Cobb P. (1987) Information-processing psychology and mathematics education: A constructivist perspective. Journal of Mathematical Behavior 6(1): 3–40. Fulltext at http://cepa.info/2968

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.

Cobb P. (1994) Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher 23(7): 13–20. Fulltext at http://cepa.info/3049

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.

Cobb P. (2011) Implications of Ernst von Glasersfeld’s Constructivism for Supporting the Improvement of Teaching on a Large Scale. Constructivist Foundations 6(2): 157–161. Fulltext at http://cepa.info/190

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.

Cobb P. & Steffe L. P. (1983) The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education 14(2): 83–94. Fulltext at http://cepa.info/2096

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

Cobb P. & Yackel E. (1996) Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist 31(3–4): 175–190. Fulltext at http://cepa.info/4586

Our overall intent is to clarify relations between the psychological constructivist, sociocultural, and emergent perspectives. We provide a grounding for the comparisons in the first part of the article by outlining an interpretive framework that we developed in the course of a classroom-based research project. At this level of classroom processes, the framework involves an emergent approach in which psychological constructivist analyses of individual activity are coordinated with interactionist analyses of classroom interactions and discourse. In the second part of the article, we describe an elaboration of the framework that locates classroom processes in school and societal contexts. The perspective taken at this level is broadly sociocultural and focuses on the influence of indlividuals’ participation in culturally organized practices. In the third part of the article, we use the discussion of the framework as a backdrop against which to compare and contrast the three theoretical perspectives. We discuss how the emergent approach augments the psychological constructivist perspective by making it possible to locate analyses of individual students’ constructive activities in social context. In addition, we consider the purposes for which the emergent and sociocultural perspectives might be particularly appropriate and observe that they together offer characterizations of individual students’ activities, the classroom community, and broader communities of practice.

Cobb P., Wood T. & Yackel E. (1991) A constructivist approach to second grade mathematics. In: Glasersfeld E. (ed.) Radical constructivism in mathematics education. Kluwer, Dordrecht: 157–176. Fulltext at http://cepa.info/5284

Our overall objective in this paper is to share a few observations made and insights gained while conducting a recently completed teaching experiment. The experiment had a strong pragmatic emphasis in that we were responsible for the mathematics instruction of a second grade class (7 year-olds) for the entire school year. Thus, we had to accommodate a variety of institutionalized constraints. As an example, we agreed to address all of the school corporation’s objectives for second grade mathematics instruction. In addition, we were well aware that the school corporation administrators evaluated the project primarily in terms of mean gains on standardized achievement tests. Further, we had to be sensitive to parents’ concerns, particularly as their children’s participation in the project was entirely voluntary. Not surprising, these constraints profoundly influenced the ways in which we attempted to translate constructivism as a theory of knowing into practice. We were fortunate in that the classroom teacher, who had taught second grade mathematics “straight by the book” for the previous sixteen years, was a member of the project staff. Her practical wisdom and insights proved to be invaluable.

Cobb P., Yackel E. & Wood T. (1992) A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education 23(1): 2–33. Fulltext at http://cepa.info/2967

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.

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