The Federal government in legislation of the past ten years has been making a commendable effort to induce the States to compose Standards for instruction in English, mathematics, history and so on, on the basis of which statewide examinations can be composed to ascertain whether their schools are accomplishing what the public wants. School education is in principle still a local or State responsibility, but it is becoming ever more clear that the nation as a whole has an interest in the results, much as it has a national and not merely local interest in communicable diseases.
This effort being recent, except for a few states which have had a statewide examination system for many years, the results of recent federal policy in this regard have yet to be seen; but we can see already some of the difficulties. The most important difficulties are associated with a desire to "leave no child behind", which in terms of examinations is mostly construed as meaning "so that everybody passes". States which have written exacting standards and examinations have been finding an excessive, or embarrassing, number of "failures", while states with the foresight to make their standards vague and their examintions trivial are rewarded with a high success rate. Yet politics seems to dictate the desire to look good while hiding the real failures either by inflating the scores, calling the failures "learning disabled" and so not subject to the examinations that count, or gradually eroding the originally exacting standards so that these devices are not necessary.
We do have at least one "national" examinaton, the NAEP, which is used only statistically, to compare state with state and year against year, but not to decide who "passes" or "fails" in some absolute sense, as for example the New York Regents mathematics A examination decides who may not get a diploma. Any mathematician studying the questions asked by NAEP is immediately struck by the low level of mathematical knowledge and understanding it asks. Furthermore, the fact that its multiple-choice format questions offer only four choices means that guessing at answers will yield a 25% correct result on average; and yet on many of its questions the percent answering correctly is really not much more than that, sometimes 30%, so that it is hard to distinguish those who know a little from those who know nothing. At the other end of the scale, the NAEP for mathematics, by not asking anything exacting, not even at the level nominally prescribed by some of the weaker State standards, does not permit superior students to exhibit their superior scholarship, for knowing half of what a State's standards might demand may well be sufficient for a 100% score on NAEP.
Such examinations yield very little useful information, and yet most State education departments, in composing their own examinations, tend to follow this model, apparently to make sure that enough students pass. And it must be said that some of these State standards and consequent examinations are composed by people with little understanding of mathematics, even on the elementary level, and are phrased in ambiguous wording, or ask questions not worth asking. Such examinations not only fail to distinguish levels of accomplishment very well, but discourage teaching. Students who are "good at math" are neglected; they learn all they need to know to succeed early in the semester, if not at home or in earlier grades, and are encouraged to tune out on further study by the boredom of hearing in class again and again what they already know, or sometimes what they know to be trivial and even false.
The problem is, what devices will encourage better and more demanding teaching of the able students, while not leaving behind those less able, or less interested, or less well prepared by home or earlier teaching? Throughout most of human history, societies were not reluctant to dismiss the problem: they were willing to call some children "smart" and others "dumb", and let the dumb ones drop out early and become farmers or laborers who didn't even need to be able to read, let alone use mathematics usefully, while school teachers could concentrate on the "teachers' pets" and bring up an elite to run the country, its industries, its government, and its intellectual life.
Not only is it unfair to dismiss the "lower half" in our schools, it is often a misjudgment that the reluctant, scornful or bored student is one of the "dumb" ones, so that dismissing them as dropouts is to lose a valuable resource as well as create an unnecessary underclass by making mistaken judgments in the schools. How can we save the non-performing students and yet retain high standards and rich offerings for students who can take advantage of them? The problem is exacerbated by the pieties of the last century, which have merely denied any differences between children, and have insisted that everyone can learn equally well, and with the proper instruction will. They don't say this about violinists and basketball players, but they do say it about mathematics.
It is important for us to reverse this doctrine and admit that some will -- for whatever reason -- learn substantially less than others, no matter what we do. In a democratic and egalitarian society like ours, however, we must do this without simply dismissing the "losers" as subhuman, but with every effort to bring them to the highest level possible for them. It is in the statewide examination system that we can find the means to do so, or at least the diagnosis needed to address the problem properly. Hiding the differences by feeble examinations and low standards is cheating; it may be likened to two common images: putting a band-aid on a cancer, or shooting the messenger.
We say the the messenger is essential if we are to defend the fortress; we must not honor him who says the enemy is not at the gates when he is, even if the news is uncomfortable. On the other hand, there is no reason to retain the tradition that 90% is an A, 80% a B, 70% a C and so on. With exacting examinatons it is quite possible to have scores run from 10 to 100, with 30 as a passing score. (Obviously not using a four-choice multiple-choice examination, but there is no reason not to offer ten choices on such an examinaton, something I have done for years in elementary calculus courses at the University of Rochester.) Students do not all have to learn the same amount, and to go on to the next grade a certain minimum should be recognized as necessary, while two or three times that skill should be recognizable in the results when they occur.
Galton once said that in comparison of intellect, Isaac Newton was to the man in the street as the man in the street was to his dog. We must identify the "villge Milton" if we can, while advancing the sturdy yeoman at the same time. Pretending there is no difference is hurtful to them both. We need examinations much more demanding at the high end than those we are seeing now, and we need minimal scores that are sufficient but not attainable by luck. There then comes a time, typically the beginning of high school, when as a matter of volunary choice, the aspiring Miltons and Hilberts will find their way into the more demanding curriculum choices, much as is now done voluntarily (though with advice) in the colleges, while those of less accomplishment will have a path that can be navigated successfully even if they know a fraction of what is possible for some people in their age cohort. We already do this with basketball players and musicians; why do we stop short in mathematics and history?
Our policy is to conceal nothing, when it comes to measuring accomplishment, and to make use to a wide differential in these measurements to make optimal future choices, or to give advice to students and parents by which they will make these choices of their own volition. Some few will fail, we know; indeed, some will drop our when they can, or even before, or go to jail for crimes. We must have the courage to recognize that not everyone will succeed, even if everyone can. We need not decide the difference is genetic, or caused by bad companions, or whatever "cause" is popular at the moment. It is results we measure, and these results, at each stage, are the diagnosis for the next application of educational expertise, according to the case. We must do what we can not to shame those who do not reach the highest ranks, but we already know that non-violinists and non-Varsity athletes can sustain their "failure" without pain. Educating children to their demonstrated maximum should be no loss and no shame. The key is to get the demonstration to be fair and accurate.
Those who deplore "high-stakes" testing on the grounds of "elitism", or psychological damage to the "losers" are only guaranteeing the increase in the number of losers, even though at the same time these advocates of so-called equality look for other means of concealing the message. The message is real, the success and failure are real, and they always have been. Our task is to reduce the pain of what is inevitable, be sure we do not actually generate failures that are not inevitable, and maximize what our next generations will in fact accomplish. They will, on average, be prouder of real accomplishment, even modest accomplishment, than of fraudulent or empty certificates and honors.
September, 2002
Ralph A. Raimi
Department of Mathematics
University of Rochester
Rochester, NY 14627
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