This article describes how fortuitous mathematical moments should be noticed, encouraged, embraced, and capitalized upon.

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### Jere Confrey, Meetal Shah, and Alan Maloney

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.

### Amanda K. Riske, Catherine E. Cullicott, Amanda Mohammad Mirzaei, Amanda Jansen, and James Middleton

We introduce the Into Math Graph tool, which students use to graph how “into" mathematics they are over time. Using this tool can help teachers foster conversations with students and design experiences that focus on engagement from the student’s perspective.

### Esther M. H. Billings and Barbara A. Swartz

For two decades, teacher educators have called for practice-based approaches in teacher education (

Excerpts from discussion threads on the online MyNCTM community

### Amber G. Candela, Melissa D. Boston, and Juli K. Dixon

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.

A cartoon exploring a problem about order of operations is coupled with a full-page activity sheet.

### Kami M. Dupree

Abandon mnemonics and make stronger connections between the operations and properties of arithmetic.

### Sonalee Bhattacharyya, Nama Namakshi, Christina Zunker, Hiroko K. Warshauer, and Max Warshauer

This activity engages students in problem solving while exploring key concepts of number theory, such as divisibility and divisibility tests, place value, fractions, and scale factors.

### Lincoln Peirce

A cartoon involving presidential birth dates is coupled with a full-page activity sheet.