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The Case For Long Division

Brian D. Rude, 2004

      The Standards published by the National Council Of Teachers Of Mathematics, in 1989 and again in 2000, have been criticized for questioning, even disparaging, the value of learning the long division algorithm, and various other algorithms, in the elementary grades. David Klein and R. James Milgram have an article on the internet, “The Role of Long Division in the K-12 Curriculum” (link at mathematicallycorrect.com), which argues that long division should not be neglected. Perhaps there have been others that have addressed the issue. I will try to add to that criticism, but from quite a different perspective.

      Klein and Milgram have made the case that an understanding of long division is a necessary, or at least valuable, precursor to some important topics in more advanced mathematics. I have a little bit of experience that speaks to that, or at least close to that. In teaching college algebra I have often told students that dealing with algebraic fractions is really nothing new, in one sense at least. Everything that applies to adding, subtracting, multiplying, and dividing algebraic fractions is, in principle, the very same as in adding, subtracting, multiplying, and dividing simple fractions in elementary school. And in explaining long division of polynomials I emphasize that we are really doing exactly the same thing as we did in long division in elementary school. These comparisons often seem to fall on deaf ears. I always suspected that these comparisons don’t help much because the students are weak in arithmetic. But only in the last year have I become aware that their weaknesses in arithmetic might be a consequence of purposeful practices by their teachers, practices that may turn out to be a neglect of arithmetic.

      Before going further I will digress just a bit into the aims of education, because I think that bears directly on my argument. I consider that there are two general reasons for studying any subject, or any topic of a subject, in school. One is the utilitarian reason - some topics will actually be used by ordinary people in everyday life. Obviously children must learn to read. They will read everyday as adults. And obviously every child ought to know enough arithmetic to handle his or her own checkbook, for people do have check books. However the utilitarian aim of education is limited. Much of what high school students study does not have a direct practical use in everyday life. And much of what people do everyday does not seem to be exactly what they learned in school.

      The other general aim of education is what I will call the “liberal education” rationale. I consider this very important, though it is a bit hard to put into words, and there are differing perspectives and opinions as to what it might mean. I will say that there are basically three parts to the idea. First is perspective, the idea that by gaining a wide range of knowledge one has a better perspective to make judgments on the issues that may arise in life. History, for example, gives one a better perspective when it comes time to vote. A second part of the liberal education aim of education is the “training the mind idea”. This idea, I understand, was very popular in the 1800’s, then was debunked by educators in the early twentieth century, but never really died. I think there is something to the idea. I don’t think it should die, or be debunked, but it’s not easily argued. It’s not easy to pin down just how the mind is trained. A third part of the liberal education idea is what I like to call the “opening doors” rationale. When a person takes a course it is like a door being opened to a world that one can otherwise know nothing of. It does not follow, after the course is over, that one will want to actually go through this door. Many times one does not, but it’s good to get a glimpse through the door. For example many parents insist that their children take music lessons for a year or so when they are young. They do this because the feel that “door” should be open to every child, but they realize that many children will drop the lessons at some point and forget most all that they learned. Opening doors to art, literature, languages, and so on is seen by society in general as a good thing. It might even be argued that it is one of the primary aims of education. It does not even follow, in the opening doors rationale, that the knowledge, or even the perspective, gained from taking the course will ever be useful. The central rationale is that only by opening very many doors, will one be able to best choose the few that one really wants to go through.

      Learning topics that are precursors to more advanced topics seems to bridge the gap between the utilitarian aim of education and the liberal education aim. We can argue that it is utilitarian to learn long division, because it will be useful knowledge, even indispensable, when one comes to those more advanced topics mentioned. But that begs the question as to whether everyone really needs to learn those advanced topics. There are many, many people who have no intention of ever learning those advanced topics. Anyone not college bound can argue that these precursors are not needed in their particular lives. The long-division-as-precursor-to-advanced-mathematical-ideas, therefore, is an important idea to some, but not to everyone.

      But I wish to argue quite a different point. My argument is that long division is just one procedure, among many, that promotes the usefulness of arithmetic, that helps to make arithmetic a life skill rather than just one more subject forgotten, one more door opened but then allowed to close again by lack of relevance to everyday life. I will make my argument for long division purely on the utilitarian aim of education. My argument is that even the person who has no intention of learning anything more than perhaps ninth grade algebra should learn long division. Perhaps especially for that person is long division, and other topics of that complexity, important. But my argument has very little to do with being able to divide one number by another when an actual division problem arises in everyday life. We’ll use a calculator for that. My argument is that we should not expect anyone to really make sense out of arithmetic if they can’t do long division. If you can’t really make sense out of arithmetic a calculator is nothing more than a paperweight. My argument is that long division, and other topics of similar complexity, are required if students are to obtain an a level of understanding that allows them to actually use arithmetic. Then, and only then, will the calculator be useful.

      Arithmetic is number relationships. I’m sure there could be other definitions, but it seems to me that number relationships are what people think of as arithmetic. And the numbers I have in mind are the numbers of everyday life for ordinary people, people who never tried calculus and forgot everything they knew about algebra. Such people still spend money, they still paint bedrooms and lay rugs, they pay interest, they buy cars, they read newspapers. All of these things require dealing with number relationships. Therefore everyone should be competent with number relationships. To be sure, some people manage to paint bedrooms and lay rugs even though they are not sure about three or nine square feet in a square yard. And some people really have no idea how the finance charge on their credit card bill is figured. And many people really can’t figure their own income tax. Such people are at the mercy of others when it comes to numbers. People get by with a weak knowledge of arithmetic, but they do not thrive. No one seriously disputes that every young person should learn arithmetic. It’s a life skill second only to reading in importance.

      But won’t calculators make knowing arithmetic obsolete? That could be only if we equate knowing number procedures with understanding of number relationships. Number procedures are something to be done when you have the number relationships figured out. A calculator will add two numbers perfectly, and it will multiply two numbers perfectly. But it is of no use whatsoever if you don’t know whether you should add or multiply. A calculator will quickly and accurately give you the square root of 97.8. It will also just as quickly, and even more accurately, give you the square root of 1. But we have to seriously consider something wrong when students actually punch in the square root of one in their calculator, as I have seen on more than one occasion. And also a few times I have seen students use a calculator to divide or multiply by one .

      Learning arithmetic can roughly be divided into three parts, acquiring a sense of quantity, acquiring knowledge of combining numbers, and acquiring knowledge of combining ideas.

      I will start with a sense of quantity. Number relationships primarily mean quantity relationships, and a sense of quantity should not be taken for granted. I have heard that there are some primitive cultures that have number words only for “one”, “two”, and “many”. I have always been skeptical of this, but I don’t know. Counting, obviously, is a beginning of a sense of quantity. But can we say that when a child can count to a hundred that he has a sense of quantity? I would argue that a sense of quantity is something that develops over a span of years. Counting to a hundred is advanced over the one-two-many stage. But it is still very limited.

      Then one must develop some sense of relative quantity. This gets into the ideas of “how much more” and “how many times more”. This requires fractions, decimals, and percents.

      Developing a sense of quantity can be promoted by putting some ideas of quantity on an immediate response level. I think this is something that is generally neglected in the teaching of arithmetic. An inch should not be an abstract idea. Any school child ought to be able to show a length of an inch immediately, either by holding thumb and finger an inch apart, or by drawing a line an inch long, or something similar. I am not sure this is appreciated by all teachers. And people ought to have a very immediate idea of how long 3 inches are, and 5 inches, and 18 inches, and all measurements in between. I have seen some evidence, both as a teacher and just in everyday life, that for many people this is not the case. Many people have a pretty good idea of how long an inch is, but may have very little idea of how long 16 inches. If it were needed to cut a piece of string 16 inches long I expect many people would be at a loss without a tape measure. You could show them a piece of string 40 inches long and tell them it was 16 inches long and they would accept it. I am guessing here. I haven’t done research. I haven’t approached all my friends, or students, with different lengths of strings and tested their immediate knowledge. I wonder if anyone has. It seems a sensible thing to do.

      But one thing I have done, when my children were young, was to cut wooden dowels, about pencil thick, into lengths of one inch, two inches, three inches, on up to about ten inches, and did enough work with my children to know that they had some idea of these lengths. I would argue that anyone should have these quantities in mind, on an immediate response level, just as a foundation in developing a sense of quantity, and therefore as a foundation for learning arithmetic. I am not arguing that every person should have a good idea of the length of 718 inches, or three million inches. Obviously it is impossible to get an idea of every possible length. But it is possible, and I think very sensible, to get a good idea of single digit multiplies of an inch.

      Instead of saying I cut wooden dowels about pencil thick to different lengths, I could have said wooden dowels about seven millimeters thick. Would that have been as descriptive? To some people it would. Some people have an intimate knowledge of millimeters and centimeters, and would immediately know I was talking pencil thick. I think for most people in America, that would not be the case. Most people, when I say seven millimeters thick, would have to think something like, “Let’s see, there are 2.54 centimeters in an inch, so that’s 25.4 millimeters, so seven millimeters is just a part of an inch. Why didn’t he just say pencil thick.” And a lot of people, I suspect, would get nowhere near that far.

      My thinking along these lines is prompted by my own realization that I don’t know the metric system that well. I think it was in the seventies that there was a big push for America to convert to the metric system. I’m all for it, but it doesn’t seem to have happened the way it was envisioned. The auto industry did switch over, so I presume mechanics know what to look for when a 17 mm socket is called for. But I expect few people appreciate it if you tell them the next town is 14 kilometers away, or if you go into a lumber yard and ask for a board 292 centimeters long, or complain about the heat and humidity when the temperature is pushing 35 degrees. My argument is not that we should become more familiar with the metric system. That will happen in due course. But my point is that we should not assume children are intimately familiar with any set of measurements. We may discover that many people are as unfamiliar with any set of measurements as most people in America are with the metric set of measurements. If this is the case then I would argue that one important base of learning arithmetic is being neglected.

      Whether we eventually stick with the English system or switch to metric, or use them both into the foreseeable future, I would argue that children need a lot of measurements on an automatic response level, in all the major dimensions. That is, children need an idea of length, in at least one system. They need an idea of volume. They need an idea of capacity (in one sense the same as volume, but in common language not quite the same thing). They need an idea of area, and of weight, and of speed, of time, and perhaps a few others that I haven’t thought of.

      The second part of arithmetic, in my rough three part division, is the idea of combining numbers. I would think that a child learning that five plus six is eleven would be thinking quantity, thinking of objects, not thinking of combining numbers. But at some point in the child’s brain, questions of “how many” must give way to ideas of how numbers go together. The idea of number itself must be conceptualized to a greater extent than needed to learn to count. I would think that to a five year old the number three would be primarily an adjective. There can be three stars, three squares, three flowers, three horses, three ducks, and so on. But at some point the concept of “threeness” must be abstracted from objects. Three must become a noun as well as an adjective. When this is done, when numbers seem concrete to the learner, then numbers can be combined. Numbers can go together by adding, and numbers can go together by multiplying. In arithmetic we really don’t go much beyond that. But if we try to cover all the basic consequences of these, including inverses and different systems of notation (fraction, decimal, and per cent), we have a tall order.

      Perhaps a final stage in the emerging mathematical intellect is to turn to questions of how ideas go together. This is logic, of course.

      Now I will turn to a different classification of arithmetic. We can divide all arithmetic learning into two categories, procedures and ideas. This is apparently done in some modern ideas of teaching math. It is sometimes taken so far as to claim that, since all procedures can be done by calculators, we are free to teach only ideas. In one teaching situation a colleague told me, “We don’t teach completing the square”. In the situation of the moment this didn’t seem to make sense. How then, I wondered, are our students supposed to do the problems in the chapter that require changing, for example, y = x2 + 6x + 10 into the form y = (x + 3)2+ 1. This was after we had covered vertical and horizontal transformations of functions. Now we can take any quadratic function and interpret it as a vertical and horizontal shift from the basic function. But that takes a bit of algebraic manipulation. So learning to complete the square seemed not only logically sensible, but totally necessary to do the problems. Yet the attitude of my colleague seemed to be that anything that can be reduced to a procedure is not worth our time. She did not put it into these words, to be sure, but that seemed to be the inevitable conclusion that I had to draw. And I still don’t know how she handled these problems in her classes.

      There are some lines of thought that support this perspective of a duality of procedures and ideas. And there is a good argument to be made that ideas are more important. When thinking about learning long division, I remembered the paper-and-pencil method of finding the square root of a number. Actually, I didn’t remember it, for I never learned it. What I do remember is that in about the fifth or sixth grade our teacher told us that such a procedure exists, but that we would not learn it. My impression was that in previous years, or generations, students were expected to learn it, but experience over time caused schools to drop it. It was sometime in the mid fifties that I was in elementary school. Is it possible that students in the 20’s or 30’s, or perhaps in the previous century, routinely learned this square root procedure? I don’t know. Should we routinely teach it now? I don’t know that either. If the square root procedure was dispensed with at some point in time, might it be that long division at some point will follow that example? I think the idea is worth serious consideration, but my opinion is that no, long division will be very important to teach in elementary school in at least the foreseeable future.

      And one might think of other topics that are left in or taken out of the elementary math curriculum. Should we teach the abacus at some point? I don’t know how to use an abacus, but it has occurred to me that it might very well be worthwhile teaching for a couple of weeks somewhere in elementary school.

      Another topic we might consider adding in the elementary curriculum is bases other than ten. I know this was a topic advanced in the “new math” of the sixties. I always thought it was a good idea, providing it can be squeezed in without knocking some other topic out.

      I don’t think we should ever consider the math curriculum fixed and immutable, but dropping long division does not seem like a good idea. Hopefully by becoming more efficient and effective in our teaching we can add topics to the curriculum, not drop them.

      I would argue that to quite an extent, procedures are the applications of the ideas. They are not the only applications, but they are indeed applications. Math is very difficult to learn in the abstract. We expect to learn by doing many applications. Procedures, at least if they are taught well, are applications of mathematical ideas. If procedures are explained, not taught by rote, then they apply the ideas that are being taught.

      I think “carrying” in the addition algorithm is a good example to consider. I put “carrying” in quotation marks because it is my understanding that the term fell out of favor at some point, probably in the sixties. I think of carrying as being a second grade topic. In first grade, I think, students get the idea of adding as a short way of counting. There are sums to be figured out, and then memorized, and the idea of addition to be understood. This is no small task. It is accomplished with a lot of explanation and practice. The basic method, I presume, is that to learn the sum of five and six, students draw five circles (or stars, or ducks, or x’s, or whatever), then six more, count the total, and thereby are led to understand that no matter how many times you figure it out, five plus six is eleven, so you might as well just memorize that five and six are eleven. This basic method, it seems to me, is indispensable if students are to actually understand addition. But eventually we must address more advanced topics, such as how much are 25 and 36. Is this question related to the previous problem of how much are 5 and 6? Obviously it is, and it is also related to the question of how much are 2 and 3.

      So carrying is not taught as a rote procedure. It is taught, I presume, by explaining how adding 25 and 36 is related to adding 5 and 6 and adding 2 and 3. This explanation would be asking the students to apply their previous knowledge to this new problem. Carrying may accurately be thought of as a procedure, but it is a procedure that is an application of previous ideas. At least it is if it is not taught by rote.

      All during my lifetime it has been the practice of advocates of any new, or not so new, educational idea to claim that previously teachers taught by rote memorization, and the new idea is in contrast to that. The new idea, of course, is based on understanding. We certainly hope the new idea is based on understanding, but it is not a good idea to think that everyone else has not taught for understanding. It is a mistake to think teachers in the eighties taught by rote memorization. Critical thinking was not invented in 1990.

      I have divided arithmetic into three parts, acquiring a sense of quantity, acquiring knowledge of combining numbers, and acquiring knowledge of combining ideas. Then I divided arithmetic, or math in general, into two parts, ideas and procedures that implement those ideas. Now I will another classification system of learning, this time a two part division, and this time it applies to learning in general, not just arithmetic or mathematics. It is a very broad classification with a lot of fuzzy middle ground, but I think it is a worthwhile classification.

      Some learning develops like the unfolding of a flower as it grows and blooms. You can divide the growth into stages, but there are no clear dividing lines. One stage blends imperceptibly into the next stage. Learning appreciation of any art form works much like this. There is not easily identifiable point at which one can say that the student “appreciates” art. And learning of many skills fits this pattern. In learning to hit a ball with a bat, or innumerable other things, one level of learning shades into the next. For this type of learning the “nurturing environment” concept seems applicable. And for this type of learning, “cramming” doesn’t seem to work.

      But there is lots of learning that does not fit this unfolding pattern at all. Some learning is more like laying bricks. Each brick is totally distinct, and the structure of knowledge must be built up one brick at a time. If the proper bricks are not available the next stage of construction can not take place until they are. The bricks, of course, are not all the same. The wrong brick will not do. And the bricks cannot be assembled in any fashion. The bricks must fit together, for the bricks are ideas, concepts, connections, etc. I think of mathematics in general as fitting this second pattern. We learn math by carefully applying one brick at a time, not by letting a flower unfold. When I help a student figure out a problem it is my job to clearly identify what bricks are needed, what bricks the students has in place in his mind , what additional bricks must be obtained, and then to help him assemble those bricks in the right way. The idea of a “nurturing environment” seems much less applicable to this type of learning, and this type of learning is more amenable to “cramming”. In other words, to learn math you have to get the details right. The details are what make up the structure of knowledge such as arithmetic.

      There seems to be a recurring theme among idealists of many kinds. They want to look for shortcuts, shortcuts for world peace, shortcuts for losing weight, shortcuts to avoid the messiness of democracy, shortcuts for explaining complex subjects. In education we always look for shortcuts to learning. Of course we should look for shortcuts, in whatever we do. Sometimes we find them - real, effective, shortcuts that constitute real progress. But it seems more often we latch on to shortcuts that are all promise and no results. In education it seems we often think that dropping the details will be a shortcut. Often this is coupled with an appeal to “naturalness” in some form. We need to just get out of the way and watch the learning happen. It’s natural! In much of the early twentieth century it was held that reading could be taught better by throwing out all those messy details about phonics. But the results were mixed at best. In the last half of the twentieth century there has been a revolt against the “whole word” method. Phonics, again, is seen to have value. We have discovered that you can’t throw out all those details of how symbols represent sounds and how those sounds and symbols go together and still do an effective job of teaching reading.

      I also have the personal experience of a college history course in which the rhetoric of disparaging details was very much expressed. We shouldn’t worry about memorizing dates, we were told. This rhetoric, I felt, was counterproductive to learning any history. I don’t know just where on the unfolding-of-a- flower versus laying-of-bricks continuum history lies. But I would certainly argue that learning of facts in history should not be disparaged. I understand that history teachers want to do a lot more than just teach facts, but by disparaging facts they do a lot less.

      I said that math is primarily a matter of laying up bricks, and most of the routine activities of teaching and learning math fit that paradigm. However I believe there are also some parts of math that do develop by the unfolding paradigm. The sense of quantity, I think, works that way. Also I think mathematical intuition develops that way. (Or maybe mathematical intuition is inborn. Maybe it’s just another name for intelligence. I don’t know.) For these types of learning, learnings that unfold rather than being laid up brick by brick, we need a nurturing environment. But what kind of environment is “nurturing” for the goal of developing a sense of quantity, mathematical intuition, and mathematical maturity?

      I believe the nurturing environment for arithmetic is one of careful direct instruction, instruction in which each idea is carefully explained and followed by exercises that apply the ideas. Details are very important in such an environment. And what’s wrong with details? Details are important in everyday life. If I’m baking a cake is the temperature of the oven a detail that can be dispensed with? If I’m driving down the highway is the speed limit a disposable detail? Or the level of gas in the tank? If I have an important interview but I forget the time, is that okay because, after all, I understand the idea of an interview, and isn’t that what’s really important?

      One should not be compulsive about unimportant details, of course, and some details qualify as unimportant. But it is equally true that many details are important. In a well run life the important details are attended to. In a well run classroom the important details are attended to.

      I will present three premises that children should be able to count on in learning arithmetic, and which I believe are consistent and conducive to all that I have argued so far in this article. I first presented these in 1996 in my article “Chicago Math”. I called them “heretical premises” then, and some will consider them exactly that. But I think they are constructive. They are:

     

      1. Each problem will have one and only one right answer.

      2. There is one correct way to get that answer.

      3. The process will make sense.

      Some would consider these premises to constitute a “recipe approach” to learning arithmetic. Without premise three, it would be a recipe approach. However with premise three, which requires constant vigilance on the part of the teacher, I think they lead to effective learning of arithmetic.

      Note that I said “arithmetic”, not “mathematics”. These premises give way to other premises at least by the high school level. They don’t work for Euclidean geometry. They sort of work for elementary algebra. I think they have some value in many fields of advanced mathematics, but are very incomplete at that level. But on the elementary school level I think they work very well.

      When I say the process will make sense, that does not mean that strict mathematical logic can be applied. The process must make sense on a concrete level, on an intuitive level. When first graders are drawing circles and counting to discover the addition combinations, they are using an appropriate level of logic and abstractness. We don’t ask them to think about inductive logic. Learning arithmetic becomes somewhat like a game, or a puzzle, in which to goal is to figure it out and get the right answer. Students derive satisfaction form their growing accomplishments and understanding. But it is not a game or a puzzle just for enjoyment. It is serious business. And today’s game is not just a repeat of yesterday’s game. The game keeps morphing , the goal keeps receding. And it is not a game that can be left to children. It must be carefully guided all along the way. But it is an engaging game. It is a healthy game. It is a productive game. When well played students turn into adults that can handle their checkbook, and can measure the bedroom for paint or carpet, and have no trouble with three or nine square feet in a square yard.

      Lastly, there are two more concepts I will discuss next that connect with details and enters into my argument for long division. These concepts are supporting structure and setback rules.

      Supporting structure refers to parts of the structure of knowledge that are not the important foundation elements of the subject. In a fifth grade history lesson, for example, the teacher may have the goal that the class learn just a few basic facts about George Washington, that he was a good general, the first president, and so on. But to accomplish this she will teach a lot more. She will talk about Washington’s boyhood, his family, his country, the Revolutionary War. Some of these learnings are pruned away for a test a week later. The test would contain the more important facts and ideas, not every little thing that was presented and discussed. More learnings disappear over the next few months as their study of history covers other topics. By sixth grade very little will have remained. But if the sixth grade teacher observes that the students have retained a few basic facts about Washington, that he was a good general, the first president, and so on, then that is success.

      In athletics, in learning to play piano, and probably lots of other things the idea of “overlearning” is common. Even the young realize that to do okay on a piano recital, or to do okay in a ball game, you’d best overlearn as much as you can. In these situations overlearning usually means drilling again and again on the same thing. But drilling again and again on the same thing is less appropriate for retaining a few basic facts about George Washington. Bringing in related learnings is much more appropriate. These related learnings act as supporting structure to the main learnings.

      In math related learnings may include a wide variety of problems for a given topic, or they may consists of related mathematical ideas. Also supporting structure in arithmetic is a mass of details on an immediate response level. When I talked about developing a sense of quantity I argued that children should learn “by heart” the approximate lengths of small multiples of an inch, and other dimensions. This collection of details makes supporting structure for arithmetic. In seventh grade I would argue that students should memorize the common decimal-fraction equivalents. This again provides supporting structure. The long division algorithm itself acts as supporting structure for the basic concept of division. Division of fractions and decimals provides more supporting structure for the basic concept of division. And in general, I think it can be said, all topics of arithmetic provide supporting structure to all other topics. Some topics are central, fundamental, germinal beginnings. Other topics are less central. Some topics are peripheral. Some topics are so far on the edge that they are not very important, brought in only as enrichment when there is time. But if we want the core to be solid, then we must spread out from the core. The more on the periphery we can bring in, the more we give support to that core. We have no right to expect the core learnings to hold solid if we attempt to teach nothing but the core. In my view the long division algorithm is closer to the core than the periphery. Bases other than ten are closer to the periphery. Probability is closer to the periphery.

      Finally, I want to talk about “setback” rules. These are just rules of thumb, not very exact, but I think of some importance. There are three of them, and they are all four year setbacks. The first is the idea that when a person totally leaves a subject, he or she will forget what they learned in the last four years. My personal experience that led me to this conclusion was my knowledge of math. My major in college was math, and I taught high school for just a few years. But after a couple of decades of disuse my college math had evaporated. I remembered that calculus had something to do with derivatives, but little more. My knowledge of math had pretty much reverted to little more than the high school level. When in the mid eighties I wanted to study physics I had to relearn a great deal of math. And in the mid nineties when I decided to get back into math I continued that relearning.

      So by this four year setback rule, if we want people to retain a useful knowledge of arithmetic we should expect them to take math every year in high school. I think that is sensible.

      The second four year setback rule is that students will appreciate a subject about four years back from where they left it. By this rule fifth graders really understand why everyone has to learn to read in our society. I doubt that this appreciation comes much before this time.

      The third four year setback rule is that people who are going to teach a subject ought to study it about four years beyond the level at which they are going to teach. When one falls short of that one’s preparation is probably inadequate. But when one goes too far beyond that there are some disadvantages of inefficiency and frustration.

      These last two four year setback rules are not very relevant to the present discussion, but the first one is. Remember my main argument in this article is that we want arithmetic to be useful, an actual tool for the person who wants to figure out his bank account, or do his income tax, or lay carpet in a bedroom. If we want arithmetic to be useful we have to go beyond it. And if we go beyond it we have to do arithmetic right.

      So let’s keep long division in the arithmetic curriculum. Sure, the details can be tricky. But the details are applications of ideas that are important, and the details act as supporting structure to the main ideas. Sure, students are going to use calculators to divide, just like you and I do. But a calculator should be more than a paperweight. Let’s teach for understanding. Let’s open doors for children, and give them perspective. Let’s give them a solid grounding in arithmetic.