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• Author or Editor: Ian Whitacre
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## Attending to Precision with Secret Messages

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

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## Relational Thinking: What's the Difference?

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

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## Informing Practice: Developing Symbol Sense for the Minus Sign

### research matters for teachers

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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## Unlocking the Structure of Positive and Negative Numbers

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

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## Engaging Students with Mathematics through Play

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 (http://phet.colorado.edu), are freely available virtual tools that promote play and exploration in mathematics and science topics for K-16 students.

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## A Cross-Sectional Investigation of Students' Reasoning About Integer Addition and Subtraction: Ways of Reasoning, Problem Types, and Flexibility

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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## Obstacles and Affordances for Integer Reasoning: An Analysis of Children's Thinking and the History of Mathematics

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.