Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced mathematics doctoral students were given a task-based interview in which they were presented with various sources of evidence in support of a specific mathematical claim and were asked how convinced they were that the claim was true after reviewing this evidence. In particular, we explore why our participants retained doubts about our claim after reading its proof and how they used empirical evidence to reduce those doubts.

### Keith Weber, Juan Pablo Mejía-Ramos, and Tyler Volpe

### Jessica Pierson Bishop, Hamilton L. Hardison, and Julia Przybyla-Kuchek

Responsiveness to students’ mathematical thinking is a characteristic of classroom discourse that reflects the extent to which students’ mathematical ideas are present, attended to, and taken up as the basis for instruction. Using the Mathematically Responsive Interaction (MRI) Framework and data from 11 middle-grades classrooms, we illustrate varied enactments of responsiveness and describe fluctuations in and relationships among different components of responsiveness. We found positive associations between different components of responsiveness, but they were not entirely predictive of one another. Individual classrooms appeared more or less responsive depending on which component was foregrounded. Our findings offer a more comprehensive characterization of responsiveness that documents the intertwined nature of teacher moves and student contributions during all whole-class instruction.

### Rebekah Elliott, Megan Brunner, Elyssa Stoddard, and Jenny White

The authors share a teacher-designed mathematical modeling routine geared to support teachers and to leverage opportunities for their students in learning important modeling practices and mathematical content.

### Dorothy Y. White

Use this activity to support students in working together, recognizing one another’s contributions, and leveraging their mathematical strengths to solve challenging problems.

### Johnnie Wilson

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.

### Larry Buschman and Introduction by: Beth Kobett

From the Archives highlights articles from NCTM’s legacy journals, as chosen by leaders in mathematics education.

### Stacy R. Jones and Carlos Nicolas Gomez Marchant

Through a composite counter-story from the perspective of fifth-grade Raza learners, the authors show how race and language play a role in the mathematics classroom.

### Angela T. Barlow

### Mollie Siegel, Cathy Sinnen, and Penny Smits

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

### Miranda L. Sigmon, Kavin Ming, and Daniel Herring

Disciplinary literacy involves reading, writing, speaking, listening, and thinking in the context of specific disciplines. Word sorts are a way for students to acquire and examine vital features of mathematics vocabulary as they process and organize new content-specific ideas.