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## Seeing Algebraic Structure: The Rubik's Cube

Few high school students associate mathematics with playfulness. In this paper, we offer a series of lessons focused on the underlying algebraic structures of the Rubik's Cube. The Rubik's Cube offers students an interesting space to enjoy the playful side of mathematics, while appreciating mathematics otherwise lost in routine experiences.

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## Positioning During Group Work on a Novel Task in Algebra II

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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## Mathematical Explorations: A New Twist on Collaborative Learning

Research on students' learning has made it clear that learning happens through an interaction with others and through communication. In the classroom, the more students talk and discuss their ideas, the more they learn. However, within a one-hour period, it is hard to give everyone an equal opportunity to talk and share their ideas. Organizing students in groups distributes classroom talk more widely and equitably (Cohen and Lotan 1997).

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## Developing the Area of a Trapezoid

When calculating the area of a trapezoid, students use a range of problem-solving strategies and measurement concepts.

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## Mysterious Subtraction

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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## Changing the Rules to Increase Discourse

A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.

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## Develop Reasoning through Pictorial Representations

Solutions coupled with drawings can illustrate students' understandings or misunderstandings, particularly in the area of proportional reasoning.

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## Capturing Thinking on the Talk Frame

This instructional tool helps students engage in discussions that foster student reasoning, then settle on correct mathematics.

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## Authentic Argumentation With Prospective Secondary Teachers: The Case of 0.999…

Engaging prospective secondary teachers in mathematical argumentation is important, so that they can begin to learn to engage their own students in creating and critiquing arguments. Often, when we attempt to engage prospective secondary teachers in argumentation around topics from secondary mathematics classes, the argumentation is not authentic, as they believe they already know the answers. I suggest that there are problems related to the secondary curriculum around which we can engage students in authentic argumentation, and I propose one of them is whether 0.999… = 1. Purposefully engaging and supporting students in discussing this problem, and others like it, can lead to productive discussions that go beyond the answer to the question, including, for instance, what counts as evidence in mathematics.

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## Developing Formal Procedures through Sense Making

Allowing students to reason and communicate about integer operations, or any idea, before these ideas are formalized can be an important tool for fostering deep understanding.