The pre-educated student

I'm Professor Superfluous

Either their ranks have increased or I've gotten more sensitive to them. My classes seem to have more than their usual complement of students who are apparently enrolled in courses they don't need. These students attest to this themselves: “I could have taken the next class in the sequence, but I decided to take this one instead.” Two students who recently used this excuse were a pair of brothers who, as I previously reported, indolently earned failing grades in an arithmetic class and then used the excuse that they should have been taking prealgebra. (Therefore their interest waned because the work was too trivial—which I guess is why they couldn't do it.) These boys followed up by enrolling in prealgebra despite having failed the prerequisite. Predictably, they failed the prealgebra, too.

Perhaps the brothers sensitized me to other examples of pre-educated students who are taking classes on subjects they claim to already know. One assumes they enjoy education so much that they do not hesitate to enroll in courses whose content they have already mastered. The journey is the reward, or something like that.

Recently I was teaching a unit on multiplying fractions and I was stressing the importance of reducing the answer to simplest terms. Some teachers and textbooks stress “cross-cancellation,” which involves seeking out and reducing any common factor that appears both in the numerator and denominator. After various amounts of crossing these factors out, the product is ready to compute in reduced form. For example, the product of 25/12 and 9/10 can be cross-cancelled as shown, with the common factor 5 cancelled from the numerator of the first fraction and the denominator of the second and the common factor 3 cancelled from the denominator of the first fraction and the numerator of the second. Like this:

I'm not a big fan of cross-cancellation. Too many students, in my opinion, go vigorously cancelling things out and later, if there's a mistake in their work, cannot figure out where it is because they've obliterated the problem. The example shown above is an exceptionally neat example of the practice and not at all representative of what I find on my students' papers in this handwriting-challenged era.

I have instead been emphasizing actual factoring of the numerators and denominators. Cancellation (or reduction) is still our goal, but I also want to make the process a bit more obvious and, perhaps, just a little neater. Hence I encourage my students to write out the factors before they go cancelling. (They don't even have to go all the way to prime factorizations, as the current example demonstrates, just far enough to ensure that all common factors have been dealt with.) The result is something like this:

Many of my pre-educated student are scandalized by my disdain for the traditional cross-cancellation and not at all inclined toward my alternative: “Do we have to do it your way?” “Is cross-cancellation wrong now?”

No. And no. I cheerfully give credit for correct answers and correct work—and cross-cancellation can certainly be done correctly—that are shown in legible form. But my favorite student response is, “Do I need to learn this? I already know how to do it!”

Then why are you here, buddy?

A few push it even further:

“That's not how I learned it!”

Well, that's how I am teaching it. Have you considered trying to learn what I'm teaching? There's some evidence you haven't been a roaring success at this in the past. One of my colleagues told me a story about a student who explained how he had learned fractional arithmetic, offering an algorithm that was guaranteed not to work. My colleague patiently explained to his student that he must not remember the technique correctly, because what he had described was doomed to failure. The teacher then led the class through an example, after which the student in question announced that he had used his own technique and his answer did not agree with the teacher's. Was it okay if he continued to use the technique he had “learned”? Yeah. Get a clue, Sherlock.

The funny thing is that I'm quite laissez-faire in terms of technique most of the time. I seldom give prescriptive exam problems that specifically demand the use of a particular technique. I normally ask for a result and allow the student to choose the best way to do it. As long as the work is coherent and the result is correct, full credit is given. Yet I have these pre-educated students who fuss and fume and take it personally that I insist on teaching techniques they haven't seen before, instead of recapitulating their prior experience. Why won't I do it their way?

It does try my patience.

And it's not just students in the more elementary courses. My calculus students have a tendency to arrive with a smattering of high school calculus, which enables them to perform the more routine tasks with a minimum of difficulty. They can differentiate a polynomial like nobody's business. A few of them therefore announce that they already know how to take derivatives and pout when I make them work out the problems from the definition of the derivative (the limit of a difference quotient). They don't realize, although I try to explain, that not all functions are neatly differentiable by means of things like the power rule. Functions in real life may be tables of values gleaned from the output of instruments in an experiment. You have to go back to the basics to estimate the rate of change because no one is going to give you a nice simple function to take the derivative of.

Nevertheless, despite the explanation, when they get to the chapter test there'll be the pre-educated cadre that insists on simply writing down the derivative when told to demonstrate the use of the definition. Oh, no, they're way beyond that.

And next semester, when they repeat the course, will they be pre-pre-educated?