Efectos sobre el animal

Questions on qualifying examinations often penalize the creative student while rewarding the student who mindlessly applies memorized "tricks." To solve a typical math or science problem, you manipulate symbols in a sequence of steps, each of which takes you closer to the solution in a way that is apparent. However, it is possible to contrive problems whose solutions require a step that appears to be arbitrary or even counterproductive, the logic of that step becoming clear only later in the problem. This crucial step is the trick. The student who doesn't know in advance the trick required to solve a test problem is extremely unlikely to discover it while working on the problem, especially if no time is allotted for that purpose, and so problems based on tricks favor the memorizer over even the creative individual who has a good overall understanding of the subject.

Some problems that can be solved without tricks are nevertheless made easier by tricks, as two examples from mathematics will illustrate. Because tricks often seem simple and obvious after they are explained, readers will better appreciate their nature by attempting the two problems now, before reading the discussion below:

1. Multiply 503 by 497.

2. The cube root of 64 is 4, because 4 X 4 X 4 is 64. What is the cube root of 1,728?

Problems that are made easier by tricks can be put into two groups. The first, represented by problem 1, are problems simplified by tricks that are useful in many situations. To multiply 503 by 497 quickly and with little chance of error, you can write the factors as (500 + 3) and (500 - 3), whose product is 250,000 - 9, or 249,991. This trick is based on the algebraic equation (a + b)(a

— b) = a2 _{— b}2 _{and is useful for multiplying any numbers that are equally spaced around a}

number whose square is easy to compute. The second group, exemplified by problem 2, are problems made easier by tricks that are useful in essentially just one problem. For example, to solve the cube root problem above,2 you can simply recognize 1,728 as the number of cubic inches in a cubic foot, which is 12 inches by 12 inches by 12 inches, and so the answer is 12. In either case the tricks are optional—you can do the multiplication the long way and the cube root by trial and error. However, with a slight change—say, a stringent time limit on the

EXAMINIG THE EXAMINATION 140

tests of arithmetic skills, but they would really be disguised tests of memory and of the ability to "psych out" the questions.

One often finds that a qualifying examination problem is impossible to solve without a special trick that is good for only that one problem. This kind of problem is usually constructed by starting with the trick and working backwards. The student who attacks such a problem with creativity, with an understanding of the subject, with insight and with the standard tricks of the trade gets absolutely nowhere. The special trick is the only approach to the problem that works. The unwitting student who uses the "wrong" approach, as logical as that approach may be and as effective as that approach may be in general, sinks deeper and deeper into increasingly

complicated calculations—a frustrating "mess," as it is often called—in an ultimately futile effort to get an answer. Be-cause only the special trick works, only the student who has seen the problem before—and memorized it—can solve it. And because the students’ creativity, understanding, insight and experience do not lead to the trick, they contribute nothing to the students score.

Students are eager to learn the standard tricks of the trade for the field they are preparing to enter. A symbol-manipulation routine that is good for only one problem will be part of this standard bag of tricks only if the problem is particularly important in the field. Otherwise the routine will remain just another obscure entry in the reference books. Qualifying examination problems that re-quire special tricks are not likely to be particularly important problems in the field, because, like the cube root problem above, they are typically written around the tricks rather than around something important in the field. Hence even the student who knows the standard tricks used by people working in the field cannot necessarily solve the qualifying examination problems that are based on obscure tricks.

What is the aim of examination problems that reward the memorization, quick recall and mechanical application of obscure symbol-manipulation routines? Problems that are disguised requests to give performances of memorized obscure routines clearly do not test the students creativity, understanding, in-sight or knowledge of the standard tricks of the trade. However, they do an excellent job of revealing whether the student is willing and able to do disciplined,

alienated work on assigned problems—the assigned problems in this case being the test preparation problems, which include problems based on obscure tricks.

I present here as an example a "quantum mechanics" problem from a physics PhD qualifying examination that was given at the University of California, Irvine. My argument doesn't depend on this example, and so readers not familiar with the technical details of quantum mechanics can skim the next few paragraphs without missing an essential part of the discussion. The problem, as it was stated:

Consider a 1 dimensional harmonic oscillator.

H = ( 2_{/2m )d}2_/dx2 _{+ (k/2)x}2

Compare your calculated value to the true ground state energy for H.

The reader wishing to appreciate this example fully should stop here for a minute or two and just list the steps to solve the problem.

The curtness of the statement of the problem—"minimize the energy"—ensures that many students will not even know what they are being asked to do. The problem is an example of a technique for calculating an upper bound on the ground-state energy of a system, but few students are likely to recognize it as such, because the technique is not central to quantum mechanics and is often not part of the curriculum. Hence, most students will not be able to decode the phrase "minimize the energy"; they will not realize that the problem is asking them to compute the energy associated with the pictured wave function and then to adjust a and b so that the energy is minimum.

Students who do decipher the question can begin to show some quantum mechanics skill by computing H_{ψ and integrating ψ}*_{Hψ over all x to get the expression for the energy. However,}

there is a catch. In most problems involving a function that has two parts that meet at a sharp point (the diagonal lines that meet at point b in this problem), physicists handle each part separately. In this problem, however, separately applying the Hamiltonian H to each part of the wave function _{ψ leads to the wrong answer. The trick here is to treat what appear to be two linear} functions as a single, absolute-value function—that is, as a single binomial with an absolute value-function term; the second derivative of the absolute-value function is twice the Dirac delta function, which will con-tribute to the energy integral.3

This crucial mathematical trick is so obscure that the only students likely to have learned it are those who have worked this particular physics problem before. The few students who somehow know the obscure mathematics from somewhere else are not much better off than those who don't know it at all, for they are unlikely to know that its use is required in this physics problem. Simple as the solution seems after it has been explained, it is certainly not some-thing a student is likely to discover while under examination.

The trick that solves a problem may reflect a principle of the subject, but use of the trick certainly does not imply an understanding or even a recognition of the principle. Due to the peculiar function used in the problem discussed here, the single point where the two diagonal lines meet actually contributes one-half of the energy. The mathematical trick that leads to the right answer accounts for this energy, but students who use the trick do so because they have memorized the need to do so, not because they recognize the need to account for the energy associated with the one point, and certainly not because they understand the quantum mechanical reasons why the one point contributes so much to the energy.

It is unlikely that the people who use this problem on a qualifying examination expect the student to understand the quantum mechanics of sharp points in wave functions; such quantum mechanics is obscure—sharp points in wave functions do not occur in nature and do not play much of a role

x a -a

b

EXAMINIG THE EXAMINATION 142

in the quantum mechanics curriculum. Anyone interested in simply measuring understanding would know better than to try to do so with this problem. Those who use this problem are not troubled by the fact that the typical student who gets credit for it probably does not really understand it, because a problem based on an obscure trick does not aim to test any such understanding; it aims to see if the student saw the problem, and memorized it, while preparing for the examination. By revealing what the student did specifically to prepare for the examination, problems like this are an excellent measure of the student's willingness and ability to do

disciplined, alienated work, the work that characterizes the preparation process. It is this willingness and ability, not knowledge of obscure tricks, that employers value most. Hence, the examination serves employers well even if students quickly forget the obscure tricks that they must memorize to pass. It is understandable that such forgetting, which is a big worry of naive students, does not trouble professional training institutions or employers. (The examination's emphasis on tricks also is part of its emphasis on speed, which favors a narrow-minded approach to problems, as I discuss in the next section.)

Standardized professional and preprofessional qualifying examinations, just like the faculty- written tests that I have been discussing, also measure the student's willingness and ability to memorize the obscure and to "psych out" the test writer's intentions. Consider the Graduate Record Examination, a multiple-choice verbal, quantitative and analytical test given to college graduates seeking admission to programs for higher degrees, and the SAT, a mainly multiple- choice verbal and mathematical examination given to high school seniors who are seeking admission to the colleges of their choice. The verbal sections of these standardized tests feature rarely used words that students typically en-counter only on such tests and in test-preparation books. The analogy, antonym, sentence-completion and reading-comprehension questions that make up these verbal sections frequently offer more than one correct answer, only one of which gives credit. (The same is true of questions on standardized science achievement tests used in admission decisions, as Albert Einstein's collaborator Banesh Hoffmann shows in very simple language in his 1962 book, The Tyranny of Testing.4) The tests' instruction to pick the "best" answer means that the successful student is the one who either shares the testers' values or senses those values and adopts them for the examination. In general, students who simply look for the answer that they think is best don't do as well as those who look for the answer that they think the test makers favor. The unconscious ideological discipline that the latter approach represents is the preprofessional's first step toward the more developed ideological discipline that characterizes the professional.