Professor Steven Strogatz of Cornell has posted another installment in his series on math at the “Opinionator” blog of the New York Times. In this new article, “The Joy of X”, Prof. Strogatz moves along from arithmetic and the properties of numbers to algebra. I know many people remember algebra with less than positive feelings, but it, along with geometry, are the first (and sometimes the only) math courses, beyond basic arithmetic, that people take.
As Prof. Strogatz points out, there are really only two basic things that algebra does:
Algebra, for example, may have once struck you as a dizzying mix of symbols, definitions and procedures, but in the end they all boil down to just two activities — solving for x and working with formulas.
Although he says he has just thought up this distinction, it seems sensible to me. In this latest article he focuses on working with formulas — solving for X is promised for the next article — and I think that is useful, since this is the area where I have seen so many students have difficulty, especially when they have to deal with so-called “word problems”.
I think one of the reasons that this troubles people is that it requires a way of thinking that is not intuitive, or instinctive. Prof. Strogatz cites a simple example that many people get wrong when they answer offhand:
… suppose the length of a hallway is y when measured in yards, and f when measured in feet. Write an equation that relates y to f.
Many people will come up with the answer:
y = 3 f
which, as Strogatz says, seems like a straightforward translation of the sentence “One yard equals 3 feet.” It is of course “backwards”; the correct equation is:
f = 3 y
I was glad to see that Prof. Strogatz also points out that, if we think about the units attached to these quantities, it can help us get the right answer. For example, the ‘3’ is really 3 feet/yard. If we substitute that value with units into the first (incorrect) equation for a hallway 15 feet long, we get:
y = 3 ft/yd × 15 ft = 45 ft² / yd
Clearly we expect an answer in yards, not in feet² per yard.
Another example (not from the article) of how intuition can mislead us is this problem:
A bat and a ball together cost $1.10. The bat costs $1 more than the ball. How much does the ball cost?
Most people (you can try this yourself) will say the ball costs $0.10 (10 cents), which is wrong. The ball must cost $0.05, and the bat $1.05, to meet the specification of the problem.
Now, these are not really particularly tough problems, so what is going on here? I think there is a significant clue in the fact that, when people are warned in some way that they are about to be asked a “tricky” question, they are much more likely to get the right answer. I think this is because getting the right answer requires what I have called “non-intuitive” thinking; warning people beforehand gives them a cue to engage that mental process. That doesn’t happen automatically, because that logical process is learned, and has to be consciously engaged.
The difference between the intuitive and non-intuitive modes of thought is not related to the intrinsic difficulty of the computational problem. Consider, for example, that every normal person learns to recognize faces, walk, and talk, without any formal education being needed; but I don’t know of anyone who has spontaneously learned algebra. Yet anyone who has tried can tell you it is much tougher to program a computer to recognize faces, for example, than to solve Algebra I problems.
It is a credit to the flexibility of our mental apparatus that we can learn to think in this non-intuitive way, but it does take work. Those word problems are not in your math book just to make your life unpleasant.