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## Teaching Is a Journey: From Apathy to Passion

This department provides a space for current and past PK-12 teachers of mathematics to connect with other teachers of mathematics through their stories that lend personal and professional support.

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## Using Number Talks to Compare Fractions

Using this strategy, a teacher facilitates a short conversation during which students verbally explain and justify reasoning. We have found that a coordinated series of number talks supports students’ reasoning when comparing fractions.

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## The Opportunities of No-Solution Problems

When students encounter unusual situations or exceptions to rules, they can become frustrated and can question their understanding of particular topics. In this article, I share some practical tips.

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## Mathematical Explorations: A New Twist on Collaborative Learning

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).

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## Mysterious Subtraction

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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## Changing the Rules to Increase Discourse

A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.

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## Develop Reasoning through Pictorial Representations

Solutions coupled with drawings can illustrate students' understandings or misunderstandings, particularly in the area of proportional reasoning.

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## Capturing Thinking on the Talk Frame

This instructional tool helps students engage in discussions that foster student reasoning, then settle on correct mathematics.

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## Developing Formal Procedures through Sense Making

Allowing students to reason and communicate about integer operations, or any idea, before these ideas are formalized can be an important tool for fostering deep understanding.

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## Informing Practice: Examples as Tools for Constructing Justifications

### research matters for teachers

Research explores how to help students build from, instead of building with, examples when justifying mathematical ideas.