# How do you know? Making thinking explicit

When you watch a typical math class, you see a lot of rapid fire questioning going on.
Teacher: Number 1?
Student: Five and ¾
Teacher: Number 2?
Student: 3 and ¼
Teacher: Number 3?
Student: 2 and ½
Teacher: Anybody get a different answer?
Student: 2 and ¼ .
Teacher: Good. Next problem?

Now in some classrooms, a teacher will ask a student to come up to the board and show their work. Sometimes a student will even be asked to explain their work. However, that often sounds like this: “First, I looked at the denominators. They weren’t the same so I made them the same by multiplying. Then I added the numerators but left the denominators the same.”

This explanation is a summary of what the student did, but it doesn’t give any insight into how the student thought about the problem. Why did the student make any of these decisions along the process? How did he know why to change the denominators and not the numerators?

For the students who didn’t get the correct answer, this explanation doesn’t help. They just heard a list of steps that they need to memorize and remember correctly next time. The process of why these steps work is a mystery. It’s a black box.

For the students who can explain why it works, they are often not given the opportunity to explain. They are never asked to make their thinking explicit. What if they were given opportunities to try to explain their thinking so that it becomes an explicit process instead of a black box? Would all the students in the class benefit from it? I think they would.

At first this process can be uncomfortable. Students aren’t used to talking about their thinking. It seems private, something that no one asks you about.

But slowly, if we ask questions like:
Why did you do that?
• How do you know that is the correct answer?

They will begin to be comfortable talking about their thinking.

Now if the students only know because they memorized a set of steps, that is all they will be able to explain. But if they are given tools to start thinking about why things work, asking them how they know can change the way a classroom operates.

If the goal in math is to figure out how rules work or why they always work, the questions in math class change, the activities change and everyone is engaged in trying to discover what is in the black box.

I’d love to hear what happens when you ask students, “How do you know?”

# Math mistakes: What are they thinking?

I’m embarrassed to admit this, but until recently I was relatively unaware of all the great blogs and websites dedicated to the teaching and learning of mathematics.   I love having so many perspectives on education a click away.

Michael Pershan has a great site dedicated to compiling, analyzing and discussing mathematical errors students make.  In a recent post, he invited a discussion about whether we need to move beyond the identification of misconceptions.

I think we do.

Being aware of the errors is a good first step.  While researchers have done a lot of work on documenting common misconceptions at various grade levels, I don’t think that means that all teachers are aware of their findings.  The learning trajectories I mentioned in my last post are one place you can go to view some of the work that’s been done.

But it’s not enough just to know what the mistakes are.  In order to figure out how to avoid the misconceptions or correct them, we need to know more about what the student is thinking.

The best resource we have in trying to understand our students’ mistakes are our students themselves.  While an incorrect answer may make no sense to us, it makes sense to a child.   For example, I gave the following task to fourth graders recently.

Which fraction of the region is shaded?

Many students answered one-fifth.   While this solution might not make any sense to us, it made sense to them.  When I asked a student why it was one-fifth, he explained that one box was shaded and there were five boxes in the rectangle.   Although those of us who understand fractions know that fifths have to be equal size pieces, this students was not thinking about the size of the pieces.  He thought about a fraction as the number of pieces shaded out of the total number of pieces, regardless of the size of those pieces.

This is a simple example but it shows that an incorrect answer often makes sense to a student.  It is not simply because he or she made a careless error.  The mistakes students make can often tell us a lot about what students understand and don’t understand.  However, it’s difficult to do this by just looking at the student’s work.  You need to ask the student about what he did.

One of the most useful things I have learned in my Ph.D. program is how to do a clinical interview.   Clinical interviews involve asking a student about what they are doing as they are trying to solve a problem.   You aren’t trying to teach them.  You are trying to get into a student’s head– to see things from his or her perspective.  The goal is to try to understand what the student is thinking, not what you expect or want the student to be thinking.

As they are working on a problem, you can ask them questions like:

• Why are you doing that?
• What are you thinking about?
• How do you know that?
• Tell me more about what you just did.

There is a temptation to try to lead them to do things differently so they arrive at the correct answer.  However, you need to ask probing questions, listen carefully and try to understand why they are solving the problem the way they are.  The teaching can come after you have an understanding of their thinking.

Now this isn’t something that you can realistically do all the time in your classroom, but I think it’s worth taking some time to interview a student one-on-one in order to try to understand how they are thinking.

Have you tried clinical interviews in your class?  What have you learned?