This study explored 10 prospective teachers’ (PTs’) understanding of the area of a rectangular region using square and non-square rectangular area-units. In an hour-long interview, each PT was first asked to explain how they would find the area of a given rectangular region in terms of a non-square notecard. For several PTs, this task prompted discussion of a square unit defined by the edge of the notecard. In a second task, PTs were presented with a rectangular array of squares and were asked to explain to a fictional child why multiplying length times width does not count the top left corner square twice. An analysis of transcripts of the interview suggested three conceptual components to understanding the area of a rectangular region—area-units, multiplication, and their connection to the length times width formula. Based on the PTs’ responses, we propose multi-tiered levels of reasoning. Each new level of understanding was built on the prior level in a hierarchical fashion. Juxtaposing tasks involving different area-units revealed PTs’ thinking about non-square and square area-units and how these related to the formula “L × W”.

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and I conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.

In the research conducted, the relationship between teaching complexity and children's mathematical thinking was investigated in 4 'reform' classes and 1 conventional elementary class (7-8 years). Forty lessons were analyzed for the type of teaching and children's mathematical thinking revealed during class discussion. The results indicate increased complexity in teaching and level of children's thinking was highly related to the kinds of interaction that distinguished three class cultures. These findings complete a previously proposed theoretical framework that integrates teaching and learning by detailing acts of teaching in relation to complexity of children's thinking. It is well documented that the concerted effort in the U. S. to change conventional mathematics teaching to forms of pedagogy that coincide with learning for conceptual understanding is more difficult than initially anticipated. One reason may be, in part, due to the fact that this requires the development of far more complex and sophisticated pedagogy than was understood or even known at the onset of the effort (Wood, Nelson & Warfield, 2001). Although research on learning over the past century has influenced our knowledge of learning, similar transformation in our understanding of the teaching practices has yet to occur. Educators, such as Darling-Hammond (1996) believe the challenge for education in this century is the advancement of " knowledge for a different kind of teaching. .. that goes far beyond dispensing information, giving a test, and giving a grade " (p. 7). From recent studies such as Askew et al. (1999) and Franke et al. (1998) we are beginning to understand what characterizes the complexity in new forms of teaching and how this relates to student learning. However, it still remains that " only a few studies exist which empirically examine teaching in these classes with the same detail and attention to theory building as found in the investigations of learning " (Wood, 1998, p. 193). In previous research we have examined class cultures for differences in pedagogy and found that teaching for conceptual understanding does not consist of a singular practice, but rather varies on two dimensions—expectations for class members' participation and the breadth of pupils' thought (Wood & Turner-Vorbeck, 2001). While these two dimensions differentiate the nature of teaching in 'reform-oriented' class cultures, the relationship of teaching to children's mathematical thinking was only theoretically conjectured. Therefore, the purpose of this research report is to present the results of an investigation into the relationship between teaching …

As part of an investigation of the mathematical and pedagogical development of prospective teachers, the second author taught mathematics to a group of undergraduate teacher candidates in a way that is compatible with current mathematics education reform principles. Initially, the lack of a shared basis for communication was evident when these students, acculturated to the practices of school mathematics, interacted with a teacher who was trying to promote inquiry mathematics. Our analysis of the data indicates that by the latter stages of the course, a classroom microculture characterized by inquiry mathematics had evolved. In this article, we examine the processes by which the participants in this classroom community negotiated norms and practices. The result of this analysis was an identification and elaboration of four categories of interaction central to the ongoing negotiation. This article illustrates how each of these categories of interaction contributed to the negotiation of new norms and practices.

This study coordinates anthropological and cognitive perspectives on one child's learning of the standard addition algorithm. Changes in the child's mathematical beliefs and constructions were analyzed as he moved from a experimental 2nd-grade mathematics class characterized by inquiry mathematics to a textbook-based third grade. Piagetian clinical interviews focusing on this mathematical understanding before, during, and after 8 weeks of instruction in third grade were coordinated with an interactional analysis of a typical textbook lesson selected from a larger microethnographic study of his classroom mathematical community. The resulting description of the social norms and mathematical practices of this community provide a background against which the child's mathematical development is examined. The analysis shows that he had abandoned his self-generated computational algorithms in favor of less understood conventional procedures.

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.