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.

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### 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.

### Gayle Millsaps

Preservice elementary school teachers (PSTs) often have difficulty understanding hierarchical (i.e., class inclusion) relationships between geometric shapes. In particular, PSTs' predisposition to place squares and rectangles in separate categories can be attributed to their concept images. Although the larger mathematics community prefers the hierarchical definitions of special quadrilaterals, the concept images of special quadrilaterals such as squares and rectangles that PSTs develop in their early experiences contribute to a preference for partitional definitions. This study examines the benefits and limitations of using the Shape Makers curriculum unit to modify preservice teachers' concept images and their definitions of special quadrilaterals.

### Heidi L. Fleharty and Carolyn Pope-Edwards

Sixty-three teachers in a K–3 mathematics specialist certificate program conducted family projects in order to improve their skills in partnering with families around mathematics. Past studies have indicated that family involvement in children's education has many positive influences on academic achievement; however, parents' discomfort with math, and teachers' discomfort with working with parents, may be obstacles. The purpose of the present study was to examine 2 years of teachers' mathematical family projects and describe the types of projects chosen, the risks and benefits of these projects, and the quality of the parent–child interaction. It was found that the teachers implemented a variety of projects that promoted parent participation in mathematics. Teachers were also able to utilize a cycle of inquiry to examine the progress of their project. The results showed that teachers were able to create a strong connection between the math classroom and the home environment of the child, as shown, for example, by findings related to the themes of *home–school connections* and *mathematics curriculum of the home*.

### Jennifer Noll and J. Michael Shaughnessy

Sampling tasks and sampling distributions provide a fertile realm for investigating students' conceptions of variability. A project-designed teaching episode on samples and sampling distributions was team-taught in 6 research classrooms (2 middle school and 4 high school) by the investigators and regular classroom mathematics teachers. Data sources included survey data collected in 6 research classes and 4 comparison classes both before and after the teaching episode, and semistructured task-based interviews conducted with students from the research classes. Student responses and reasoning on sampling tasks led to the development of a conceptual lattice that characterizes types of student reasoning about sampling distributions. The lattice may serve as a useful conceptual tool for researchers and as a potential instructional tool for teachers of statistics. Results suggest that teachers need to focus explicitly on multiple aspects of distributions, especially variability, to enhance students' reasoning about sampling distributions.

### Aki Murata, Laura Bofferding, Bindu E. Pothen, Megan W. Taylor, and Sarah Wischnia

This study investigated how elementary teachers in a mathematics lesson study made sense of student learning, teaching, and content, as related to using representations in teaching multidigit subtraction, and how changes occurred over time in their talk and practice. The lesson-study process paved a group talk path along which teacher talk shifted from superficial to deeper consideration of student learning. By providing a context in which interactions of diverse ideas drove teacher learning, lesson study facilitated teachers making connections between the craft knowledge of teaching and scholarly knowledge. Individual teacher talk paths varied within the group path, and one teacher's learning path and the interaction of different learning paths is discussed.

## Connecting Research to Teaching: Why Did You Do That?

### Reasoning in Algebra Classrooms

### Daniel Chazan and Dara Sandow

Secondary school mathematics teachers are often exhorted to incorporate reasoning into all mathematics courses. However, many feel that a focus on reasoning is easier to develop in geometry than in other courses. This article explores ways in which reasoning might naturally arise when solving equations in algebra courses.