Incompatible Expectations

Now that the semester is almost over, I've identified one of the main causes of my frustration when teaching the course for pre-service teachers. (My other class, the gen-ed freshmen, are doing awesome.) It's that we have completely incompatible expectations for how much math the typical person should know and what fraction of the material from a math class a student is expected to know.

I am not so naive to expect that everything that I teach on day n will be familiar to the students on on day n+1. Yet, there are certain things that do build on the previous material. About a week ago, I did an activity that illustrated why the formula for finding the area of a circle is plausible; we cut cardstock circles into sectors and arranged them into a parallelogramesque shape. (Did I get this activity out of a 6th grade textbook? Yes, I did.) Unfortunately, the activity was a flop, as they didn't know how to find the area of a parallelogram (taught the previous class) or the circumference of a circle (taught two classes previously).

Their working model for learning mathematics seems to be based on these assumptions:

1. They have been admitted to college; therefore, they know enough K-12 math.
2. They know about the same amount of math as their friends. Any more math would be the realm of the math-super-geniuses.
3. You can pass a math class by ignoring the parts that you don't already know. Nothing you think is hard will be essential for scoring well.

Now, for some people, these are good working assumptions. I was able to fake my way through undergraduate topology by using many of the same techniques that my students are using. ("Well, a topological group is really just a group, so if I use what I know about group theory....") Unfortunately, while it is possible to make the argument that the Urysohn Metrization Theorem is an obscure fact belonging in the realm of the math-super-geniuses, it is much harder to make a similar claim about finding the area of a rectangle.