I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

Joan M. Kenney

**Ch 5: Discourse in the Mathematics Classroom**

*I really really liked this chapter so I'm pretty much typing the whole thing. But oh well!*

For the purposes of this chapter, I am defining

*discourse*as the genuine sharing of ideas among participants in a mathematics lesson, including both talking and active listening.

**What Discourse Looks Like**

*Traditional*. Prompts from the teacher generally lead to a preplanned answer.*I do this.**Probing*. Stem from the teacher's desire to hear about students' thinking, rather than from a need to move students along a planned route.*I do this too.**Discourse-Rich*. "Students must...learn to question and probe one another's thinking in to clarify underdeveloped ideas.*Not this.*

When the classroom climate fosters genuine student discourse, students react to their classmates; ideas, asking questions and checking for understanding. Conversations continue without the need for teacher participation or supervision.

**Creating Discourse-Friendly Classrooms**

- Arrange furniture so that students can easily turn to see each other. They must be able to speak and listen to classmates.
*How can I do this with 24 desks?* - Encourage students to direct questions and explanations to the class, rather than to the teacher.
*I usually redirect the question back to the class but have not actually told to ask the class instead of me.* - When recording ideas on chalkboard or chart paper, use the students' words as much as possible. This is a matter of respecting their ideas.
- Try not to repeat or paraphrase everything students say. Paraphrasing can give the impression that the student is being corrected and may indicate to others that they don't need to listen unless the teacher speaks.
- Remind students that conversation is a two-way operation requiring both talking and listening.
- Stand in a variety of spots. As students turn to look at you, their views of the classroom and their positions relative to classmates will shift.
- Give students time to think. Wait time or brief writing moments help students to solidify ideas and formulate good questions.
- Arrange lessons so that students have a product to share as they explain their thinking. They might illustrate ideas on chart paper or overhead transparencies or demonstrate by using manipulative materials.

*This is exactly what I am afraid of!*

Here are several ways to let student ideas take the lead in class:

- Involve students in engaging and challenging problems.
- Ask open questions to stimulate student thinking. (Examples: "What does this make you wonder about?" "Are there patterns?" "Is this logical?" "Can we estimate a solution?")
- Listen carefully to student responses.
- Train students to listen to their classmates; observations by asking questions that engage.
- Honor diverse ideas, methods, and example from varied sources.
- Honor ideas even if they're incorrect. Do not quickly agree or disagree. Students will come to realize that you are giving them time to think and to justify. Often, as students explain erroneous thinking, they uncover their own errors or classmates step in to clarify or correct them.
- Encourage mathematical arguments between students.
- Remember, confusion is okay. Some of the best learning happens when we sort out what it is that has pushed us a bit out of balance. Be sure students know you are deliberately letting them be confused and that this is based upon your knowledge of how people learn, as this tactic may not match what previous teachers have done.
*And they will think you are not doing your job!* - Take time to let students share different problem-solving methods. Even when a correct solution has been shown, ask if there are other ways to do the problem. This helps to deepen understanding and makes students more willing to work with their own strategies, rather than thinking there is only one correct method.
- Capture teachable moments. Tangents are good.
- Decide how much leadership your students need. Let students' ideas lead but this doesn't mean that the class just moves without teacher direction.

**Tips for Facilitating Classroom Discussion**

- Focus students attention on a problem, puzzle, figure, process, question, or set of numbers. Stimulate discussion by asking the following types of questions:
- What do you notice?
- Do you see any patterns?
- What is similar?
- What is different?
- How do you think this works?
- Why does this work/look this way/give this result?
- What questions do you have?
- What can we do with this information?
- What do you want to know?
- Rather than rephrasing their responses, ask, "How do you want to say this?"
- When observations or questions are brought up by one student, ask, "What do the rest of you think about this idea? Does it make sense?" Encourage them to consider other examples that would show that the observation is or is not always true.
- Motivate students to search for patterns, delve deeper, and generalize.
- If students are making mistakes or doing something awkwardly, ask them "Is there an easier or more efficient way?" or "In what other ways could this be done?" rather than telling them how to do it.
- If students have difficulty thinking about a concept, suggest examples to consider or play devil's advocate. Ask "What if?" questions.
*I love playing devil's advocate!* - Counter questions with questions instead of explanations.
*(Students hate this!)*People tend to blank out when one person asks a question and the teacher immediately gives an explanation. - Even when a solution is successful, take time to ask whether anyone did the problem a different way or discarded an idea. Help students to build confidence in their own ideas , knowledge, and insights by showing that problems can be solved in a variety of ways.

**Discourse and Problem Solving**

As students explain their thinking, others can see connections and the usefulness of different methods.

Summarizing and labeling the strategies make them memorable, as does naming them in honor of the students who came up with them (e.g., Mary's Method, Theo's Theory, Pedro's Plan).

**Discourse and Vocabulary**

During lessons in which students first encounter a new concept, teachers should encourage them to describe ideas in their own words before introducing the specialized terms.

**Summary**

As we change our teaching, it is important to realize that we expect students to change with us; they also have new responsibilities.

Students may well be uncomfortable shifting from passive observers to active learners. How can we help them learn these new skills?

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