The constructivist teaching experiment is used in formulating explanations of children’s mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time – anywhere from 6 weeks to 2 years. The explanations we formulate consist of models – constellations of theoretical constructs – that represent our understanding of children’s mathematical realities. However, the models must be distinguished from what might go on in children’s heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children’s construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher’s understanding of the children. Relevance: Constructivist teaching experiment, Model building, Clinical interview. Teaching episode, Counting scheme, Teacher as researcher

The representational view of mind in mathematics education is evidenced by theories that characterize learning as a process in which students modify their internal mental representations to construct mathematical relationships or structures that mirror those embodied in external instructional representations. It is argued that, psychologically, this view falls prey to the learning paradox, that, anthropologically, it fails to consider the social and cultural nature of mathematical activity and that, pedagogically, it leads to recommendations that are at odds with the espoused goal of encouraging learning with understanding. These difficulties are seen to arise from the dualism created between mathematics in students’ heads and mathematics in their environment. An alternative view is then outlined and illustrated that attempts to transcend this dualism by treating mathematics as both an individual, constructive activity and as a communal, social practice. It is suggested that such an approach might make it possible to explain how students construct mathematical meanings and practices that, historically, took several thousand years to evolve without attributing to students the ability to peek around their internal representations and glimpse a mathematically prestructured environment. In addition, it is argued that this approach might offer a way to go beyond the traditional tripartite scheme of the teacher, the student, and mathematics that has traditionally guided reform efforts in mathematics education.

A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget’s postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.

Key words: philosophy of mathematics, cognition, development

Key words: cognition, development

Key words: education

Radical constructivism is currently a major, if not the dominant, theoretical orientation in the mathematics education community, in relation to children’s learning. There are, however, aspects of children’s learning that are challenges to this perspective, and what appears to be “at least temporary states of intersubjectivity” (Cobb, Wood, & Yackel, 1991, p. 162) in the classroom is one such challenge. In this paper I discuss intersubjectivity and through it offer an examination of the limitations of the radical constructivist perspective. I suggest that the extension of radical constructivism toward a social constructivism, in an attempt to incorporate intersubjectivity, leads to an incoherent theory of learning. A comparison of Piaget’s positioning of the individual in relation to social life with that of Vygotsky and his followers is offered, in support of the claim that radical constructivism does not offer enough as an explanation of children’s learning of mathematics.

In their response to my (1996) article, Steffe and Thompson argued that I have taken an early position of Vygotsky’s and that his later work is subsumed in and developed by von Glasersfeld. I argue that the two theories, Vygotsky’s and radical constructivism, are, on the contrary, quite distinct and that this distinction, when seen as a dichotomy, is productive. I suggest that radical constructivists draw on a weak image of the role of social life. I argue that a thick notion of social leads to a complexity of sociocultural theories concerning the teaching and learning of mathematics, a perspective that is firmly located in the debates surrounding cultural theory of the last 2 decades.

Piagetian theory describes mathematical development as the construction and organization of mental operation within psychological structures. Research on student learning has identified the vital roles two particular operations – splitting and units coordination – play in students’ development of advanced fractions knowledge. Whereas Steffe and colleagues describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students’ psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students’ development with a structure that is isomorphic to the positive rational numbers under multiplication – “the splitting group.” We use a quantitative analysis of written assessments from 59 eighth grade students in order to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students’ constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students’ constructions of three levels of units coordination. Because three levels of units coordination also plays a vital role in other mathematical domains, such as algebraic reasoning, implications from the study extend beyond fractions teaching and research. Relevance: The paper uses constructivist theories of learning, including scheme theory and Piaget’s structuralism, to study how students construct mature conceptions of fractions.

William of Ockham responds from the dead to an article appearing in the January 1992 issue of the Journal for Research in Mathematics Education, in which Paul Cobb, Erna Yackel, and Terry Wood propose a “constructivist alternative to the representational view of mind. ” Ockham, now a convert to Platonism, argues three points. First, that by opposing construction to representation, Cobb et al. misinterpret the postepistemological perspective of Richard Rorty’s 1979 influential book, Philosophy and the Mirror of Nature. Second, that by opposing mathematics in the students’ mind to mathematics in the environment, and, in particular, by attempting to argue that the representational theory of mind opens the “learning paradox, ” Cobb et al. misinterpret Carl Bereiter (1985), confuse ontological and epistemological issues, stumble into the perennial philosophical problem of universals, and indicate that they might be interested in discussing the philosophy of mathematics. Third, that in arguing for a relatively pure, “radical” constructivism, Cobb et al. mistake the pragmatic force of the constructivist argument, confuse matters of value with matters of taste, and attempt to fashion too dogmatic a connection between theory and practice in mathematics education.

Excerpt: Piaget is not a realist, for each individual constructs his own reality. In contrast with the title of Rotman’s book, Piaget is not a psychologist of the real but of the concept of the real. What he has studied for almost 70 years is not reality but the construction of reality. When he reports his findings, when he explicates his theory, however, he is compelled to use more or less ordinary language-and ordinary language is, of course, rife with the ontological implications of naive realism. Nevertheless, we believe that he has made his position quite clear. To Rotman’s assertion that he is a “psychologist of the real” he responds, “Je m“en fous de la realite."

Key words: reality, evolution, knowledge, mathematics, Jean Piaget