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## Angling for Students' Mathematical Agency

A group of fourth graders went from overreliance on protractors to relying on their own reasoning and understanding of how to measure angles.

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## Reflecting on the Cloud Clock problem

This problem scenario explores the analog clock, a rich source of tasks associated with angles and angle measures. The Cloud Clock problem is an opportunity for students to deepen their understanding of analog clocks, angles, and time and angle measurement. To access the full-size activity sheet, go to http://www.nctm.org/tcm, All Issues. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics.

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## 5 Strategies for Scaffolding Math Discourse with ELLs

Eliminate obstacles to effective classroom communication with these research-tested suggestions.

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## Going Over the Test

To address student misconceptions and promote student learning, use discussion questions as an alternative to reviewing assessments.

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## Angle detectives

This problem scenario presents how a fifth-grade class used logical thinking and spatial reasoning to find the angle measurements of certain polygons without using a protractor. To access the full-size activity sheet, go to http://www.nctm.org/tcm, All Issues. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics.

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## Investigations: EEK—a cockroach!

Why would a person who is terrified of cockroaches use them in a math lesson? The idea for this investigation did not occur to me until I read a newspaper article that described Italian scientist Paolo Domenici's research about cockroaches' escape trajectories. In particular, he found that cockroaches have preferred escape trajectories of 90, 120, 150, and 180 degrees from the source of danger (Domenici et al. 2008). Because this real-world information presents a unique problem-solving context for fifth graders to explore angles formed by clockwise and counterclockwise rotations, I overcame my fear of the creatures to develop this investigation.

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## Effective and Efficient Use of Math Writing Tasks

Almost twenty years ago, the National Council of Teachers of Mathematics (NCTM) published Principles and Standards for School Mathematics (2000), which recommended that teachers should incorporate more writing into their math lessons, claiming that writing helps students “consolidate their thinking” (p. 402) by causing them to reflect on their work. In recent years, various studies point to the many benefits that can be gained by writing in mathematics class (e.g., O'Connell et al. 2005; Goldsby and Cozza 2002). Much research suggests that writing activities, if implemented effectively, can help students enjoy class more (Burns 2005) and can also help them deepen their understanding of the content (Baxter et al. 2002). In addition to benefiting students, student writing benefits teachers as well by providing a clear picture of what their students understand and even deepening understanding of the content for teachers themselves (Burns 2005; Pugalee 1997).

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## Writing Code to Assess Geometric Reasoning

Engage students and promote thinking in a student-centered environment that is rich with technology.

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## Solutions: Mirror, mirror

Each month, this section of the problem solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics. In the problem from the December 2015/January 2016 issue, the task that integrates students' understanding of shapes and their properties and reflections. Students must determine which shapes can be reflected over a line so that the original shape and its reflection form specified figures.

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## Keep on Rollin'

Here is a simple way to turn an ordinary whiteboard into an interactive tool that allows students to design and build pathways along which a sliding object will flow—within certain constraints—to reach its final destination. Students must reason, conjecture, test, conjecture again, and then retest their design features to determine a solution to the presented investigation.