard Number Representations in the Teaching of Arithmetic

The paper describes the specific approach towards the choice and use of number representations as developed by the project Mathe 2000.

Developing Mathematics Education in a Systemic Process

The aim of this paper is to make a concrete proposal for bridging the gap between theory and practice in mathematics education and for establishing a systemic relationship between researchers and teachers as well as to explain the background and the implications of this proposal.

Teaching Units as the Integrating Core of Mathematics Education

How to integrate mathematics, psychology, pedagogy and practical teaching within the didactics of mathematics in order to get unified specific theories and conceptions of mathematics teaching? This problem—relevant for theoretical and empirical studies in mathematics education as well as for teacher training—is considered in the present paper. The author suggests an approach which is based on teaching units (Unterrichtsbeispiele). Suitable teaching units incorporate mathematical, pedagogical, psychological and practical aspects in a natural way and therefore they are a unique tool for integration. It is the aim of the present paper to describe an approach to bridging the often deplored gap between didactics of mathematics teaching on one hand and teaching practice, mathematics, psychology, and pedagogy on the other hand. In doing so I relate the various aspects of mathematics education to one another. My interest is equally directed to teacher training and to the methodology of research in mathematics education. The structure of the paper is as follows. First I would like to make reference to and characterize an earlier discussion on the status and role of mathematics education; secondly, I will talk about problems of integration which naturally arise when mathematics education is viewed as an interdisciplinary field of study. The fourth and essential section will show how to tackle these problems by means of teaching units. The present approach is based on a certain conception of mathematics teaching which is necessary for appreciating Sect. 4. This conception is therefore explained in Sect. 3.

The Mathematical Training of Teachers from the Point of View of Education

The paper describes an approach to integrating the mathematical and educational components in teacher training which is based on elaborating educational and psychological aspects inherent in “good mathematics”.