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## Defining Supports Geometry

The process of developing definitions is underemphasized in most mathematics instruction. Investing time in constructing meaning is well worth the return in terms of the knowledge it imparts.

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## Coming Full Circle

Teacher candidates develop processes and proficiencies articulated in the Common Core State Standards for Mathematics by investigating connections between and among geometry concepts.

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## Become an Escher Sleuth

Students discover why certain regular polygons tessellate and how M. C. Escher translated and rotated modifications of polygons to create tessellations.

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## If Only Clairaut Had Dynamic Geometry Tools

Using Clairaut's historic-dynamic approach and dynamic geometry tools in middle school can develop students' conceptual understanding before they encounter formal proof in geometry.

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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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## 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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## Photon Translations

“Photon Translations” is a piece of installation art that incorporates all the disciplines of science, technology, engineering, art, and mathematics (STEAM) and is on display at the Stoneleigh-Burnam School in Greenfield, Massachusetts. Installation art is just what it implies: art that is created for the purpose of being installed at a particular location. Sometimes the artwork can be interactive, allowing audience members to participate in the experience in some dynamic way. Terry Marashlian of Northfield, Massachusetts, created this artwork. The author collaborated with Marashlian on creating the musical chimes.

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## Harnessing Students' Misconceptions and Fragile Understandings

Use the Reversibility, Flexibility, Generalization (RFG) questioning framework to develop robust, multifaceted, interconnected, and lasting mathematical comprehension.

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## August 2016 Problems

A monthly set of problems is aimed at a variety of ability levels.

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## October 2016 Problems

A monthly set of problems is aimed at a variety of ability levels.