Characteristics of Classroom Mathematics Traditions: An Interactional Analysis

Abstract

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.

Levels of introspection in mathematics instruction

“…Their flexibility could be due to the fact that, in responding to children's thinking, teachers adjusted their plans to accommodate that thinking. This study is also related to previous studies linking teacher knowledge of children's thinking to learner outcomes (e.g., Fisher et al, 1980;Carpenter et al, 1989;Yackel, Cobb & Wood, 1992). In our study, findings from each of the case studies revealed a positive correlation between a teachers' dominant level of introspection and the quality of children's thinking.…”

“…Recent research (e.g., Carpenter, Fennema, Peterson, Chiang & Loef, 1989;Cobb, Wood, Yackel & McNeal, 1992) has encouraged teachers to consider on-going assessment of children's thinking as the basis for instructional decision making. The knowledge children bring to instruction significantly influences what they learn (Carpenter & Peterson, 1988;Nesher, 1986).…”

“…The knowledge children bring to instruction significantly influences what they learn (Carpenter & Peterson, 1988;Nesher, 1986). Moreover, teachers' understanding of children's abilities and thinking has been positively linked to levels of student achievement and standardised tests (Fisher et al, 1980), to measures of conceptual understanding and ability to apply mathematics (Yackel, Cobb, & Wood, 1992), and to measures of addition-subtraction problem solving .…”

Abstracting in the context of Spontaneous Learning

“…Creative mathematical activity during the spontaneous development of new mathematical concepts has been identified in secondary and elementary classrooms (Barnes, 2000;Cobb, Wood, Yackel, & McNeal, 1992) and linked to positive affect (Barnes, 2000;Liljedahl, 2006;Williams, 2002a). In these studies, students were working above their present conceptual level on a self-set intellectual mathematical challenge that was almost out of reach (Csikszentmihalyi & Csikszentmihalyi, 1992;Williams, 2002a).…”

Applying constructivist theory to practice in a technology-based learning environment

“…One direction was towards the publicly accepted knowledge (Noddings, 1990) that would allow them to answer Applicable Mathematics examination questions. Another possible direction, suggeste d by Cobb, Wood, Yackel, and McNeal (1995), was towards becoming a mathematical community. This would have involved allowing students to devise their own agreed modes of working (e.g., devising their own measures of central tendency) and was, in my view, not an option in the circumstances.…”