Friday, March 17, 2006

Cyborg students

Clutching calculator crutches

It is the lull before the storm. Algebra students take joy in the ease with which they are solving our most recent exercises. What could be simpler than factoring monic polynomials? All you need to factor x2 + bx + c is to find paired factors of c that add up to b.

For any constant c, the list of its factors is finite. The student can scan the list for a suitable pair and, upon finding them, simply write down the factorization of the original polynomial. If the search fails, he or she simply writes down “prime.”

Today I gave a quiz with four monic polynomials to factor. They embodied all the permutations on the signs of b and c, beginning with the instance where both are positive. Most of my students sailed through the quiz, but I noticed that they scrambled for their calculators while working the problems. This puzzled me. I've banned the calculators that can actually do the factoring for the students (like the TI-89), so it seemed to me that they were mostly wasting their time. What could they be doing?

After the quizzes were all collected, I asked for four volunteers to demonstrate the solutions on the board. With a little nudging, four students duly approached the board and dispatched the problems. Two of them carried their calculators up to the board with them! In the first case, the student was facing a problem where c was equal to 28. She needed the calculator to find its divisors, even though she had just recently done the problem at her desk! The other student was in a similar fix with the awesome c value of 50! (That's an exclamation point, not a factorial symbol.)

My algebra students cannot factor 28. They cannot factor 50.

Moebius Stripper over at Tall, Dark, & Mysterious often waxes profane in her disdain for calculators. Today drove home the point that MS is not exaggerating: our calculator-dependent students are mathematically crippled. Numbers have neither structure nor texture to them. Instead numbers are merely inert blocks that must be processed through electronic machinery. Punch some numbers, read a display, write down a result. The magic box holds the answers.

Woe is we.

Note: The “Evil Bad Ass Calculator” in the illustration can be found here.


ulg said...

Arithmetic is just rote regurgitation.

You memorize addition tables, times tables, etc, and some easy
cookbook steps that you roboticly follow thereafter. Where's the
thinking in that?

Sure, it would really be cool if there was some elitist reason to be
proud of having completed my calculus, differential equations, physics
and electrical circuits classes with no calculator (no slide rule
either; most the makers had stopped making them by the time I was in
school), but, well, seriously, what use is such an ability?

Zeno said...

You are certainly missing the point, ulg. Did you reach for your calculator during your calculus and physics courses every time you had to multiply two single-digit numbers? I doubt it. In fact, you would probably never have finished any quiz or exam on time because every problem would have taken you way too long.

I happen to agree that arithmetic is largely rote regurgitation, and I'm arguing that it should be. If you never memorize the multiplication table, then you are doomed to plod along at a snail's pace in every course beyond arithmetic. If the basic skills never become reflexive, then the elementary calculations keep getting in the way of the stuff that requires actual thinking.