• 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. 

• 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.

• 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.

• 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.

• Technical symbols such as ∑, ≤ , or Δ. These signs, also known as logograms, stand for whole words but have no sound-symbol relationship for students to decode.
• Technical vocabulary- words such as rhombus, hypotenuse, and integer, which are rarely used in everyday conversation.
• The assignment of special definitions to familiar words such as similar and prime.
• Subtle morphology (one hundred, hundreds-place, hundredths) and the use of "little words" (prepositions, pronouns, articles, and conjunctions) in a technical syntax so precise that meaning is often obscured rather than clarified.  

• In what other ways might the students have attempted to model the problem if the teacher hadn't offered the initial suggestion? Let students try first, then offer suggestions?
• How might using actual three-dimensional models (i.e., a set of nested measuring cups) before using two-dimensional representations alter the transfer of learning? Always teach from concrete to abstract? 
   

• How can the teacher help [William] move beyond a single conceptual image and experiment with new metaphors and visual models?
• When are models useful, and when do they get in the way of new learning?
  
  

• For how long, and to what depth, should a teacher continue to probe in order to get to the logical and intuitive root of confusion?

• How can teachers become more aware of the mathematical locutions embedded in their classroom conversations? Pre-teach all mathematical terms, especially if they are also used in a nonmathematical context? Think literally.
  

• Would this particular mathematical conversation have occurred if the students didn't have a graphic "prop"?
• How did the visual representation act as a catalyst for student discourse?
• How might conversations about a graphic display encourage students to put their mathematical perceptions into writing?
• Now that a conversation had begun on the topic of exponential decay, how might the teacher build on the richly descriptive terms that the students created to press for more precise mathematical terminology?
  

• What distinction was Benjamin making between the terms thirds, three folds, and dividing in to three parts?
• How might students conceptualize fourths differently, depending on whether they are asked to label the strip's segments (1/4, 2/4, 3/4, 4/4) or its folds (1/4, 2/4, 3/4)?
• What consideration might the teacher need to give to precision of language when providing directions for this task?
• What role might peer-to-peer discourse have played in helping Benjamin test his conjecture?
• What questions might a teacher ask to check more deeply for understanding if, at first glance, a student's thoughtfully done work is apparently correct? Ask them how they would explain it to their grandma or ask them to create a new example?
  

• From what assumptions might Benjamin have been working when he approached this new concept?
• Where was there a language-based component in his confusion?
• What do the phrases "three-tenths" "three and tenths", and "thirds and tenths" mean mathematically? What do they seem to mean to Benjamin?
• In a roomful of 20 students, how difficult is it to hear the subtleties of different word forms?
• What might Benjamin's drawing imply about what he heard?
• How could the teacher use Benjamin's drawing to encourage him to express his personal understanding in words?
• How might making additional sketches have helped Benjamin communicate what he head and understood?
  

• What windows to Sarah's thinking do her cocoa, milk, and cheese drawings provide?
• How did Sarah seem to understand division in general? What about dividing in situations when the quotient is less than one?
• What did Sarah believe about the decimal point?
• What was Sarah hearing, and how did this influence the manner in which she conceptualized decimal fractions?
• What is appropriate calculator use for students beginning to work with fractions and decimals?
• Would language-based cues, such as the teacher's asking how to divide a 12-pound wheel or chunk of cheese among 24 families, suggest useful visual models? Or might metaphors create further confusion? If I was Sarah and the words wheel or chunk were mentioned, I would have drawn wheels or chunks instead of dots but I still would have drawn 5 wheels or chunks in each box.
• What might the teacher do ti reinforce the use of proper terminology when students are working with decimal fractions?
  

1. Make the students more aware that they're "speaking mathematics" in class,
2. Show a need for greater precision in the students' use of mathematical language, and 
3. Suggest areas in which directions and explanations should be more clearly phrased.
   

• Combine verbal with visual.
• Monitor your language for words with double meanings.
• Assume positive intent to understand even from silly questions or offhand remarks.
• Ask students to move from drawing detailed pictures to simpler shapes, another step toward abstraction.
• Consider the sequence of representations students select. Does it promote depth of mathematical thinking?
• Actively point out connections to other representations so that students become fluent in translation from one representation to another.
  
  

• Articulate and enunciate!
• Keep writing utensils available if you expect students to draw.
• Extra time: putting ideas on paper takes more time than talkingg.
• For open-ended questions, suggest drawing a diagram first in order to have something concrete to write about.
• Make copies of important textbook pages so that students can highlight, draw on, underline, and write notes in the margin to better interact with the text.
• Have students use graphic organizers that incorporate both words and pictures. 
• Designate a section of notes/worksheets for drawing, showing that you value and expect to see visual thinking.

• The problem must be appropriate for the students who are going to be writing about it.
• The students must know how to use blocks, diagrams, pictures, or grids to work out their solutions before writing about them.
• The students must have confidence in their ability to respond to the problem as individuals. They must think of themselves as successful mathematics learners.
• The students must feel comfortable sharing their answers without fear of being ridiculed. This means that the teacher and other students have to accept all responses as worthy of discussion.
• The problem must be discussed with the whole class, and all strategies must be reported. 

...Allowing students to grapple with their ideas and develop their own informal means of expressing them can be an effective way to foster engagement and ownership. (p. 63)

• The words of and off cause a lot of confusion in solving percentage problems, as the percent of something is quite distinct from the percent off something.
• The word a can mean “any” in mathematics. When asking students to “show that a number divisible by 6 is even,” we aren't asking for a specific example, but for the students to show that all numbers divisible by 6 have to be even.
• When we take the area “of” a triangle, we mean what the students think of as “inside” the triangle.
• The square (second power) “of” the hypotenuse gives the same numerical value as the area of the square that can be constructed “on” the hypotenuse.

• What is the major concept?
• How can I help students connect this concept to their lives?
• Are there key concepts or specialized vocabulary that needs to be introduced because students could not get meaning from the context?
• How could we use the pictures, charts, and graphs to predict or anticipate content?
• What supplemental materials do I need to provide to support reading?

• What would you be doing in that situation?
• Does this make sense?
• What does the picture/graph/chart tell you?
• How does the title connect to what we're reading?
• Why are these words in capital letters?
• Why is there extra white space here?
• What does that word mean in this context?

Figure 2.4. Guided Reading Example