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May 1997, Volume 28, Issue 3


Reflective Discourse and Collective Reflection
Paul Cobb, Ada Boufi, Kay McClain, Joy Whitenack
The analysis in this paper focuses on the relationship between classroom discourse and mathematical development. We give particular attention to reflective discourse, in which mathematical activity is objectified and becomes an explicit topic of conversation. We differentiate between students' development of particular mathematical concepts and their development of a general orientation to mathematical activity. Specific issues addressed include both the teacher's role and the role of symbolization in supporting reflective shifts in the discourse. We subsequently contrast our analysis of reflective discourse with Vygotskian accounts of learning that also stress the importance of social interaction and semiotic mediation. We then relate the discussion to characterizations of classroom discourse derived from Lakatos' philosophical analysis.

The Changing Role of the Mathematics Teacher
Doug M. Clarke
This case-study research investigated changing teacher roles associated with two teachers' use of innovative mathematics materials at Grade 6. Using daily participant observation and regular interviews with the teachers and the project staff member responsible for providing in-school support, a picture emerged of changing teacher roles and of those factors influencing the process of change. One teacher demonstrated little change in either espoused beliefs or observed practice over the course of the study. The second teacher demonstrated increasing comfort with posing nonroutine problems to students and allowing them to struggle together toward a solution, without suggesting procedures by which the problems could be solved. He also increasingly provided structured opportunities for students' reflection on activities and learning. Major influences on this teacher's professional growth appeared to be the provision of the innovative materials and the daily opportunity to reflect on classroom events in conversations and interviews with the researcher.

Young Children's Intuitive Models of Multiplication and Division- FREE PREVIEW!
Joanne T. Mulligan, Michael C. Mitchelmore
In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children's intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it.

Facilitating Student Interactions in Mathematics in a Cooperative Learning Setting
Rosa Leikin, Orit Zaslavsky
This article discusses effects on different types of students' interactions while learning mathematics in a particular cooperative small-group setting. Three low-level ninth-grade classes engaged in learning mathematics in an experimental cooperative method. Data were collected through classroom observations, students' written self-reports, and an attitude questionnaire. A group of four students from one of the classes was observed more closely. Analysis of classroom observations and students' self-reports focused on students' activeness, interactions, and attitudes toward the experimental method. Findings for the cooperative small-group setting indicated (a) an increase in students' activeness, (b) a shift toward students' on-task verbal interactions, (c) various opportunities for students to receive help, and (d) positive attitudes toward the cooperative experimental method. A fourth class served as a control group to ascertain whether student achievement increased or decreased as a result of the experimental method.

Listening for Differences: An Evolving Conception of Mathematics Teaching
Brent Davis
The question of how to teach mathematics has become increasingly problematic in recent years as critics from diverse perspectives have offered wide-ranging, and often seemingly incommensurate, challenges to conventional conceptions of the teacher's task. This article represents an effort to "bring into dialogue" some of the varied commentaries on mathematics teaching, using an enactivist framework to interpret and to propose an alternative way of framing mathematics teaching. In this report, the manner in which the teacher listens is offered as a metaphoric lens through which to reinterpret practice, as a practical basis for teaching action, and as a means of addressing some of the critics' concerns. The report is developed around an extended collaborative research project with a middle school mathematics teacher.