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Richard Lehrer and Megan Loef Franke

Personal construct psychology provides a coherent theoretical and methodological framework for the examination of teachers' knowledge. We report case studies of two teachers who varied in their knowledge about fractions and mathematical pedagogy. We used personal construct psychology and the logic of fuzzy sets to elucidate the content and organization of the teachers' knowledge of fractions. The approach proved especially useful for describing conditional relationships among content, general pedagogical, and pedagogical content knowledge frames. We also explored associations between teachers' personal constructions and their classroom teaching. These associations suggested that personal construct psychology shows considerable promise as a way of addressing issues of teacher knowledge in the context of the classroom.

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Marta Kobiela and Richard Lehrer

We examined the codevelopment of mathematical concepts and the mathematical practice of defining within a sixth-grade class investigating space and geometry. Drawing upon existing literature, we present a framework for describing forms of participation in defining, what we term aspects of definitional practice. Analysis of classroom interactions during 16 episodes spanning earlier and later phases of instruction illustrate how student participation in aspects of definitional practice influenced their emerging conceptions of the geometry of shape and form and how emerging conceptions of shape and form provided opportunities to develop and elaborate aspects of definitional practice. Several forms of teacher discourse appeared to support students' participation and students' increasing agency over time. These included: (a) requesting that members of the class participate in various aspects of practice, (b) asking questions that serve to expand the mathematical system, (c) modeling participation in aspects of practice, (d) proposing examples that create contest (i.e., monsters), and (e) explicitly stating expectations of and purposes for participating in the practice.

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Elizabeth Nitabach and Richard Lehrer

The original sense of geometry, “earth measure” suggests that measurement is an important way in which children can come to understand and develop a mathematics of space.

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Richard Lehrer and Carmen Curtis

Describes how children searched for Platonic solids and constructed a precise definition to classify these solids.

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Elizabeth Penner and Richard Lehrer

Shape and form are often used as mathematical models of situations. For example, teachers explain that light travels in a line or that the shadow cast by a person is related by similar triangles to that cast by a flagpole. Yet despite the common use of mathematical models in the sciences and in design professions, children rarely have the opportunity to participate in this form of mathematical thinking. In this article, we describe how first and second graders modeled a “fair” playing space in a game of tag called “Mother, may I?” The children modeled the playing space by using a succession of different forms, such as lines and squares, to represent a fair game, discovering along the way the properties of each of the forms that made them less-than-ideal models of fairness. Participation in the game gave the children many opportunities to think about important concepts in measuring length and the idea of using form to model a situation.

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Cathy Jacobson and 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.