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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### Shiv Karunakaran, Ben Freeburn, Nursen Konuk and Fran Arbaugh

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.

### Heather Lynn Johnson

This article explores quantitative reasoning used by students working on a bottle- filling task. Two forms of reasoning are highlighted: simultaneous-independent reasoning and change-dependent reasoning.

### David A. Yopp

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.

Qualitative and technical considerations for the preparation of manuscripts from submission to MT.

This call solicits manuscripts emphasizing the themes of NCTM's Reasoning and Sense Making series.

### 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.

### Jennifer Noll and J. Michael Shaughnessy

Sampling tasks and sampling distributions provide a fertile realm for investigating students' conceptions of variability. A project-designed teaching episode on samples and sampling distributions was team-taught in 6 research classrooms (2 middle school and 4 high school) by the investigators and regular classroom mathematics teachers. Data sources included survey data collected in 6 research classes and 4 comparison classes both before and after the teaching episode, and semistructured task-based interviews conducted with students from the research classes. Student responses and reasoning on sampling tasks led to the development of a conceptual lattice that characterizes types of student reasoning about sampling distributions. The lattice may serve as a useful conceptual tool for researchers and as a potential instructional tool for teachers of statistics. Results suggest that teachers need to focus explicitly on multiple aspects of distributions, especially variability, to enhance students' reasoning about sampling distributions.

### Perla L. Myers and Colleen N. Pelak

Implement this professional development workshop designed to help teachers excavate the concept of area, uncover misconceptions, and cultivate understanding.

### Matt B. Roscoe

Having prospective teachers find the inscribed angle theorem for themselves can foster mathematical reasoning.