Thursday, May 10, 2012

Fill in the blanks

Template tests

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.

I noticed that I had solved the resulting quadratic equation on the quiz's solution key by completing the square. The polynomial had been monic with an even linear coefficient, so completing the square gave a quick and easy solution ...

... 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....


khms said...

As I was unfamiliar with the term, I looked up "completing the square" following your link. Ok, fine, all familiar (if making a bit much noise over a simple thing, with appeals to geometry and all, when it's just a simple application of a common binomial formula). Then the quadratic equation.

Ok, so that's simply what you get when you do "complete the square", fine, except a bit too complicated an equation to keep in mind when it's so simple and painless to derive it.

Umm, people actually prefer the second version?

An you base what's better on stuff like factors being odd, and avoiding dividing-by-two?

Are you that afraid of fractions?

I'm boggled.

I mean, I could perhaps see it if you'd have terms up to x^7, but x^2 is so simple ...

(No sup code?)

Zeno said...

Actually, khms, an unfortunately large number of math students are terrified by fractions, but that's not the point. It's about efficiency and accuracy.

For example, x^2 - 8x - 10 = 0 is quickly dispatched by completion of the square:

x^2 - 8x = 10
x^2 - 8x + 16 = 26
(x - 4)^2 = 26
x - 4 = ±26^(1/2)
x = 4 + ±26^(1/2)

It's actually a little faster than plugging into the quadratic formula. Completing the square, however, is clumsier than the quadratic formula with something like this:

x^2 - 7x - 15 = 0
x^2 - 7x = 15
x^2 - 7x + (3.5)^2 = 15 + (3.5)^2
(x - 3.5)^2 = 109/4
x - 3.5 = ±(109/4)^(1/2)
x = 3.5 ±(109/4)^(1/2)

[with additional simplification to x = (7 ± 109^(1/2))/2, which is how it falls out of the quadratic formula, so you might as well have used the quadratic formula itself and saved yourself some time.

But, like I said, the real point of the post was that students are operating on autopilot instead of picking and choosing.

Kathie said...

Sez a long-ago first-year Algebra teacher: Isn't one of the principal reasons for teaching "completing the square" because students need to understand it in order to learn the algebraic derivation of the quadratic formula?

Anonymous said...

I have no idea why completing the square is even in the curriculum. Nobody uses it in practice - if you can't see the factoring right away, you go with the quadratic formula and find the roots. In my opinion, it is more important to spend time on the meaning of roots, how many do we expect and what kind they are rather than completing the square. We tend to emphasize procedures too much and meaning too little.

Kathie said...

Next time you teach the quadratic formula, why not give your students a few additional combined-work problems in that unit, so they have both the the quadratic formula AND completing the square as templates?

Unknown said...

Normally, I would preface what follows with an apology for letting my geekiness show (and for being a bit off topic) but in this case I figure it is pretty safe to confess my sentimentality for an equation to a math teacher.

I do remember the first time I learned the quadratic formula, and it really did feel like a pinnacle of my education at the time. I remember coming home from school with a newly adopted affectation of being more mature, of being a grown up now because I was doing grown up math. I honestly doubt I ever factored another square after that- even when it may have been more expedient to do so- because I thought the quadratic formula was just too cool not to use.

I imagine now that many of my teachers probably shook their heads a bit when grading my overly formulated tests.

PhysicsPhdStu said...

Completing the square is cool! Also useful in many places including Gaussian like integrals (therefore path integrals & QFT etc)

We know how students "learn" from examples

kai said...

There’s even a name for this kind of behaviour: The Einstellung effect. There’s a quite interesting book by Luchins & Luchins: Rigidity of Behavior (University of Oregon Book, 1959), where they study this phenomenon applied to water jar filling problems. Quite amazing: they have repeated the same experiment, varying the parameters in every direction they can think of, for several decades, trying to figure out why people behave this way.

The Ridger, FCD said...

Carl Zimmer reports on experiments showing that's what people do - though not chimps.

"The children could see just as easily as the chimps that it was pointless to slide open the bolt or tap on top of the box. Yet 80 percent did so anyway. "It seemed so spectacular to me," Mr. Lyons said. "It suggested something remarkable was going on." ... Mr. Lyons sees his results as evidence that humans are hard-wired to learn by imitation, even when that is clearly not the best way to learn. If he is right, this represents a big evolutionary change from our ape ancestors. Other primates are bad at imitation. When they watch another primate doing something, they seem to focus on what its goals are and ignore its actions."

So you probably need to be more explicit in telling your students not to take something as a template, not just not telling them to.

Zeno said...

Indeed, I rail against mindless plugging into formulas and rote obedience to algorithms, but my students clearly think old Dr. Z is being wry and amusing. I must make it clearer.

And, yes, completing the square is a common tool in inverse Laplace transforms and certain antidifferentiation techniques. It's not completely obsolete once the quadratic formula is introduced. Furthermore, it is the means by which the quadratic formula is derived, even if only a few students appreciation the demonstration.

Finally, there are plenty of quadratic equations that cannot be factored but are swiftly dispatched with completion of the square (in which case the quadratic formula is overkill and requires additional simplification and reduction before reaching the solutions).

Kathie said...

Oh Zeno, I always made my algebra "appreciate" the derivation of the quadratic formula: they had to do it on the next exam from memory, and it counted for a good 25-30% of the test score!

Kathie said...

Oops! "...always made my algebra STUDENTS "appreciate" the derivation..."

Elipson said...

I am somewhat puzzled by this method, as I have never seen it before. Is it primarily an US thing? And is it really much of a time saver when doing Laplace transformations??

Zeno said...

I'm not aware, Elipson, whether it's primarily a US thing or not. However, it can be useful when computing inverse Laplace transforms and I've used it that way. Check out the examples provided in Paul's Online Notes, where completing the square is used more than once.

Tualha said...

I'm confused. You say this occurred in "joint effort" problems. Do you mean problems such as this?

"Alice takes 3 hours to weed the garden. Bob takes 5 hours to weed the garden. If they both work at the same time and don't affect each other, how long will it take Alice and Bob, working together, to weed the garden?"

I don't see any need for a quadratic equation to solve that. Alice weeds 1/3 garden per hour, Bob weeds 1/5, together they weed 8/15, so they take 15/8 hours.

Perhaps you mean some other sort of problem?

Zeno said...

Yes, Tualha. That kind of problem. It comes out quadratic when it's framed in this way: "Working alone, it takes Bob two hours longer to weed the garden than it takes Alice. If they work together, they get it done in three hours. How long does it take each of them, if they were to work alone?"

Tualha said...

Ah, thank you. Don't remember if we got it in that form when I took algebra (aka when dinosaurs ruled the earth). Alice takes $2+\sqrt{10}$ hours and Bob takes $4+\sqrt{10}$. (Done with the formula.)