"basic laws" for young children

"Basic laws" for young childrenAuthor(s): JO PHILLIPSSource: The Arithmetic Teacher, Vol. 12, No. 7 (NOVEMBER 1965), pp. 525-532Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185234 .

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"Basic laws" for young children*

JO PHILLIPS University of Illinois, Urbana, Illinois Dr. Phillips is a member of the staff of the University of Illinois Committee on School Mathematics.

1 have heard that the first thing a person should do in a discussion like this is define his terms. My title is "'Basic laws' for young children." What do I mean by "basic laws"? The commutative, associative, and distributive principles which are characteristic of the number systems used by young children. What do I mean by "young children"? Up to at least the equivalent of average sixth graders.

Now, you know what I'm going to talk about; next, you may like to know why. The reason is that I am a missionary, a missionary for a maximum amount of good mathematics in the minds and lives of our elementary school children. I am fully convinced that there is a body of fundamental mathematics - concepts and skills associated with number relations and spatial relations - which every citizen needs to get through even one day without a keeper in a culture such as ours. I believe that every child who is not mentally handicapped can learn these things, and furthermore, that he must learn them to avoid becoming so- cially handicapped. In the matter of helping children learn these necessary concepts and skills, I am convinced that the real thing in the "revolution in school mathematics" is not only desirable but also necessary, and I am worried sick by some of the things I see going on in the name of "modern math." My worry is that the modern math movement will fall into the same sort of disrepute that befell the Pro- gressive Education movement in the thirties, and for the same reasons. The

♦ Adapted from an address given at the NCTM Annual Meeting, Detroit, April, 1965.

sound principles and pronouncements of Dewey and Kilpatrick were distorted and prostituted by certain eager proponents, some of them genuinely well-meaning, until no one with a modicum of common sense could subscribe to them. A sense of perspective toward our modern math movement is long overdue, but there are signs of a far more drastic reaction to it. Should the whole movement be debunked, that would be tragic indeed.

I have said that I am a missionary. Most of the time I am a teaching missionary, but today I am preaching, so I need a text. My text is taken from the scripture according to Albert Schweitzer: "Only those who respect the personality of others can be of real use to them."

To whom do all of us wish to be of real use? The children, of course. We think we are helping young children by dealing with what we call the structure of mathematics. What do we mean by "structure"? What does what we mean (if we know what we mean) by "structure" mean to the child? Some of us want to start teaching children to give proofs early. What sort of proofs, and to whom are we proving something?

It seems to me that a good way to measure maturity in children or adults is to check on the subject's idea of what con- stitutes a proof. Does he consult his heart to tell whether somebody is right? Does he prove universal generalizations by citing one or two cases? Does he believe that the only way to prove something is to reduce the situation to some sort of deduc- tive system? If so, what levels of maturity

November 1965 525



do these, and other behavior patterns you could describe, indicate?

I believe that everyone should test his conclusions, validate his answers, convince himself that something he has done is correct or incorrect. I believe that no one is truly convinced by the application of standards which are not really his own standards, however impeccable said standards may be from another point of view.

Consider what a young child means when he says, "OK, I'll prove it to you." The first thing he means is that he wants to prove it to you. The second thing he means is that he believes it himself. Consider also what he means when he uses the word "why." He is not asking for (or giving) reasons that a certain thing is so ; he is asking for (or giving) evidence which would make him believe it. Like the jurist, and everyone else I suppose, the young child deals in likelihood. You must have had an experience comparable to this: A young- ster looks out the window and says, "It's been raining." "It has?" "Uh-huh. You know why?" "No. Why?" "The street's wet." The street's being wet is not the reason it has been raining; it is the reason the child believes it has been raining. He knows that when it rains, the street gets wet. He hasn't seen the street get wet any other way. Therefore, since he has no evidence to the contrary, a wet street tells him it has been raining. Furthermore, it is quite likely that the reason the street is wet is that it has been raining, but if you say to the child, "It hasn't been raining because the street is wet. The street is wet because it has been raining," he'll feel sorry for you. He knows that's what he said.

Similarly, a young child does not believe that 5+3 = 3+5 because of the commutative law for addition; it's the other way around. He believes the commutative law (or whatever he calls it, if he calls it anything) because 5+3 = 3+5. Given the proper guidance, he'll realize eventually that every pair of numbers behaves in this way.

Occasionally I see, or hear about, some-

thing going on in a classroom which causes me to lose several nights' sleep. Last week I heard about a first grader who said to his teacher, "Oh, 4+3 = 3+4, and it always works like that." The teacher replied, "If you wouldn't try to be so smart, you might learn something." If the teacher had just said "Why?" think what all of us could have learned from a film with sound track of what might have happened next. In my view, her response would have been only slightly less outrageous if she had said, "That's right. Addition of whole numbers is commutative."

The more I study how young children learn the primitive and absolutely essen- tial mathematical concepts, the more un- easy I become in my growing suspicion that we are hindering rather than helping them by the way we are teaching some of these concepts nowadays. We are so eager to enrich their early experiences that we are starting with the sophisticated end products and actually working backward.

We know very little about how young children think. We know a little bit about how adults think young children think. We need better answers to our current questions, and in many cases we need new, or revised, questions. But lack of definitive answers should not prevent us from taking a hard look at our current practices.

We must treat children with respect. This does not mean that we treat them as miniature versions of adults who are woe- fully inexperienced and thus need help from us. It means that we recognize their unique ability to think and to feel, even though we do not fully understand either their thought processes or their feelings. It means that we try to help them learn things which make them functionally com- petent and happy in their current role, and we try to help them learn to learn more things with little or no help from us as their fund of understandings increases. Substituting new bags of tricks for the old ones we used to dispense shows the same lack of respect for the personality of the

526 The Arithmetic Teacher



children that made our old programs less c than spectacularly successful. c

It is true for all learning areas that the c ceiling is low for the number of isolated c facts and skills a learner can master. A i principle which holds for additive struc- g tures in mathematics - the whole is equal í to the sum of its parts - does not hold for ( most areas of human activities. In a i poorly conceived instructional program, i the whole is less than the sum of its parts ; i in a well-conceived instructional program, 1 the whole is greater than the sum of its 1 parts. When the learner can see how the ^ parts fit together, he not only has less i work to do in learning each of these parts, 1 but he also is less likely to lose some of the parts without knowing that he has lost < something, and he can use relationships he ¡ sees among these same parts to construct larger wholes.

Some children, and some adults, do not habitually relate ideas or events. They interpret a phenomenon as if it were a simple concrete object not connected in any way with anything else. These are what some sociologists call content- oriented people. The same sociologists speak of a content-to-structure continuum. Structure-oriented people see phenomena as multiple and interrelated, with con- comitant elaboration of qualities.

Think about your own thought processes for a moment. When you are learning something which is brand-new to you, you have to be content-oriented at first. You need at least two pieces of content before you can see a relationship, and hopefully, the more content you know, the more relationships you can see. But if you stay content-oriented for very long, you will not acquire more than a superficial knowledge of the topic, a small collection of facts.

The success of a discovery sequence de- pends upon the learner's being to some ex- tent structure-oriented. I am teaching a class this year (seventh grade) in which at least half of the pupils drown me in utter frustration because, to them, each exer-

cise is an island. They do fairly well now in ;lass discussions where I can keep prod- ling, but in written work, my beautiful in- luctive pedagogy falls flat on its face nuch of the time because I have not yet lucceeded in changing their habitual thought patterns toward more structure mentation. They are annoyed with me arhen I don't just tell them how something s "s'posed to come out." They had 'modern math" before I ever saw them, so rïiey had heard of the "basic laws/7 but bhey view these laws as content, as facts which their teachers especially liked the sound of, I guess. They think it beneath their dignity to be asked whether 4X37 = 4X30+4X7, and if so, why. (That's the distributive principle. They've learned it, and now they're through with it.) But they do not see that "4X2± = 4X2+4XÍ" has anything to do with the same relation, even when the two examples are written together on the chalkboard. What do we mean by "structure" in "modern math"?

Suppose we consider quite carefully how a child might be led to genuine com- prehension of the commutative principle for addition. What we shall describe does not take place in a day, or even a week or a month. We must remember that the growth curve is not a vertical line. Fur- thermore, I doubt that anyone would assert that the activities I am about to describe should be done exactly as stated in the order given. We'll assume that the child understands counting, thereby skip- ping a five-hour discussion. A person who understands counting knows more than just how to count the members of a set and get the correct number; he knows, among other things, what the number means after he gets it. An activity he can do next is "count together."

He should manipulate objects to see that:

3 goats and 2 goats are 5 goats 3 pennies and 2 pennies are 5 pennies

November 1965 527



3 horses and 2 horses are 5 horses 3 boys and 2 boys are 5 boys 3 chairs and 2 chairs are 5 chairs

Pretty soon, he should see that 3 and 2 of anything aie 5 of that thing. Those who are slow to arrive at this conclusion may be helped by the use of words they cannot read on worksheets, or words they do not know in oral discussion. 3 zedmos and 2 zedmos are zedmos. What's a zedmo? Do you have to know what a zedmo is in order to fill in the blank correctly? "3+2 = 5" can be introduced as a general statement which takes the place of an endless list of statements. When it is introduced, the child should be taught to read it: three plus two is five. Why should he believe this? He can show you with his counters. "Look. I have 3 counters here and 2 counters here. If I put them together, there are 5 counters. I can play that the counters are anything I want. They can be horses, or lollipops, or chairs - just anything. I'll have five of them, no matter what." Now he ]nnderstands this addition fact. We cannot yet assume he has mastered it. Mastery requires both practice and application.

Now we can start with 2 objects and 3 objects.

We should note which children always start from the beginning in counting exercises like this and which ones count on from the number in the first group. The latter is a more mature response, of course, and some second graders who can reel off a glib recital of addition facts have not yet arrived at this stage. We go through more activities, and it should take us less time to arrive at our next general statement: 2+3 = 5 (Two plus three is five).

At this point we are by no means on the verge of discovery of the commutative principle for addition, not even when we have a long list of related facts similar to those just developed. It may be astonish-

ing, but it is the case that not everyone is equipped with a built-in awareness of the skew transitivity of equivalence relations. I was shocked when I found this out. It seems obvious to me. How does "3+2 = 5 and 2+3 = 5" become "3+2 = 2+3"?

First, it may be important to establish the symmetry of equality. A child who knows that 3+2 = 5 does not necessarily know that 5 = 3+2, and vice versa. If he interprets "3+2 = 5" in terms of some sort of action, as I believe he should do when he first encounters this fact, how should he interpret "5 = 3+2"? (Piaget's findings about a child's notion of conservation of quantities probably have implications here.) One thing he can do is start with 5 objects and partition in a variety of ways the set which has those objects as members. From this activity, he can judge that 5 is 4+1, 5 is 2+3, 5 is 1+4, 5 is 3+2, and even 5 is 5+0 and 5 is 0+5. 1 have not yet encountered a child who does not see that if 2 ducks together with 3 ducks is 5 ducks, he can start with 5 ducks and separate them into 2 ducks and 3 ducks, or 4 ducks and 1 duck, etc. But I have encountered children who do not see that if 3+2 = 5 and 5 = 2+3, then 3+2 = 2+3. Reading the equals sign as "is" should help in this regard.

Figure 1 illustrates my favorite way to show a concrete analogue to the mathematical sentence "2+3 = 3+2." Use a wire coat hanger and some clip clothespins, preferably the plastic kind which come in different colors (use 2 of one color and 3 of another). Rotate 180°.

¿in - h++^ ¿Íh - ii^ Z +• 3 IS 3+2

Figure 1

Another thing which may help is an imaginary machine which can run forward and backward, or a game, or a story, all variations on the same theme. Appropriate actions should accompany the tale.




y~^w Figure 2

For example, we could use a picture such as Figure 2 with the following story. " This is a barn. It has two doors. Here are some goats going into the barn to sleep. 3 goats go in this door and 2 goats go in this door. The same goats are coming out again in the morning. How can they come out? Can each goat come out through the same door he went in? Can each of them come out the other door? How else might they come out? (Keep a record.) Try different totals. Can the animals always come out the same way they went in? Can they always come out just the opposite? If you were watching when the goats went in and again when they came out, how could you tell without looking in the barn whether they were all out? . . . 3+2 (bunch of goats) is 2+3 (bunch of goats), and however many goats there were at first, 'it always works like that/ "

Suppose a child really believes that 2+3 is 3+2, 1 + 2 is 2+1, 4+3 is 3+4, and a few more similar assertions. Does he know that it always works like that? Not necessarily. He can learn these statements as bits of content with no thought of structure automatically involved. If you ask him to check one instance, such as "4+2 = 2+4," and he chooses the requisite number of objects to show how 4 in his left hand and 2 in his right make the same total as 2 in his left hand and 4 in his right, you cannot be sure whether he is not seeing a pattern here or whether he is being polite in responding specifically to your request. Ask him to check some more. If he still does each one separately, give him one where the counting problem

(if you think that is what he is actually solving) is laborious or even impossible with the objects he has at hand: 19+37 i 37+ 19. If he is blocked by this, he does not understand the commutative law for addition; he understands a few of its instances. A first grader who says, "Look. I have some pennies in this hand and some more in this hand. It doesn't matter how many. I'm going to give all of them to you. Do you care which hand I give you first?" has proved to me that he understands the commutativity of addition. If a fifth grader says, "If you fill in the frames in this sentence

'd+a - A + n'

according to the rules for frames, you'll never get one which isn't true," he has almost convinced me that he understands it. If I say to him, "Suppose I choose '1492' for the box and '1776' for the triangle," I hope he'll say, "That's OK. The only way you can mess this up is to break the rules, and that doesn't count."

What good is it for young children to understand the commutativity of addition? For one thing, it reduces by almost half the learning load associated with the addition facts. For another, a child who forgets a fact can sometimes recall the re- verse fact, and if he cannot do that, he can choose the easier way to figure it out (e.g., 9+2 opposed to 2+9). Furthermore, he knows the reason the usual method of checking addition works (for two addends).

The associative principle for addition is far more important to mathematicians than it is to young children. Seldom is the associative principle useful, by itself, in the problems or exercises the children wish to do. What children need is a general rearrangement principle, or "any order" law. Using the same sort of maneu- vers on objects that we used in suggesting commutativity, we can arrive at the associative law. At some point, the coat hanger

November I960 529



and clothespins are useful again. The clothespins can be slid along the wire to show the indicated groupings (see Fig. 3).

rLMJJ 1 lì> Til I I I ||J^ (2 + 3) + 4. 18 2 + C S + 4Ì

Figure 3

Even the simple matter of adding down a column of three figures and checking by adding up involves the associative law once and the commutative law twice, and no young child should be expected to prove in this way that the check is valid. What he needs to understand is that he can add any way he chooses as long as he uses each addend exactly one time, and he can prove that he understands this by resort- ing to physical analogy. He might say, "Look. I have three little bags of marbles. They are a nuisance to carry around this way, so I want to dump them all into one large bag. Any way I dump them in, as long as I don't lose any, I'll have the same number all together."

The commutative law for multiplication can be important to young children for reasons similar to those which make the commutative law for addition important to them.

However you start leading up to multiplication, or however you define or describe it, when you are ready to teach the multiplication facts in earnest, you use an array, a rectangular arrangement of rows and columns in which the number of rows is the first factor and the number of columns is the second factor. Figure 4 is a

• • ♦

• • •

Figure 4

2X3 array. The product of 2X3 tells how many elements there are in the array. Have the children make an array with their counters on top of a piece of paper on

their desks. Direct them to rotate the paper 90°.

Figure 5

Now they see a 3X2 array (Fig. 5) with the same elements and therefore the same product.

Annex some more rows to this array and/or some more columns. Rotate the paper. Each time the youngsters see that the same counters in an mXn array can make annXm array.

At some point, sometimes very soon, a few children will stop bothering to turn the paper. If you ask them why, they'll say something to the effect that they can just look at it and see it either way. Good for them. Other children who may be listening will try it, too, and eventually almost everybody will be willing to assert that 2X3 is 3X2, 5X4 is 4X5, and so on. Again, do not assume that a child who is convinced of 25 instances of the commutativity of multiplication is aware of the principle. Give him an example such as "296X473 = 473X ," or "9X7 = 7X ," depending on his grade level, and see how much work he has to do to find the missing number.

The commutative law for multiplication is the justification for the way multiplication is usually checked. When learning multiplication facts, a child who is aware of the commutative principle has only about half as many facts to learn. In working a problem like finding the cost of 137 air-mail stamps at 8 cents each, the child will feel secure in multiplying the easier way. He knows the product is the same re- gardless of the order of the factors.

With the associative law for multiplication, the young child also needs a general rearrangement, or "any order/' principle. He needs enough practice with this to learn that, since he can use factors in any




order as long as he uses each factor exactly one time, he can often make a computation easier (sometimes easy enough to do mentally) by judicious grouping of factors. For instance, "8 X43 X 125" is readily seen to be "43000" by someone who knows 8X125=1000 and the "any order" principle. Again, both the commutative and associative principles are involved in this rearrangement of factors, but the teacher who requires the young child to give de- tails of these applications of the laws can be sure the child is learning content (largely by rote), not structure. This teacher would be well advised to have the child use his time in learning to multiply mentally by powers and multiples of 10 and by other selected numbers, such as 25 and 125, instead. Fortunately, by the time it is appropriate to develop this "any order" principle for multiplication, the children are sufficiently adept at working with numbers so that physical models are less necessary. The physical model has to be some sort of three-dimensional edifice, for which it is sometimes difficult to find materials and which may be unwieldy to handle and difficult to build.

The distributive law is another matter. Life is much easier for the young child who really grasps the implications of this law. As soon as the child begins to deal with multiplication as such, he should be led to the discovery that he can find the product of 2X7 if he knows the products of 2X3 and 2X4, and how to add (Fig. 6).

2X3+2X4 = 2X (3+4) = 2X7 Figure 6

He should have a great many experiences of this kind. When he finally becomes aware of the distributive principle, it should be associated in his mind with a rectangular array (or region) partitioned appropriately. In too many classes the

only applications of the distributive law the child sees are those in which one of the addends is a multiple of 10. "4X37 = 4X30+4X7" exemplifies one class of important applications of the distributive law, of course, but children should realize that this is by no means the only way the distributive law can be applied. They should realize that "4X37 = 4X25+4 X12," "4X37 = 4X22+4X15," "4X37 = 2X37+2X37," etc., are also applications of the distributive law, and even that "4X37 = 4X40-4X3" probably has something to do with it.

Precise thinking requires us to speak of two distributive principles, either of which is readily derived from the other, but this distinction is seldom made for young children. Perhaps it should be. Division distributes over addition in one direction but not the other, and sixth graders can have some fun finding this out. If they are bright sixth graders, they may also find out why division behaves in this way. Their answer for "why" should be in terms of the way they understand the question, and the ultimate step in their verification should probably be a diagram of a rectangular region partitioned in par- ticular ways.

The distributive law furnishes the ra- tionale for the standard way of multiplying multidigit factors, but it has many other applications. I think a two-credit college course could easily be developed which treats nothing but simple applications of the distributive law. Such a course would also include counter examples to show that not every situation in which both addition and multiplication are used involves the distributive law.

The young child, at some point, needs to understand that "4+3 = 3+4, and it always works that way." Why? Not "because addition is commutative." He needs a reason he can believe - joining lengths, perhaps, or manipulating objects. He needs to understand that "4X3 = 3X4, and it always works that way." Why? Again, a reason he can believe, perhaps

November 1965 531



the general rectangular array viewed from two adjacent corners. He needs an "any order" principle for addition and for multiplication, justifiable on his own terms. He must understand that he will never get a false instance of

D* (A + ô)= (a*A)4-(D«ô).

This should be explained correctly in a way which makes sense to him, perhaps by partitioning rectangular regions.

In the culture of today and tomorrow the people who will be able to function well enough to avoid being public charges, to say the least, are structure-oriented people who are able to see relationships. The mathematics we teach to young children is a wonderful vehicle for devel- oping structure orientation, but we are in danger of accomplishing just the opposite by some of our current practices. We con- tribute nothing to structure by teaching structural elements of number fields as new content. We must keep firmly in mind that the words "old" and "new" have no relevance for children. Some of the new mathematics is new to the teachers, perhaps, and to the principals and to some of the parents, but it is no more new to the children than the old mathematics used to be. It is a very good thing that teachers, administrators, and parents have become excited about the new mathematics, because enthusiasm rubs off on the children. Every experienced teacher can tell by the second day of school what attitude toward arithmetic was held by teachers his pupils had last year. This is no more true today than it was twenty

years ago. Some teachers, principals, and parents have become hysterical about the new math. They don't know enough about it to recognize the real thing when they see it, and being impelled to "do something," they are subjecting the children to ersatz variations which do nothing more than increase the strain on a young child's memory. This never pays off. You know the old saying about a blotter: it gets everything - backward.

Real mathematics makes sense. Real mathematics for the young child comes out of his world, it makes sense to him, he sees its interrelationships, and he can use it in the very place it came from, his world. He does not think you get one answer to a problem if you do it on paper and a different answer if you do it with pennies. He does not think that a problem which comes up in arithmetic class must be done in one certain way to be correct there, and if the same problem comes up in a science class, it must be done in another certain way to be correct there.

The "basic laws" are structural elements of the number systems the young child uses, and they furnish a strong ad- hesive for some of the structure the child should be building for himself if he really understands them on his own operational terms. Rote learning of the applicable vocabulary and a few selected instances contributes nothing to the child's forward progress along the content-to-structure continuum.

"Only those who respect the personality of others can be of real use to them." We enrich ourselves, the children, and the future of our society when we truly respect the personality of young children.

Since each learner is unique and learns in relation to his uniqueness, we will need to change our schools so that they will be human-centered instead of "lessons-centered. - Earl C. Kelly in Educational Leadership.




"basic laws" for young children

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