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Geometric Reasoning about a Circle Problem

Gloriana González and Anna F. DeJarnette

An open-ended problem about a circle illustrates how problem-based instruction can enable students to develop reasoning and sense-making skills.

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Positioning During Group Work on a Novel Task in Algebra II

Anna F. DeJarnette and Gloriana González

Given the prominence of group work in mathematics education policy and curricular materials, it is important to understand how students make sense of mathematics during group work. We applied techniques from Systemic Functional Linguistics to examine how students positioned themselves during group work on a novel task in Algebra II classes. We examined the patterns of positioning that students demonstrated during group work and how students' positioning moves related to the ways they established the resources, operations, and product of a task. Students who frequently repositioned themselves created opportunities for mathematical reasoning by attending to the resources and operations necessary for completing the task. The findings of this study suggest how students' positioning and mathematical reasoning are intertwined and jointly support collaborative learning through work on novel tasks.

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Leading Classroom Discussions

Gloriana González and Anna F. DeJarnette

Students develop ownership and increase their understanding of mathematics when they are allowed to discuss alternative perspectives.

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Seeing a Colleague Encourage a Student to Make an Assumption While Proving: What Teachers Put in Play When Casting an Episode of Instruction

Talli Nachlieli, Patricio Herbst, and Gloriana González

This article reports on an investigation of how teachers of geometry perceived an episode of instruction presented to them as a case of engaging students in proving. Confirming what was hypothesized, participants found it remarkable that a teacher would allow a student to make an assumption while proving. But they perceived this episode in various ways, casting the episode as one of as many as 10 different stories. Those different castings of the episode make use of intellectual resources for professional practice that practitioners could use to negotiate the norms of a situation in which they had made a tactical but problematic move. This collection of stories attests to the effectiveness of the technique used for eliciting the rationality of mathematics teaching: By confronting practitioners with episodes of teaching in which some norms have been breached, one can learn about the rationality underlying the norms of customary teaching.

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Shadow Puppets: Exploring a Context for Similarity and Dilations

Anna F. DeJarnette, Sahid L. Rosado Lausell, and Gloriana González

Turn a typical geometry problem into a great task that promotes students' reasoning and sense making.

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Learning What Works: Promoting Small-Group Discussions

Anna F. DeJarnette, Jennifer N. Dao, and Gloriana González

Elicit productive discourse from students as they work through a bicycle rate problem.