Another day in Ypsi, and another substitute teacher. Mr. Tuttle typically does a few warm-up problems on the board at the beginning of every class, just to get the juices flowing. So, for the past three days that Mr Tuttle has been out, I've been doing the warm-ups.
Early in the year I noticed that number of the students just copy the warm-up off the board and turn it in to get their warm-up points. Just so I'm not giving them the answers, I end up asking the class what to do at almost every conceptual juncture, which often leads to a lot of uncomfortable dead space. For example, this past week, we were doing areas and circumferences of circles, and we had a circle with a known diameter and we wanted to know the area. I asked the class, "OK, if we have the diameter of a circle, how can we get the radius?" No one answered. Knowing from my time as a TA at U of M that you cannot give them the answer to a question, or else they will just learn to wait you out, I said, "Well, I can wait." One student, who was facing the wrong way, turned around and asked incredulously, "You're really going to wait?" Eventually someone gave me the answer, I wrote down r = d/2, and we went on. Then I did something that blew them away, and I tried it again in three other classes to the same effect. We were solving for the area of the circle and we had the diameter, so I put this up on the board:
A = pi * r^2 = pi * (d/2)^2 = pi * d^2 / 4
Usually the response I got was, "Whoa, whoa, how did you get 4?" but one student said, "Ok, stop doing college math, we're in high school here." I told her reassuringly, "Don't worry, I'm sure you can do this." I'm pretty sure what threw them was I used symbols instead of plugging in numbers right away. What surprised me was these were some of the better math students who got lost. Many of them got it when they focused on what I put on the board, but I think they are used to understanding things with just a quick glance, so it jolted them when I did something in a new way.
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Ben,
I think you are correct in your assumption that students are used to us doing the work for them. They glance at the pattern and complete the rest of the problems. By rephrasing the problem using more symbols, the students needed to think through the problem in order to see how you solved.
I think we need to get students to do more of their own thinking and require them to explain it to others. Ideally others would challenge the thinking and the answer would be accepted or not accepted depending on the logic and quality of the proof.
Good Job!
Carol Cramer
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