Discusses the implications of information processing psychology for mathematics education, with a focus on the works of schema theorists such as D. E. Rumelhart and D. A. Norman and R. Glaser and production system theorists such as J. H. Larkin, J. G. Greeno, and J. R. Anderson. Learning is considered in terms of the actor’s and the observer’s perspective and the distinction between declarative and procedural knowledge. Comprehension and meaning in mathematics also are considered. The role of abstraction and generalization in the acquisition of mathematical knowledge is discussed, and the difference between helping children to “see, ” as opposed to construct abstract relationships is elucidated. The goal of teaching is to help students modify or restructure their existing schema in predetermined ways by finding instructional representations that enable students to construct their own expert representations.

A distinction is made between weak and strong research programs in cognitive science, the latter being characterized by an emphasis on the development of runnable computer programs. The paper focuses on the strong research program and initially considers situations in which it claims to have advanced our understanding of mathematical activity. It is concluded that the program’s characterization of students as environmentally driven systems leads to: (a) a treatment of mathematical activity in isolated, narrow, formal domains; (b) a failure to deal with relevance, common sense, and context, and (c) a separation of conceptual thought from sensory-motor action. Taken together, these conclusions imply a failure to deal adequately with the issue of mathematical meaning. In general, the program’s primary focus appears to be on programmable mechanisms rather than fundamental problems of mathematical cognition. The purview of the discussion is then widened to consider the strong program’s difficulties in dealing with social interaction, intellectual communities, and the hidden curriculum. It is noted that instructional implications derived from this program typically involve the organization of mathematical stimuli that make explicit or salient the relevant properties of a propositional mathematical environment. Finally, it is argued that some members of the strong program have recently acknowledged that it has limitations. The possibility of a rapprochement in which the strong program is supplanted by a form of social constructivism is discussed.

Excerpt: The first five contributions to this Special Issue on Theories of Mathematical Learning take a cognitive perspective whereas the sixth, that by Voigt, takes an interactionist perspective. The common theme that links the six articles is the focus on students’ inferred experiences as the starting point in the theory-building process. This emphasis on the meanings that objects and events have for students within their experiential realities can be contrasted with approaches in which the goal is to specify cognitive behaviors that yield an input-output match with observed behavior. It is important to note that the term ‘experience’ as it is used in these articles is not restricted to physical or sensory-motor experience. A perusal of the first five articles indicates that it includes reflective experiences that involve reviewing prior activity and anticipating the results of potential activity. In addition, by emphasizing interaction and communication, Voigt’s contribution reminds us that personal experiences do not arise in a vacuum but instead have a social aspect.

Currently, considerable debate focuses on whether mind is located in the head or in the individual-in-social-action, and whether development is cognitive self-organization or enculturation into established practices. In this article, I question assumptions that initiate this apparent forced choice between constructivist and sociocultural perspectives. I contend that the two perspectives are complementary. Also, claims that either perspective captures the essence of people and communities should be rejected for pragmatic justifications that consider the contextual relevance and usefulness of a perspective. I argue that the sociocultural perspective informs theories of the conditions far the possibility of learning, whereas theories developed from the constructivist perspective focus on what students learn and the processes by which they do so.

In this chapter, I focus on one of the aspects of constructivist theory that Glasersfeld (Ch. 1) identifies as in need of further development. This aspect of the theory involves locating students’ mathematical development in social and cultural context while simultaneously treating learning as a process of adaptive reorganization. In addressing this issue, I illustrate the approach that I and my colleagues currently take when accounting for the process of students’ mathematical learning as it occurs in the social context of the classroom. In the opening section of the chapter, I clarify why this is a significant issue for us as mathematics educators. I then outline my general theoretical orientation by discussing Glasersfeld’s constructivism and Bauersfeld’s interactionism. Against this background, I develop criteria for classroom analyses that are relevant to our interests as researchers who develop learning environments for students in collaboration with teachers. Next, I illustrate the interpretive framework that I and my colleagues currently use by presenting a sample classroom analysis. Finally, in the concluding sections of the chapter, I reflect on the sample analysis to address four more general issues. These concern the contributions of analyses of the type outlined in the illustrative example, the relationship between instructional design and classroom-based research, the role of symbols and other tools in mathematical learning, and the relation between individual students’ mathematical activity and communal classroom processes.

Excerpt: In inviting me to write this chapter on philosophical issues in mathematics education, the editor has given me the leeway to present a personal perspective rather than to develop a comprehensive overview of currently influential philosophical positions as they relate to mathematics education. I invoke this privilege by taking as my primary focus an issue that has been the subject of considerable debate in both mathematics education and the broader educational research community, that of coping with multiple and frequently conflicting theoretical perspectives. The theoretical perspectives currently on offer include radical constructivism, sociocultural theory, symbolic interactionism, distributed cognition, information-processing psychology, situated cognition, critical theory, critical race theory, and discourse theory. To add to the mix, experimental psychology has emerged with a renewed vigor in the last few years. Proponents of various perspectives frequently advocate their viewpoint with what can only be described as ideological fervor, generating more heat than light in the process. In the face of this sometimes bewildering array of theoretical alternatives, the issue I seek to address in this chapter is that of how we might make and justify our decision to adopt one theoretical perspective rather than another. In doing so, I put philosophy to work by drawing on the analyses of a number of thinkers who have grappled with the thorny problem of making reasoned decisions about competing theoretical perspectives.

Problem: Ernst von Glasersfeld’s radical constructivism has been highly influential in the fields of mathematics and science education. However, its relevance is typically limited to analyses of classroom interactions and students’ reasoning. Methods: A project that aims to support improvements in the quality of mathematics instruction across four large urban districts is framed as a case with which to illustrate the far-reaching consequences of von Glasersfeld’s constructivism for mathematics and science educators. Results: Von Glasersfeld’s constructivism orients us to question the standard view of policy implementation as a process of travel down through a system and to conceptualize it instead as the situated reorganization of practice at multiple levels of a system. In addition, von Glasersfeld’s constructivism orients us to understand rather than merely evaluate policies by viewing the actions of the targets of policies as reasonable from their point of view. Implications: The potential contributions of von Glasersfeld’s constructivism to mathematics and science education have been significantly underestimated by restricting the focus to classroom actions and interactions. The illustrative case of research on the application of these ideas also indicates the relevance of constructivism to researchers in educational policy and educational leadership.

Key words: mathematics education, science education, policy, leadership

Key words: Jean Piaget, radical constructivism