Given the prominence of group work in mathematics education policy and curricular materials, it is important to understand how students make sense of mathematics during group work. We applied techniques from Systemic Functional Linguistics to examine how students positioned themselves during group work on a novel task in Algebra II classes. We examined the patterns of positioning that students demonstrated during group work and how students' positioning moves related to the ways they established the resources, operations, and product of a task. Students who frequently repositioned themselves created opportunities for mathematical reasoning by attending to the resources and operations necessary for completing the task. The findings of this study suggest how students' positioning and mathematical reasoning are intertwined and jointly support collaborative learning through work on novel tasks.

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### Anna F. DeJarnette and Gloriana González

### Patricia F. Campbell, Masako Nishio, Toni M. Smith, Lawrence M. Clark, Darcy L. Conant, Amber H. Rust, Jill Neumayer DePiper, Toya Jones Frank, Matthew J. Griffin, and Youyoung Choi

This study of early-career teachers identified a significant relationship between upper-elementary teachers' mathematical content knowledge and their students' mathematics achievement, after controlling for student- and teacher-level characteristics. Findings provide evidence of the relevance of teacher knowledge and perceptions for teacher preparation and professional development programs.

### Katherine E. Lewis

Mathematical learning disability (MLD) research often conflates low achievement with disabilities and focuses exclusively on deficits of students with MLDs. In this study, the author adopts an alternative approach using a response-to-intervention MLD classification model to identify the resources students draw on rather than the skills they lack. Detailed diagnostic analyses of the sessions revealed that the students understood mathematical representations in atypical ways and that this directly contributed to the persistent difficulties they experienced. Implications for screening and remediation approaches are discussed.

## Informing Practice: A Hybrid Perspective on Functions

### research matters for teachers

### Eric Weber

Formal notions of function, which appear in middle school, are discussed in light of how teachers might complement the input-output notion with a covariation perspective.

### Valerie N. Faulkner, Lee V. Stiff, Patricia L. Marshall, John Nietfeld, and Cathy L. Crossland

This study is a longitudinal look at the different mathematics placement profiles of Black students and White students from late elementary school through 8th grade. Results revealed that Black students had reduced odds of being placed in algebra by the time they entered 8th grade even after controlling for performance in mathematics. An important implication of this study is that placement recommendations must be monitored to ensure that high-achieving students are placed appropriately, regardless of racial background.

### Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

### Lawrence M. Clark, Jill Neumayer DePiper, Toya Jones Frank, Masako Nishio, Patricia F. Campbell, Toni M. Smith, Matthew J. Griffin, Amber H. Rust, Darcy L. Conant, and Youyoung Choi

This study investigates relationships between teacher characteristics and teachers' beliefs about mathematics teaching and learning and the extent to which teachers claim awareness of their students' mathematical dispositions. A professional background survey, a beliefs and awareness survey, and a teacher mathematical knowledge assessment were administered to 259 novice upper-elementary and 184 novice middle-grades teachers. Regression analyses revealed statistically significant relationships between teachers' beliefs and awareness and teachers' mathematical knowledge, special education certification, race, gender, and the percentage of their students with free and reduced meal status. This report offers interpretations of findings and implications for mathematics teacher education.

### Jessica Pierson Bishop, Lisa L. Lamb, Randolph A. Philipp, Ian Whitacre, Bonnie P. Schappelle, and Melinda L. Lewis

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one's learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6-10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

### Kevin C. Moore

A growing body of literature has identified quantitative and covariational reasoning as critical for secondary and undergraduate student learning, particularly for topics that require students to make sense of relationships between quantities. The present study extends this body of literature by characterizing an undergraduate precalculus student's progress during a teaching experiment exploring angle measure and trigonometric functions.

### Kathleen Lynch and Jon R. Star

Although policy documents promote teaching students multiple strategies for solving mathematics problems, some practitioners and researchers argue that struggling learners will be confused and overwhelmed by this instructional practice. In the current exploratory study, we explore how 6 struggling students viewed the practice of learning multiple strategies at the end of a yearlong algebra course that emphasized this practice. Interviews with these students indicated that they preferred instruction with multiple strategies to their regular instruction, often noting that it reduced their confusion. We discuss directions for future research that emerged from this work.