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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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## Investigating Home Primes and Their Families

The process of prime factor splicing to generate home primes raises opportunity for conjecture and exploration.

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## Algebra Homework: A Sandwich!

Given two slices of bread—a problem and the answer—students fill in the fixings: their own mathematics reasoning.

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## Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

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## Working the System

Two teachers discuss the planning and observed results of an introductory problem to help students nail a conceptual approach to solving systems of equations.

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## Using KenKen to Build Reasoning Skills

Through KenKen puzzles, students can explore parity, counting, subsets, and various problem-solving strategies.

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## Sound Off! Don't Sacrifice Geometry on the Common Core Altar

Although high school geometry could be a meaningful course in exploring, reasoning, proving, and communicating, it often lacks authentic proof and has become just another course in algebra. This article examines why geometry is important to learn and provides an outline of what that learning experience should be.

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## Are All Infinities Created Equal?

This historical—and classroom friendly—approach to the concept of infinity uses Cantor's proofs of cardinality.

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## Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

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## Geometric Reasoning about a Circle Problem

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