Last week I decided to watch a sampling of YouTube videos about the calculus of rotations of curves about an axis. The technique is used to find the surface areas and volumes of objects such as round flower vases, rocket cones, or parabolic mirrors. Some videos were made by math tutors. Others were made by college math professors or high school calculus teachers. I found that most videos were intended to help students prepare for traditional exams and were strictly about practicing the steps in a recipe to carry out the computation. The speakers presented a formula, into which they substituted values to find a solution. Only one out of seven videos provided any intuitive information about how we got the formula, and even that video discussed the path to the solution before explaining the logic behind it. That speaker claimed that he followed that order because that’s what students want: Assistance to get through the exam. Students have evolved to favor this learning pathway, and traditional exams encourage this preference because they favor exercises that assess computation skills. Most students don’t realize how they are being swindled by this approach. They memorize a formula, pass the exam, and then forget about it. Why take the class in the first place? Math courses should express more courtesy in their use of student’s time.
Yesterday, my wife was pulled over by a state policeman, who informed her of a dead headlight. Instead of simply accepting that she would follow his instructions and respond to her own commitment to safety by replacing the bulb, he presented her with a formal ticket. This ticket meant that in addition to replacing the bulb, we would need to show the repaired light to another state police officer, who would provide affidavit that the problem was corrected, to be presented in court within a month. She was required to enter a guilty plea with the magistrate, who would then table the charges after viewing the second officer’s affidavit. Now, don’t get me wrong, if a citizen who is pulled over for a missing headlight is belligerent and insists that he will not comply, the officer has a duty for public safety to follow through with the formal ticketing process. Yet, the vast majority of motorists in this situation are not aware of the problem until the officer tells them. Sometimes lights, even those that were recently replaced, burn out while on the road. In a misplaced effort to increase public safety, the system has become unnecessarily complicated, wasting the time of the officers, the magistrate, and the citizen. I found that the problem was that the electrical connection to the bulb had jiggled loose. It took me less than a minute to resolve the mechanical problem. I was subjected to more than an hour of run-around Saturday morning, to get the second officer’s affidavit, and the whole process will not even be complete for at least another couple of weeks as the court processes the paperwork. We also were subjected to an unnecessary sense of guilt and risk from having signed a guilty plea, as if we committed a crime because normal vibrations shook the cord loose.
The time we wasted in this quirk of the legal system pales in comparison with time wasted in math education. I think that students, parents, and teachers waste countless hours favoring the processes of computation. Some focus on computation is needed to generate broader understanding and to build full practical ability to apply the techniques, but I think it should occupy less than 20% of learning time. Time should be more heavily weighted toward exercising mathematical intuition: It is more important to understand the concepts behind algorithms than to learn to use the algorithms themselves. When students build intuition around a concept, they are more apt to innovate with the concept, a trait that would better prepare them to solve the presently unknown problems of the future world.