Anna Sierpinska

Epistemologies of Mathematics and of Mathematics Education

This chapter addresses issues concerning epistemology, as they relate to mathematics and education. It commences with an examination of some of the main epistemological questions concerning truth, meaning and certainty, and the different ways they can be interpreted. It examines epistemologies of the ‘context of justification’ and of the ‘context of discovery’, foundationalist and non-foundationalist epistemologies of mathematics, historico-critical, genetic, socio-historical and cultural epistemologies, and epistemologies of meaning.

Historical formation and student understanding of mathematics

The use of history of mathematies in the teaching and learning of mathematics requires didactical reflection. A crucial area to explore and analyse is the relation between how students achieve under standing in mathematics and the historical construction of mathematical thinking.

On Practical and Theoretical Thinking and Other False Dichotomies in Mathematics Education

I owe much of my understanding of the difference between synthetic and analytic thinking in mathematics to my reading of Michael Otte’s papers and the conversations we had with him within the BACOMET group. One of the first sources of inspiration for me has been his work on arithmetic and geometric thinking. In the paper I shall outline the consequences of the distinction for analyzing processes of mathematics teaching and learning in my own research. 1 shall further use this distinction to look critically upon the recent trend in mathematics education of considering mathematics as a kind of “discursive practice.”

Continuing the Search

In this chapter, we attempt to draw together the principal arguments made in the preceding chapters and at the ICMI Study Conference so as to provide the reader with a retrospective view of the clearly very complex issue, ‘What is research in mathematics education, and what are its results?’. To that question, we are compelled to add: What does it mean to be a researcher in mathematics education? Do we have a common identity?

Some Reflections on the Phenomenon of French didactique

The paper aims at sharing with the reader some remarks on mathematics education research originating in France which is related to Brousseau’s theory of didactic situations, the concepts of didactic contract and didactic transposition, and the methodology of didactic engineering. It brings forth and explains the background of some features of French didactique such as its specific view of what is mathematics, where research in didactique should be practiced institutionally, as well as its sharp distinctions between research and teaching practice, and research and pedagogical innovation. The paper contains a discussion of the differences between didactic engineering and instructional design as understood in North America. It also explains the background in French didactique of the research program in mathematics education related to the concept of epistemological obstacle which incited the author’s personal interest in the past.

Analysis of tasks in pre-service elementary teacher education courses

This paper presents some results of research aimed at contributing to the development of a professional knowledge base for teachers of elementary mathematics methods courses, called here ‘teacher educators’. We propose that a useful unit of analysis for this knowledge could be the tasks in which teacher-educators engage pre-service teachers, and that the tasks could be indexed by a system of nested categories of actions required in the tasks and the analytic tools for carrying them out. The paper develops the seed of such an indexing framework, based on analyses of tasks in two methods courses. We argue that this seed is robust enough to be expandable, in the sense that categories could be added to it without destroying the existing ones. An expanded index, accompanied by a bank of tasks, would facilitate the development of a unified discourse to bring together, represent, and communicate, the vast experience of teacher educators.

Perspectives sur les recherches en didactique des mathématiques

The paper is a review of chosen approaches to research in mathematics education in several countries: Germany, France, United States, Russia, Poland, Canada. The review is done in the literary form of a satire, in which a character is taken on a voyage to a variety of “islands” respresenting different research interests and methodologies in mathematics education. The story is a parody of Homer’sOdyssee, and the main character is called Odysseus. Odysseus’ role is played by the famous arithmetic problem about a team of an unknown number of scythers who are given the task of scything two meadows one of which is double the size of the other. As the problem travels from one “island” to another, mathematics educators do different things to and with the problem and it is solved is a variety of ways. The main text of the paper reads as a story and there are no explicit references and names of authors, whose work is only alluded to. However, the solution to all allusions, i.e. explicit references, can be found in the footnotes.

Networking of Theories as a Research Practice in Mathematics Education, by Angelika Bikner-Ahsbahs and Susanne Prediger (Eds.)

This is an unusual book. Two editors are named, but the book is not a collection of independent papers on a common theme, as is usually the case with edited volumes. The chapters depend on each other and have been written by a “collective”: the Networking Theories Group. The book recounts the story and takes stock of a truly cooperative, long term (2006–2013), enterprise. A very brave enterprise, too, considering that members of the group represented five theoretical perspectives in mathematics education, which they had to confront, compare, and, if possible, reconcile in interpreting a single classroom episode. They had to leave the comfort of their own familiar ways of doing research in mathematics education where researchers understand each other without words and are able to take countless things for granted. They had to communicate their research culture to outsiders and make the effort of understanding the foreign cultures of these outsiders. The five theoretical perspectives were Action, Production, and Communication (APC) (Arzarello, Paola, Robutti, & Sabena, 2009); the Theory of Didactical Situations (TDS) (Brousseau, 1997), the Anthropological Theory of the Didactic (ATD), (Barbé, Bosch, Espinoza, & Gascón, 2005); the Abstraction in Context approach (AiC), (Schwartz, Dreyfus, & Hershkowitz, 2009); and the Theory of Interest-Dense Situations (IDS) (Bikner-Ahsbahs, 2005). Sixteen authors contributed to the book’s 17 chapters. Here they are, in alphabetical order: Michèle Artigue; Ferdinando Arzarello; Angelika Bikner-Ahsbahs; Marianna Bosch; Agnès Corbin-Lenfant; Tommy Dreyfus; Josep Gascón; Stefan Halverscheid; Mariam Haspekian; Ivy Kidron; Alexander Meyer; Susanne Prediger; Luis Radford; Kenneth Ruthven; Cristina Sabena; and Ingolf Schäfer. Two of these authors, Kenneth Ruthven and Luis Radford, position themselves somewhat apart from the rest of the group, commenting on the research that the others have done without engaging in this research themselves. They are presented in the Preface as “critical friends of the group,” although Kenneth Ruthven has been one of the “founding members” of the Networking Theories group. Kenneth Ruthven’s chapter, “From networked theories to modular tools” (pp. 267–280), is written in the style of a regular book review and could well be published as such in any scholarly journal in mathematics education, as a stand-alone piece. It is rather unusual for a book to include a review of itself within its covers. Images of self-referentiality—such as Escher’s lithograph of a hand drawing itself— come to mind, but the impression of self-referentiality comes not only from this particular chapter but also from the character of the book as a whole: authors writing about themselves writing this book. Luis Radford’s “Heideggerian commentary” on theories and their networking discusses the meanings of such fundamental concepts as “theory,” “practice,” “observation,” and “mathematics,” and proposes to look at them not as finished products of human activity but as human activities, or indeed “forms of life.” These are, respectively, a theoretical form of life, a practical and productive form of life, a contemplative form of life, and a mathematical form of life understood as “a reckoning that, everywhere by means of equations, has set up as the goal of its expectation the harmonizing of all relations of order” (Heidegger, 1977, p. 282, in the reviewed book). This perspective on theories as