Please don’t be intimidated by the calculations below! Occupy Math needs them to make a point — part of which is that lines two through six are completely not needed. This post was written directly after grading a midterm in a first-year university calculus class. What that grading taught Occupy Math is that the students have been deceived about the very nature of math!
One of the questions on the midterm was “Find a sixth degree polynomial with exactly two roots”. Occupy Math thought that this would be a freebie question. There are many correct answers, one of which is the first line of the calculations above. About 20 of the 100+ students at the exam took this correct answer and drove it right off a cliff by trying to multiply it out. Occupy Math has multiplied it out to show how much work that is! Here’s the punch line: not one of the students got the arithmetic correct. As far as Occupy Math can tell, they have been indoctrinated to believe they have not answered a question until they have done a good deal of calculation? They feel that math is not math without the busywork!
Mathematics is the art of avoiding calculation!
Occupy Math has said in class several times that the factored version of a polynomial is the “simplest” version for most purposes — but the problem seemed too easy so the students performed a bunch of additional calculations. They also lost some points because those additional calculations (while not needed) did not make the answer wrong until they were done incorrectly. Another of the problems was to compute:
Since we learned, in class, that the function being integrated is the derivative of the inverse sine function, it follows that the following is a complete, correct answer:
A fairly distressing number of students instead used a difficult technique (trig-substitution integration) and spent 8-15 steps getting an answer — which was only correct occasionally. A large part of the problem is that they were unwilling to believe that a math problem could have a simple answer.
Occupy Math has ranted quite a bit in the past about the use of ineffective techniques and bad curricula, but this is a more fundamental issue. About one-fifth of Occupy Math’s students have learned that if there is an easy, obvious solution to a math problem, then, in the words of Admiral Akbar:
In addition, the issue of math as ritual magic remains present. Occupy Math knows that many of the students were taught in high school that trying to learn and understand math is vigorously punished (no, really!) and that what they should do is memorize the standard rituals and, confronted with a problem, apply the correct ritual. Occupy Math warns the students, repeatedly, that they are supposed to understand the principles and that attempting to continue ritualistic behavior will not work — but in their previous education they have not been taught any other way!
Occupy Math as Greek Chorus
In plays in the ancient world, the Greeks would have a chorus, a large group of people that would chant and sing background information (including warnings) to the protagonist. They would literally tell him his actions were putting him in peril. Occupy Math feels like a one-man chorus. You warn people that there will be traps on a particular path. You show how the traps work. You then set the traps with markers and sign-posts next to them. BLAM! KAPOWEE! SMACK! The traps go off. The bad teaching in high-school is very hard to reverse.
This wildly counter-productive style of teaching, training students in rituals and avoiding actual learning, has its roots in high-stakes standardized tests. It’s somewhat ameliorated by discovery learning (except that that is being done wrong too — see the “ineffective technique” link above). Conditioning students instead of teaching them math is a declaration by the educational policy establishment that they cannot teach math, but must instead cheat to get good institutional scores on their standardized tests. What would you think of a student that designed their entire educational strategy around cheating?
What can we do?
The single greatest predictor of success for a student, and not only in math, is parental involvement. Occupy Math is picking up the debris, he is down-stream of the problem. Parents and teachers need to work together through their PTAs and in community interest groups to insist that their children be helped to get good scores on tests by learning the material rather than by operant conditioning techniques more appropriate to places like North Korea. This is a serious and ongoing issue, and Occupy Math spends a lot of time deprogramming his students — an extremely non-fun job.
This is not a new topic for Occupy Math, but this post does give some new examples and a somewhat different perspective. Watching a student correctly answer a question and then mangle that correct answer beyond recognition because no one ever taught them math in their math class is tragic. Occupy Math is ready to help where he can, and he has friends. Got any thoughts on this topic? Please comment or tweet!
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics