Research focused on learning mathematics in a 2nd language is generally located in individual 2nd-language contexts. In this ethnographic study, I investigated mathematics learning in 4 different second-language contexts: a mainstream classroom, a sheltered classroom for Indigenous students, a welcome class for new immigrants, and a French-immersion classroom. The study was framed by a view of learning as socialization and the Bakhtinian notion of centripetal and centrifugal language forces. I present 7 socialization events that were particularly salient in 1 or more of the classrooms. For each socialization event, I identify various socialization practices. Based on a comparison of socialization practices in the 4 classrooms, I propose a distinction between language positive and language neutral mathematics classrooms. In language positive mathematics classrooms, students’ socialization into mathematics and language includes explicit attention to different aspects of language use in mathematics. In language neutral mathematics classrooms, the role of language in mathematics tends to be implicit.

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### George J. Roy, Jessica S. Allen and Kelly Thacker

In this paper we illustrate how a task has the potential to provide students rich explorations in algebraic reasoning by thoughtfully connecting number concepts to corresponding conceptual underpinnings.

### 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.

### Jen Munson

Understanding mathematics teacher noticing has been the focus of a growing body of research, in which student work and classroom videos are often used as artifacts for surfacing teachers’ cognitive processes. However, what teachers notice through reflecting on artifacts of teaching may not be parallel to what they notice in the complex and demanding environment of the classroom. This article used a new technique, *side-by-side coaching*, to uncover teacher noticing in the moment of instruction. There were 21 instances of noticing aloud during side by side coaching which were analyzed and classified, yielding 6 types of teacher noticing aloud, including instances in which teachers expressed confidence, struggle, and wonder. Implications for coaching and future research on teacher noticing are discussed.

### Salvador R. Vazquez, Bradley A. Ermeling and Gerardo Ramirez

Productive struggle—expending effort to make sense of something beyond one’s current level of understanding—aids in learning mathematics concepts and procedures. In this study, we surveyed 197 parents with children in the 1st to the 5th grade on their beliefs about productive struggle. Beliefs were assessed via questionnaire and rating of a recorded lesson involving productive struggle. Parents also reported how often they helped with math homework and their child’s ability in math. The results show that parents had diverse beliefs about the efficacy of productive struggle, with fathers favoring it more than mothers. A significant relation was found between parents’ beliefs about productive struggle and reports of their child’s ability in math. The findings of this study suggest that for productive struggle to be effective, parents must intentionally facilitate experiences through student-centered approaches. Programs for parents should emphasize specific evidence-based behaviors rather than broad generalizations about increased involvement with homework. Schools and educators should also provide guidance for parents to explain the potential harmful effects of gender stereotypes and parents’ own math anxiety and to teach methods for limiting homework interaction while students grapple with difficult problems.

### Joe F. Allison

When I was in graduate school, my math professor, using a straightedge and a compass, marked off a unit distance and then halved it. He said he could halve the exact ½ again and exactly get ¼. He was leading up to infinite series.

### Steve Ingrassia and Molly Rawding

These are the March 2020 P2P problems from Steve Ingrassia and Molly Rawding.

### Emilie Wiesner, Aaron Weinberg, Ellie Fitts Fulmer and John Barr

Textbooks are a standard component of undergraduate mathematics courses, but research shows that students often do not view textbooks as productive resources to support learning. This article seeks to understand the factors affecting how individuals engage in reading a calculus textbook excerpt and what they learn from reading. To better understand the separate roles of background knowledge and other reading practices, we compare 2 readers: a 2nd-semester calculus student and a nonmathematics STEM professor. We employ the concepts of *sense making* and the *implied reader* to analyze each reader’s experience and a disciplinary literacy perspective to explain the similarities and differences we find between the 2 readers. We propose the concept of *didactical disciplinary literacy*—an adaptation of disciplinary literacy applied to didactical texts—to describe the ways that the professor drew on his identity as a teacher to shape his reading practices.

### Dan D. Meyer

Students use computers outside and inside of math classes and they enjoy them immeasurably more outside of math class. That's because, outside of class, they use their computers in ways that are creative and social. The same can and must be true about computers inside of math class.