Book Review: Connecting Mathematics and Mathematics Education: Collected Papers on Mathematics Education as a Design Science

Connecting Mathematics and Mathematics Education (CMME) by Erich Christian Wittmann is a timely resource on the attempts to combine mathematics and its pedagogical realm. Wittmann proposes his “design science approach” (DSA) concept for this connecting purpose. As stated by Wittmann himself, DSA is “born from the intention to assist teachers in these tasks, that is, to provide them with first-hand knowledge for organizing learning processes in the form of elaborated teaching units (later called substantial learning environments)” (p. 4–5). Wittman then draws on Brousseau’s (1997) “didactical situations” in providing a teaching model based on his DSA concept. Compared to another model by Lesh and Sriraman that utilized a phylogenetic approach to study the growth of ideas (Lesh and Sriraman, 2010). The book is a collection of Wittman’s published papers from 1982 to 2019 in various reputable peer-reviewed journals and publications, such as Educational Studies in Mathematics and proceedings of a mini-conference at the International Conference of Mathematics Education (ICME) held in Adelaide in 1984.

The monograph consists of 14 chapters beginning with an introductory chapter that provides the rationale behind the writing of this book. This chapter has several sections. The first section shows how challenging the teacher is with their tasks. The toughest one is how to get students actively involved by their intrinsic means of motivation. In this section, Wittmann explores the best way to develop mathematics teaching using the learning environment approach. The author ends this section with a design of a teaching model that was inspired by Guy Brousseau’s theory of didactical situations. In this book (p. 5), Wittmann showed five “didactical situations” (Instruction, Action, Formulation, Validation, and Institualization) by Brousseau’s theory in the form of a table between teacher and students. For instance, for Instruction, the teacher is “explaining the objectives and the problem(s), providing students with material” while students are “paying attention, listening, asking for further explanations, and ‘joining in.’” The second section provides a concrete example focusing on the teaching materials for the beginning of grade 5. Starting with an objective, mathematical background, teaching materials, and following several steps make this sketch a valuable guide for teachers. The following section explains the beauty of mathematics as the natural and valuable source for designing learning environments. How mathematical courses and design science approaches are needed for teachers and the consequences are explained in the last section. The highlight is the idea to accelerate progress in mathematics teaching at all levels.

The second to the eleventh chapters in this volume are Wittmann’s articles published from 1982 to 2002 in several reputable journals. Since Wittman wanted to focus on elaborating the concept of the design science approach down to the most applicable theories, he took a “silent” period of working on the approach in the form of a textbook for more than a decade. From the text, Wittmann received feedback from teachers and believed that his design was adequate. To better understand the concept, the author began publishing other articles from 2014 to 2019 as a follow-up process in coordinating other ideas to the main one.

Wittman divided all the papers in this book into four categories: First, the methodological framework that is demonstrated in “Teaching Units as the Integrating Core of Mathematics Education'' (p. 25), “Mathematics Education as a Design Science'' (p. 77), and “Understanding and Organizing Mathematics Education as a Design Science'' (p. 265). At the same time, the elaboration of mathematics education as a systemic evolutionary design science is included in “Developing Mathematics Education in Systemic Process” (p. 191) and “Collective Teaching Experiments: Organizing a Systemic Cooperation Between Reflective Researchers and Reflective Teachers in Mathematics Education” (p. 239). Second, “Structure-genetic didactical analyses—empirical research ‘of the first kind’” (p. 249) and the paper “Designing Teaching: The Pythagorean Theorem” (p. 95) explain the primary method for designing environments. Third, the articles that deal with teacher education were “Clinical Interviews ‘Embedded in the Philosophy of Teaching Units’” (p. 37), “The Mathematical Training of Teachers from the Point of View of Education” (p. 49), and “The Alpha and Omega of Teacher Education: Stimulating Mathematical Activities” (p. 209). Fourth, the essential factor for designing a learning environment, non-symbolic means of representation, was clearly illustrated in the other papers. However, in order to facilitate readers to gain a complete understanding of the process and the ideas, the chapters were sequentially arranged by the year.

Chapter 2 tells the readers about problems of integration when mathematics education (ME) is seen as an interdisciplinary field of study. In the next chapter, Wittmann depicts interaction as well as the gap between theory and practice in ME. First, Wittmann examines John Dewey’s perspective on the link between theory and practice. Second, within the “concept of teaching units,” Wittmann demonstrates how clinical interviews may be utilized to build teachers' attitudes and skills. The following chapter outlines a method for combining mathematical and educational components in teacher education focused on developing educational and psychological features inherent in “good mathematics.” This approach leads to the idea of informal, problem- and process-oriented elementary mathematics presentations. The chapter finishes with a sketch of a “mathematics research program for primary students” in ME.

In the fifth chapter, Wittmann contributes to the deconstruction of formalism in ME. Wittmann proposes two ways for combating the formalistic viewpoints. One of them entails citing works by prominent mathematicians that provide an accurate description of their work. Another approach is to develop informal mathematics as a separate mathematical reasoning. Wittmann depicts ME as a “Design Science” to retain the particular position and relative autonomy of ME in the sixth chapter. Wittmann’s thoughts in this chapter are meant to be a critical assessment of the current situation and an attempt to convey the uniqueness of ME. Wittmann portrays the views as a form of “thinking aloud about our profession,” in which they are presented in complete subjectivity and a succinct manner. While the focus of the study in this chapter is on mathematics didactics, the reasoning may be applied to other courses and education in general. In this chapter, Wittmann also discusses a unique approach to empirical research, namely, empirical research centered around teaching units. In the next part, Wittmann takes a fundamental activity founded on an integrated understanding of mathematics and pedagogy in order to be successful. The Pythagorean theorem, a well-known issue in geometry, is used to illustrate this integrated approach to student teachers. The focus of the article is less on the Pythagorean theorem itself but more on basic ideas of a teacher’s “design kit” that may be applied to a variety of topics.

Chapter eight by Wittmann defines and demonstrates a specific aspect of the project based on a conception of ME as a design science, namely the development of a “grammar of non-symbolic representations.” The topic chosen is arithmetic, one of the most fundamental topics of mathematics teaching. The next paper aims to present a natural solution for bridging the gap between theory and practice in ME, and to explain the context and consequences of this idea. Chapter 10 presents and explains the concept behind an introductory mathematics course for primary student teachers. Wittmann argues that mathematical training of any category must be placed into their professional context. In this chapter, the context of primary education in Germany is sketched. Here Wittmann also introduces the “O-script/A-script method,” a unique teaching/learning format for motivating student teachers' mathematical activities based on the principle of discovery learning. The concept of proof in primary teacher education is given significant consideration. Concerning professional context, the idea of proof in the context of primary teacher education is given important consideration. The author explains the conceptual and practical approach to “operative proofs” more specifically in the eleventh chapter. Chapter eleven is about operative proofs developed in the Mathe 2000 project. This chapter also discusses some learning with operative proofs, operative proofs concepts, and operative proofs theoretical background.

The last three chapters, Wittmann’s works after the “silent” period, contain how to bridge the gap between didactical theories and practice (i.e., collective teaching experiments). In Chapter twelve, Wittmann offered options to link didactical theories and practice. There are empirical studies about design science, structure-genetic didactical analyses, and collective teaching experiments. Chapter thirteen discusses structure-genetic didactical analysis as a balance of teaching and learning mathematics and mathematics. Finally, Chapter fourteen tells about: (1) revisit the concept of ME as a design science and (2) describe in some detail recent developments in connection with conceptual and practical issues.

The book’s core strength lies in the long-life discussion of connecting mathematics, a pure and separate field of science or a research field, and ME. Mathematics educators worldwide have been conceptualizing how to bridge both fields of science. Among other proposals is Wittmann’s DSA proposed within the current book. DSA is viewed as the best approach for bridging purposes. Another strength of the book is the way the chapters are arranged. Wittman cleverly arranges the chapters chronologically from the first paper (Chapter 2) in 1982 to the last one (Chapter 14) in 2019. It is generally challenging to connect papers written in different topics and contexts like those in this book, but Wittman succeeds in doing so. The book may also be limited to a single context, Germany, yet it is still applicable to other contexts. While the book addresses Wittman’s concept of DSA in an attempt to link mathematics and ME, some chapters provide some quotations from John Dewey, a well-known philosopher, psychologist, and educational theorist. As stated by Wittman, instead of being a disadvantage to the book, these quotations serve as connections between chapters. However, the book might have been better if the author provides a separate chapter as a conclusion, so readers are not left behind to create their conclusions.

Overall, Erich Christian Wittmann’s Connecting Mathematics and Mathematics Education is a valuable resource, particularly for those interested in ME. It definitely contributes to the studies of mathematics and ME. It is a must-read for students-teachers, researchers, and practitioners in ME.