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Jonathan A. Supovitz, Caroline B. Ebby, Janine T. Remillard, and Robert Nathenson

In this article, we use a two-dimensional assessment to examine the experimental impacts of a mathematics learning trajectory–oriented formative assessment program on student strategies for problems involving multiplication and division. Working from the theory that the development of students’ multiplicative reasoning involves improvements in both problem-solving accuracy and sophistication of strategies used to solve problems, we designed an assessment instrument to measure both dimensions of student learning. The instrument was used to measure the impact of the Ongoing Assessment Project (OGAP), which develops teachers’ capacity to regularly assess student thinking in relation to a learning progression to develop instructional responses that are based on evidence of student thinking. The results showed significant impacts of OGAP on both students’ problem-solving accuracy and the sophistication of their strategy. The findings suggest that capturing both dimensions of students’ multiplicative reasoning offers important information for researchers and program designers who seek to understand different dimensions of student mathematics performance.

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Amanda L. Cullen and Rick Anderson

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Cheng-Yao Lin and Aviva Hamavid

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

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Jonathan D. Bostic, Brooks Vostal, and Timothy Folger

All students have strengths that can be leveraged through universally designed instruction.

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Danny Bernard Martin and Introduction by: Robert Q. Berry III

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

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Jo Boaler, Tanya LaMar, and Cathy Williams

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Darrell Earnest and John Chandler

This article investigates the interplay of time words with how children position hands on an analog clock. Using a mathematics discourse framework (Sfard, 2008), we analyzed how students interpreted precise (e.g., 2:30) and relative (e.g., half past 11) times, finding that particular words are dynamically interwoven with activity. Interviews with students in Grades 2 and 4 revealed that different prompts led to different narrative descriptions about time on the clock, with precise times leading to whole-number descriptions and relative times to part-whole descriptions consistent with fractions. Subsequent analysis of assessment performance for students across Grades 2–5 corroborated that specific time prompts led to particular clock interpretations. Implications for theory and the K–12 treatment of time measure are discussed.

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Nicholas H. Wasserman and William McGuffey

This article explores secondary teachers’ opportunities to learn from an innovative real analysis course, as reflected in their actual classroom teaching. The course used cases of teaching as a site for applying mathematics and developing pedagogical mathematical practices. This article explores particular teaching moments in (N = 6) secondary teachers’ classrooms, and the attributions they gave for why they engaged in those teaching practices. Teachers engaged in instructional practices that exemplified course objectives, and their attributions for their actions contribute a teacher perspective on opportunities to learn in teacher education from (advanced) mathematical coursework. Results highlight cases of teaching and modeled instruction as catalysts of change and as opportunities to develop pedagogy from mathematical activity, and vice versa.

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Megan Holmstrom and Megan Korponic

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.