Part 1 of A Mathematical Manifesto
by Ralph Raimi for NYC HOLD
1. On Memory
In mathematics as in almost every other endeavor, memory is indispensable. We begin by learning our native language, and certainly we have memorized thousands of words as well as some of the rules for putting them together into thoughts by the time we are five years old. Does having "memorized" all those words, and those rules, mean we do not understand them?
The present orthodoxy in mathematics teaching, as represented by the National Council of Teachers of Mathematics 1989 Standards, and its successor, the PSSM of 2000, downgrades "mere" memory, sometimes called "rote-memory", and exalts what it calls "understanding" instead. This opposition is most clearly seen in the PSSM denigration of established algorithms and theorems (of which more below), in favor of student discovery of methods unique to the problem at hand. But there is no conflict between memory and understanding. The public knows this even though the incessant propaganda from the colleges of education insists otherwise.
Memory and understanding are more than not in opposition, they are necessary to each other. It is plainly impossible to understand something you cannot even bring to mind. In the other direction, while it is possible to memorize some things without understanding, a Latvian poem, perhaps, for someone who has no knowledge of the Latvian language, it is extremely difficult to do. No sensible teacher of mathematics will ask such memorization, unless the thing to be remembered is some- how linked to the rest of our lives, and to other things already in our memories, in which case understanding often follows.
Thus most people have memorized the fact that 8*7=56, early in life. When it is necessary to use this fact, they simply call it up without thought. If asked by a child how they know it is true, however, they won't say they learned it in a book somewhere, but will probably explain with a picture of a rectangle measuring 8 by 7, and point out that partitioning it thus and so shows that 56 is the answer. But they do not picture the rectangle in their own minds every time they use the fact. If they had to go back to such first principles all the time they would never be done with their daily tasks.
When children first learn that 8*7=56, they should be given the model of the rectangle to contemplate, and other illustrative models of course. This ancient pedagogical insight, that one needs to couple understanding with the memory where possible in the early stages of learning, has caused some "modern" or "reform" professionals in mathematics education to forget that the picture, in this case, the rectangle exhibited for understandings' sake, is mainly a learning device, and sometimes an application of the fact to be memorized, but not something to carry around along with the thing memorized. Once a person has learned the rectangle interpretation of multiplication, that model is only needed when one must use multiplication in connection with measuring rectangles, or when teaching someone else what multiplication means; it is not needed as an immediate part of the recall of the multiplication table entries themselves. If the rectangle were a necessary adjunct to calculation, every multiplication would become a new problem in counting, and the computation of 345*678, say, would use unacceptable amounts of paper and time.
The purpose of memory is convenience. Memory is not a burden; it is a way to make things easier than having constantly to go back to first principles. But memorizing the products of all possible pairs of numbers being manifestly impossible, mankind has invented devices by which the necessary memorization may be minimized, with logic supplying the means of obtaining the rest. Such a general scheme is the goal of all mathematics: to make precise what is being spoken of, to learn the essential definitions and assumptions, and then to apply reason to determine other such facts as needed. These last facts, called "deductions", can, if forgotten, be recovered at will, though a good number of them, those needed repeatedly, should also be memorized for convenience in use. Thus while one can easily deduce 6*7 = 42 from 3*7 = 21, the two facts being far from independent of each other, it is convenient to have memorized them both, and one should demand it of children, even if they protest.
1. We therefore ask school mathematics programs, their textbooks, their teachers and their examiners, to make all prudent use of memory in pacing and simplifying the curriculum, and to demand that students remember and bring to mind easily those facts most repeatedly used, even when some are deducible from others or from first principles.
Ralph A. Raimi
Department of Mathematics
University of Rochester
Rochester, NY 14627
Return to the NYC HOLD main page or the NYC HOLD News page.