My degree is in Computer Science. But I am part of a combined mathematics/computer science department. This means that I also teach the odd math service course, so far always Calculus I. It is profoundly strange to be a Calc teacher when you are pretty sure that it is 1s and 0s at the bottom and not infinitesimals. But, aside from that, it has become clear to me that Calculus, while not generally important for daily life, is, as taught, a fantastic way of figuring out whether people are good at pattern recognition.
Learning to take a derivative is essentially an exercise in repeated pattern recognition. This becomes even more clear when we hit this first big cognitive barrier in Calculus: the chain rule. Here in the department, we gave a test at the beginning of the second term of Calculus. (The second term is basically all the hard parts of integration). I then took a nod from Titus Winters and decided to mine the test data to see if anything could be learned from it, and I found out a fact that is obvious to mathematicians, maybe, but was wonderful to see as a stranger in these parts. To analyze the test, I constructed a decision tree using ID3. Each node of the decision tree represented “did the student get question X right?”, and we built the tree to try and built the most-accurate classifier of students into the categories “Got 70% or more on the test” and “Got less than 70% on the test”.
The most important question on the test – the one that, if you got it wrong, you were highly likely to have gotten a lot wrong, and if you got it right, you were highly likely to have gotten a lot right – was the first chain rule question on the test. The question that was second best at determining student performance was, wait for it, the second of the two chain rule questions on the test. This means that we now have quite good evidence that we really have two kinds of student: the ones who can do the chain rule and will do ok in the class, and the ones who can’t and won’t.
What we don’t know, and I’m testing it this term, is whether we can cheat this – i.e. if I spend more time teaching the chain rule (to the detriment of spending as much time on the other subjects on the test), will more people develop more skill in the course overall? We don’t know, but it sure is fun to experiment on your students.
Obligatory disclaimer: these results are currently being written up, and may change if our analysis is revealed to have flaws. Nothing has been peer-reviewed yet, and we’re a bunch of math and cs people looking at data that involves people. However, the analysis was really trivial, and the result was really strong, so I’m pretty confident in it.
