Part 3 of A Mathematical Manifesto
by Ralph Raimi for NYC HOLD
3. On the Algorithms of Arithmetic
One dispute between mathematicians and school mathematics educators, or some of them, has to do with the algorithms of arithmetic, of which "long multiplication" and "long division" are today the most divisive examples. A third example, the division of fractions by the "invert and multiply" rule, does not come up as often in these debates, probably because it is so much easier to remember, but probably also because it is a rule that doesn't achieve much prominence in one's imagination until its appearance in algebra, where it is essential. Not many people use algebra in daily life, but everyone has to multiply and divide numbers written in decimal form.
Now an algorithm is not just a method of solving a problem; it is a systematic method which prescribes explicitly, step by step, what the user must do with the information at hand, to get to a desired answer. The first step is prescribed explicitly, and the algorithm states exactly what has to be done with the result of the first step to perform the second step, and so on until the end. The number of steps needed to solve the problem might turn out to be very large, but this is not necessarily to say that the rule itself is difficult or lengthy in description, for the essence of an algorithm is its recursive nature: Given the result of any given step, it prescribes the next. The user of the algorithm may have to do this many times, but the inventor need only specify the first step and the recursive rule for all succeeding steps. An algorithm is entirely definite and can be taught to a machine. Indeed, computer and calculator programs are nothing but algorithms, though not necessarily in the realm of arithmetic.
To subtract 38 from 72, children learn that one starts with the 8 and the 2, but since 8 is greater than 2 one "regroups", or "borrows a one from the 7" and subtracts the 8 from 12 rather than from 2, bringing down a 4. This is followed by subtracting the 3 from the 6 that remains on the top line's "tens position", bringing down a 3 in the tens place, and the answer is visible as 34. This procedure (with slight elaboration) is applicable to any decimally expressed subtraction problem, and since it is not very hard to learn most children learn it, though not all. Among those who do learn it, there has always been a rather large number who managed to do it with no understanding at all of what is going on. After all, if a machine can do it, it cannot require understanding. What does a machine understand?
In truth, the description of the algorithm as applied to the subtraction problem above (72-38 = 34 via regrouping) has omitted much that could make its working comprehensible. Knowing how to perform an algorithm of this sort does not automatically imply knowing what the algorithm is actually accomplishing, for one virtue of an algorithm is its efficiency, and this efficiency requires the omission of explanatory details. By definition the algorithm is a prescription for mechanical, "mindless" actions only.
A less efficient algorithm for such problems may in fact carry with it a more obvious clue to its validity. For the subtraction 72-38 one might ask the student to mark off 72 intervals of equal length on a line, and then, starting at the same beginning (or zero) point, mark off 38 such intervals with a crayon of a different color superimposed on the 72 earlier markings. Then one counts, beginning at the end of the shorter set of markings, as if that were a new zero, "1,2,3,..." until reaching the last mark of the original 72. The count will conclude at 34, the answer.
This second "algorithm" is in fact a possible definition of subtraction, yet it is an algorithm nonetheless, and it never fails. It has the advantage that it can be used in all cultures, including those which use Roman numerals, provided they have a language rich enough to name all numbers. But it has the disadvantage that it is convenient only for the smallest whole numbers. Even so ordinary a computation as that CMLVII - DXVIII = CDXXXIX, solved by laying out lengths, will consume a lot of time or paper, while we are able to apply the decimal system algorithm to obtain 957 - 518 = 439 in a moment, even mentally, if we understand the standard algorithm well enough.
How well is "well enough"? Mathematics educators, having observed over literally hundreds of years that many children in fact do learn this algorithm without understanding what it is actually doing, and even then often making the enormous mistakes that even the slightest accidental departure from the computation might engender, have spent much time devising alternate ways of presenting it in school; yet somehow there was no more skill shown by American children in performing such subtractions in 1970 than there had been in 1920, and maybe even less.
Furthermore, these same educators have observed that the effort exerted by the average elementary school teacher in teaching these algorithms usually displaced everything except the mechanics of the procedure, sometimes obscuring the very meaning of subtraction in a child's mind, which focuses on "getting the answer". A dutiful child might learn mechanically how to subtract 5.78 from 10 to get 4.22 in thirty seconds (something a Roman Senator might consider miraculous), but when it came to a real-world problem about a profit margin, or whether a ten dollar bill would be enough to cover the cost of a movie after a certain list of objects had to be paid for first, this same child would ask, in class, "Teacher, do we have to add here, or subtract?"
About twenty-five years ago, the mathematics educators' prayers seemed to have been answered. The small hand calculator was invented. The engineers and computer scientists could teach the necessary algorithms to the electronic circuits, and so the children no longer had to learn them. Every sensible person uses a calculator these days for computations of more than trivial difficulty, so why agonize over teaching children what they will never need or use?
How attractive, to be able to ignore the whole question of how the algorithm does what it does, when one doesn't need the algorithm at all! "Is it not better to spend the time on real-world problems involving subtraction, and learn to understand subtraction in principle, than to learn to make calculations one does not know how to apply?" Here a straw man has been erected; this dichotomy does not exhaust the alternatives.
We must certainly answer the child's question about whether addition or subtraction is needed in the movie-ticket problem, and why, and indeed cause children to see for themselves by making models and playing games what the meaning of the operations are in simple cases. For most adult purposes we will indeed leave "doing the math" to the machine. But this does not yet imply the futility of learning the algorithms. Since understanding require some acquaintance with examples, teachers have been instructed to let children "discover their own algorithms" for simple problems, and praise them when they discovered one that illustrates the meaning of subtraction. The usual discovery that earns praise tends to be allied to the second "algorithm" described above, which lays out numbers along what might be a terribly long number line, for such a procedure certainly exhibits understanding of the meaning of subtraction. But that is not enough; laying out lines does not make use of our marvelous decimal system of notation, a triumph of the modern world, with uses that go far beyond making sense of CMLVII - DXVIII = CDXXXIX, uses and understandings no child will discover without having been taught the algorithm mathematicians call "standard", the algorithm that not only gets the answer but teaches the system -- provided the teacher teaches it. If children could discover such things, why could not adults in ancient Rome do as well?
A child bereft of the "regrouping" algorithm for multidigit subtraction has lost something deeper than a quick answer to the subtraction problem CMLVII - DXVIII, as recent experience shows. The first casualty has been the understanding of the nature of the decimal system itself, a system of notation that includes decimal fractions and "mixed" numbers such as 345.031, very small numbers such as .00003244, and very large numbers whose expression is aided by exponent notation (as are very small numbers, for that matter). It is important to be able to order numbers decimally written, to understand that while 450 is ten times as much as 45, 4.50 is not ten times as much as 4.5, to know that 1.03 is not larger than 1.2, and so on. Children brought up on calculator computation are weak in such understanding. They can be drilled in the meaning of the decimal notation, yes, but if in addition they learn the standard algorithms of arithmetic, with due care to the exposition that explains the meaning of "carry the two" in terms of the place value of the "two" in question, which might be 20 or .002 according to the case, they will have arrived at an internalization of the sense of our number system that no other way has been shown to accomplish.
As adults they will surely use calculators, and indeed computer programs, to prepare tax returns. The check-out clerk at the drug store has a cash register that computes change without any arithmetic skill required of the operator, and it is clear that the needs of commerce no longer require the lengthy drilling in the columns of figures children were given (and accountants were pondering) in the 19th Century. In 19th Century teaching, in fact, the rationale for the algorithms was generally ignored, so pressing was the need for speed and accuracy of execution. As a result, schools were often negligent in pressing for understanding of the rationale that made the algorithms so illuminating an exercise. Today we are fortunate not to need so much skill and speed in their use, though we still must make approximate calculations every day of the year, to decide restaurant tips and the like. We must rather take our present electronic conveniences as an opportunity to omit the extra tedium, not to abolish learning of these algorithms, not even the difficult ones such as "long division", but as an opportunity to teach the rationale with the algorithm, and to reap other benefits: the understanding of the number system and its notations that the present age increasingly calls for.
None of this discussion is meant to denigrate the small devices for simplification of computation people sometimes use in daily life. Surely one does not find the price of 6 melons costing $1.98 each by going through the ritual of "6*8=48; write the 8 and carry the 4, ...", but rather we say it is $12 less 12 cents, or $11.88. Estimation and mental arithmetic are valuable, and some such calculations even illuminate the decimal system or some other valuable understanding of mathematics (the distributive law, in the present case), but though these devices are sometimes called algorithms in mathematics education literature, they generally are not algorithms at all. Six melons at $1.73 each does not yield so easily to this method, and 6.05 x 1.73 not at all. The machine cannot understand instructions that begin, "See if you can find a conveniently roundish nearby number ...", even though some children can, especially in modern "reform" mathematics curricula where they are taught to denigrate algorithms by being fed problems artificially generated to succumb to such short cuts as will not work in general.
Again, a problem like subtracting 97 from 356 looks complicated using the subtraction algorithm described above, but a child can easily be taught to discover (with a little prodding) that this is the same as subtracting 100 from 359, with the easy result 259. Such observations are heralded by some of today's novel curricula as "student-discovered algorithms", yes, and often trotted out in evidence of the superiority of today's pedagogy; but this is not an algorithm, it is a clever observation (by no means contemptible, of course) concerning these particular numbers. If the problem had been to subtract 178 from 3562, the old tried-and-true subtraction algorithm would have produced the answer without thought and without fail, while children "discovering their own algorithms" have been seen floundering for hours trying to find convenient nearby numbers that produce a visible answer, coming home without learning a thing except a great dislike of math class. Even in the case of the put-up job, substituting 359 - 100 for 356 - 97 might come to mind in a class where such things are looked for, but the literature of child-centered mathematics education research does not record children having invented the corresponding device for 35.6 - 9.7. The celebration of the "student-discovered algorithm" for 356 - 97, in short, is a fraud.
In all matters of choice, "drawing the line" has been the bugbear. We can drive cars instead of walk, but still have to walk sometimes. Experience tells us when one is more convenient than the other. Perhaps we drive many more miles over our lifetimes than we walk; does this mean we should cut down on the time we spend learning to walk? In the case of the algorithms our answer is this: Children should learn all the algorithms, just as they learn to walk many distances, including some they will ordinarily be driving in future years. They should then revisit the algorithms for fuller understanding as they grow older as well. (Adults owning cars run for exercise, for that matter, and sometimes in sports competitions.) Anyone who has to reach for a calculator when asked to multiply 57 by 10 has been robbed of his patrimony, and we have been raising a generation of such children.
Some parts of the mathematical rationale for the algorithms will probably not be fully grasped by many of the children learning and using them in grades 5 and 6, but they will be better understood when revisited with care, provided the teacher and the textbook understand them. This last proviso is not gratuitous; in the elementary schools especially we do not have specialist teachers of mathematics, and the explanations such people sometimes give -- and sometimes instructed to give -- are far from optimal. Indeed, more than "explanation" is needed for something as complicated as long division; a textbook is needed that has graded exercises causing the student to go through the steps in problems of increasing complexity, in such a way as to cause the rationale to uncover itself as the procedure goes on.
Understanding a routine surely helps to remember it, but it often happens that merely remembering it comes first. The thing remembered (or reviewed) is what the mind needs to work on as it goes about the business of connecting its steps with each other and with the rest of human experience. Memorizing a Latvian poem might not teach the average American child anything about either the Latvian language or the subject of that poem, but memorizing the Preamble to the Constitution of the United States is a good start in learning what the Constitution is all about. A ten year old can memorize the Preamble with only partial understanding, and then at age 15 find in his own memory the material enabling him to appreciate the problems that faced the Framers. "The general welfare" might at first have sounded like a phrase concerning military rank and a monthly check from the government, but the words stick in the mind to fuel that extra understanding that comes with further education. We don't learn everything at once, but this is no reason not to begin.
Why the algorithms of arithmetic, in particular?
1. Those of us who are experienced mathematicians and scientists know that we make approximate and mental calculations all the time. Experience with arithmetic algorithms have given us an intimacy with the decimal system of notation that is prior to these essential daily skills. Calculators will give the more exact answers, to be sure, or answers easier arrived at, but they will not give us the feeling for structure in the path to the answer that having gone the road of the familiar algorithms has given us. Mentally calculating fifteen percent of a restaurant bill is really quite trivial compared to the mental estimates an engineer makes daily, to decide whether to start a serious calculation in one direction or some other. Even reading the newspapers, voting, investing and buying a house require a good sense of number; those with a firm background in arithmetic algorithms do better at all such things.
2. People do not carry calculators with themselves at all times. There is no telling when a pretty good answer will be needed even by ordinary people, pronto, as when deciding whether to buy the economy size laundry detergent.
3. Some children will become mathematicians or scientists, economists, engineers or physicians, and later must learn their chemistry and physics using laws of nature codified in mathematical terms. Algebraic notations are essential in the expression of these laws, and in making numerical deductions when the time comes to be practical. Some of these notations concern polynomials, or power series, which are a generalization of polynomial. Decimal arithmetic is a disguised and condensed polynomial arithmetic of a special sort, and a valuable prelude to learning the more general formulations used in the sciences. The "invert and multiply" rule for fractions is also a rule with applicability in every part of an algebraic formulation, whether involving polynomials or not. It is not "just a rule", either, for it carries with it the very meaning of "fraction" as it lives within an algebraic system such as the real number system, or the complex. The fact that there is no such rule in matrix algebra, and that with some exceptions the very idea of "fraction" cannot be found there, is illuminated by experience with what fractions say and do in elementary arithmetic. Every part of school arithmetic prefigures something in more advanced uses of mathematics, and without the earlier practice it is difficult, and in most cases turns out to be impossible, for the student to appreciate what is going on in calculus or quantitative chemical reactions. It is true that only a minority of the population will suffer from this lack if the beginning is insufficient in the early grades, but to say "only a minority" is to dismiss the entire leadership and technical expertise of the future. It is, in fact, not "only" a minority, but it is a large minority, unless we are to be a society built on physical labor alone. To dismiss attention to this population as "elitist" is the reverse of the truth, for to deprive all children of public school access to the skills underlying advanced scientific learning is to decide from birth what the status or caste of a child is, and separate by heredity the future scientists from the ordinary children, for today's elite will surely find the means, whether by paid tutoring, home schooling or private schooling, to assure access to knowledge for their own children, while the pearls in the rest of the population will remain hidden. A curriculum such that absolutely everyone can and must learn everything on offer does not in fact serve the commendable ideal of equal opportunity, let alone the impossible ideal of equal outcome. It is merely a curriculum with too little on offer, hence undemocratic. There is no surer way to widen the separation of classes than by diluting education so that "everyone passes".
4. The algorithms of arithmetic were invented by Renaissance mathematicians for use by adults, bankers and merchants, but they could not have been invented by people who used them only for trade and carpentry. Teaching of these algorithms today can and should not only be for their value in computing taxes and measuring lumber, their original uses, but must include attention to their inner structure, the structure that made their invention possible. Knowledge accumulates with the centuries, and our children, if they are to have an intellectual future as adults, must now learn much of what in Renaissance times was known only to the few, much as they now learn chemistry and physics unknown to anyone in Seventeenth Century Europe.
Mathematics is part of our general culture. While most people understand the value of music and architecture, theater and poetry, even though appreciation of these arts contributes nothing practical to their wealth or health, there are distressingly few with a corresponding appreciation of the values of science and its mathematical formulations. Yet mathematics is not notably less practical than music. There is no more excuse for the general ignorance of mathematics than there would be for a correspondingly general ignorance of music or art. It is a marvel of the past few centuries that we now have a numeration system and a set of algorithms of so polished a form as we do. Knowing only the addition and multiplication tables up to 9 by 9, these algorithms permit us to calculate the sums, differences, products and quotients of any two numbers whatever, howsoever large or small, negative as well as positive, simply through the knowledge of these few algorithms. Is not this system comparable to Johann Sebastian Bach's well-tempered scale, and the perspectives of Albrecht Dürer? Whoever thinks not should reconsider, see what he has been missing, and make sure his children are not as deprived as he has been.
3. We therefore ask school mathematics programs, their textbooks, their teachers and their examiners, to emphasize in the early grades the rational teaching of the standard algorithms of decimal arithmetic and of fractions, for this teaching uncovers the structure of our modern systems of numeration and prepares for its extensions in higher mathematics such as are needed by all who aspire to a technical career, or to a rounded understanding of modern culture in general.
Ralph A. Raimi
Department of Mathematics
University of Rochester
Rochester, NY 14627
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