Paul Cobb

Design Experiments in Educational Research

In this article, the authors first indicate the range of purposes and the variety of settings in which design experiments have been conducted and then delineate five crosscutting features that collectively differentiate design experiments from other methodologies. Design experiments have both a pragmatic bent—“engineering” particular forms of learning—and a theoretical orientation—developing domain-specific theories by systematically studying those forms of learning and the means of supporting them. The authors clarify what is involved in preparing for and carrying out a design experiment, and in conducting a retrospective analysis of the extensive, longitudinal data sets generated during an experiment. Logistical issues, issues of measure, the importance of working through the data systematically, and the need to be explicit about the criteria for making inferences are discussed.

Where Is the Mind? Constructivist and Sociocultural Perspectives on Mathematical Development:

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.

Cognitive and Situated Learning Perspectives in Theory and Practice.

In their recent exchange, Anderson, Reder, and Simon (1996 Anderson, Reder, and Simon (1997) and Greeno (1997) frame the conflicts between cognitive theory and situated learning theory in terms of issues that are primarily of interest to educational psychologists. We attempt to broaden the debate by approaching this discussion of perspectives against the background of our concerns as educators who engage in classroom-based research and instructional design in collaboration with teachers. We first delineate the underlying differences between the two perspectives by distinguishing their central organizing metaphors. We then argue that the contrast between the two perspectives cannot be reduced to that of choosing between the individual and the social collective as the primary unit of analysis. Against this background, we compare the situated viewpoint we find useful in our work with the cognitive approach advocated by Anderson et al. by focusing on their treatments of meaning and instructional goals. Finally, we consider the potential contributions of the two perspectives to instructional practice by contrasting their differing formulations of the relationship between theory and practice.

Participating in Classroom Mathematical Practices

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students’ diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.

Characteristics of Classroom Mathematics Traditions: An Interactional Analysis

In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewder’s (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of the other classroom indicated that the teacher and students constituted mathematical truths as they coconstructed a mathematical reality populated by experientially real, manipulable yet abstract mathematical objects. These and other differences between mathematical activity in the two classrooms characterize two distinct classroom mathematics traditions, one in which mathematics was learned with what is typically called understanding and the other in which it was not.

The Constructivist Researcher as Teacher and Model Builder

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.

Situating Teachers’ Instructional Practices in the Institutional Setting of the School and District:

In this article, we describe an analytic approach for situating teachers’ instructional practices within the institutional settings of the schools and school districts in which they work. In doing so, we draw on an ongoing collaboration with a group of teachers in an urban school district to illustrate both the approach and its usefulness in guiding the development of analyses that feedback to inform such collaborations. The approach involves delineating communities of practice within a school or district and analyzing three types of interconnections between them that are based on boundary encounters, brokers, and boundary objects.

Change in Teaching Mathematics: A Case Study

The purpose of this case study was to examine teacher’s learning in the setting of the classroom. In an ongoing mathematics research project based on constructivist views of learning and set in a second-grade classroom, the teacher changed in her beliefs about learning and teaching. These alterations occurred as she resolved conflicts and dilemmas that arose between her previously established form of practice and the emphasis of the project on children’s construction of mathematical meaning. The changes that occurred as the teacher reorganized her practice were analyzed and interpreted by using selected daily video recordings of mathematics lessons along with field notes, open-ended interviews, and notes from project meetings. The analyses indicated that changes occurred in her beliefs about the nature of (a) mathematics from rules and procedures to meaningful activity, (b) learning from passivity to interacting and communicating, and (c) teaching from transmitting information to initiating and guiding students’ development of knowledge.

A Constructivist Approach to Second Grade Mathematics

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. It appears that we have had some success in satisfying the institutional constraints. The achievement test scores did rise satisfactorily, the parents were all universally supportive by the middle of the school year, and the administrators developed a positive opinion of what they saw. As a consequence, we are currently working with 18 teachers from the same school system. In general, we hope that our on-going work constitutes the beginnings of a response to Brophy’s (1986) challenge that “to demonstrate the relevance and practical value of this point of view for improving school mathematics instruction, they [constructivists] will need to undertake programmatic development and research – the development of specific instructional guidelines (and materials if necessary) for accomplishing specific instructional objectives in typical classroom settings” (p. 366). Thus, we concur with Carpenter’s (1983) observation that “If we are unable or unwilling to provide more direction for instruction, we are in danger of conceding the curriculum to those whose basic epistemology allows them to be more directive” (p. 109). Constructivism as an epistemology is, for us, a general way of interpreting and making sense of a variety of phenomena. It constitutes a framework within which to address situations of complexity, uniqueness, and uncertainty that Schon (1985) calls

Interaction and learning in mathematics classroom situations

An analysis of one ten minute episode in which three seven year-old students engage in collaborative small group activity is presented to explore the relationship between individual learning and group development. Particular attention is given to the establishment of a taken-as-shared basis for mathematical activity and to the attainment of intersubjectivity. From a perspective which treats communication as a process of active interpretation and mutual adaptation, learning as it occurs in the course of social interaction is characterized as a circular, self-referential sequence of events rather than a linear cause-effect chain. In addition, the relationship between individual learning and group development is such that the students can be said to have participated in the establishment of the situations in which they learned.