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The Mathematics Curriculum

Some Personal Observations And Preliminary Thoughts

Brian D. Rude, 2006

      What mathematics should be learned, and when? When I was young these questions did seem not very important, for the answers were obvious. In elementary school you learn arithmetic. In high school you learn algebra, geometry, and perhaps trigonometry. In college, if you continue in math, you learn calculus. Arithmetic means learning to add, subtract, multiply, and divide whole numbers, fractions, and decimals. Algebra means learning to solve equations, and hopefully before you are done you realize it means abstracting arithmetic. Calculus means you learn about the derivative and all that follows from that.

      If mathematics may be visualized as a tree, then I would argue that the sequence of arithmetic-algebra-calculus forms the main trunk. Thus the tree of math is a tree of numbers. For mathematicians this sequence is only the beginning. The tree grows up and out in many directions. There are many, many topics that for one reason or another may be considered mathematics, and may or may not have anything to do with numbers. Perhaps in this extended structure of knowledge number does not form the main trunk. But it could still be the case that number is the best way to start building this tree of knowledge. Indeed a main point of this article is that we should start teaching math with numbers, and stick with numbers through all of elementary school, most of high school, and quite a bit of college. Where math goes after calculus is also an important question, of course, but not the subject of this article.

      We should not only start with numbers, but we should teach numbers well, and that means a number of things. It means, at least in my opinion, giving a lot of careful explanation. It means assigning homework and grading it. It means giving quizzes and tests. It means using drill when needed. It means trying to develop a sense of quantity in the students’ minds. It means trying to develop a high level of fluency in working with numbers, symbols, and ideas. It means building a firm and permanent structure of knowledge of numbers.


Curriculum, Methods And Objectives

      Things have changed from when I was young. The answer to the question “What mathematics should be learned, and when?” is not at all obvious now. There is a ferment in the world of mathematics education. Perhaps this is a good thing, perhaps not. It is a good thing if it leads to genuine progress in the teaching of math. But there seems plenty of evidence that much of the activity in math education is not progress. It is more along the line of wishful thinking and undisciplined good intentions producing confusion and frustration. Much of new math promoted by the National Council Of Teachers Of Mathematics (NCTM) seems to be a continuation of the basic premises of progressive education that has dominated educational thought for most of the twentieth century. Perhaps the most basic premise is that we don’t need to teach subjects directly. Rather we can engage students in “activities” or “projects” and they will learn. This is sometimes called a “child centered” or “learner centered” approach. Using group projects is currently a favorite theme among many who favor this approach. The emotional appeal of this approach is apparently very strong.

      My goal in this article is to discuss curriculum, not methods, and not the politics or ideals of what we might call the “math ed establishment”, or of their opponents. However curriculum and methods have a sometimes nebulous border between them, and curriculum and methods interact to some extent, and much of the current thinking in math education is shaped by methods, ideals, and objectives, so I will discuss all of these to some extent.

      I would argue that in progressive education curriculum has been very much affected by considerations of method. If we don’t have to teach math directly, if we can just set students going on projects and get out of the way, then we must question whether we can even set a curriculum. Perhaps, if the “activities method” works, curriculum becomes meaningless. Indeed I believe some people would advocate exactly that. We should let the learning go wherever it may go. A corollary of this approach, I would argue, is that we cannot test. A test implies that we have predetermined what the students should learn, and how well they should learn it.

      Some other people, I presume, would argue that we can limit the “activities method” to being just a method. We can still have predetermined objectives of what students should learn, and we can still test for that learning. It is not clear to me just how this view is reconciled with the idea of letting the learning go wherever it may go. My view is the basic premise of progressive education is wrong. I believe there is no substitute for direct instruction. We should not just choose activities and just let the learning go wherever it may. It will most likely go nowhere, and even when it goes somewhere it will not go far. I believe we should use direct instruction and work toward predetermined ends.

      A discussion of curriculum must also consider objectives. Curriculum is obviously affected by objectives. Indeed one might say that they are the same, or at least inextricably bound, because curriculum is determined by objectives. By this idea it makes sense to say that we must first decide on our objectives. We can’t decide on curriculum until we agree on what we want to accomplish. Then we must choose those subjects or topics that lead to the accomplishment of those chosen objectives.

      Or one might take a different approach. One can argue that there are multiple curricular choices that would lead to a given objective, therefore choosing an objective may limit curriculum to at least some extent, but it does not determine it. I favor this perspective. I do not subscribe to the idea that objectives have to be chosen first and then curriculum chosen to implement the objectives. I would argue that the connection between objectives and curriculum will always be rather loose. If an objective is not highly specific then many topics may be said to contribute to that objective, but normally no particular topic will guarantee the attainment of that objective. Certainly, for example, we would want one objective of math education to be something to the effect of promoting critical thinking. But what topics of mathematics would accomplish this? Or what methods would accomplish this? I do not think there is any easy and obvious answer to this. I think it makes more sense to let objectives remain rather generalized and focus discussion on curriculum.

      By this perspective objectives do not determine curriculum, at least not in detail. A choice of a given broad objective suffers from imprecision of language. We should not let this imprecision of language color our judgment of the value, or lack of value, of various topics that might be included in the math curriculum. If we choose a certain topic for inclusion in the curriculum, it is because that topic has value in some way. Other topics may have value in other ways. We may discuss these differing values of various topics at least somewhat independently of objectives.

      But this is not to say that objectives are not important. In fact I want to discuss one particular issue of objectives. I would argue that there have been subtle shifts in objectives in the past half century. One shift in objectives, in particular, I feel is important. I will describe this shift as I see it, though it is not the sort of thing that is subject to easy proof. At the beginning of the twentieth century, I believe, an important objective of mathematics instruction was to build a structure of mathematical knowledge in the learner’s mind.1 By the end of the century this objective has been displaced by the objective of simply having the students engage in mathematical thinking, whether or not this mathematical thinking contributes to mathematical knowledge. This shift in objectives may not seem substantial, at first, but I believe it is. And the shift may not have been conscious, or even intentional, but I think it is worth analyzing. I believe it is not a good shift.

      Perhaps at the beginning of the twentieth century the objective of building mathematical knowledge was secondary to the objective of having fluent computational skills, for at the beginning of the century that would certainly seem important. But I would conjecture that it was common knowledge that understanding - which I would interpret as having a firm structure of knowledge - is an important basis for becoming fluent in computational skills. By the end of the century the objective of having fluent computational skills certainly has become less important. That is understandable, though it may not be wise. We’ll use calculators and computers for routine computation in everyday life. But to argue that we therefore don’t need to teach computation would be to argue that the development of computational fluency is not connected to understanding of mathematical ideas. I would argue that that is wrong. They are connected. More specifically I would argue that developing a sense of quantity is important to understanding number operations, and that computational facility aids in developing a sense of quantity.

      Here is a hypothetical example to illustrate the shift in objectives that I am talking about. A class of fifth graders is given the project to analyze the game of tic-tac-toe. (Start with a grid of nine boxes, one player uses x’s , the other o’s, they take turns, the goal being to get three x’s or o’s in a row. This is the name I remember for this game, though perhaps it has other names.) They are given some guidance. They are told to start out with the first player putting an x in the center. Then catalog where the next player could put his first o. Decide how many different second moves there could be. After devoting several math classes to this effort the students begin to get the idea of where they are heading. After being given more guidance, over the course of perhaps six or eight school days, they are led to a conclusion, and most students seem to understand both the conclusion and the process that brought them there. The conclusion is that a knowledgeable player cannot lose, but the only way he can win is if the other player makes a mistake.

      Is this mathematics? Is it mathematical thinking? Is it a good use of students’ time in the fifth grade? My opinion is that if done right it is indeed genuine mathematical thinking, but it is definitely not good use of a fifth grade’s time. I have a personal interest in this problem. A number of years ago this problem caught my interest and I devoted a number of hours, over a number of days, to figuring it out. My method was simply to catalog every possible variation of the game, the method of exhaustion. It turned out to not be very hard, for it’s a very simple game. I did arrive at the conclusion that a knowledgeable player cannot lose, but he can win only if his opponent makes a mistake. It was an interesting exercise, and I thought at the time that it was indeed mathematics, but so trivial as to have no value other than it entertained me for a few days. But if this problem is indeed mathematics and, and if students pursuing this problem are indeed engaging in mathematical thinking, then does that make it worthwhile? At the beginning of the twentieth century, I think, it would not seem worthwhile to most teachers. But at the end of the century, with the objective simply to engage in mathematical thinking, I expect a fair number of teachers would say that it is worth while.

      I think it has been characteristic of progressive education through much of the twentieth century to attach the label “educational” to any activities of which we approve, or which seem good in some way. When students appear to be engaged in an activity, goal directed, using their abilities and talents, it seems tempting to label the activity educational. I consider it possible that a good teacher could indeed spend a week or so of math time leading a class through the analysis of tic-tac-toe. If well done the students would indeed become engaged in the problem. They would be doing genuine mathematical thinking. But is it educational? Maybe it is just play. Or maybe it is just busywork.

      Play is important in a child’s life, of course, and some play is genuinely educational. But some play is not. I do not think play needs to be educational. If play is enjoyable, that is quite enough justification for it. But we don’t send children to school for play. We send them to school to become educated.

      I believe we need to revert to a greater emphasis on the objective of building a structure of mathematical knowledge, and, as I have already pointed out, that structure of mathematical knowledge will be mostly about numbers. The objective of engaging in mathematical thinking, without learning mathematics, does not give students much of lasting value (unless one believes that the mind is a muscle - it improves with exercise - but this idea has been out of favor for many decades). Of course this latter objective, simply to engage in mathematical thinking, certainly fits well with the “activities method”. But again there is a problem in testing. If a structure of knowledge is not to be built and tested for, then what is to be tested for? It might be argued that we would test for the ability to think mathematically. How would this be different from an intelligence test?2

      Of course, many leaders in mathematics education would argue that I have it all wrong. More specifically they would argue that in old mathematics instruction the emphasis was on rote memorization. In the modern mathematics instruction the emphasis is on understanding. I find this argument to be unconvincing. The argument that old math consisted of rote memorization is never more than a simple assertion. I have long argued that the first requirement of any science is simple description, and I have long argued that the field of education is very deficient in that. I don’t think the educational literature has any plain, simple, accurate description of what goes on in real classrooms, in either 2006 or 1906. How do we know how teachers taught in 1970, or in 1950, or in 1910? To progressive educators the answer is very simple. Back then they taught by rote memorization, without understanding, and that is the end of the story. They have nothing more to add to that assertion. They have no descriptive accounts of classroom activity in 1910, or 1950, no statistical analysis of classroom activity in those times, no chains of reasoning based on what we know of those times, not even any anecdotal evidence.

      And when did the change occur to the new thinking, the way that leads to understanding? Was critical thinking invented in 1995, just in time for the update of the NCTM standards in 2000? Or was critical thinking invented in 1985, after “A Nation At Risk” was released in 1983? Or in 1970? Or in 1955?

      The charge that we used to teach by rote memorization does not ring true to me. I don’t remember much about my elementary education in the 1950’s, but I remember a few things, some good and some not so good. But very little of what I remember could be described as “teaching by rote memorization”. Of course we did drill in the flash cards to learn the multiplication tables. We “went to the board” for practice in computation. We had weekly spelling tests of a list of 20 words. We had to memorize the presidents in order in eleventh grade history. But to say that instruction in the 1950’s included some memorization is a long way from saying it consisted only, or even primarily, of memorization. That was certainly not the case in my experience. What I do remember of math in elementary and high school was that it made sense. We solved problems, because the math made sense. Just what kind of sense it made, and on what basis it made sense, are very important concerns, which I will deal with in this article.

      I also have a more personal reason to reject the claim that the new “reform math” is based on understanding - my experience teaching at NDSU in 2003-4.3 We claimed to be teaching for understanding, but in reality, it seemed to me, our emphasis on “technology” - meaning the graphing calculator - turned college algebra away from a course of understanding into an unfocused course of problem solving techniques. “Understanding”, apparently, is interpreted by advocates of NCTM math to mean doing math without structure, addressing problems one at a time, but not developing mathematical ideas. Again this fits with the idea that our objective is to engage in mathematical thinking, rather than building a structure of mathematical knowledge, but it seems a stretch to call this approach “understanding”. What I would consider an “understanding” approach would be to develop a structure of knowledge, which would primarily be a structure of implication4, with the implications understood by students.



      There are many, many possibilities of what could be labeled as “mathematics” and taught in the third grade, or eighth grade, or in high school or college. But surely some choices of what to teach are much better than other choices. This article is an attempt to investigate some of the possible bases for making these choices. I will also try to develop the argument that what we think of as “traditional” math is traditional for good reason. It is not good because it is traditional. Quite the contrary, it is traditional because it is good. It is good because it builds in the learners’ minds a structure of mathematical knowledge that has real long term value, both for the individual and for society.

      It is easy to jump to the conclusion that something that is “traditional” is accepted purely on emotional grounds. Obviously sometimes this is the case. But it is also very possible that it is accepted on purely practical grounds. My favorite example is this. If I plan a trip from Fargo to Omaha I will take the “traditional” route through South Dakota. Do I make this choice for emotional reasons, just because I revere the traditional? I have roots in South Dakota. My wife and I lived there twenty-five years and raised our children there. Someone who wants to accuse me of choosing that route on emotional grounds has some facts to argue with. However a glance at a map will show the purely practical reason for choosing this route. Interstate 29 goes through Fargo, down through South Dakota, Iowa, and approaches rather closely to Omaha. Any other route would be less direct. I am a pragmatist, not a traditionalist.

      If A is correlated with B, it may or may not be true that A causes B, or that B causes A. That is perhaps the most basic idea to learn about any research involving statistics. Yet it is also perhaps the most difficult lesson to learn. Indeed, the general public never gets it. If we read in the paper that use of a certain drug is positively correlated with crime, the vast majority of the general public immediately jumps to the conclusion of causation. And if I conclude this article by favoring a math curriculum that sounds in many ways traditional, advocates of what we might call “NCTM math” will immediately jump to the conclusion that I am a “traditionalist”, motivated by blind adherence to tradition. But I consider myself a pragmatist, not a traditionalist. I’ll do what works. Therefore I will proceed to discuss what I think are the ideas and concerns that ought to shape the math curriculum with little concern about whether or not I am a “traditionalist”, any more than I would worry about being a traditionalist if I choose the traditional route to go from Fargo to Omaha.


The Beginnings Of Math


      It would seem to make sense to give students the most basic elements of math first. In a story we start at the beginning, do we not? So in teaching math why not start at the beginning? Why not give students the basics of mathematics right from the start? Isn’t this an important beginning principle in designing a math curriculum? We must decide what is the beginning of math and start there, or so it would seem at least. If functions are basic to mathematics, then why not teach functions right form the start? If proofs are important in math, why not start proofs in kindergarten? If logic is important to arithmetic, why not talk about logic before we talk about numbers and addition?

      To many people math means numbers, and nothing else. Some people have had a good course in Euclidean geometry, and when reminded of it can appreciate that math does not have to have numbers. Some of those people, I think, wonder why geometry is math. But of course math includes a lot more. There are many important topics in mathematics that have nothing to do with numbers. So must we start with numbers? Could we start first grade children out in some mathematical topic that would better prepare them for learning about numbers? Do they have to learn about numbers? Can we learn algebra concurrently with arithmetic? Must arithmetic come before algebra?

      There has been some experimentation with the math curriculum in years past. In the “new math” of the late 1950’s we thought that set theory was so fundamental to math that it should be introduced to school children. So a generation of youth learned a little about union and intersection of sets. However set theory has not withstood the test of time. It has not been totally abandoned in school mathematics, but I would argue that it never really had much impact on the mathematical knowledge of students. Why didn’t it? Isn’t it basic to mathematics?

      Why don’t we start math at the very beginning? I believe there are very good reasons why not. Back a few paragraphs I said, “In a story we start at the beginning, do we not? So in teaching math why not start at the beginning?” That was a rhetorical question. We do not really start every story at the beginning. Indeed, I would argue that we don’t start any story at the beginning. It is impossible. What we think of as the beginning of any story is really just some convenient place in the middle. History shows this very well. If we want to tell the story of America we might very conveniently start in 1492. But obviously a lot happened prior to 1492 that has to be considered part of the story of America. Could we argue that the “beginning” of the story of the Civil war is the election of Lincoln in 1860? Could we argue that the beginning of the story of the pilgrims in Massachusetts is in 1620? Could we argue that the beginning of the story of space travel is in 1957 when Sputnik was launched? We could make all of these arguments, perhaps, but some would disagree. I would say that it is impossible to begin any story at the absolute beginning.

      So it is impossible to start the story of mathematics at the very beginning. It is not a first principle in choosing a math curriculum that we must start at the beginning of math. I don’t know just what the very beginning of mathematics is, or might be, or could be. But I am confident that whatever, or where ever, that beginning is, it is not the place to start teaching it. I will next argue that the mathematics of numbers is the best topic to begin math with. Then I will argue that what we might call “conformity to reality” is the type of proof that we should start out with.


Starting With Numbers

      It has been observed that learning about objects is one of the important developmental tasks of very early childhood. In particular “object constancy” is important. There is an age, as I understand it, when a baby does not realize that an object out of sight must continue to exist. But at some point the child realizes that an object cannot just cease to exist. An object has constancy. If it is no longer in the child’s sight then he may look for the object, as he realizes that it must be somewhere. Once this stage of development is reached by the child then it would seem that the concept of number could start to form.

      An important corollary to object constancy is number constancy of objects, which would have to come sometime after object constancy. If there were three blocks in a box, but now there are two, the child knows the third one must still exist. Once number constancy is established in the child’s mind then counting makes sense.

      It might be said that “object constancy” is really just a “theory of object constancy”. A child learns to assume that an object out of sight continues to exist. Experience leads him to trust in this assumption. The assumption is shown by experience over a period of time to conform to reality. This is not to say that the child consciously thinks about such things, of course. Rather experience builds up. It is learned that objects have constancy, and therefore number constancy, long before the child’s language is advanced enough to verbalize this concept.

      I understand there is a stage in which children can count, but don’t quite have the concept of counting. They may point to successive objects while reciting “One, two, three . . . . ”, but there is not a one-to-one correspondence between the objects pointed to and the numbers recited. They may count to ten while pointing to seven objects, or three, or twelve. Only after some time in this stage does a child realize that there is more to counting than that, and then the idea of number really begins to make sense. Once the idea of number is established, then the idea of combining numbers in various ways can develop. Arithmetic may then be learned.

      Numbers arise early in the life of a child, and numbers are extremely important in the lives of virtually all adults. So math education surely must include a thorough study of number. And that is arithmetic. But should arithmetic be the only math studied in the 1st grade? Or in the fifth grade? Or in high school?

      At this point one might ask for a definition of math. There can be a number of answers to this, but to me the primary meaning of math is a set of interrelated ideas that derive logically from a small set of beginning ideas. However it does necessarily follow that this definition gives a lot of guidance about how to teach math to ten year olds. Euclidean geometry works very well in high school when taught by this perspective. A list of postulates is presented and at least briefly discussed. Then a logical system is built from them. The degree of mathematical rigor may vary, of course. There is no benefit in trying to address every little point that an advanced mathematician might want to address. The understanding by the students may suffer from either too much or too little mathematical rigor. In a well taught high school geometry course the students fully understand that problems are solved by applying theorems, that theorems are general problems that can be logically solved by applying previous theorems and postulates, and postulates are starting points arrived at by common sense, but without proof.

      But experience does not show that this mathematical pattern is the best way to teach every mathematical topic. What works well in Euclidean geometry may not work well at all for learning fractions in the fifth grade or calculus in college. I did have one experience in trying to teach algebra to high school freshman by a pure mathematical approach. I was very young, inexperienced, and idealistic. But the results were not good. The class was frustrated, and I abandoned the effort after about two months.

      Proof is very important in mathematics. Shouldn’t we therefore start out in first grade teaching the idea of proof? I would argue that we do, but I am not talking about the formal proof of advanced mathematics. The “proof” we use through all of elementary school I think is best called “conformity to reality”, and I think this deserves very careful consideration. It is a type of proof that works very well with numbers and small children. It also works very well in science.


Proof By Conformity To Reality

      Once counting makes sense to a child then addition can start to make some sense. If a child has four blocks in one pile and seven blocks in another pile, how many blocks does he have in all? Of course he has eleven blocks in all. We know that four plus seven is eleven. But the child, before addition is well established, knows that the way to find the number of blocks is to count them. And he knows that there is a number of blocks, and that number cannot magically change. From this situation he can be told that the sum of four and seven is eleven. With more examples he can eventually abstract out the concept that the sum of four and seven is eleven. The “proof”, of course that four plus seven is eleven is that it conforms to reality. Every time he tries it, with blocks, or soda straws, or checkers, or buttons, or anything else, it is the same. The answer is eleven. We can say that he is using inductive reasoning, which is true enough, but to the child, I would argue, the “proof” of its correctness is by referring to reality. He can count, and a pile of four blocks and a pile of seven blocks will indeed make a single pile of eleven blocks.

      Conformity to reality is the basic proof in science. Science seeks to learn about reality. Any idea that does not conform to reality is rejected by science. This is not true in mathematics, and that is very important in mathematics. However that is advanced mathematics. I think the mathematics that young children learn is like science. They need to learn how the world works. In the real world two and two are four, always. Reality is the ultimate test. And in the world of numbers of the young child, any mathematical idea is easily put to the test of reality, simply by counting.

      Addition is established by observing reality, and is tested, or proved, by conformity to reality. But addition is not confined to reality. It becomes abstracted, separate from reality. After using blocks to learn that three plus five equals eight, and drawing circles to learn that three plus five equals eight, and a lot of other procedures to show that three plus five equals eight, the child does not need any more objects. He or she abstracts from reality. At some point it makes sense to simply say that “three plus five equals eight”. Numbers are abstract, and addition is abstract, but young children of five or six are quite capable of handling this abstraction. (Actually, I might be wrong about the age when the abstraction really makes sense, as I have limited experience. Perhaps seven or eight is a more accurate age at which the actual abstraction takes place and makes sense.)

      Subtraction seems a little more abstract than addition, but young children somehow manage to understand it. Subtraction can be explained as “take away and how many are left”, or as backwards addition, or as “how many more or less” and perhaps in other ways. Once the idea of subtraction is understood the child can learn the subtraction facts. The proof of these facts is once again conformity to reality. If you have seven blocks and take away three of them you can count how many are left, and indeed there will be four blocks left, every time you do it. The statement, “Seven minus three is four” is an abstraction, but it conforms to reality, and it always will. It may be very easily tested, as often as one wishes.

      Carrying and borrowing are methods to be learned. But of course they are to be learned with careful explanation. The explanation consists of demonstrating that the processes, and the results, conform to reality. One method I used with my own children to help them with carrying and borrowing was with pennies. I would use clear tape to bind ten pennies into one stack. I would make a number of these stacks. Then 23 cents could be represented by two stacks of ten and three single pennies. Of course it was more than just representation. Indeed two stacks of ten and three single pennies actually are 23 cents.

      In demonstrating the subtraction problem 23 - l9, we can take a pile of twenty-three single pennies and take away nineteen. There will be four pennies left. This shows conformity to reality of results. But to demonstrate borrowing this method also allows us to show conformity to reality of the process of borrowing Instead of starting with a loose pile of twenty-three single pennies we can start with two stacks of ten pennies and three single pennies. Then to take away nineteen pennies we have to make a trade. We can take one stack of ten and trade it for ten single pennies, so that the twenty-three cents is in the form of one stack of ten and 13 single pennies. Then we can take away nineteen by removing one stack of ten and nine single pennies. With enough repetitions of this sort of thing the whole process makes sense to the child. We have not “proved” anything from the usual mathematical practice of using deductive logic on a set of premises. But we have proved it on the child’s common sense basis of showing that everything we do, and every result we get, conforms to reality.

      Conformity to reality is not always an easy standard. As one advances in arithmetic there are many ways to make mistakes, and as a result our conclusions may not seem to conform to reality. In my article, Chicago Math,5 I mentioned three “heretical premises” that apply to the learning of a great deal of arithmetic. These premises are: 1) There is one way to do the problem, 2) There is one correct answer, and 3) it will make sense. These premises, I argued, are very beneficial in learning arithmetic, though certainly not all of mathematics. To meet this ultimate test of conformity to reality everything must be right. There are many sufficient conditions for the result of a problem to fail to conform to reality. In other words, students make many mistakes and don’t get the right answer. The only sufficient condition, short of dumb luck, to get the right answer is to get everything right.

      I will digress a moment here to consider motivation. The test of conformity to reality can work both ways in motivation. Consider forms of children’s play for a moment. Pretend play is common in childhood, and I think it is considered by experts in child development to be a healthy and beneficial form of play. However it is true than in pretend play a gross approximation to something is just as good as the real thing. To be a truck driver and child needs only make a few steering motions with his arms and he is indeed a truck driver. To be a mother in pretend play a child needs only make cradling motions with a pretend baby and she is indeed a mother. In pretend play there is no ultimate test of conformity to reality. This makes for certain benefits and certain liabilities. One benefit is that pretend play is easy. Anyone can do it. Another benefit is that a child may put ideas together in a way in which negative consequences are also only pretend. But a disadvantage of pretend play is that it demands little and consequently returns little in satisfaction of accomplishment.

      Now consider another form of play, learning to hit a ball with a bat. In this activity there is indeed a reality test. It is very plain whether or not the ball is actually hit. This type of play is also common in childhood, though perhaps not at the same ages as pretend play, and it is also, I presume, considered healthy and beneficial to children. And of course it has benefits and liabilities. One important benefit is satisfaction of accomplishment when the child has success. An important liability is frustration when the child fails. Learning math is more like learning to hit a ball with a bat than like pretend play. The possibility of failure, apparently, is seen by some as of prime importance. There have been various proposals to shield children from failure. This is understandable, but has to be limited. We don’t withhold balls and bats from children because they might experience failure and frustration. For most children the possible frustration of failure is far outweighed by the possible enjoyment of success, even though that success will not be perfect. Children want to use their abilities. They want to accomplish. They want to stretch themselves and test themselves, in mind as well as body. In a well taught math class children enjoy math. In a well run class of any subject children enjoy learning. Learning may never be made totally free of failure and frustration, any more than sports can be, but we still want to learn.

      So the test of conformity to reality is a good thing, for motivation as well as achievement.

      “Collaborative learning” and group projects are a favorite theme of the idealists for teaching math. Many students, however, and many teachers don’t like group projects. Part of the reason why can be seen in the above paragraphs. Collaborative learning tends to move the activity to something more like pretend play than like learning to hit a ball. Ideally all members of a group contribute equally to the project. Indeed they may include a signed statement in their written report to that effect. But reality is far different, at least in my experience, and I have had a bit of experience.6 Typically a group of four students will have one or perhaps two people who, by either ability or temperament, do the work and understand the math. But another member or two of the group, again by either ability or temperament or both, don’t do much and don’t understand much. For them a gross approximation is as good as the real thing. Pretend play may have an important place in the growth of a child, but it does not contribute to learning to hit a ball with a bat and it does not contribute to learning math.


Deductive Proof

      The “proof” of some very basic ideas, such as the addition facts, is only by conformity to reality. But very quickly that form of proof is augmented by genuine deductive proof, though that proof may not be formally stated. I will give a few illustrations.

      The proof that three times four equals twelve may not seem to use deductive reasoning, but the proof that seven times eight equals fifty-six may use it. Drawing fifty-six stars or circles may be a bit tedious. Hopefully by this time the idea of multiplication has made sense to the child. Multiplication is probably understood by the learner as finding the total when you have several groups of objects with a given number of objects in each group. This is a definition of multiplication, though it may not be presented as such. Then instead of drawing stars or circles or whatever, the child may just take forty-eight, the result of six times eight, and add another eight to it to get seven eights. The result, fifty-six, comes by deductive logic. We know the definition of multiplication. We know that six eights are forty-eight (previously established). We apply that definition by adding one more eight to get fifty-six as the product of seven times eight.

      Once a student gets into fractions and decimals there is a great deal of deductive logic used. Everything learned about working with integers can be taken as established, as conforming to reality, and thus may be used when needed to deal with fractions and decimals. A decimal is a particular kind of fraction, so anything known about fractions can be used to figure out decimals. The same is true for per cents. The logic may not be formal, put into statement and reason columns, but it is still deductive reasoning.

      To make this informal deductive logic into formal deductive logic we would have to explicitly identify and justify each step along the way, and be able to relate it all back to an accepted set of postulates. There is not much benefit in trying to do this when children are first learning multiplication. But we informally do it all the time. When we explain, review, or quiz the learner we are explicitly concentrating on one step of the process at a time. When a step in reasoning is not understood we go backwards, relating it to the previous steps that are understood. If need be we can go all the way back to counting. We use the idea that what we do conforms to reality, but we also use the idea that we can rely on all the mathematical knowledge that we have already established, because we already know that it will conform to reality. We use deductive logic to lead back to our basic proof by conformity to reality.

      In teaching traditional Euclidean geometry we emphasize deductive logic, and we do it formally. But that does not mean that we totally get away from proof by demonstrating conformity to reality. The postulates of Euclidean geometry are not just drawn out of a hat at random. They are common sense postulates, and they conform to reality. To say that “through any two given points one and only one straight line can be drawn”, conforms, as far as we can tell, to reality. In the third grade that would be enough. It would be self evident. By high school we can recognize that being self evident doesn’t quite “prove” it. But by the end of the school year, students recognize that the set of postulates that include this statement make a very good set of premises from which to reason further. The structure of geometrical knowledge that we build on this basis has value, and the explicit recognition of the logic that we use has value. But until we get into non-Euclidean geometry, we still expect all of our results and processes to conform to reality.


Definitions In Proofs

      Deductive proof in arithmetic depends heavily on definitions. However definitions in arithmetic are usually not formally stated, and even when they occasionally are formally stated I think it’s fair to say that it is the informal definitions that count. Definitions are normally developed operationally, not definitionally (as I have discussed elsewhere7). Fractions, for example, perhaps could be formally defined. 1/x could be defined as the multiplicative inverse of x, and a/x could be defined as the product of a and 1/x. But for fifth graders this probably is not the best way to do it. But that does not mean that definitions are not important. To a fifth grader, I would argue, the fraction a/x should mean that we are talking about dividing something into x parts and taking a of those parts. This understanding would come by a lot of examples and problems. After this process of explanation, examples, and problems, the learner may not be able to recite a definition of fractions. In fact the teacher would probably never ask him to. But it is very important that the learner know what we mean by a fractions. This meaning may not be called a definition, but it is, at least informally.

      We should always be aware that definitions, whether formal or informal, of basic concepts may not really be understood. In learning to work with fractions it is always possible that a student who is conscientious may get a lot of right answers, but still may not understand much.8 To what extent this basic understanding of the meaning of fractions is conveyed to students by teachers in general I cannot say. I would expect quite a bit of variation. I would expect to find some teachers who do not really understand this idea of fractions themselves, some teachers who understand it themselves but do not explain it very well, some teachers who explain it well enough at the beginning but do not reinforce the definition when needed (and I expect it’s needed many times in the development of the topic), and some teachers who understand it well and explain it well and explain it again and again as needed. The charge that traditionally we teach arithmetic by memorization without understanding, I expect, holds true for some of those teachers, but certainly not all.

      When definitions are understood, informally at least, they can be used. For example consider this problem:

      “In making salads a restaurant may waste as much as 3/7 of the lettuce that it begins with. If this is true how much lettuce would be wasted if the restaurant starts out with 6 1/2 pounds of lettuce?”

      Suppose the teacher explains as follows:

      “First let’s think about quantity. And remember what a fraction means. Three sevenths means divide something into seven parts and take three of them. Can you figure out 1/7 of 6 1/2 pounds. It would be a little less than one pound, wouldn’t it? Then take three of those amounts and you’d get a little less than three pounds. So shouldn’t you expect the answer to come out a little less than three pounds?”

      This explanation, so far, does not use any standard algorithm. It is simply setting the stage for testing the answer for conformity to reality. But it does use deductive logic. It uses the definition of three-sevenths and what the students already know about numbers and quantity to arrive at a logical conclusion. Then if the standard algorithm for multiplying fractions gives an answer of a little less than three pounds the students have at least an approximation of the proof that makes sense to them. The answer seems to conform to reality.

      Let us assume that in this example the students have learned the standard algorithm for multiplying fractions, as well as understanding the meaning (definition) of 3/7. It still may be unclear in their minds why they multiply. The answer to this again uses a definition and deductive logic. But at this point it is important that the definition of multiplication has changed. Many definitions in arithmetic change as a topic is developed, and this needs to be recognized and well understood. The teacher may explain as follows:

      “Now, how do we know to multiply? Well, what is multiplication for? Is it a short way of counting, like it was in the third grade when you memorized the multiplication table? No. Remember that we have changed that a little. What multiplication means now is that one number tells you how many of the other number you have. We start with the number 6 1/2, but do we expect an answer of more than that. No, the three-sevenths tells us how many of those 6 1/2’s we are talking about. And we’re talking about less than one of them, so of course the answer must come out to be less than 6 1/2. We only want a part of that 6 1/2. Three-sevenths of something is only a part of something.”

      This bit of explanation again uses a definition and deductive logic. The definition of multiplication is “one number tells you how many of the other number you have”. It may not be formal logic with postulates, definitions, and theorems, and a formal proof with statements and reasons, but it is genuine logic and it is understandable to fifth graders. And it leads to a demonstration that the result obtained seems to conform to reality. It is important that since we use only informal definitions developed operationally, students may not realize that the definition of multiplication has changed. The old definition could be stated:

      Multiplication means to find the total when you have so many groups of objects with so many objects in each group.

The new definition could be stated:

      One number tells you how many of the other number you have.

      The old definition works only for positive whole numbers. The new definition works for whole numbers, both positive and negative, and for fractions and decimals. Therefore this switch in definition is important. The teacher must be sure students understand it.


Appropriate Logic

      I have argued that conformity to reality is the proof that makes sense to a young child. I have also argued that when well taught geometry can be learned in high school with formal deductive logic applied to unproven postulates. And I have argued that genuine deductive proof is often used in learning arithmetic, but only informally. All this leads to the general idea that at any stage in the teaching of mathematics the type of logic used, and the level of logical rigor used, may vary, but is important. Logic is always the basis of any mathematics, but different things may go under the label of logic. I consider it an important principle of teaching mathematics that the logic used must be appropriate to the topic and the abilities of the learners. I consider this “Principle of Appropriate Logic” to be one of the more important principles of teaching math.

      Some of the new math programs which claim to stress understanding, it seems to me, fail badly. Important ideas are often stated, but not grounded in either conformity to reality, as in arithmetic, or in deductive logic, as in Euclidean geometry. What are we to make of a math book that states the distributive law, but calls it neither self evident because it is easily shown to conform to reality, nor a postulate to be assumed without proof? The distributive law can be very easily “proved” by conformity to reality. Just imagine three baskets each containing two apples and four oranges. The result is a total of six apples and twelve oranges. That is easily understood. One can draw pictures and count up the apples and oranges. If this common sense “proof” is not utilized, but we wish to make it a postulate, then the students need to have some sense of the complete set of postulates that we are going to use. Similarly the associative and commutative laws can be easily demonstrated to conform to reality, or they may be taken as postulates. To simply state them and use them seems very inadequate to me.

      A good example of this is in my article on “Chicago Math” concerning the area of a rectangle.9 The seventh grade textbook presents the “area model of multiplication” stating that the area of a rectangle is given by A = bh, where b is the base and h is the height. They do not explain why this works - or whether it is to be taken as a postulate. It can be made very sensible to elementary school students by explaining that area means the number of squares can be fit into an area, and if the area is a rectangle then we can put squares in rows and columns in the rectangle, and multiplication is a quick way of counting up the total number of squares, because arranging objects in rows and columns has already been well established as a way to visualize multiplication. This explanation shows conformity to reality, and it also uses informal deductive logic.


Understanding And Fluency

      This article is meant to be primarily about curriculum, not about methods, but I do want to say a few words about practice. Mathematics always has a structure primarily of implication10, but that does not mean that nothing needs to be memorized. It only means that it must be understood as well as memorized. This is true for most any subject. A child may understand perfectly well what multiplication means, but he still has to memorize the product of nine times eight. As another example, a child may perfectly well understand that the surrender at Yorktown must come after the Declaration of Independence, not before, but still dates must be learned. Acknowledging that memorization is important, necessary even, in no way detracts from the importance of understanding.

      It is common among advocates of “NCTM math” to argue that we should teach for understanding. Few would disagree with that. However often they go further. They disparage practice. Phrases like “drill and kill”, or “worksheet teacher”, has been used at times to disparage any sort of practice. This is very unfortunate. Practice is important in just about any kind of learning.

      In some learning situations practice takes the form of intense drill. This would be true in many athletic activities, or in learning to play any musical instrument. Intense drill has a place in learning the multiplication facts and the spelling of words. In other situations practice takes less intense and stereotyped forms. A psychology class discussing case histories, relating ideas and concepts to real lives, would be an example of this. “Going over the lesson” in a history class would be another example. Attempting to do problems after listening to an explanation in a math class would be yet another example.

      It is useful, I believe, when considering any subject to ask what should be on an instant-recall level. I discussed this in another context.11 I divided responses into three basic levels. Responses that should be memorized and immediately available are on the “immediate response” level, which I also call the “look-say” level (A fluent reader can look at a word and say it immediately). Responses that require some thought, a procedure, an algorithm, or a computation, are on the “ciphering level”. Responses that do not need to be either memorized or computed, but can be looked up, are on the “reference level”. This type of analysis can be beneficial in analyzing the teaching of many topics.

      In most topics or subjects there is a “bank” of memorized responses that must be on a fluent immediate response level. Consider language as an example. To obtain even a basic level of fluency in a language one must put thousands of words on a immediate response level. No one can convincingly say, “I know the language - I have a phrase book”. Vocabulary must be available immediately. In arithmetic the addition, subtraction, multiplication, and division tables must be put on a look-say level. It is not enough to say, “I can’t remember six times nine, but I can look it up, or I can figure it out if I have to.” In music, if one is to be a competent note reader, one must have all the notes on a look-say level. To claim to know something about geography one must have a large number of basic maps in one’s head. In chemistry one quickly learns that the sulfate ion is SO4-- and the calcium ion is Ca++, and this learning must remain on an immediate response level. All these learnings take time and practice, of course. Often the practice is informal and not greatly concentrated. At other times, especially in the lower grades, it takes the form of drill.

      The “ciphering” level of response is important in many subjects, and very important in math. No one is expected to have an instant response to the problem, “divide 435 by 23”. But students should be able to figure it out. They should know the procedure and be able to efficiently apply it. Becoming efficient at computation and routine problem solving again takes time and practice, and is sometimes put in the form of drill. Again any algorithm or procedure learned should be understood, but understanding is not enough. Algorithms and procedures must be practiced enough to gain a high degree of fluency. Fluency does not detract from understanding, and understanding does not detract from fluency. Indeed, in most cases they reinforce each other.

      Sometimes we say that the student must “master” one level before moving on to the next. I think this is generally not true, depending, of course, on how “mastery” might be defined. But it is true that a certain degree of proficiency and fluency must be attained in any given topic before moving on. Otherwise one has an insufficient basis on which to build. Analyzing a topic into learnings that must be put on each response level can be very helpful in identifying the level of proficiency and fluency to require in any given topic before moving on.

      The typical pattern of problem solving in math is that the student uses mathematical knowledge to plan a solution to a problem, then uses a procedure to solve the problem, and then uses mathematical knowledge to judge the accuracy and appropriateness of that answer. Understanding is essential to the first and third step in this process, and these steps should not usually be rushed. The second step is a little different. Understanding is no less important in this step. Everything in math should be understood. But something more is required. That something more is fluency in the procedures needed.

      In teaching college freshmen in recent years I got the impression that the solving of simple equations was always treated as a process of exploration by many students, rather than as a small set of techniques that must be practiced enough to become efficient. This is illustrated in the common terminology used. Consider the equation, 3x + 7 = -3 - 2x. When I was young we “transposed” the +7. Now days, apparently the term “transpose” has gone out of favor (like “carrying” and “borrowing”) because it does not promote understanding. Instead we say, “subtract seven from both sides”.

      It indeed may be true that the term “transpose” does not promote understanding. But that is irrelevant at this point. Understanding should come earlier. It should come by careful explanation by the teacher and conscientious thought and practice by students. The term “transpose” does not promote understanding, but it does not detract from understanding, and it is a very convenient term to use. Perhaps it does not really promote fluency, but it is convenient language to use when working on the fluency that must be developed. It is a useful term, as are “carrying” and “borrowing”. When I first learned algebra the explanation, “subtract seven from both sides and the resulting equation will have the same solution as the original equation”, was given and carefully considered during the first month of learning algebra, but not thereafter. After this initial period we just transposed the +7. It required no more thought than plugging in a lamp.

      Should plugging in a lamp require understanding? In one sense at least, it should. We should understand something about electricity, energy, electric circuits, and so on, just as a part of basic science. And we should have some understanding of the safety considerations involved with using electricity. But we must also be able to fluently do the procedure with little thought. If we had to review our understanding each time we plugged in a lamp, we could not accomplish much. If we had to go through the explanation, “If I put this plug on the cord connected to the lamp into this receptacle in the wall, then the lamp will have a source of energy and we can have light” every time we wanted light we would be in sorry shape indeed. Understanding is important. That is a very basic purpose of all education. But when one is plugging in a lamp understanding should be in the background. We should be able to just do it. And when solving a routine equation understanding should be in the background. We should be able to just do it.

      I learned, after a few semesters of teaching college algebra, to give a five minute “pep talk” early in the semester about equation solving. The gist of this talk was that if you’ve never gotten the solving of simple equations down to a routine, do it now. “Streamline the process” I would tell them. “Practice enough so you can do it mindlessly. Understanding is important, of course. But by this time you should understand. The next step is to become fluent in doing it”.

      All the standard algorithms should reach a high level of fluency. All the basic concepts of any subject should reach a high level of fluency. Fluency in no way detracts from understanding. And understanding in no way detracts from fluency. And most importantly, understanding and fluency reinforce each other.


Possible Topics In The Math Curriculum

      If we grant that numbers deserve to be first in math education, meaning we will continue to teach arithmetic in elementary schools and try to teach it well, it still does not follow that the math curriculum is set. And it certainly does mean that only numbers can be taught. Perhaps there are some nontraditional topics, or non-number topics, of mathematics that could or should be included in elementary school or high school.

      I believe the math curriculum should continue to be primarily numbers. The main trunk of the tree of mathematics should continue to be arithmetic-algebra-calculus. But within that basic framework there is room for a lot of adjustment. Sometimes a minor change in curriculum consists only of concentrating a bit more or less on a specific topic, or bringing in or deleting a particular concept. A not-so-minor change might be bringing in a topic that was previously left out, or taught only at a more advanced level.

      So the math curriculum will always be open to discussion, and it should be. With this perspective I would like to discuss some other possibilities of what could be taught in elementary school or high school. In particular I will discuss some topics that the NCTM, in their 2000 edition of their standards, advocates. I think these topics are worth serious consideration, but they should not be accepted indiscriminately.

      There are several questions that can be asked about any topic proposed to be part of the math curriculum. Perhaps the most important question is the relation of the proposed topic to the structure of current curriculum. Does a proposed topic for the elementary grades act as a framework support for arithmetic? Is it useful supporting structure? Or does it depend on arithmetic? Is it totally irrelevant to arithmetic? If it is totally irrelevant to arithmetic, then it would have to be valuable indeed to be worthy of inclusion in the curriculum, such as geometry is in high school.

      To illustrate these concerns consider the topic of areas and volumes. Traditionally in arithmetic we get through fractions, decimals, and per cents, and then, in seventh and eighth grade, we teach a number of applications. One applies arithmetic in order to compute areas and volumes. But the topic is not just application of arithmetic. There are new concepts to learn about areas and volumes. Learning about areas of rectangles leads to the idea of areas of other shapes. The area of a circle leads to the concept of pi, and a whole world of math that follows.

      Perhaps it could be argued that the topic of areas and volumes depends on arithmetic, but it does not add to arithmetic, other than being an application. But it does seem to be a very good application of arithmetic. It is an application that has utilitarian value in everyday life of ordinary people. Ordinary people paint rooms and plant gardens. They need to be able to make sense out of instructions on cans of paint and sacks of fertilizer, so they need to understand areas and volumes. But I would argue that there is more to it than simple utilitarian value. It does add to arithmetic. It is more than just a topic for practicing arithmetic. It adds to, as well as reinforces, what the students have previously learned in arithmetic, and arithmetic reinforces what the students learn about areas and volumes. Thus the topics of areas and volumes acts as supporting structure for arithmetic, and arithmetic acts as both supporting structure and framework structure for the topic of areas and volumes.

      Additionally, and perhaps most importantly, the topic of areas and volumes it leads to other things. It leads to both algebra and geometry. It is useful to have formulas for the relationships discovered when investigating areas and volumes. Thus it is useful to develop the idea of formulas. And formulas are close to equations. Formulas use letters to stand for quantities, as do equations. So a good through study of areas and volumes, leading to a thorough study of the idea of formulas, works very well as a transition between arithmetic and algebra. The considerations about other shapes lays important groundwork for geometry.

      Thus the topic of areas and volumes fits very nicely at the end of arithmetic, just before algebra. Are their other topics that would fit as nicely somewhere in the curriculum of numbers?

      Data analysis and statistics are advocated by the NCTM Standards to be included in the math curriculum from elementary school. Certainly they are important topics in our world today. Should they be integrated into arithmetic? Should they be integrated into algebra? I have a bit of experience that speaks to this.12 I come away from the experience with the opinion that though statistics depends on arithmetic, it does not add to it, at least it does not add to it much, and it does not aid in understanding arithmetic. Certainly the idea of the arithmetic mean ought to be taught at some point in elementary school. It is an application of arithmetic, and it is a part of everyday life for both children and adults. The median and mode might be introduced along with the general topic of graphs (line graphs, bar graphs, pie charts) at some point. But what comes after that - standard deviation?

      Standard deviation can be taught without algebra or calculus. The general idea of standard deviation, I think, can be made understandable to students perhaps even before high school. Examples can be cited in which two sets of data have the same average and range, yet differences in their variation are important. These examples can be understood by students with very little mathematical ability beyond arithmetic. However for these students the computation of the standard deviation remains a rote procedure. If we want to really understand the standard deviation we have to have calculus. To understand statistics beyond standard deviation we need probability, which I will discuss in the next few paragraphs. Therefore I do not favor giving too much stress to statistics in elementary or high school.

      Is probability suitable for teaching in elementary school? Or in high school? I’m not sure. I have a bit of experience teaching probability to college freshman.13 The results are mixed. I feel one important consideration that applies here is the concept of “minimum teachable structure”. For any given topic there is a minimum amount that should be learned or the whole effort is wasted. This minimum amount may not be easy to define, and may be subject to considerable disagreement. Some people, for example would argue that if you don’t take four years of a foreign language you might as well take none. You’re just wasting your time with less. Others might argue that even a week spent on a foreign language in the sixth grade is worth while. My experience teaching probability leads me to believe that the minimum teachable structure is fairly well defined. It would include the “and rule” (the probability of both of two independent events is the product of their separate probabilities) and the “or rule” (the probability of either of two mutually exclusive events is the sum of their probabilities), and enough practice to enable one to actually apply this knowledge. This is quite a tall order, but it seems to me if we do any less we might as well do nothing. In my experience teaching probability to college freshmen it seemed we never quite made this minimum. Thus, I would guess, with very few exceptions everything about probability has evaporated from my students’ minds within a couple of months of the final exam.

      Teaching probability in elementary school does not seem promising to me. I don’t think we can get close to what I have described as a minimum teachable structure. But some might disagree with me, saying that teaching just a few ideas of probability could be beneficial (in which case my idea of the minimum teachable structure of probability is wrong). However the results of a probability problem are not amenable to testing by conformity to reality, at least not like children are used to. When you do a problem like “What is the probability that the total of two dice will be seven?”, how can you check for conformity to reality?

      There is plenty of evidence that the average adult has trouble with the basic idea of probability. People may understand phrases like “There’s a 30% chance of rain this afternoon” in an intuitive way. And they understand phrases like “one chance in a million”, or “a 99% chance” in an intuitive way. But even this intuitive understanding is often absent. People often don’t seem to understand the idea of relative risk, which requires at least an intuitive understanding of probability. When someone asks “So what are you saying? Is it safe, or is it dangerous? Just give me a straight answer!”, they are indicating that the basic concept of a risk pool is not well formed in their mind.

      I would certainly not totally reject the teaching of probability in some form at some level. The potential rewards, to both the individual and to society, are great. A society that thinks no deeper than “safe or dangerous” is doing very shallow thinking. A society that recognizes that “safe or dangerous” is not a good question is doing much better. But where should probability fit into the curriculum? In the fifth grade? In the tenth grade? It seems to me that probability would have no lasting value at all for students weak in fractions and decimals. Therefore I cannot see any argument for giving probability in the eighth grade or lower. Once algebra is started I would think it should not be interrupted for probability. If given in high school I would think that probability would be a topic only for advanced high school math courses, after algebra is well established. This would mean there is no good way to give probability to every student.

      I would like to be wrong in this. The NCTM standards call for giving bits and pieces of probability in every grade. Perhaps the child that understands what the fraction 3/7 means, such as in the statement “My gas tank is 3/7 full”, can also understand what a probability means, such as in the statement, “I have a 3/7 chance of being elected”, and perhaps that is worthwhile learning. If this is true then my opinion of the minimum teachable structure of probability is wrong. But at this point I am not an advocate of trying to teach probability in elementary school.

      What we might call “descriptive geometry” is often seen as a desirable part of the math curriculum. In fact sometimes it seems to have been taken for granted that students should learn some bits of geometry before they get to Euclidean geometry in high school. I think this comes out to be a “bits and pieces” approach, for until you start Euclidean geometry, taught rigorously enough to be genuine mathematics, bits and pieces is all there is. Certainly one can learn a bit of terminology - triangle, parallelogram, quadrilateral, and so on - but that doesn’t really seem to add up to a topic of much substance. And certainly one ought to learn angles before high school. Measuring angles with a protractor and drawing angles of a given size take explanation and practice. But other than that I cannot see what we might call “preliminary geometry” as having much importance in the elementary math curriculum.

      Just where these topics of “preliminary geometry” ought to be placed in the elementary school math curriculum, I’m not sure. I do not subscribe to the NCTM approach that a little bit of every topic should be given in every grade. I would think a few days learning terms and angles in perhaps the fifth grade would be sufficient. If well taught I would think the learning would be permanent for most students. Returning to the topic for a few days in the seventh grade would probably be worthwhile to cement the learning in place.

      There are some modern geometric topics that are interesting. Symmetry and transformations in two dimensions - rotations, reflections, and translations - come to mind. I enjoyed teaching these topics in a freshman college math course.14 But they proved difficult for some students, and I suspect it is not at all clear to all students why they are mathematics, or why they should be taught.

      This topic brings up a phenomenon that educators sometimes run into. A teacher or a school tries to teach a topic to students for enrichment, but parents object, saying something like, “My Johnny is struggling hard to learn to do numbers, and the teacher is playing stupid games with shapes!” This phenomenon can be important and is worth careful analysis.

      In the history of American education15 a common theme is that parents worry about their children not getting a good education, so they do what they can to prevent this. Thus early in the century many parents wanted their children to take algebra and Latin even though ideals of progressive educators thought these subjects were a waste of time for most students. Many parents pushed for Latin and algebra as a way to make sure their children were not being left behind. Another common theme is that many parents only want a very basic utilitarian education for their children. These two themes are opposite each other in general. The first theme is pro-intellectual. The second theme is anti-intellectual. But sometimes both themes come together. This can happen when a parent is concerned about his children getting a good education, but a topic that the teacher and the school consider enrichment is seen by the parent as frivolous and thereby detracting from a good basic education. For want of a better term I will call this the “unappreciated enrichment” phenomenon. I heard it most forcefully one time from a parent who resented the fact that her child was being given Spanish in elementary school . By my perspective a foreign language in the elementary school is clearly enrichment. But she rejected this. By her perspective time and effort spent on Spanish was time taken away from reading, writing, and arithmetic. By her perspective Spanish was clearly detracting from her child’s getting the important basics. Was her attitude pro-intellectual or anti-intellectual?

      A topic such as symmetry and geometric transformations is clearly wide open to this “unappreciated enrichment” phenomenon. It may not be seen as enrichment by some parents. It may be seen as a distraction from more important things that children should be studying. A topic even more vulnerable to this perception is bases other than ten. Bases other than ten was a popular part of the “new math” of the sixties. I have no personal experience teaching it. I would think it would have the potential of being very valuable, of bringing into relief some basic ideas of our number system. But it is also obviously wide open to the unappreciated enrichment phenomenon, especially so because of its potential for confusion. A parent who has seen her child struggle to memorize the multiplication facts and learn to do long division may be disturbed and incensed when her child comes home and announces, “The teacher said six plus eight is sixteen. I know it’s fourteen. I don’t get it.”

      Should the schools lead in setting curriculum? Or should they follow? I think the answer is “yes” on both counts. Society tends to be conservative in the sense of passively expecting continuity. Parents expect their kids to get basically the same education as they got. We elect school boards to give us the schools we want, and we complain when they don’t. So there is a lot of continuity in schools from decade to decade. But obviously there is also a case to be made for change. We want schools to anticipate the future needs of our children. I don’t think there are any easy answers to this problem.

      Should we teach set theory sometime in elementary school? Should we teach graph theory? Number theory? Formal logic? I don’t have an answer for these topics. But I would argue that the issues I have discussed form the basic framework for discussion. How does a proposed topic fit in with the current curriculum? Will the topic spark resentment by parents - the unappreciated enrichment phenomenon? Do the students have a background such that we can reach the minimum teachable structure? Does the topic have any particular benefit not found in any other topic?

      Does the proposed topic promote critical thinking? This would be a very important question if indeed some topics promote critical thinking more than other topics. I am not sure of this, one way or the other. We could argue that geometry is valuable primarily, even only, because it promotes critical thinking, while more mundane topics are taught for their utilitarian value. Or we could argue that any mathematical topic taught well promotes critical thinking. I think I hold to this latter view. Critical thinking cannot be taught directly, any more than art appreciation can be taught directly. But critical thinking is a very valuable by-product of teaching math, just as appreciation is a very valuable by product of teaching art.

      Finally, I want to give one example from my personal experience of a minor change in curriculum. When teaching in a prison school in the 1970’s16 I noticed that the “third type” of percent problem was not well understood. Type one per cent problems, to explain my terminology a bit, are problems in which you take a percent of a number. An example would be, “Experience shows that only 38% of seeds will germinate after one year of storage. A scientist has fifty-five seeds. Find the number expected to grow after one year of storage.” Type two per cent problems are when you find the percent. And example would be, “John had $940 saved up when he used $314 to buy a television. What per cent of his savings did he spend?” Type three per cent problems are when you know the part, and the percentage, but do not know the whole. An example would be, “Mary went to Las Vegas and lost $150 gambling. She claimed this was only 11% of her savings. If this is true how much did she have in savings when she started?”

      Students had trouble understanding this third type of problem. One can tell them to divide by the percent, but that hardly explains the situation. I would tell them to first simplify the problem to, “11% of what number is 150?” Then it would still not be obvious that you divide by .11.

      I made up a worksheet and inserted it earlier into the course. This worksheet explained that every operation has its opposite. Subtraction is the opposite of addition. Division is the opposite of multiplication. If you have an addition problem, but instead of having two addends you have one addend and the sum, then how do you find the other addend? Subtraction is the opposite of addition, and subtraction is a way to find a missing addend in an addition problem. In fact subtraction may be defined, among other ways, as a way to find a missing addend. Similarly division may be defined as a way to find a missing factor. If three times some number is 12, what is that number? The number is 4, because three times four is twelve. Type three per cent problems may be thought of as a multiplication problem in which you have the product, but are missing one of the factors.

      My worksheet had a paragraph or so of explanation, much like the preceding paragraph, and then some problems. As I recall before the problems I put in a few fill-in-the-blank questions, such as, “If you have a multiplication problem in which one factor is 24 and you know that the product is 72, then you find the answer by _________.” Problems would be something like, “If you subtract 27 from a certain number you get an answer of 28. Find the number.”

      I can’t remember just where in the course I inserted this worksheet, but I do remember that it seemed to work out well. Students apparently were able to understand it, for with a little study they could do the exercises. And, most importantly, a month or so later I could explain type three per cent problems in these terms. “This problem says that 5% of some number is 14. If you knew the number you would multiply it by .05 and you’d get 14. But you don’t know the number. It’s a missing factor in a multiplication problem. How do you find a missing factor in a multiplication problem?” Usually at this point the student would have to think just a bit, but would then get it.

      After some months I began to realize that this type of lesson, and this type of thinking, ought to be a regular part of arithmetic for everyone. However I could not remember ever being exposed to this topic as a child. Why was I not? Or did I just forget it? I’m not sure, but I think part of the answer, and a very important part, is that the arithmetic curriculum has never been given a whole lot of thought.

      Like many things humans do, I believe, the mathematics curriculum primarily just evolved. I would argue it that it evolved in a pretty good form. Lots of conscientious teachers over many years made choices that seemed right at the moment, and over decades and even centuries produced a pretty good curriculum. Textbooks perhaps sometimes led this process, and sometimes perverted it. Progressive education tried to pervert it for many years, but with limited success. Most teachers paid little attention to what they learned in education courses and did the best they could with their own good sense. Primarily the curriculum was passed on by imitation. Teachers taught what they knew. For the most part they taught what they had themselves been taught, and it worked out pretty well. But I think we can do better than that. We can analyze what we are doing and on the basis of that analysis make improvements. But we can not substitute unrealistic ideals and wishful thinking and expect to make progress. This article is an attempt in the direction of that analysis.


1. I think “structure of knowledge” is a basic idea of teaching that is not appreciated as much as it should be. I have developed the idea to some extent in chapters two and five of my proposed book on teaching and learning on my website,

2. I addressed this issue, at least indirectly, in my article NDSU Math on my website. I felt we taught a bit of math, and then tested for intelligence. The issue of separating achievement from intelligence is by no means simple in almost any subject, and is perhaps particularly difficult in math.

3. See “NDSU Math” on my website.

4. See chapters two and five in my book on principles of teaching and learning on my website. In Chapter Five I develop the idea of structures of implication versus structures of accretion.

5. See “Chicago Math” on my website.

6. See “NDSU Math” on my website.

7. See Chapter Thirteen in my book on my website for a more thorough discussion of operational versus definitional concept formation.

8. I discussed the “conscientious algorithmist” in Chapter Three in by book mentioned above. The opposite type of student is the “intelligent detail dropper”. Most students are neither of these, but I believe the concepts are beneficial.

9. See my article “Chicago Math” on my website.

10.See chapters two and five, as previously mentioned, in my book on my website.

11. See Chapter Fourteen in my book for a discussion about response levels.

12. Some elementary statistics were included in the courses I taught at both SCSU and NDSU. See “Thoughts On My Teaching At SCSU” and “Thoughts On My Teaching At NDSU” on my website.

13. See “Thoughts On My Teaching At SCSU” and “Thoughts On My Teaching At NDSU” on my website.

14. See “Thoughts on My Teaching At SCSU”.

15. See the book “Left Back” by Diane Ravitch. I have not read a great many books on the history of American education, so I don’t have a good basis for comparison, but I strongly suspect this book stands out for being relevant and readable.

16. A description of this situation can be found in chapters three and four of my book on teaching and learning on my website.