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## Postscript: Knock 'em down

This simple dice game supports students' development of flexibility with numbers, the properties of the four operations (+, −, ×, ÷), and the order of operations. It requires only dice and a game board for each player.

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## Building the Optimal Learning Environment for Mathematics

Teachers can use Universal Design for Learning (UDL) in their math classroom to anticipate potential barriers, know which tools will engage students, and provide safe spaces for learning.

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## Solving Equations: Exploring Instructional Exchanges as Lenses to Understand Teaching and Its Resistance to Reform

Many research studies have sought to explain why NCTM's vision for mathematics classrooms has not had greater impact on everyday instruction, with teacher beliefs often identified as an explanatory variable. Using instructional exchanges as a theoretical construct, this study explores the influence of teachers' institutional positions on the solving of equations in algebra classrooms. The experimental design uses surveys with embedded rich-media representations of classroom interaction to surface how teachers appraise correct solutions to linear equations where some solutions follow suggested textbook procedures for solving linear equations and others do not. This paper illustrates the feasibility of studying teaching with rich-media surveys and suggests new ways to support changes in everyday mathematics teaching.

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## Technology-Powered Assessments for Learning

Making student thinking visible improves the feedback loop for the learner, the teacher, and the entire class.

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## What my thinking looks like

Challenge your students to create a visual representation of their thinking that can be captured in a single photograph. First, pose a task based on a real-world situation that is familiar to your students.

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## Build those numbers

Students are challenged to use numbers rolled on three dice to write expressions for values 1–12, using as few rolls as possible. Postscript items are designed as rich grab-and-go resources that teachers can quickly incorporate into their classroom repertoire with little effort and maximum impact.

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Several detours prompted me to find time in an overcrowded school day to incorporate important, powerful, daily, whole-class application of fact strategies.

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## Arithmogons

Easy-to-design puzzles that encourage mathematical reasoning and promote numerical fluency, arithmogon puzzles are simple: Add the numbers in two circles to get the number in the square. Every month, this final page of the journal highlights a quick game, puzzle, activity, or instructional strategy and suggestions for teachers of different grade bands to use the idea in the classroom.

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## Digital Date Equations

This article shares ideas for using the calendar date to increase students' mental mathematics and problem-solving skills. Postscript items are designed as rich grab-and-go resources that any teacher could quickly incorporate into his or her classroom repertoire with little effort and maximum impact.

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## Oki-Doku: Number Puzzles

Consider using these problems to help students develop number and operation sense in a simple and fun way.