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January 2000, Volume 31, Issue 1

FEATURES

Enhancing Prospective Teachers' Knowledge of Children's Conceptions: The Case of Division of Fractions
Dina Tirosh
In this article I present and discuss an attempt to promote development of prospective elementary teachers' own subject-matter knowledge of division of fractions as well as their awareness of the nature and the likely sources of related common misconceptions held by children. My data indicate that before the mathematics methods course described here most participants knew how to divide fractions but could not explain the procedure. The prospective teachers were unaware of major sources of students' incorrect responses in this domain. One conclusion is that teacher education programs should attempt to familiarize prospective teachers with common, sometimes erroneous, cognitive processes used by students in dividing fractions and the effects of use of such processes.

Knowledge Connectedness in Geometry Problem Solving
Michael J. Lawson, Mohan Chinnappan
Our concern in this study was to examine the relationship between problem-solving performance and the quality of the organization of students' knowledge. We report findings on the extent to which content and connectedness indicators differentiated between groups of high-achieving (HA) and low-achieving (LA) Year 10 students undertaking geometry tasks. The HA students' performance on the indicators of knowledge connectedness showed that, compared with the LA group, they could retrieve more knowledge spontaneously and could activate more links among given knowledge schemas and related information. Connectedness indicators were more influential than content indicators in differentiating the groups on the basis of their success in problem solving. The tasks used in the study provide straightforward ways for teachers to gain information about the organizational quality of students' knowledge.

Developing Concepts of Sampling
Jane M. Watson, Jonathan B. Moritz
A key element in developing ideas associated with statistical inference involves developing concepts of sampling. The objective of this research was to understand the characteristics of students' constructions of the concept of sample. Sixty-two students in Grades 3, 6, and 9 were interviewed using open-ended   questions related to sampling; written responses to a questionnaire were also analyzed. Responses were characterized in relation to the content, structure, and objectives of statistical literacy. Six categories of construction were identified and described in relation to the sophistication of developing concepts of sampling. These categories illustrate helpful and unhelpful foundations for an appropriate understanding of representativeness and hence will help curriculum developers and teachers plan interventions.

Teacher Appropriation and Student Learning of Geometry Through Design
Cathy Jacobson, Richard Lehrer
In 4 Grade 2 classrooms, children learned about transformational geometry and symmetry by designing quilts. All 4 teachers participated in professional development focused on understanding children's thinking in arithmetic. Therefore, the teachers elicited student talk as a window for understanding student thinking and adjusting instruction in mathematics to promote the development of understanding and used the same tasks and materials. Two of the 4 teachers   participated in additional workshops on students' thinking about space and geometry, and they elicited more sustained and elaborate patterns of classroom conversations about transformational geometry. These differences were mirrored by students' achievement differences that were sustained over time. We attribute these differences in classroom discourse and student achievement to differences in teachers' knowledge about typical milestones and trajectories of children's reasoning about space and geometry.

Grade 6 Students' Preinstructional Use of Equations to Describe and Represent Problem Situations
Jane O. Swafford, Cynthia W. Langrall
Grade 6 Students' Preinstructional Use of Equations to Describe and Represent Problem Situations

Exploring Situated Insights Into Research and Learning
Jo Boaler
In this report I offer an exploration of the insights that may be provided by a situated perspective on learning. Through an extension of my previous analysis of students learning mathematics in two schools (Boaler, 1998), I consider the ways in which a focus on the classroom community and the behaviors and practices implicit within such communities may increase one's understanding of students' mathematical knowledge production and use. The implications of such a focus for classroom pedagogy and assessment as well as for research in mathematics education are considered.

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