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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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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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Victoria R. Jacobs, Lisa L. C. Lamb and Randolph A. Philipp

The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.

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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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Annette Ricks Leitze, Pamela Cintas, Erica L. Granger, Lisa Knauff, David Ryan Lamb, Jodi A. Morrow and Heather Dockter Wells

This department features collections of short activities focused on a monthly theme. These articles aim for an inquiry or problem-solving orientation that includes at least four activities each for grade bands K–2, 3–4, and 5–6. This month's sports theme is designed around many mathematical strands, including problem solving, mathematical communication, geometry, and measurement.

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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.

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Randolph A. Philipp, Rebecca Ambrose, Lisa L.C. Lamb, Judith T. Sowder, Bonnie P. Schappelle, Larry Sowder, Eva Thanheiser and Jennifer Chauvot

In this experimental study, prospective elementary school teachers enrolled in a mathematics course were randomly assigned to (a) concurrently learn about children's mathematical thinking by watching children on video or working directly with chil-dren, (b) concurrently visit elementary school classrooms of conveniently located or specially selected teachers, or (c) a control group. Those who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not. Furthermore, beliefs of those who observed in conveniently located classrooms underwent less change than the beliefs of those in the other groups, including those in the control group. Implications for assessing teachers' beliefs and for providing appropriate experiences for prospective teachers are discussed.