Problems to Ponder provides 28 varying, classroom-ready mathematics problems that collectively span PK–12, arranged in the order of the grade level. Answers to the problems are available online. Individuals are encouraged to submit a problem or a collection of problems directly to mtlt@nctm.org. If published, the authors of problems will be acknowledged.

# Browse

### Rick Anderson and Peter Wiles

Recognizing the complex nature of students’ geometric reasoning, we present guidelines and suggestions for implementing a Guess My Shape minilesson that focuses students’ attention on properties and attributes of geometric shapes.

### Derek A. Williams, Kelly Fulton, Travis Silver, and Alec Nehring

A two-day lesson on taxicab geometry introduces high school students to a unit on proof.

### 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.

### Lindsay Vanoli and Jennifer Luebeck

Engaging mathematics students with peers in analyzing errors and formulating feedback improves disposition, increases understanding, and helps students uncover and correct misconceptions while informing opportunities for targeted instruction.

### Susan Ahrendt, Debra Monson, and Kathleen Cramer

Examine fourth graders’ thinking about the unit, partitioning, order, and equivalence on the number line and consider ways to orchestrate mathematical discussions through the Five Practices.

### Josephine Derrick and Laurie Cavey

Challenging to learn, proof can be equally challenging to teach. Insights gleaned about students’ conceptions of proof from 10 high school students who completed four proof-related tasks during one-on-one interviews led to a few instructional takeaways for teachers.

### Nicholas J. Gilbertson

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.

### Nat Banting and Chad Williams

This article examines the mathematical activity of five-year-old Liam to explore the difference between the mathematics games designed for children and the children's games that emerge through playful activity. We propose that this distinction is a salient one for teachers observing mathematical play for evidence of mathematical sense making.