In math thinking conversations the teacher attempts to learn as much as possible about the child's thinking. The teacher presents problems, asks questions, observes carefully, listens, and then, in reaction to what the child says and does, revises the questions in order to clarify the problem or to probe more deeply into the child's reasoning. The teacher’s questioning is flexible, attempting to respond to the individual child's needs and idiosyncrasies. The goal is to find out how the child arrives at the answer and how the child thinks. Being aware of the child's right and wrong answers is not enough; the teacher needs to understand why the child gets the right and wrong answers. It may be that the right answers are superficial and that the wrong ones are based on a sensible logic. And of course the ultimate goal is for the teacher to use the results of the math thinking conversion to guide and improve instruction.

It is hard to conduct a math thinking conversation without a plan, so your conversation will not be completely spontaneous. Say, for example, that you want to learn more about the child’s counting above 20. You then need to have in mind some ideas about how you are going to determine that. Before you have a math thinking conversation with an individual child or a group of children, you may wish to prepare a few tasks or problems that can help you. Here are a few ideas:

To establish a mood of confidence, start with a task that is reassuringly easy.
Let the child begin by doing well, by demonstrating his competence. If the child seems to have trouble with the initial task, change it right away; make it simpler so that the child can succeed without too much trouble.

Try to make the task as concrete as possible. 
Don't just ask the child how we can figure out the answer to 3+2; present the child with manipulatives like blocks, chips, paper and pencil, or other materials which could be used to solve the problem. Young children unable to work with an abstract task often can achieve success when the same task is represented with concrete objects.

Focus on what the child can do.
Begin with the assumption that the child is capable of doing some interesting math. Even if the child has had difficulty with some topics, don't assume that he or she knows nothing. Try to locate the child's strengths; focus on what the child does know.

Communicate that the focus is on thinking, not merely on correctness of the answer.
You need to reassure the child that you are interested in how she solves problems, not only on whether the answer is right or wrong. You can do this by saying things like: “That’s an interesting way to solve that problem” or “Can you think of another way to solve the problem?”

The heart of the math thinking conversation is finding out what the child knows and how the child thinks. Doing this requires several strategies:

Observe carefully what the child is doing as he or she solves a problem.
You can get some hints about the child's thinking by observing how she is using her fingers, or what she says when she whispers to herself. Also notice the child's pauses, pace, gestures, and facial expression. Children's behavior often reveals a good deal about their thoughts.

Ask the child how he or she solved the problem and wait for the answer.
The fundamental question of thinking conversations is: "How did you do that?"  There are of course many variations on the question.  "Can you do it out loud?"  "How would you help a friend do it?" "How could you explain to a friend?" "How did you figure that out?" "Can you show me how you did it?" Of course, some children find it hard to do this; and it is often easier for older than it is for younger children to talk about their thinking. You have to be patient and let children feel comfortable by giving them time to develop an explanation; it may take some time before children are prepared to tell you much about their thinking. But practice makes perfect: with your guidance, children can learn to become aware of and express their thinking. Promoting this kind of talk is an important goal for early mathematics education.

Ask the child to explain why the answer is right.
Sometimes the child has difficulty talking about a method of solution, or may not even realize that there is a method. For example, a child may say that two and two is four, but does not know how he got the answer or that there could be a sensible way of getting the answer aside from memorizing it. When this happens, you can ask the child to justify the answer, to prove that it is correct. "How do you know that two and two is four? How could you prove it to me?  How can you show me that you are right?" This type of questioning helps the child to talk about his method if he has one and to reveal his understanding.

Be responsive and flexible.
One key feature of the math thinking conversation is to follow the child's response wherever it leads. If the child uses an unusual method to solve a problem, explore the child's method; don't teach the "right" way to do the problem. Use the child’s language too (such as using and rather than plus); this will foster communication. Responsiveness to the child may not only lead you to the underlying thought; it also may contribute to the child's positive motivation.

Always think about why the child did what she did.
While listening to the child's responses and observing her behavior, always be thinking about the underlying thinking. Did the child get the wrong answer because she did not understand your instructions? Did she get the answer by counting or by simply remembering the number facts? And once you have an idea about the child's thinking, test it. If you think the child was counting, then give a problem where that is required. Analyzing the child's behavior is one of the most interesting, but also one of the hardest parts of having these conversations. Just remember that anything really interesting or worthwhile is not likely to be easy, at least at first.

Suspend the tendency to correct and "teach".  
Your ultimate goal, of course, is to help the child learn. But you must often suspend that goal during math thinking conversations. You must refrain from correcting and "teaching." Don't say, "No, you're wrong; this is how to do it." Don't show your disapproval by grimacing or scowling at the child. Instead, you should convey the idea that you are interested in what the child said, even if it is wrong, and that you want to find out more about it. If you wish to have deeper understanding of the child's thinking processes, you must allow erroneous beliefs and unproductive strategies to be expressed freely and without concern for your response. Later on, you will have an opportunity to use what you learned in the interview to help the child.

It often takes more work beyond the initial question to draw out a child's thinking. You can try these approaches to help you capture the details of the child's thinking more fully:

Rephrase the question; change the task. 
Suppose the child has not been able to solve the problem or explain her answer. Is there some way to get beyond the initial response to uncover her true thinking? First of all, don't assume that the child's incorrect response necessarily indicates a true lack of knowledge or skill. It is possible that you have not yet succeeded at finding a good way to communicate with her. Perhaps you haven't yet found the right question or task. Try again. Rephrase the question. Use different language, even the child's "incorrect" words. Try changing the task. Make it different, or more specific, or more concrete. You can never know what will work until you try; be flexible enough to try almost anything.

Challenge the child's response.
Sometimes when a child gives a correct answer, you have the feeling that he nevertheless does not really understand what he is doing. Many children simply parrot what adults say and do not understand the reasons for what they do. Getting the right answer is not enough. Children need to understand why the answer is correct. If you suspect that the child's response is superficial, challenge it. You may even suggest a wrong answer in order to challenge the strength of the child's beliefs. The child who really understands will stick with his answer and even explain it; the child whose answer is based on superficial understanding may well accept the wrong answer merely because you suggested it.

You will never feel that you have finished the conversation. You will always wish that you had said something different. You will always think of something that you could have said. And you almost never learn as much as you would like about the child in a single conversation. But even if your interview is imperfect, as it is likely to be, it is worth doing because you are sending the child a message that you are interested in her thinking.