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                      THE GREAT MISTAKE! A Teaching Story

Author: Kari Augustine 

  TEACHER'S NOTE: This story is from the internet newsletter THE ALGEBRA
  TIMES. I think it is a good focus point for the beginning of the school

                       CREATIVE APPROACH TO MISTAKES

        When I was about eight years old, my father had a 'creative' 
        way of handling the mistakes I made in math.

        Once he got home from work, he would ask me how I did on
        that day's quiz. I would sheepishly tell him that I got a 'D'
        (something about fractions didn't agree with my young mind).

        He would become enraged and start parading around the
        family room, wondering aloud how a child of his could make
        so many 'dumbbell' mistakes.  He even sent me to my room
        a few times, evidently believing that would cure me of my

        As I said, this was 'creative.' He was creating anxiety in me 
        and futile rage in himself. I think that being branded the child 
        who makes 'dumbbell' mistakes also created in me a fear of math 
        and a sense that I would never excel at it.

        Somehow in college, I regained my interest in math and even
        ended up majoring in it. But I never stopped thinking about
        my dad's way of handling mistakes. Not only that, but as a
        teacher and a tutor, I have seen how students freeze up when
        they find they are making mistakes. Surely, I thought, there must
        be a better way to deal with students' mistakes.

        Here are some thoughts on how I try to treat mistakes in a more
        positive way, a way that helps students actually learn to 
        appreciate their mistakes.


        When I'm working with a student on some problem and he/she 
        makes a mistake, I congratulate him/her.

            "Hey, way to go! You just made an excellent mistake."

        The student usually shoots me a sidelong glance as if to check
        whether or not I just dropped down from Neptune.

        In this example, let's say the student just told me that:
        - 3 - 5 equals + 8 

        "No, I'm serious," I continue. "You just made a mistake that
        will help you understand what's really going on in this problem."

        Now, grudging seriousness from my young student.
        "What I want you to do," I say, "is tell me why you think the
        answer should be positive 8."

        "Well, doesn't a negative and a negative make a positive?"

        "Good question," I say. "When does a negative and a negative
        make a positive?"

        "When you're multiplying, right?" the student replies.

        "True," I say. "A negative number times another negative number
        gives you a positive number."

        "So don't two negatives always make a positive?" the student asks.
        At this point, I would break out of the Socratic mode and simply
        point out that addition is different than multiplication. I would 
        use a quick analogy to explain why:  - 3 - 5 = - 8

        [ - 3 means you owe $3.  - 5 means you owe $5. 
        If you owe $3, and you also owe $5, altogether you owe
        $8, so the answer is - 8 ]

        When it appears that the student understands, I ask him three 
        key questions:

                1)  What was your mistake?

                2)  What made you think this was true?

                3)  But what is in fact true?

        Once the student can answer these three questions clearly and
        completely, I do a little role-playing activity to reinforce the

        The student and I switch roles. I pretend I am a student making 
        this mistake, and the student is now the tutor. I make the mistake,
        and s/he must catch it and then lead me through the same process
        to clarify my thinking. Once a student can not only catch a mistake
        but can teach it, s/he really has nailed it down.

        Here, in a nutshell, are the five steps I go through to help 
        students work through their mistakes in a constructive way.

        P -  Point out the mistake and congratulate the student.

        E - Explore the mistake. Help the student do some detective
        work to figure out how or why s/he came to make this
        particular mistake.

        T - Teach the concept.

        A - Ask the student the three key questions.

        R - Role-reverse to reinforce the concept.

        You can recall the process with this acronym:


        Why go to all this trouble? Two main reasons:

        1) First of all, this approach is gentle. It avoids creating
        scars that lead to math anxiety, and it keeps the learning process
        fun and light.

        2) But just as important, this approach helps students become
        learners for life. We as adults are always learning. How? From
        our mistakes. Does anyone beat up on us when we learn in this
        way? Hopefully not! So if we teach children to learn from their
        mistakes, we help them develop the mature habit that will allow
        them to be lifelong learners.

        If you want to remember the gist of this whole discussion,
        just memorize this one little phrase: