Wednesday, April 02, 2008

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?


Anonymous said...

Requisition pallet of bricks and bucket of mortar. Construct wall. Wait until mortar dries. Bang head repeatedly against wall.

It really does feel so, SO good when you stop.

Gary said...

I think that many students at all levels have lost touch with what it means to become educated. They just want to finish a test which earns them a grade in the course which counts towards their degree. Learning, to these students, is unimportant, while completing the requirements of the course, in however a half-assed manner, is what is important.

I just don't get it. As a substitute teacher, I often try to help high school kids with their work, and they are just unwilling to learn anything. They'll demand answers, even when the instructions clearly tell them to show work. If I tell them not to copy off each other because I'm there to help them, and try to help them through a problem they've copied the answer to, they'll get angry because the problem is already "done," even if it's wrong. It's really depressing when I have an entire class full of kids like this.

Jokermage said...

I like your factoring method better than cross canceling. I'll probably still use cross canceling for most of the basic problems (where the greatest common factor is obvious) I'll run into, but it is nice to have an alternative for the more complex fractions.

I have a method for polynomial multiplication where I use long multiplication instead of FOILing, mostly because I hated FOILing ever since I first saw it. Some people who I've shown long polynomial multiplication to have reacted to it the same way some of your pre-educated students reacted to factoring fractions.

Anonymous said...

I have met more than a few students who like to obliterate their work as they go, leaving non-equations and crossed-out expressions in their wake. If all goes well, they get the right answer with untrue work, harder to diagnose or to grade.

Now, some other students write arrows instead of equal where there is clear equality. I'd guess they'd seen this before but forgotten the reason why = was not used, like

Jokermage, I'd guess that you're referring to using the distributive property to multiply polynomial expressions. This is the fundamental connection between addition and multiplication which lets us say, for example, that multiplication is "repeated addition". That "FOIL" business is really just the special case of multiplying two binomials, though that is a common case.

Of course, in practice perhaps you're using the shortcut where you just multiply the terms in each possible pairing, as in (a + b + c)(d + e) = ad + ae + bd + be + cd + ce, which is just applying distributivity several times.

I think clinging to such things is a consequence of a reluctance to really explore mathematics and work on understanding, instead trying to follow recipes exactly, sighing in relief and tossing the concepts out into the "done" pile afterward. Of course, once you start having trouble in math, it can be hard to convince yourself to spend even more of your time exploring the source of your misery.

As you note, some people absolutely swear by FOIL, but I think most people cling tightly to such things because they don't really play with mathematics, just do homework following exact recipes and then sigh in relief if their execution was correct. Mind you, I have tendencies to retreat into what I know too; it seems that I try to make a great many things into elementary topology if I can.

P.S. Zeno, why didn't I think of that? It could be because I don't technically teach arithmetic, but in practice it comes up all too often. I am so stealing your educational technique!

P.P.S. Gary, it's definitely no fun to have that attitude in the great majority, but I've seen my share of it at my Prestigious University which is Hard to Get Into. I suppose, in their defense, ours are particularly stressed by the demands on their time, but whatever the reasons they are just trying to get credit. This is a particularly unfortunate attitude to adopt in a precalculus course when you plan to follow up with the calculus sequence.

Jokermage said...


My long polynomial multiplication is just distributive property, but I organize it visually like long multiplication because it is easier to understand what is going on (for me at least). Here is a basic example of how I multiply polynomials. I find that with more complex polynomials it is easier to keep track of like terms by putting them into the same columns. I don't expect this system to be easier for all cases for all people.

Anonymous said...

Part of the not-wanting-to-learn attitude is that Math Is Hard. Really. Even talking Barbie says so. So why bother?

I found math difficult even well into college; I remember differential equations in particular were just so much magic. Then I began taking physics and engineering classes that really used the math I'd been struggling to learn, and suddenly math was worthwhile... and amazingly, it got a lot easier.

Anonymous said...

Multiple distribution and long multiplication use the same principles, but organize the intermediate steps differently on the page. For high school students I ban FOIL, for college students I come close to banning it.

There are other places where multiple approaches are available, and I teach only one. I solve equations by transposing terms (moving from one side to the other, and changing signs) rather than what I call "pendant subtraction" (hanging -7x under each side of the equation). My experience says that the latter leads to more confused mistakes.

I don't insist, I strongly recommend. "Do we have to do it that way?" No. But I will only provide assistance this way. If you are genuinely good with this particular skill, wonderful, keep doing what works. But if you need help, you need to switch.

Cheri said...

I really struggled with math in high school. Teachers have a way of teaching "their way" without teaching WHY their way works (some may not even know why). When a student has a method they learned in the past that doesn't work, I guarantee it came from a teacher who expected the student to memorize, not understand.

My daughter is in a community college math class now, and is having difficulty integrating this teacher's way with the last teacher's way. She is going home and having her fiance explain why both ways work. She'll be better for it in the long run.