In an essay first published in The American Scholar (Winter, 1980-81), Clara Claiborne Park, a professor of literature, celebrates Mnemosyne (Memory), the mother of all the Muses. She observed that even in her day, eighteen years ago, the word "memory" seemed to have been transmogrified into one beginning with 'R' -- "rotememory" -- at least in the world of Education. And she considered the attitude to be a bad one, that associated the new syllable to the name of the Goddess. So do I.
Without memory there is no knowledge, however many encyclopedias and computers one might own, or even be able to carry around with him. I have an old friend, Henry Geller, who still lives in Washington, D.C., though his days as Counsel to the FCC there ended long ago with a change of administration. I knew him in high school and college, but mostly by reputation since 1943, though I looked him up in Washington some years ago and talked over old times. Geller had a photographic memory ("eidetic memory", it is called by psychologists) which he found convenient in school and in his Christmas season Post Office job every year, where he earned a lot of money sorting mail by addresses into boxes that were in effect ZIP code classifications. He had memorized all the streets of
Detroit by address numbers, with the code appropriate to each, and therefore earned a high wage for those few weeks, that was otherwise only earned by sorters with many years of experience behind them. In college he majored in chemistry, and on exams could call up the proper page of the textbook, that had the formulas he needed. It would shorten his work if he remembered, e.g., that the formula in question was on an upper left page, but even if he didn't know such things at once he could leaf through the book until he found what he needed. And even in law school, he said, it sometimes came in handy.
We diverged in 1943 when we were drafted, and when we next met, forty years later, I asked him if he still could do this sort of thing. He laughed, and said he hadn't used it in many years, though he supposed he still could. It wasn't really very useful, he said, since it took so much time in most applications. It was like having a not-very-complete encyclopedia without an index. It was better to remember things. He had argued cases before Federal courts in behalf of the FCC, and at the Appeals Court level there was limited time to make one's case. He had written briefs, and clerks galore, but when he stood up to argue it all had to come from the 'random-access' memory, not the eidetic memory.
These days encyclopedias are found on ROM disks and are more convenient than they used to be, and the calculator is much better, quicker and more accurate, than the log tables or slide rule, but all this is analogous to eidetic memory, not Mnemosyne. Anyone rich enough can buy a computer and encyclopedia, and even a copy of Shakespeare, and thereby have a lot of knowledge around the house, but if he knows nothing but what he can look up on demand he is what I call ignorant. His ability to use this stuff via the "understanding" he might have learned – or 'developed', as they say -- in a course in literary criticism or mathematics is zero if he has no part of this material already in his accessible memory. For otherwise, what will he know to look up, or why, or when?
I was reminded of all this when I spent two days in Albany, NY last week (July, 1998), as a member of a committee of math educators working on the syllabus for the Regents' Exam "B" for mathematics. The new regime in NY will require a Regents' Exam "A" for all students, for graduation, in the essential subjects: math and English at least, and probably others but I don't know. These will test what should be known by the end of the 9th grade, maybe middle of the 10th, in math at any rate, and so will not be at all comparable with the famous old Regents' Exams which covered more territory, except that some primitive statistics is now included which wasn't there twenty years ago.
But for students who intend college, and some who don't, of course, the more advanced "B" exams, optional, will be administered some time in the 12th grade, and should be something like the end part of the old Regents'.
The syllabus for math is pretty well outlined in the New York Standards (not a very good, or demanding, document) and the more discursive booklets named Math I, Math II, and Math III, intended for 9th, 10th and 11th grades for ordinary students. ("Accelerated" students will do these things a year earlier, giving time for "pre-calculus" and maybe "statistics" in the 11th grade and AP Calculus in the 12th.) So the Regents' "B" exam is not going to be quite as demanding, at least as to analysis ("pre-calculus") as the old Regents'. Very well; those are the rules; now for the exam.
The actual exam questions will be written by some other committee than the one that met last week in Albany. I fear it will be entirely composed of exam experts, who will know how to make the April test give the same results as the preceding January test even though the questions are different, and will know how to write questions so that a predetermined number test "procedural knowlege" while some other fractional parts of the examination test "conceptual understanding" and "problem-solving". Our committee was actually asked for our opinions as to the proportions we considered desirable, and most did have an opinion. This was done by email after our meeting in Albany was over, and I was amazed at the ability of the people to distinguish these three aspects of mathematical competence. On this question I had to pass, saying that (whatever they meant) I believed every exam question should test at least two of them, and 90% test them all.
Other aspects of the exam were for our discussion, and indeed voting: should calculators be available, and what sort; should there be multiple-choice questions and how many; how many questions should there be altogether; what fraction of the test should be "contextual". (A contextual" question is one that invokes a real-world situation.) The results will be unimportant to the present discussion of memory, however, except for one matter on which we took a vote:
Should there be a "formula page" for the student's use, as there will be for the "A" exam? The answer was yes, and the page will include such things as cos(A-B) = cos(A)cos(B) + sin(A)sin(B) and cos(A+B) = ditto - ditto. I suggested that just one of these formulas should suffice, for if the students were expected to know even the elements of trigonometry they surely would know how to derive the difference formula from the sum formula, but I was voted down. Of course the corresponding formulas for sin(A+B) and sin(A-B) will be on the crib sheet, and – for all that I can remember -- the fact that tan(A)=sin(A)/cos(A).
Perhaps the sheet will contain tan(B) = sin(B)/cos(B) too? B might not equal A, after all, and so a separate formula for B would relieve those who found it in the formula page only for "tan(A)" and therefore couldn't use it. I was so distressed at the attitude displayed towards memorization here that I stopped keeping track of what will or will not be included. It probably doesn't matter much to the results of the exam, but it does matter to the morale of the test-takers, according to experienced teachers, and maybe to the morale of the teachers as well, according to me. It seems to be expected that if students were not assured of having these formulas printed there with the exam they would spend months in drill during preparation for the exams, wasting time on mere memorization that they now have free for conceptual learning.
My own view is that it is the more honorable of the math educators who believe this, and probably have *seen* it, too, and that the less honorable simply want to make things as easy as the public will let them get away with, so as to get higher "Regents'" scores and be seen, thereby, to be doing their jobs.
To the degree the honorable educators are right, and that depriving students of the comfort of knowing that these things need not be memorized will save them the agony and waste they imagine to be the alternative, I must say the nation has come to have an altogether diseased notion of the function of memory. And, while this is only part of a larger disease concerning education in general, whose description is too long for the margin of this page, we shall have enough to do attacking the particular problem of the place of memory in mathematical education.
I believe I will call for a campaign to restore memory to the position of respect it had up to about a hundred years ago, when school children memorized orations of Abraham Lincoln, scenes from Shakespeare, The Wreck of the Hesperus and Casey at the Bat, not to mention the procedures of arithmetic. The fact that some teachers drilled children in routines that were not given sense does not mean that drill as such creates a vacuum in the brain. I have known actors who memorized scenes from Shakespeare and also knew what they were about. I have known chemists who knew the size of the dihedral angle in a regular tetrahedron (do you?) and understood organic chemistry too.
There is no harm in knowing things, and much value. Some of the actors who memorized their parts in Shakespeare (I have known them quite well, all my adult life) did not in fact understand much of what they were saying during the first few read-throughs, but would never have got on to their characters if they hadn't first had the words in their minds and ready to their tongues. Why is a technique thousands of years old, and still considered valid in the teaching of music and theater, reviled in school mathematics?
Ralph A. Raimi