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These examples illustrate some of the elements of a classroom-whether it be elementary, middle, or high school-in which high-quality mathematics instruction and learning are taking place.

**What Are Students Doing?**

- Actively engaging in the learning process

- Using existing mathematical knowledge to make sense of the task

- Making connections among mathematical concepts

- Reasoning and making conjectures about the problem

- Communicating their mathematical thinking orally and in writing

- Listening and reacting to others' thinking and solutions to problems

- Using a variety of representations, such as pictures, tables, graphs, and words, for their mathematical thinking

- Using mathematical and technological tools, such as physical materials, calculators, and computers, along with textbooks and other instructional materials

- Building new mathematical knowledge through problem solving

**What Is the Teacher Doing?**

- Choosing "good" problems-ones that invite exploration of an important mathematical concept and allow students the chance to solidify and extend their knowledge

- Assessing students' understanding by listening to discussions and asking students to justify their responses

- Using questioning techniques to facilitate learning

- Encouraging students to explore multiple solutions

- Challenging students to think more deeply about the problems they are solving and to make connections with other ideas within mathematics

- Creating a variety of opportunities, such as group work and class discussions, for students to communicate mathematically

- Modeling appropriate mathematical language and a disposition for solving challenging mathematical problems

Although the content changes as students progress through the grades, the statements mentioned above for teachers and students are common characteristics that you should see in any mathematics classroom. In all these scenarios, a climate has been created that supports mathematical thinking and communication. Students are accustomed to explaining their ideas and questioning solutions that might not make sense to them. Students are not afraid to take risks and know that it is acceptable to struggle with some ideas and to make mistakes. The teacher responds in ways that keep the focus on thinking and reasoning rather than only on getting the right answer. Incorrect answers and ideas are not simply judged wrong. Teachers help students identify parts of their thinking that may be correct, sometimes leading students to new ideas and solutions that are correct.

From *Administrator's Guide: How to Support and Improve Mathematics Education in Your School*. Copyright © 2003 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be distributed electronically without written permission from NCTM.