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Courtney Starling and Ian Whitacre

Introduce your students to a fun and innovative game to encourage precise communication

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Ian Whitacre and Susan D. Nickerson

This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of partial products and to reason flexibly about products. We document how the class overcame the challenge of accounting for partial products in multidigit multiplication, including particular activities and ways of reasoning that facilitated progress. The results provide insights into how prospective elementary teachers' understanding of multidigit multiplication can develop during a content course and how a sequence of instructional activities and practices can productively leverage the resources that they bring in support of that development.

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Ian Whitacre, Robert C. Schoen, Zachary Champagne and Andrea Goddard

Instructional activities designed to encourage relational thinking in primary-grades classrooms can give students advantages when they reason about subtraction.

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

A story problem about borrowing money may be represented with positive or negative numbers and thought about in different ways. Learn to identify and value these different perspectives.

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Karina K. R. Hensberry, Ian Whitacre, Kelly Findley, Jennifer Schellinger and Mary Burr Wheeler

Mathematics teaching that provides opportunities for play embodies many of the Mathematics Teaching Practices described in Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014). PhET interactive simulations (or sims), developed by the PhET Project at the University of Colorado Boulder (, are freely available virtual tools that promote play and exploration in mathematics and science topics for K-16 students.

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

Research on how students make sense of and use the minus sign indicates that students struggle to understand the multiple meanings of this symbol. Teachers can support students in developing a robust understanding of each interpretation.

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

Reasoning about integers provides students with rich opportunities to look for and make use of structure.

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

Find zero minus four? Pascal argued it was impossible! Twenty-first-century students, given the right tools, can solve counterintuitive problems.

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

In a cross-sectional study, 160 students in Grades 2, 4, 7, and 11 were interviewed about their reasoning when solving integer addition and subtraction open-numbersentence problems. We applied our previously developed framework for 5 Ways of Reasoning (WoRs) to our data set to describe patterns within and across participant groups. Our analysis of the WoRs also led to the identification of 3 problem types: change-positive, all-negatives, and counterintuitive. We found that problem type influenced student performance and tended to evoke a different way of reasoning. We showed that those with more experience with negative numbers use WoRs more flexibly than those with less experience and that flexibility is correlated with accuracy. We provide 3 types of resources for educators: (a) WoRs and problem-types frameworks, (b) characterization of flexibility with integer addition and subtraction, and (c) development of a trajectory of learning about integers.

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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.