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

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### Alice Aspinall

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

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

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

### Linda L. Cooper

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

### 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

### Rick Anderson

A few years later, I had the opportunity to be his colleague during the early years of my own teaching career. By then graphing calculators were ubiquitous in

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.