An unsolved problem gets elementary and middle school students thinking and doing mathematics like mathematicians.

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## Challenging Students to Experience Mathematics as Mathematicians

### Jenna R. O’Dell, Cynthia W. Langrall, and Amanda L. Cullen

## Problems to Ponder

### The Math Learning Center Content Development Team and J. Michael Shaughnessy

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.

## 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.

## “Counting” on Quantitative Reasoning for Algebra

### Lori Burch, Erik S. Tillema, and Andrew M. Gatza

Use this approach to developing algebraic identities as a generalization of combinatorial and quantitative reasoning. Secondary school students reason about important ideas in the instructional sequence, and teachers consider newfound implications for and extensions of this generalization in secondary algebra curricula.

## High School Students’ Understanding of Proof

### 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.

## Envelope Curves Unify Sinusoidal Graphing

### Christopher Harrow and Ms. Nurfatimah Merchant

Transferring fundamental concepts across contexts is difficult, even when deep similarities exist. This article leverages Desmos-enhanced visualizations to unify conceptual understanding of the behavior of sinusoidal function graphs through envelope curve analogies across Cartesian and polar coordinate systems.

## GPS: Working Backward with Data

### Matt Enlow and S. Asli Özgün-Koca

This month's Growing Problem Solvers focuses on Data Analysis across all grades beginning with visual representations of categorical data and moving to measures of central tendency using a “working backwards” approach.

## Modeling a Bouncing Ball with Exponential Functions and Infinite Series

### Tim Erickson

We modify a traditional bouncing ball activity for introducing exponential functions by modeling the time between bounces instead of the bounce heights. As a consequence, we can also model the total time of bouncing using an infinite geometric series.

## Students' Understanding of Counterexamples

### Rebecca Vinsonhaler and Alison G. Lynch

This article focuses on students use and understanding of counterexamples and is part of a research project on the role of examples in proving. We share student interviews and offer suggestions for how teachers can support student reasoning and thinking and promote productive struggle by incorporating counterexamples into the classroom.

## Technology Continues to Evolve

Over the past 100 years, technology has evolved in unprecedented fashion. Calculators, computers, and smart phones have become ubiquitous, yet school mathematics experiences for many children still remain without many powerful technological tools for the exploration of mathematics. We consider the evolution of some tools as we imagine a future.