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?

4 thoughts on “Math mistakes: What are they thinking?

  1. Michael Pershan (@mpershan)

    I don’t know how I only came across this post now.


    I agree with you that, as far as helping kids goes, identifying misconceptions is definitely not enough. So, yes!

    When I wrote that post, I was feeling a bit down about the site. I was feeling like I had become the “mistake” guy, and that was feeling a bit confining. After all, identifying misconceptions is not enough. Had I accidentally locked myself into a (relatively) uninteresting project?

    Since then, I’ve spent a bunch of time thinking about what I personally get out of my site. What I’ve come around to is that misconceptions do a really good job of getting my brain working on interesting teaching problems. What I love is reflecting on student work, and using that as a jumping off point for thinking about learning, teaching or whatever.

    All of this is to say that the site, right now, isn’t really about identifying misconceptions. It’s more about offering little provocative snapshots. If anything particularly interesting comes up from that provocation, then that’s part of our project too.

    There’s more to say about the direction of the site. I’m not sure how many more years this project will go on, but for now, it still feels vibrant and interesting to me.

    1. Nicora Placa Post author

      Michael– I hope this post didn’t come across as an attack on your site. I really enjoy looking through the mistakes and reading the thoughtful comments. When I wrote this, I was just trying to say that we need to push ourselves as educators to do more than identify mistakes. I think we both agree that why students make the mistake is so interesting to think about. And I think you are absolutely right in saying that misconceptions engage us in thinking about teaching and learning. Your site does a great job of starting the conversation. I’m exciting to see how it continues to evolve.

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