Thursday, May 10, 2012
Fill in the blanks
I was flummoxed. Under normal circumstances, algebra students abandon the complete-the-square technique for solving quadratic equations as soon as they meet the quadratic formula. It is by a significant margin the least-favored of the solution techniques, trailing badly after formula and factoring.
Why, therefore, were so many of my students diligently completing the square when they didn't have to? Even worse, they were doing it on an exam problem, when time is at a premium. Worst of all, they were completing the square to solve a quadratic equation where its use was clearly contraindicated! I was at a loss.
As you may know, the solution of the quadratic equation is the great pinnacle and climax of your traditional introductory algebra class. The end of the semester wraps up with the astonishing revelation that one can now solve any quadratic equation. No exceptions! Such universality is rare, and I try to engender a little appreciation in my students for so powerful a conclusion, the big finish of Algebra 1.
Of course, I also try to get them to approach quadratic equations thoughtfully and methodically. First of all, does the equation factor easily? Then go for it! Is it (or does it appear to be) prime? Then one can apply the never-failing quadratic formula or—in certain specific cases—resort to completing the square. The specific case, naturally, is one in which the quadratic polynomial in question is monic (has a lead coefficient of one) and possesses a first-degree coefficient that is even (making it easy to take half of it and square the result, as required for completing the square).
Otherwise, don't even think of completing the square.
The problem that was puzzling me was monic, all right, but its middle term had an odd coefficient, making it a quite unsuitable candidate for square completion. Why, then, did so many of my students plow right in and start juggling fractions and slogging through more and more complicated expressions? They didn't know and couldn't tell me why they had done it.
The reason finally came to light while I was paging through my collection of quiz keys. I paused to consider the quiz containing the combined-work problem (or “joint effort”—computing the time a job takes if two or more people pitch in and you know how long it takes each person to do the job alone). This was exactly the kind of problem that had caused so much square-completion grief on the exam.
... and my students had learned the lesson that combined-work problems are solved by completing the square! After all, the teacher had demonstrated this in a quiz solution key that he had posted on the course website. Did he not constantly encourage them to emulate his example? Follow his lead? Write solutions like he did? Indeed! Indubitably!
They learned a lesson I wasn't teaching. They had studied my solution to a particular combined-work problem and then followed it slavishly when next they encountered a problem of the same type—even though the resulting quadratic equation had different characteristics and argued for a different solution technique.
I failed to banish the template problem. My fault!
You know what a “template problem” is, don't you? I'm sure you do. Lots of books are full of them. It occurs when a section of the text presents a carefully worked-out problem in Example 1, you turn to the homework section, and Exercises 1 through n follow the prompt “See Example 1.” And then all of the problems are exactly like Example 1 except that the numbers got tweaked a little. Or maybe Example 1 was a word problem about Sally and Exercise 1 is about Sam. Trivial changes. You can copy the solution of Example 1 as a template and go through filling in the old numbers with the new numbers.
Hardly any thought necessary.
I don't want to be too harsh. Routine drill problems are useful for building basic skills. They are, however, too bland for a steady diet and do not do much (if anything) for building conceptual understanding. Students, however, often prize them for their dull predictability and lack of challenge. They even ask for more, as when they beg for a “practice test” before a big exam. The most favored practice tests are those full of templates for the real thing. Woe betide the instructor who gives in to the pleas for a practice test and then changes the problems too much in the actual exam! Students will feel betrayed.
I refuse to give practice tests. I decline to channel my students' attention too narrowly to specific kinds of problems solved in specific kinds of ways. I want them to consider each problem independently, with a minimum of prompting, examining their knowledge of solution tools and picking the most appropriate one to apply.
The complete-the-square affair demonstrates, I'm afraid, that I have discouraged template thinking less than I had hoped. Perhaps I should ask my colleagues how they avoid it and then do exactly what they do....