You’re in the classroom. You gave students a task to complete with their group. You notice that one group of students have completed the task: all the answers are there, and they seem correct when you glance at their pages – the sequences of steps looks the same on each student’s paper! They are “done talking,” and waiting for the next thing to do.
So what’s going on?
Whenever I encounter smiles, nods, and virtual silence, I start getting suspicious. They might all have really reasoned through the mathematics and constructed their own pathways, but more often, I am probably seeing the work on one dominant student being accepted as proxy for the other students’ thinking.
But I don’t really know, and I don’t want to reach any false or unfair conclusions. What should I do?
This story illustrates two crucial questions we are faced with many times a day:
What do you want for/from your students right now? Do I want to see more problem solving? Do I want to see more evidence of collaboration? Do I want to better understand their process? Do I want to help them build confidence?
How will I make that happen? What kinds of prompts work when I want them to “dig deeper?” What if I want to encourage a student to “own” their reasoning?
Here’s a start. This bullet list is entitled Tips For Developing Mathematical Thinking with Effective Questions. Or read below. It’s from PBS’s TeacherLine site, but I really got it from my work with peers at the Park City Mathematics Institute in 2009.
Developing Mathematical Thinking with Effective Questions
To promote problem solving (first steps), ask…
• What information do you have? What do you need to find out?
• What strategies are you going to use?
• Will you do it mentally? With pencil and paper? Using a number line?
• What tools will you need? Will a calculator help?
• What do you think the answer or result will be?
To move problem solving farther along, ask…
• How would you describe the problem in your own words?
• What facts do you have?
• What do you know that is not stated in the problem?
• How did you tackle similar problems?
• Could you try it with simpler numbers? Fewer numbers? Using a number line? What about putting things in order?
• Would it help to create a diagram? Make a table? Draw a picture?
• Can you guess and check?
• If you compared your work with anyone else’s, what did they try?
To make connections among ideas and applications, ask…
• How does this relate to…?
• What ideas that we have learned were useful in solving this problem? • What uses of mathematics did you find in the newspaper last night?
• Can you give me an example of…?
To encourage reflection, ask…
• How did you get your answer?
• Does you answer seem reasonable? Why or why not?
• Can you describe your method to us? Can you explain why it works?
• What if you had started with… rather than…?
• What if you could only use…?
• What have you learned or found out today?
• Did you use or learn any new words today? What did they mean?
• What are the key points or big ideas in this lesson?
To help students build confidence and rely on their own understanding, ask…
• Why is that true? How did you reach that conclusion?
• Does that make sense?
• Can you make a model to show that?
To help students learn to reason mathematically, ask…
• Is that true for all cases? Explain.
• Can you think of a counterexample?
• How would you prove that?
• What assumptions are you making?
To check student progress, ask…
• Can you explain what you have done so far? What else is there to do? •
Why did you decide to use this method?
• Can you think of another method that might have worked?
• Is there a more efficient strategy?
• What do you notice when…?
• Why did you decide to organize your results like that?
• Do you think this would work with other numbers?
• Have you thought of all the possibilities? How can you be sure?
To help students collectively make sense of mathematics, ask…
• What do you think about what ____ said?
• Do you agree? Why or why not?
• Does anyone have the same answer but a different way to explain it? • Do you understand what _____ is saying?
• Can you convince the rest of us that your answer makes sense?
To encourage conjecturing, ask…
• What would happen if…? What if not?
• Do you see a pattern? Can you explain the pattern?
• Can you predict the next one? What about the last one?
• What decision do you think he/she should make?