Thursday, May 25, 2006

Prechewed mathematics

Disagreeing with Feynman

Word problems are the domesticated beasts of burden of school mathematics. They would never survive very long on their own in the wild, but they do a good day's work in the service of giving students practice in basic problem-solving techniques. Does it matter that classroom word problems are so weak and denatured?

Permit me to lay out some of the difficulties that led me to the question I posed. The problems we give our students in math classes are highly idealized. Their answers are typically whole numbers or, at worst, simple rational numbers with small denominators. There's the rub. It means that many of them (most of them?) can be readily dispatched with a few desultory rounds of guess-and-check. Does x = 1 work? No? Then how about x = 2? Lots of students know this approach and even get instruction in guessing strategies in their elementary math classes.

This makes me unhappy. Students end up wanting full credit for answers that they snatched out of thin air. It's the right answer, damn it! Give me my points! I tell them that we can make a little deal: Either I will devise problems with fairly nice answers (like 2, 5, or 3/2) and they will show me their work in careful steps, or I will devise problems with less nice answers (like –17 and 41/29) and they can find the answers however they damned well please. With a certain amount of grumbling, a large majority then concedes that they prefer to show their work and find rather “nice” answers.

Process & product

Unfortunately, we often train our students to focus too much on product (the answer!) and neglect the process (the solution algorithm). I've tried very hard, but with only mixed success, to persuade my students that we care how the answer is found as much as we care about what the answer is. This position produces cries of dismay from students who want to do things their own way. One student complained at about a teacher who insisted on “showing your work”: “The last time I checked, math was all about finding the right answer!”

Sorry, kid. I don't know where exactly you “checked,” but it's not that simple. Sure, answers are important and you won't get full credit without a correct answer, but I expect more from you than just answers. I expect a demonstrated ability to apply the processes that I teach you.

It might appear that the answer-focused students have an ally in the late Richard Feynman, the famous Nobel-laureate physicist. In a delightful video interview (at the 9 minute mark), Feynman relates his cousin's unhappy experience with algebra:
My cousin at that time—who was three years older—was in high school and was having considerable difficulty with his algebra. I was allowed to sit in the corner while the tutor tried to teach my cousin algebra. I said to my cousin then, “What are you trying to do?” I hear him talking about x, you know.

“Well, you know, 2x + 7 is equal to 15,” he said, “and I'm trying to figure out what x is,” and I says, “You mean 4.” He says, “Yeah, but you did it by arithmetic. You have to do it by algebra.”

And that's why my cousin was never able to do algebra, because he didn't understand how he was supposed to do it. I learned algebra, fortunately, by—not going to school—by knowing the whole idea was to find out what x was and it didn't make any difference how you did it. There's no such a thing as, you know, do it by arithmetic, you do it by algebra. It was a false thing that they had invented in school, so that the children who have to study algebra can all pass it. They had invented a set of rules, which if you followed them without thinking, could produce the answer. Subtract 7 from both sides. If you have a multiplier, divide both sides by the multiplier. And so on. A series of steps by which you could get the answer if you didn't understand what you were trying to do.
Hmm. Feynman was a lot smarter than I am, so should I now stroll away, whistling casually, as if I had never argued against the primacy of the value of x? Uh, no.

I believe that Feynman and I are talking about rather different things. Or different contexts, at least. I share Feynman's disdain for the blindly memorized algorithm, which is guaranteed to generate the correct answer whether the student understands the process or not. I want my students to understand why an equation remains valid when you add the same quantity to both sides, or divide both sides by the same nonzero number. (I like content in addition to process and product.) On the other hand, I'm dismayed when students (college students, no less) refuse to learn how to follow instructions. Carefully rehearsing algorithms and practicing problem-solving processes should permit almost any student to achieve the minimum level of expertise that we require for a passing grade. In reality, however, approximately half of the students who take elementary algebra at the community college level fail the class. What's going on?

Too bad there's no simple answer. But I think part of it lies in Feynman's story about his cousin's problem. He was able to tell at a glance what the value of x had to be, provoking his cousin into accusing him of using mere arithmetic instead of the required algebra. Feynman seems to agree that he did not solve the problem in an “algebraic” way. Let's consider that for a moment. What do you think passed through Feynman's head when he saw 2x + 7 = 15? Here's my guess. First, he saw that 2x had to equal 8, because that's what you add to 7 to get 15. Second, with almost no elapsed time at all, he knew that x had to be 4 because that's the number you multiply by 2 to get 8.

Does that seem right to you? If that's what occurred in a split-second in Feynman's head, he could readily agree that he didn't need algebra to solve the problem. However, my imagined first step is nicely equivalent to subtracting 7 from both sides, the rote algorithmic process that Feynman cited with disdain in his interview. As for the second step, it matches with the process of dividing both sides of the equation by the multiplier in front of the variable. To Feynman's cousin, however, Feynman was just blurting out a number without doing any work, but algebra by any other name is still algebra. I don't believe for even a second that Feynman found the answer by running through lists of numbers until one happened to work. Yeah, he was doing algebra.

If you know anything about Feynman, you know that he was a prodigious problem solver. He puzzled over problems both great and small. (In the video interview he recounts the famous story of the spinning dinner plate that eventually led him to the work that won the Nobel prize. That started out as a very small problem indeed.) Feynman was not interested in rote processes, although he used them subconsciously over and over again whenever he was making computations. For him, the subroutines of calculation were submerged at a very low level while the novel aspects of each new problem remained uppermost in his mind. Algebra students, by contrast, exist at the level of those basic subroutines and may be puzzling over them quite as much as Feynman did with the problems at his much higher level. Who's to say?

Earlier I asked whether it mattered that the problems in our math classes are so lame. Part of the answer lies in the fact that most students meet these problems at a very elementary level of mathematics. We are nowhere close to the Feynman level of relativistic physics in our applications exercises. Heck, we're not even at the level of Newton's basic law of gravity. It's all very well to be told that a thrown object traces a parabolic trajectory, but a real-life projectile problem would have to factor in the aerodynamic properties of the thrown object as well as the effects of any wind. You can be quite sure that the result would not be a problem suitable for introductory algebra. Yes, we draw the teeth of the application problems before we give them to our students. When they suspect that I'm holding something back, I cheerfully admit that there are always additional complications that can be thrown in later if they ever grow up to be a range officer at an artillery field. In the meantime, we are clearing the playground of dangerous obstacles so that they can run and jump safely. Sometimes, unfortunately, the result is an extremely flat and boring space. I promise to stay on guard against overdoing it.

It's a conversation

As an algebra teacher, I frequently fear that I am in the position of punishing creativity. That can happen when one is dealing with a highly prescriptive syllabus for a course that is a prerequisite for practically every health, science, and technology class on campus. I take some refuge in my practice of showing alternative approaches to problems, giving students some flexibility in finding the method that works best for each individual. I'm not very prescriptive on exams either, sharing my students' negative attitude toward problems that demand a specific technique. Thus my students are free to solve their quadratic equations by factoring (if possible), completing the square (always possible), or applying the quadratic formula (after it's been introduced, of course: no fair using it before it's been presented to your classmates!). But even if I try to keep algebra from being a straitjacket, it is nevertheless a tight fit. Not a lot of wiggle room.

Sometimes I invoke Feynman's name when I want to make a point about problem solving, and I particularly recommend his compilation of autobiographical anecdotes (“Surely You're Joking, Mr. Feynman!”) as a wonderful introduction to the life of one of the twentieth century's greatest thinkers. I don't know, though, if I'll risk sharing with my algebra students his remarks about how their class is a “false thing” invented to permit the clueless to solve math problems. It's an intriguing thought. I will, however, continue to work on my answers to those students who want to do things “their own way.” My usual answer is to apologize in advance if they are creative geniuses whom I have failed to recognize and to suggest it will be amusing to report my myopia in their autobiographies after they are famous. When they stop laughing (and if they don't laugh then you have completely misread the class and made a huge mistake!), I move on to my next point: Human endeavors don't exist in a vacuum. Even math is a form of communications that can be used to convey information if applied in ways that other people will understand. Here I can invoke Feynman once again, who recounts in the first chapter of his autobiography how he once went astray with his highly personal and idiosyncratic approach to trigonometry:
While I was doing all this trigonometry, I didn't like the symbols for sine, cosine, tangent, and so on. To me, “sin f” looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath.... I thought my symbols were just as good, if not better, than the regular symbols—it doesn't make any difference what symbols you use—but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, “What the hell are those?” I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.
See, kids, sometimes even Feynman had to go mainstream.


wegrit said...

I find this a fascinating post because I'm beyond terrible at math, but I did very well at algebra because it was logical and in steps. I find it interesting that what everyone else seems to denigrate about algebra is the one thing that made it make sense to me.

Reasonably Cranky said...

Feynman missed that a small number of students make the intuitive leap from "just sets of rules" to fundamental concepts in which they can derive and prove their own rules. As the anonymous they said, "You can leave a horse to water...".

Sometimes we try to beat that intuition into our kids with hundreds of rote arithmetic problems in grammar school, but that just turns math into a drudgery for the brighter ones. By the time they get to college they've been permanently damaged.

I detested grade school because of the immense amount of repitition. Why can't the United States teach algebra in grade school, as in other countries, rather than waiting to put some frustrated college teacher through the wringer?

John Morgan said...

I happened on this post rather late in its history. It's 3 Jul 2009 as I write, but the problems outlined still remain.

In teaching simple algebra, I still consider that numerical methods (ie doing arithmetic on the problem) is the natural bridge to doing full analysis. The trick is to introduce the two in parallel. For quadratic equations, start by solving one of the factorisables, say x² + x = 12 which easily yields three by inspection. Only the brightest students will see that -4 is also a result, and then the need for analysis becomes apparent to them. One can continue by constructing the original quadratic multiplying out the binomials and arrive at the notion of factorising quadratics to solve them.

Numerical helps when you have analytically intractables like 2^x + ln x - sin x = 5 ( I am prepared to be counted wrong about this one as I don't know for sure if it can't be rendered down algebraically) and makes for an interesting intermission when students get bored by routine. Give them simple order 5 equations to play with and tell them how no amount of algebra will solve the equation - unless you happen to believe, like some do, that you can trisect an angle on the Euclidean plane :-(

Anonymous said...

Careful, John. Trisection of angles is tangentially related to solvability of the general quintic, but not quite how you're thinking. Trisection is all a matter of which tools you're allowed to use.

You can't trisect angles with ruler and compass. If you have a magic wand that computes cube roots, however, you can trisect angles all day long. Solving the general cubic also requires being able to take cube roots. But solving the general quintic is impossible no matter what roots you're allowed to take.

John Morgan said...


Yes. I should have specified ruler and compass. But isn't your magic wand for cube roots rather different to the magic wand for square roots. Pythagoras' theorem works algebraically and on the Euclidean plane plus higher dimensions, doesn't it? But is there an equivalent wand for X^3 + Y^3 ... ? I only ask out of semi-ignorance about such things. It's so damned irritating not to be able to know everything about everything :-)

Anonymous said...

The problem is that you are teaching stupid people. In order to make it so that they can do some minimal amount of math just to be able to survive in the real world, you aren't teaching basic principles. You're teaching esoteric rules based on, and several orders removed from, the basic mathematical principles that you're teaching. The intelligent people in your classroom are left to their own devices to figure out that these rules come from somewhere. If you could teach the basic laws of mathematics from which your limited problem-solving techniques are derived, you could give them extremely complex problems with complex answers that they could do however they saw fit. Just like Feynman did in his head; because he had already figured out the basic laws behind solving for an unknown quantity without needing to be taught the dumbed-down techniques like subtracting from both sides.

Anonymous said...

I don't know if there is such a physical object like there is for ruler and compass. But that's not really the point. What I mean to say is that if you had it, you could trisect any angle, but you still couldn't solve the general quintic.

Incidentally, for anyone reading who doesn't know, ruler and compass constructions allow you to perform any arithmetic operations you want, and also square roots. Given one reference length to serve as a "unit", you can use a lot of what's in Euclid (mostly in book I) to make geometric analogues for addition, subtraction, multiplication, division, and square roots.

So the question is: what numbers can you get from these operations starting with the natural numbers? It turns out you can't get things like the cube root of 2, which you would need to double the cube. Trisecting angles also turns out to require arbitrary cube roots. You can't square the circle because you can't construct a length pi. All these classical puzzles come down to reinterpreting Euclidean geometry in light of abstract algebra.

And the captcha today: "soisi". So I see...

Robert Furber said...

There are physical objects that can be used to trisect angles. A carpenter's square is one, and the more usual one is the neusis construction, which is using a marked ruler. These are described in Underwood Dudley's book "The Trisectors". I don't know what the correct Galois theoretic descriptions of these constructions are, in general (i.e. what kind of field extensions of the rations you can get. Ruler and compasses gets you compositions of quadratic extensions).

Anonymous said...

I don't see why students should be punished for seeing a better solution. If the problem has been made so easy they can immediately see the solution, this is the fault of the teacher and not the student.

I realise that you are teaching algorithms that work in general and trying to assess the students ability at implementing these algorithms, but there is no reason why students should only try to solve solution by the algorithm taught to them. (In fact this is precisely the problem in higher mathematics: here is a problem that there is no known algorithm to solve - invent one).

I also strongly disagree with the statement "no fair using it before it's been presented to your classmates!" Legend has it that as a student Euler was set the problem of adding a large quantity of sequential numbers together, and he immediately saw the problem reduced to a multiplication if you add from both ends at once. I think any teacher who would punish a student for seeing such a clever solution (that was not taught because the teacher did not know it) has no business teaching.

Zeno said...

I realise that you are teaching algorithms that work in general and trying to assess the students ability at implementing these algorithms

Haven't you answered your own question, physjam? I can't assess the student's ability to use the general algorithm unless they actually use it. There is nothing to prevent them from also trotting out some clever shortcut (if they have the time and inclination).

A similar problem arises if they run ahead of their classmates. If, for example, they drag out a previously-learned quadratic formula before we've covered it in class, they're inclined to use it exclusively. It's not always the best solution technique (although it always works) and students who jump the gun cannot be assessed on their mastery of alternatives if they use nothing else.

The anecdote you cite concern Gauss, who is said to have added up the numbers from 1 to 100 by discerning a clever shortcut. His teacher was impressed and supposedly provided the young Gauss with a more advanced math text to keep him happy and occupied. If I find a young Gauss in any of my classes, I will try not to punish him (but I will wonder why he's taking algebra in college).

Varyag said...

From the first grade, I had very good math teachers. My main purpose in taking math was not to get the right answer nor to show that I knew how to perform the operations. I wanted to discover new methods.

In plane geometry, most students would memorize proofs and spit them back on tests. I tried to use the axioms and postulates and previously proven examples to arrive at n answer.

My geometry teacher told me how glad he was to have someone like me in his class and that there was only one.

Varyag said...

From the first grade, I had very good math teachers. My main purpose in taking math was not to get the right answer nor to show that I knew how to perform the operations. I wanted to discover new methods.

In plane geometry, most students would memorize proofs and spit them back on tests. I tried to use the axioms and postulates and previously proven examples to arrive at n answer.

My geometry teacher told me how glad he was to have someone like me in his class and that there was only one.