In response to Howard Penn's complaint that multiple-choice exams confound the material we wish to teach with a lot of other test-taking skills, and encourage teachers to teach to the exam:
The Japanese exams which I believe are roughly equivalent to SAT exams in the US, seem to represent a good solution. A sample was reprinted in FOCUS about a year ago. In essence, each answer consists of a template with a number of blanks. For instance, one common format had a fraction with two blanks in the numerator and two in the denominator, and one blank in front. The students answered the question by filling in one bubble from a row for each of the blanks, indicating a digit for the first four I mentioned, and a plus or minus sign for the blank in front. This format preserves the ease of computer grading which is the only virtue of multiple-choice exams, while reducing the level of information provided by the form of the answer to the point where students really cannot work toward the answer.
The questions I recall all had answers which were numbers. It may or may not be difficult to adapt this format to suit the questions we ask in a calculus class. The coefficients and exponents of a polynomial are easy to code in this fashion, if we are willing to reveal that the solution is a polynomial (which seems a minor concession). On the other hand, if we ask students to solve a differential equation, we certainly would not want to reveal the form of the answer. One work-around might be to ask the students to evaluate their solutions at a particular point, rather than give the function.
One of the root questions here is what are we trying to teach. The simplistic view of the reform effort is that the "old" courses sought to teach students to solve equations, evaluate derivatives, and to answer other clearly-posed questions, while the "new" courses try to teach students to think, and focus on such "higher-level" activities as mathematical modelling and the verbal communication of results. Of, course, both kinds of learning are necessary, and some amount of rote learning is, in my opinion, a prerequisite for creative work. A difficulty arises when we try to test both kinds of learning in compatible ways. I am unwilling to let exceptional performance at the computation of integrals substitute for any understanding whatsoever of what integrals means. And vice-versa. Yet this is what we do if we give an exam in which both kinds of learning are awarded points, and those points are then added together.
One solution I have been using for the last three semesters is to separate the two kinds of learning I want to test. In my first-term calculus courses I give either three or four "Skills Test," to test the computational skills with limits, inequalities, differentiation, and sometimes knowledge of the trig. functions. A passing score on each skills test is 9 out of 10, and students must pass all the skills tests in order to pass the course. I make up several versions of each test. One is given in class (I usually allow 20 or 25 minutes), and students who do not pass the test come to my office hours when they feel they are ready, and take another version of the same test. They cam repeat the test as many times as necessary, and the only thing I mark in my grade book is a check to indicate that they have passed a given skills test.
If a student passes all the skills tests, then they get a grade determined by exams, homework, class participation, etc. I try to keep the computations involved in the exams very straightforward, so that the exams test the understanding of ideas with as little interference as possible from questions of computation.
It takes some guts to stick to it, and this approach causes some students considerable agony, but I have not yet failed a student for inability to pass the skills tests who would otherwse have passed the course on the strength of his/her work on exams, homework, etc. I feel that this grading scheme reflects fairly well my conviction that _both_ computational skill _and_ conceptual understand are indispensible.
Michael Kantor Guilford College Greensboro NC 27410 910-316-2280