Two classic hands-on tasks address conceptual understanding of functions. The tasks center student discourse and rough draft mathematics as students grapple with the relationship between input and output.

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## Filling Vases and Making Tanks

### Jana Dean

## My Funny (Mathematical) Valentine

### Joanna Papakonstantinou

Students create clever mathematical Valentine’s Day cards.

##
S^{3}D: Small-Group, Student-to-Student Discourse

### Sarah Quebec Fuentes

Learn about strategies and tools to examine and improve your practice with respect to fostering equitable small-group, student-to-student discourse.

## Math Chat Opportunities Abound!

### Alice Aspinall

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

## Student Engagement with the “Into Math Graph" Tool

### 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.

## Asked & Answered

Excerpts from discussion threads on the online MyNCTM community

## Discourse Actions to Promote Student Access

### 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.

## Mathematical Explorations: A New Twist on Collaborative Learning

### classroom-ready activities

### Stephanie M. Butman

Research on students' learning has made it clear that learning happens through an interaction with others and through communication. In the classroom, the more students talk and discuss their ideas, the more they learn. However, within a one-hour period, it is hard to give everyone an equal opportunity to talk and share their ideas. Organizing students in groups distributes classroom talk more widely and equitably (Cohen and Lotan 1997).

## For ELLs: Vocabulary beyond the Definitions

### Nancy S. Roberts and Mary P. Truxaw

A classroom teacher discusses ambiguities in mathematics vocabulary and strategies for ELL students in building understanding.

## Improving Student Reasoning in Geometry

### Bobson Wong and Larisa Bukalov

Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof.