Testing the Logic

By Dr. Eugene Maier

When George Bush speaks it’s not always clear to me what he’s saying. The following quote comes from his February 27 address to Congress: “Critics of testing contend it distracts from learning. They talk about ‘teaching to the test.’ But let’s put that logic to the test. If you test a child on basic math and reading skills, and you are ‘teaching to the test,’ you are teaching math and reading. And that’s the whole idea.”

When he says “that logic,” I wonder, “What logic?” Making a statement like “teaching to the test” doesn’t entail any use of logic. What he seems to be doing is casting slurs upon those who oppose his testing program while attempting to offer a logical argument of his own to support it. An argument that runs something like this: If the teacher teaches to the test and if the test tests knowledge of basic math, then, ergo, the teacher is teaching basic math.

The logic is impeccable. But there’s a hitch. Impeccable logic doesn’t guarantee the truth of the conclusion one reaches. All that impeccable logic guarantees is that the conclusion of the argument is true provided the premises are true. If one bases an argument on a false premise, impeccable logic notwithstanding, the conclusion one reaches may well be false. (Here’s an example of a logical argument which has a false premise and arrives at a false conclusion: All Texans are ten feet tall. President Bush is a Texan. Therefore, President Bush is ten feet tall.)

Whether or not one agrees with Bush’s agenda for testing the “basics” isn’t as much a matter of logic as it is belief. If you don’t agree with his premises, no amount of logic compels you to accept his conclusions.

I, for one disagree with his premises. I don’t believe it’s possible to construct a test to be administered on a large scale that adequately measures knowledge of basic math. In the first place, I don’t believe there is any consensus on what constitutes basic math, and secondly, I believe test scores are more a measure of test-taking ability than of knowledge.

During a half century of teaching math, I’ve been engaged in lots of exercises to list the “basic” mathematical skills. Generally these attempts begin with some kind of litany about adding, multiplying, subtracting and dividing whole numbers, fractions, decimals, etc. The discussion turns to what ought to be known about these things. Should it be algorithms for computing; if so, which ones? There are lots of algorithms, are any of them basic? Does one need to know formal mathematical definitions of all these operations and a formal list of rules governing their behavior? Or is it sufficient to have a good intuitive understanding—good “number sense”—something I recognize when I see it but have a hard time describing definitively. In the end, “basic math” takes on the aura of an undefined term—not an entirely unforeseen circumstance; undefined terms are at the root of any mathematical discourse.

In reality, when a test is composed that supposedly tests basic skills, what’s basic doesn’t determine the test but, conversely, the test determines what’s basic. Put a test in the hands of teachers, tell them it’s a test on the basics, and whatever is in the test becomes the basics. And that’s what gets taught as the basics, regardless of its appropriateness or importance.

Also I, and many others, question whether any test given repeatedly on a large scale—even if versions change over the years—measures anything but the most superficial knowledge. Rote memorization and drill—solving lots of problems like those known to be on the test—carry the day on such exams, and require little in the way of profound understanding or working knowledge of the subject at hand.

The American public is never going to be convinced of the appropriateness of a nation-wide testing program on the basis of logic. As in any situation where there are strongly held and widely diverse opinions and beliefs, logic won’t carry the day. One is not going to gain consensus on a common set of premises from which to proceed. So the President will never achieve acceptance of a nation-wide school testing program by logical argument. If he achieves it at all, it will either be by persuasion, at best, or, at worst, by demagoguery.