Posted by: Gary Ernest Davis on: January 18, 2010

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Some years ago I taught mathematical methods to prospective elementary teachers at Washington State University in Pullman, Washington. In a class of 30 or so students there might be 2 or 3 young men, the rest being young women, principally from the Seattle area.

Many of these prospective elementary teachers would volunteer they were math phobic. When someone mentioned math phobia, I would ask people in the class to raise their hand if they were math phobic, and to only leave it raised if that meant shaking at the thought of algebra or possibly wanting to vomit if they had to do fraction problems. Usually 10-15 hands out of 30 were raised and stayed raised. Many of these students had very bad experiences in mathematics, and were afraid and sickened at the thought of having to teach mathematics to young children.

Apart *about* mathematics, how to *teach* mathematics, and how to *become* mathematics teachers, I also had to inject a large dose of therapy in the form of making mathematical experiences light, enjoyable, fun, empowering and enlightening.

One semester, in 2003 as I recall, a young women, Jennifer, told me about the second week that although she liked me, and liked my class, she was not going to learn anything about mathematics or about teaching it from me. She was, she told me, such a hopeless case that I would not be able to impart any such knowledge to her.

Now I love a challenge! So I asked Jennifer if she would agree to a bet: I would bet her I could, contrary to her expectations, not only teach her relevant mathematics but enable her to teach it to others. She, of course, bet I could do no such thing. We agreed to put up $5 each, winner take all, on this bet. We each gave $5 to one of the other students to keep for us, and the class went on as usual.

As a mathematician teaching mathematical methods to prospective teachers I had been a bit leary of getting students to practice arithmetic in different bases: a common practice among teacher educators. My initial thoughts were that this is a time-wasting exercise. As I thought about it, however, I came to the conclusion that working in a different base forced prospective teachers into the position of naive learners, and stimulated them to think about the deeper aspects of our base 10 system.

One morning, after mid-semester, I came to class with a bunch of cartoon cutouts. Students were to work in groups. One group would be the octopus group and they would do arithmetic in base 8. Another group was the Mickey Mouse group and they also would do arithmetic in base 8. The ostrich group would have to do arithmetic in base 4. Jennifer’s group was the pirate group, and with the cartoon pirate having only one hand – the other being a hook – the pirate group would have to do arithmetic base 5.

My condition for doing arithmetic – addition, subtraction, multiplication and division of whole numbers – was that translation back into the familiar base 10 was not allowed.

I explained what I wanted by using a language analogy. It was as if they were transported to a Filipino community where everyone spoke Tagalog, and no-one spoke English. They, the students, would have to rapidly learn to speak and think in Tagalog, without translating in their heads into English.

Of course they found this difficult, especially when, doing addition, they had to carry digits. The different bases were forcing them to think what the digits meant in a place value system.

Jennifer’s group were having a particularly hard time of it. There were three other young women in this group and addition and subtraction in base 5 was proving hard for them. As I watched, Jennifer explained to one of the others how the carrying went. Jennifer seemed to have cottoned onto the idea. As she explained her understanding to the others in the group there were frowns and looks of puzzlement. Jennifer was a little frustrated and repeated her own understanding, telling the others how easy it was.

At that point I stopped the class and announced that people were making good progress, but that there were still some points that needed clarification, particularly with regard to carrying. I feigned looking around the class for someone who might explain this and lighted upon Jennifer.

“Jennifer, you seemed to know how to do this. Could you explain it to the rest of the class? ”

She looked a little surprised, but went to the board and wrote her explanation and talked it through as she wrote. Other students asked questions and she answered them patiently and with clarity. Several times she stated how obvious this was.

As Jennifer looked around for other questions she saw me at the back of the room, smiling.

“Oh no!” she said. “I’ve just lost my bet!”

I couldn’t have been happier to take her money.

This story sticks in my mind for several reasons, not the least of which is the sense of achievement Jennifer felt in class that day. She now knew, with certainty, what it felt like to learn and teach mathematics. It wasn’t always hard, it wasn’t mysterious: it just took attention and thought. And it could actually be quite easy.

I have also thought often, of how relatively rare are those moments in class, and what I can do to stimulate them. I’ve had other similar experiences, but not enough that I could say with any assurance, that I knew how to remedy prospective teachers mathematical phobias in one semester.

I do know that creating an atmosphere in which these experiences might take place is a necessary condition to resolving student fears. I also know that some students need a lot longer than a semester for this to happen.

I hope, but do not know, that these young prospective teachers will find a caring mentor when they are in schools teaching.

I hope, but do not know, that they will be able to reach back into some positive experiences they had and say that’s how they would like their classes to be.

I hope, but do not know, that they will pass on an attitude of digging deeper and deeper into mathematics, in such a way as to encourage light, enjoyable, fun, empowering, and enlightening experiences.

Hi Jason,

I thought I had Mickey working base 8 (?)

I wondered about the pirate’s hook at the time, but thought it was too pointy and sharp to use for counting. Well, that was my rationale at least!

or maybe with the eyepatch the pirate has trouble seeing two hands at once…

1 | Jason Baldus

January 19, 2010 at 2:20 pm

Shouldn’t Mickey be working in base 8 (because he has 8 fingers all together)? For that matter shouldn’t the Pirate be base 6 (five fingers and a hook)? I liked the idea of using the cartoons to motivate working in different bases. I have gone back and forth on the idea of teaching arithmetic in different bases. Thank goodness I have tons of other things to “cover” for the big test. Whew! That’s one tough decision I don’t have to make. Bureaucracy to the rescue!