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

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

### Megan Holmstrom

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.

### S. Leigh Nataro

Ear to the Ground features voices from several corners of the mathematics education world.

### Michael S. Meagher, Michael Todd Edwards, and S. Asli Özgün-Koca

Using technology to explore a rich task, students must reconcile discrepancies between graphical and analytic solutions. Technological reasons for the discrepancies are discussed.

### Percival G. Matthews, Patricio Herbst, Sandra Crespo, and Erin K. Lichtenstein

### Meghan Shaughnessy, Nicole Garcia, and Darrius D. Robinson

Using cases from early childhood, elementary, and secondary classrooms, we showcase the work that teachers do to support students in building a collective argument and critiquing an individual’s argument. We identify four areas of work central to teaching students to build and critique mathematical arguments.

### Kevin J. Dykema

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.

### Elise Lockwood, Zackery Reed, and Sarah Erickson

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.

### Paul Christian Dawkins and Dov Zazkis

This article documents differences between novice and experienced undergraduate students’ processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices’ reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting*. Alternative task models* refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. *Accrual of premises* refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of *warrants*, which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form.

### Erin Smith, Jo Hawkins-Jones, Shelby Cooley, and R. Alex Smith

Teachers can use shared story reading with interdisciplinary lessons to simultaneously advance students’ mathematics, literacy, and social-emotional competencies. In this article, we use the book, *Two of Everything*, to illustrate how this routine can be used in K–2 classrooms.