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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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## Supporting Probability Understanding through Area Models

Moving beyond memorization of probability rules, the area model can be useful in making some significant ideas in probability more apparent to students. In particular, area models can help students understand when and why they multiply probabilities and when and why they add probabilities.

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## Teacher Noticing of Mathematical Thinking in Young Children’s Representations of Counting

Counting is fundamental to early mathematics. Most studies of teaching counting focus on teachers observing children count. The present study compares mathematical ideas that 12 PK, transitional kindergarten (TK), and kindergarten teachers noticed from observing their own students count during a classroom session of Counting Collections with ideas that they noticed outside class time in the same students’ representations of counting on paper. Inviting teacher noticing in representations (a) drew attention to distinct conceptions that children required to represent counting; (b) increased the number of mathematical ideas that participants perceived in students’ thinking; and (c) helped participants perceive different levels in, and their own uncertainties about, students’ understanding. This study suggests that teacher noticing in children’s representations of counting can deepen teachers’ understanding of students’ mathematical thinking.

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## What's in a Name? Language Use as a Mirror into Your Teaching Practice

Language is key to how we understand and describe mathematics teaching and learning. Learning new terms can help us reflect on our practice and grow as teachers, yet may require us to be intentional about where and how we look for opportunities to expand our lexicons.

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## Working Across Contexts: Scaling Up or Replicating With Variations?

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## You Must Be Joking!

May 2020 For the Love of Mathematics Jokes

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The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions related to mathematical depth in preschool, spiral review in the upper elementary grades, ideas for differentiation in middle school, and projects for high school algebra.

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## Discourse Actions to Promote Student Access

We discuss how discourse actions can provide students greater access to high quality mathematics. We define discourse actions as what teachers or students say or do to elicit student contributions about a mathematical idea and generate ongoing discussion around student contributions. We provide rubrics and checklists for readers to use.

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## Envelope Curves Unify Sinusoidal Graphing

Transferring fundamental concepts across contexts is difficult, even when deep similarities exist. This article leverages Desmos-enhanced visualizations to unify conceptual understanding of the behavior of sinusoidal function graphs through envelope curve analogies across Cartesian and polar coordinate systems.

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## Giving Birth to Inferential Reasoning

Too often, statistical inference and probability are treated in schools like they are unrelated. In this paper, we describe how we supported students to learn about the role of probability in making inferences with variable data by building models of real world events and using them to simulate repeated samples.