Alphametics, the SAT and the theory behind math tests

I once saw an alphametic in an SAT question -- simpler than this one but with the same basic principle. My first thought (after, "Was that an alphametic?") was what a great question.

Of course, solving alphmetics is a completely useless skill. No one has ever or will ever actually needed to do one of these. It is that very frivolousness that makes it such a good question for a college entrance exam. It requires sophisticated mathematical reasoning but it comes in a form almost none of the students will have seen before.

For comparison, consider a problem you would not see on the SAT*, factoring a trinomial that wasn't the square of a binomial (this is another skill you'll never actually need but it's not a bad way for students to get a feel for working with polynomials). Let's look a two students who got the problem right:

Student one hasn't taken algebra since junior high but understands the fundamental relationships, finds the correct answer by multiplying out the possibilities;

Student two was recently taught an algorithm for factoring, doesn't really understand the foundation but is able to grind out the right answer.

Obviously, we have a confounding problem here, and a fairly common one at that. We would like to identify understanding and long term retention but these can easily be confused with familiarity with recently presented information (particularly when certain teachers bend their schedules and curricula out of shape to teach to the test). The people behind SAT have partly addressed this confounding by including puzzle-type questions that most students would be unfamiliar with.**

All too often, the people behind other standardized tests deal with the issue by pretending it doesn't exist.




* Not to be confused with the SAT II, which is a different and less interesting test.

** The type of kid who reads Martin Gardner books for recreation would generally do fine on the SAT even without the familiarity factor (though the prom may not go as well).

An alphametic pre-footnote

Light posting today, but since I'll be mentioning alphametics in an upcoming post I thought I'd give you something to chew on while you're waiting.

In 1924, the great Henry Dudeney published the following puzzle in the Strand Magazine:

S E N D

+ M O R E
__________________

M O N E Y

Each letter here represents a digit 0 through 9. If you substitute in the proper digits, the first two numbers (words) should add up to the third.

No one who reads this blog should have any trouble with this, but if you do, Wikipedia has a good step-by-step solution (though it does assume a passing acquaintance with number theory).