# 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?