Erna Yackel

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.

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.

What we can learn from analyzing the teacher’s role in collective argumentation

Abstract In this paper, I use analyses of collective argumentation in a variety of classroom settings, from elementary school to a university-level differential equations class to illustrate various roles the teacher plays. These include initiating the negotiation of classroom norms that foster argumentation as the core of students’ mathematical activity, providing support for students as they interact with each other to develop arguments, and supplying argumentative supports (data, warrants, and backing) that are either omitted or left implicit. We gain two important insights from these analyses. First, an emphasis on argumentation can be used productively to provide openings in mathematical discussions for new mathematical concepts and tools to emerge. Second, the analyses demonstrate that teachers need to have both an in-depth understanding of students’ mathematical conceptual development and a sophisticated understanding of the mathematical concepts that underlie the instructional activities being used.

Beliefs and Norms in the Mathematics Classroom

The central purpose of this chapter is to demonstrate that by coordinating sociological and psychological perspectives we can explain how changes in beliefs might be initiated and fostered in mathematics classrooms. In particular, we examine: 1) the coordination of students’ beliefs about mathematical activity and classroom social norms and 2) the coordination of specifically mathematical beliefs and classroom sociomathematical norms. Examples from a university level differential equations class are used for purposes of clarification and illustration.

A Follow-up Assessment of a Second-Grade Problem-Centered Mathematics Project.

Five second-grade classes in two schools participated in a project that was generally compatible with a constructivist theory of knowing. At the end of the school year, the students in these classes and their peers in six non-project classes in the same schools were assigned to ten textbook-based third-grade classes on the basis of reading scores. The two groups of students were compared at the end of the third-grade year on a standardized achievement test and on instruments designed to assess their conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computation performance on familiar textbook tasks were comparable, but former project students had attained more advanced levels of conceptual understanding. In addition, they held stronger beliefs about the importance of working hard and being interested in mathematics, and about understanding and collaborating. Further, they attributed less importance to conforming to the solution methods of others.

Didactising: Continuing the work of Leen Streefland

In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by elaborating how the constructs chains of signification and models were used to design an instructional sequence involving linear growth. The second and third cases illustrate vertical didactising by developing argumentation analyses and generative listening, respectively, as instructional design tools. In the second case, argumentation analyses emerge as a tool that other designers can use to anticipate the quality of conversations that can occur as students engage in tasks prior to implementing the instructional sequence. The third case develops the notion of generative listening as a conceptual tool within the context of designing differential equations instruction to gain insights into what are, for students, experientially-real starting points that are mathematical in nature and to provide inspirations for revisions to instructional sequences.

Chapter 5: The Relationship of Individual Children's Mathematical Conceptual Development to Small-Group Interactions

This chapter elaborates our understanding of small-group problem solving in elementary school mathematics by focusing on the relationship between childrens individual mathematical conceptions and the nature of their social interactions as they work together in small-group problem-solving settings. In particular, the manner in which childrens individual mathematical conceptions influence the nature of the social interactions that take place is explained. Previously we have focused on the importance of the social interactions that occur in small-group settings in proving unique opportunities for learning (Yackel et al., 1991) and on the negotiation of classroom social norms that facilitate collaboration. As we analyzed small-group interactions, we were able to trace the development of cooperative activity between partners. Initially, children typically divided the labor, taking turns as they completed the activities. Under the guidance of the teacher, social norms for small-group activity were negotiated that resulted in children reconceptualizing their understanding of work cooperatively from division of labor to working together to complete the instructional activities. Working together came to be understood as explaining your thinking to your partner, listening to your partners explanations, and jointly developing a solution. Thus, situations in which children failed to work collaboratively were interpreted as situations where the participants were violating the social norms. However, as we continued to analyze the small-group data, we found that some social interactions that we had previously explained only as violations of social norms could be further interpreted by taking the childrens individual mathematical conceptions into account. The analysis shows that childrens conceptions

Guided Reinvention: What Is It and How Do Teachers Learn This Teaching Approach?

In this chapter, the theoretical construct of guided reinvention is extended to include desirable pedagogical practices for teachers implementing RME sequences. First, we explain what a guided reinvention teaching approach looks like and how it evolved out of over 25 years of research. We then articulate the planning and teaching practices of guided reinvention teachers and describe how those practices move beyond what many call “inquiry approaches” to mathematics teaching. We end the chapter by offering a set of learning goals that professional developers might use when mentoring aspiring guided reinvention teachers.