The Art of Asking Good Questions

Written by: Dr. Volker Ecke with Dr. Christine von Renesse

In the blog "Our Inquiry-Based Classroom", I invited you into one of my IBL classrooms for students in the Liberal Arts. We saw students working in groups on a series of Investigations, discussing their thinking. As I walk around listening to the student groups grappling with making sense of the mathematics on their own, how can I encourage and support their efforts without just giving them "the answers"? How to engage them in mathematical conversations that will make their thinking visible?

What does that sound like?

If you want to jump right in, take a look at the following two minute video showing a group of students working on figuring out for themselves how to solve the top layer of the Rubik's Cube. What do you notice about what I say? What do you notice about what I'm not doing? What impact do you see on the students?

Listening more closely

Now I would like to ask you to join another classroom where a group of students has just started to look at the series 1 + 1/2 + 1/4 + 1/8 + … =? (from "The Infinite" book). Perhaps you have listened to students grappling with related concepts in your own classes.

Max: Well, all the numbers are added so it will be more and more. I think it will never stop!

Tina: But aren’t they getting smaller and smaller? Like ½ is less than 1 and ¼ is less than ½…

Max: Yes, but I am adding them up, see? (Pointing to the plus sign between the numbers)

Tina: Hmmm (clearly thinking hard). Ok, I agree, I add them up, but they are still getting smaller and smaller, so it should stop sometime, right?

Max: It can’t stop, since there are always more numbers being added, the numbers never stop. So I still think it will never stop.

Kim: So wait, you can’t both be right, so who is right now? I am so confused!

Hearing this conversation, what are my options as the teacher? What do I say?

Imagine that you are the teacher who has been standing quietly with the group, listening to their thinking. I invite you to

• Find three responses that you like and explain why you like them. Be specific.
• Find three responses that you don’t like and explain why you don’t like them. Be specific.

Before you read on, take a moment to jot down some ideas. It would be exciting if you could share them with us (email to ). Which are easier for you to think about: responses you do like or responses you don't like?

How to respond?

Back to the classroom. As I walk around listening to such a brief snapshot, I have to assess the students' understanding of and confusion about the investigation and decide what action to take (if any). A few options that you may have thought of: check how they understand the problem, help them devise a plan to approach the investigation, ask a clarifying question to make their language more precise, ask them to explain their thinking in more detail, or do nothing. In reality, of course we usually have a lot more information about students from interactions in prior classes, which can help decide on a good course of action (or at least our best bet).

First Ideas:

Let me first share a few responses that I find myself reflexively grabbing onto as the "Sage on the stage":

Jumping Ahead:

1. So would the sum be a finite number or would it be infinite?
2. So is the answer smaller or larger than, say, 10?

The Big Idea of these investigations is for students to discover that adding infinitely many things can sometimes "add up to" a finite number, and to begin to develop tools and intuition about when and how that happens. Jumping ahead to the Big Idea before the students have grappled with the conceptual foundations will leave them adrift

Alternatively, in my efforts to be "helpful" to the group, I may find myself tempted to

Evaluate Thinking:

1. Tina, you have the right idea, keep going.

Students become critical thinkers only when it is their responsibility to evaluate everybody's ideas, to ask questions when something is unclear, to state their agreement or disagreement with other's ideas (and give reasons for their position). There is no room for the students when the teacher occupies that role. Many students have developed sophisticated methods for fishing for evaluation. It is so much easier getting the teacher to do the thinking! We see students fishing for evaluation in the following two videos: Calculus 2 Conversation and Powers of i. How to respond in ways that refrain from biting the hook, and instead return the power and responsibility to the students?

I may also discern an underlying confusion in this group and intervene by

Telling the Answer:

1. You have to distinguish between the sequence and the sum to make sense of this problem.
2. Let me show you how to think about this…
3. No, it doesn't keep going, the answer is actually a number.
4. No, it doesn't keep going, the answer is 2.

While it is important that the teacher knows the answer, is telling the answer really the best way to support the students' learning? I might rob students of the opportunity to construct the underlying conceptual foundation for themselves. I might reinforce their belief that only "other" people can make sense of mathematics, forcing them to continue their dependency on an outside authority.

What would it be like to have students' thinking take center stage? And how to bring that about?

Further Ideas:

Let me also give you some responses that we feel can empower students to persist in making sense of the mathematics.

Say nothing, stay and listen.

It is very difficult to "sit on your hands" but it may be very rewarding for you and your students.

Understanding the Problem:

1. Can you remind me again which question you are trying to answer?
2. What are you looking for?

In their attempts to express their understanding of the problem, the students may clarify for themselves some of the concepts they need to keep separate.

Devising a Plan:

1. What would happen if the numbers were ½+ ½+ ½+ ½+ …?
2. Can you draw a picture to show me what you are thinking?
3. (If the previous idea doesn’t work) I wonder if a picture would help. What if we look at a square of area 1 and then add ½ of a square and then add ¼ of a square (drawing the picture of a 2x1 rectangle being filled in)…
4. Can you compute some of the additions to show me what’s happening?
5. (If the previous idea doesn’t work) How about looking at parts of the sum, I think that would be less confusing… like 1 + 1/2 = 1.5, 1 + 1/2 + 1/4 = 1.75, … I wonder what will happen to these sum numbers? (They don’t know the term “partial sum” yet)

These responses provide tools to focus on limited aspects of the problem, or to bring in additional tools.

Ask a clarifying question:

1. So, Max and Tina, you are both talking about an “it” that is stopping or not stopping. Can you tell me what you mean by that “it”?
2. So Max, you are saying it can’t stop. Can you tell me what the “it” is that you are thinking about? (If nothing happens, add in: Do you mean the sequence of numbers, the sum itself, or something else? )
3. So Tina, I heard you say that it should stop sometime. What is the “it” you are talking about? (If nothing happens, add in: The sum? The sequence of numbers? Something else?)
4. Hold on, I am confused, can you show me what you mean by “something is stopping”?

Precision with language is important when communicating ideas, especially in mathematics. Part of the confusion in the group may stem from the fact that they don't make explicit what "it" they are talking about.

Explaining:

1. Kim, you say you are confused about who is right? How about Max and Tina both try to convince you? And you get to ask them lots of questions and you can then decide what makes more sense to you?

If one of the students in the group can step into the role of the person asking questions and helping the members of the group clarify their thinking, the teacher is ready to move on to visit the next group.

Analyzing your own classroom dialogues

What are some brief classroom dialogues that you have experienced in your classes? What are some Good Questions you have found valuable in supporting your students?

You might find it valuable to videotape or audio record a small segment of your group interactions with students. What do the students actually say? What do you think is going on? How do you respond? In retrospect, what else could you have said?

Members of our group are available to mentor and support you in this investigation of your teaching practice (stipend). Contact us at to find out more!

Resources

A collection of "Good Questions" is included here for reference. Please extend and share.
Inquiry-Based Learning and the Art of Mathematical Discourse.