Mathematical activity

Mathematical activityAuthor(s): DAVID M. CLARKSONSource: The Arithmetic Teacher, Vol. 15, No. 6 (OCTOBER 1968), pp. 493-498Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187429 .Accessed: 17/06/2014 11:15

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Mathematical activity DAVID M. CLARKSON

New York State Education Department, New Paltz, New York

Mr. Clarkson is a mathematics consultant for the Board of Cooperative Educational Services of the New York State Education Department. He is chairman of a joint school-college committee on elementary mathematics teaching that is working on a cooperative program of teacher education in elementary mathematics.

IVLathematicians are constantly warning the public that mathematics is not a spec- tator sport - that you must get in there with pencil and paper and fight if you are to profit from its study. But even pencil- and-paper work may be essentially pas- sive. For example, many teachers today make use of the technique of programming instruction, yet little research is available to predict the effect on children of an ex- tensive amount of this kind of teaching. There is something narrow, perhaps even crippling, about a long string of questions like "A triangle has sides."

While it may be true that some routine skills may be learned in this manner, it is unlikely that any mathematician would regard these chains of leading questions as good examples of mathematical activity. What a mathematician may have in mind is some kind of structural response to a situation - perhaps a reaction to an observed pattern, or to something odd that breaks up a pattern. The mathematical activity here is that of forming a hunch, test- ing it, and trying to confirm it by some kind of mathematical proof.

Until quite recently, mathematical proof was not introduced before the tenth-grade geometry course. However, there are certain simple ways of proving conjectures which are accessible to even young children in elementary classes. After all, a proof is whatever convinces you that a conjecture is true, so a mathematical proof

is whatever convinces mathematicians that something is true in mathematics. Take, for example, the following "proof" that the sum of any two odd numbers is an even number (Fig. 1). Now, if you are an alge-

~*~1| 1 . 1

Figure 1

bra teacher, and if you want to insist that your students use expressions like 2n + 1 to indicate odd numbers, you may not want to regard the illustration above as a "proof." It is certainly not a typical alge- braic proof, but at the level of a third- or fourth-grade child, I think it is a very good mathematical proof.

Of course, a good deal depends on how a child understands his own "proof." In the case above, it would not be a very good proof if a child thought, "Seven and three make ten. Seven and three are odd numbers and ten is even, so whenever you add two odd numbers you get an even one." However, he might be thinking, "Any odd number has that extra dot, so when you put two odd numbers together, no matter which ones, the two extra dots fit together and you get an even number as the sum." If the latter is the case, then I would say you have a pretty good proof. It is not always easy to tell whether a child has a good proof or not from just looking at his

October 1968 493


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paper. A teacher may have to question the chUd, sometimes at some length, in order to find out just what the "proof is.

Not all conjectures likely to be made by active, exploring, elementary school students will be as susceptible to "proof" as the conjecture that two odds make an even sum. Should we deny children the oppor- tunities to make guesses based on their observations just because they are not ready to formulate proofs? Of course not! Un- proven conjectures abound in mathematics as they do in all fields. But if we are to throw children into mathematical situations and encourage them to mathematize, we should also be willing to attempt proof where suitable, and to discuss the failure of proof where appropriate.

One pitfall of attempting mathematical proof with elementary-age children is their, and our, tendency to "jump to collisions" when observing regularities. On the one hand, we should encourage children to ob- serve patterns, but, on the other, we should caution them against "proofs" based only on observations. If you think that seven- teen examples of a certain pattern are more convincing than three examples, well, you are right, psychologically; but you are wrong mathematically. While it takes only one counter example to kill a conjecture, no finite number of examples "prove" a conjecture if that conjecture refers to in- finitely many cases. A classic example of this mathematical fact of life is the pattern generated by counting the number of regions formed in the interior of a circle

Figure 2

when the maximum number of line seg- ments are drawn between points on the circle's circumference. When two points are connected, the interior of the circle is divided into two regions. With three points

there will be four regions, as shown in Fig- ure 2. The four-point situation produces eight regions. Form a conjecture about this pattern and test it with five points on a circle. Does the experiment confirm your conjecture? Do you now feel that you have something close to a "proof?

If you are convinced that you have found a general pattern for this situation, try counting the number of regions when six points are connected. The result should con- vince you that you can never really trust a pattern. It is extremely important that we present mathematical situations to children in such a way that they feel free to look for patterns and form conjectures; but our atti- tude, short of proof, should always be tenta- tively approving. Such expressions as "That looks good," "This seems to be the case," "Probably that pattern will continue," should be accompanied by more probing questions like "Do you think it would always hold?" "What about really big numbers?" or "Can you find a counterexample?" It is truly surprising that one of the famous conjectures of Fermat, dating from about 1640, went unproved and more or less un- contested until 1732, when Euler found a counterexample in the fifth example of the conjecture. For almost a hundred years no one had tested Fermas conjecture for N = 5! (For those who like to check things for themselves, Fermat guessed that all numbers of the form 22* + 1 were prime numbers. Euler showed that when n = 5, F5 = 282 + 1, which can be factored as 641 X 6,700,417.)

Conjectures and conjecturing should be encouraged in a free classroom and school atmosphere, but teachers should always be careful to distinguish between conjecture and proof. A good way to motivate an elementary school child to appreciate the distinction is to expose him to patterns that don't work out and to train him to look for counterexamples. Suppose, for example, you ask a child to find two odd numbers whose sum is odd. In this case every example he finds will be a counterexample, and he may be led by his search to form, first a conjecture, and then a

494 The Arithmetic Teacher



Tangram Puzzi Work

(Sixth graders, Highland, N.Y.)

proof, that the sum of two odd numbers is always an even number. The credulity with which children accept patterns should always be tempered with a healthy skep- ticism about the extent of the pattern.

Is it reasonable to expect that all, or even most, of the children in a class will form the same conjectures from a mathematically rich situation? Certainly not, nor is it likely that more than a few will be ready to discuss the "proof of a conjecture as it is discovered by one of their classmates. And why should we require that children discover in unison anyway? Much is to be gained from the laboratory- classroom environment, where children choose their daily tasks from a wide vari- ety of carefully prepared situations, and from some that are not carefully prepared at all!

What is this classroom environment called the mathematics laboratory? There are many kinds of math labs, but one of the most common is simply the provision, by the classroom teacher, of materials and some time when children may choose to

work on a mathematical problem that in- terests them, either alone or with a partner or small group of children sharing a similar interest. The distinguishing features of such lab periods are the independence of t children from large group- or teacher-directed lessons, and the possibilities for individ- ualistic and active solutions to a wide vari- ety of problems. Needless to say, independence and active participation produce an extremely high degree of motivation in the children. Anyone who has observed laboratory sessions can attest to this fact!

Many teachers who have never con- ducted lab classes may find it difficult to get started. This is natural, and it is probably a good idea to begin with a small group during that part of the school day when a teacher may be easygoing with her class. It is also probably a good idea to do a moderate amount of preplanning for materials acquisition. A lab period can be very trying if you run out of squared paper, tape, felt-tipped pens, or string before you even start! The best source of ideas for getting started that I know of is the Nuf-

October 1968 495



field booklet, / Do and I Understand [3],* but the list of sources is growing longer every day as worldwide experience with laboratory methods expands and deepens.

Motivation, individualized instruction, activity: these are all good-sounding words to an elementary teacher's ear. But what about the mathematical content? There still seems to be some lack of agreement among specialists about this question. Some experts speak out strongly for a heavy science orientation. They argue that mathematics is the language and servant of science, and that very few elementary

How Wide 1$ the Gymnasium?


school children will go as far in mathematics as they will in the uses of mathematics. The math lab should be a science lab, they say. Measure things, perform simple sci- entific experiments with mathematical over- tones; don't be afraid to try out experiments that don't come out "evenly" or "nicely"! Emphasize empirical graphing, sampling methods, structural effects of rigid geometrical figures.

* Numerals in brackets indicate references at the end of the article.

Contrary to the natural scientists, some "pure" mathematicians advise us to emphasize mathematical structures, often through games. These purists assert that young children, indeed people of any age, are fascinated by puzzles and patterns and that the math lab is one of the best places for this activity because of its informality. Geometers tell us to study geometric forms and relationships in the laboratory classroom. And so we receive advice on content from all quarters!

I think that all these advisers are right; we should do all of these things in our math labs. No facet of elementary mathematics instruction should be exempt from laboratory treatment, nor should we avoid whole- class discussions of these topics and treat- ments as a follow-up to the individual explorations. Indeed, the picture that may be emerging is one of a kind of classroom organization that will permit an easy transi- tion from class discussions to laboratory work and back again. The answer to the question "How can I get started?" is: Start when you feel like it, and build up as much laboratory activity as you want and the children can take.

There are some special advantages to the lab methods worth mentioning. First of all, Piaget has made us aware of the developmental needs of children. We are now much more aware of the importance, for example, of the ideas of conservation in the scheme of young thoughts about quantities. While this subject is still quite controversial, wide agreement can probably be obtained for the thesis that children should have a very active experience with, say, measurement concepts before formal instruction begins. This means that, for example, a good deal of play and work with water and bottles and containers of various sizes and shapes should precede formal discussion of the idea that two pints make a quart of liquid measure, etc.

Piaget emphasizes further, although his critics seem often to ignore this, that children develop, in their understanding of quantitative studies, very individually. It




Counting a Quartful of Lima Beans


may be the case that the typical five- or six-year-old schoolchild in Geneva, Switz- erland, does not yet appreciate the conservation of number, but it is by no means a Piagetian assertion that "if a child is less than seven years old he will not know that seven beans is seven beans no matter how they are spread out on the table." Piaget never said, and never implied, anything like that! And if development is highly individual, then this is one more reason why the laboratory situation, which provides an easy opportunity for children to choose tasks appropriate to their stage of development, is a good one.

Furthermore, the laboratory setup enables the teacher to engage in that one-to- one correspondence with her students so necessary to, say, a discussion of a "proof." Even more important, it enables children to communicate easily and naturally with each other. Children are really great at explaining even highly complex sets of rules to each other. Have you ever over- heard an office secretary explain the situation to a new girl? Well, children are equally expert in conveying information

to their peers in equally elegant language! There is nothing pedantic about it.

To sum up: Establishing a math lab in your classroom is neither expensive nor risky. You can do it in your own sweet time and with whatever budget is at your disposal. There are many materials already available to help you select content, and more are on the way. As you come to appreciate the benefits of laboratory methods, you may wish to expand the time de- voted to this type of instruction. One thing can be guaranteed: your children will be highly motivated in their mathematical studies.

Annotated Bibliography 1. Association of Teachers of Mathematics.

Notes on Mathematics in Primary Schools. New York: Cambridge University Press, 1967. Paper.

This book, written by elementary school teachers, contains an extraordinary wealth of ideas for use in a laboratory situation.

2. Davis, Robert B. Discovery in Mathematics. Reading, Mass.: Addison- Wesley, 1964.

. Explorations in Mathematics. Read- ing, Mass.: Addison-Wesley, 1967.

These Student Texts, together with the Teachers' Guides that accompany them, are a distillation of the "Madison Project" supplementary materials. In particular, the introductions to the Guides express a philosophical point of view that would be extremely valuable to any teacher con- sidering instituting laboratory procedures.

3. Nuffield Foundation. / Do and I Under- stand; Pictorial Representation; Beginnings ], Mathematics Begins 1; Shape and Size 2; Computation and Structure 2. New York: John Wiley & Sons, 1967 and 1968.

These pamphlets are the first important publications of the Nuffield Mathematics Teaching Project, which bases its work on laboratory methods.

4. Polya, George. Mathematical Discovery. 2 vols. New York: John Wiley & Sons, 1962.

This classic, for teachers with a reason- ably strong background in high school mathematics, conveys the flavor of true mathematical discovery, conjecture, and "proof."

5. Sawyer, Warwick W. Vision in Elementary Mathematics. Baltimore: Penguin Books, 1964. Paper.

This book provides a philosophical basis for a concrete approach to elementary mathematics instruction.

October 1968 497



6. (ed.). Maihex. Toronto: Encyclopae- dia Britannica of Canada.

Monthly bulletins, published in six levels, containing practical suggestions for using laboratory techniques in elementary class- rooms.

Films

1. Mathematics and the Village. 16mm, color. Greenford, Middlesex, England. Rank Film Library, Aintree Road, Perivale

Three short films illustrating the use which may be made of a typical school environment to bring to children an aware- ness of the mathematical nature of their surroundings.

2. Maths Alive. 30 min., 16mm, color. Na- tional Council for Audio- Visual Aids in Education, 33 Queen Anne Street, London, W. 1.

Illustrates laboratory techniques as de- veloped in England.

Letters to the editor

Dear Editor:

The article "Humpty-Dumpty's Lesson in Arithmetic" by Bryce E. Adkins in the February 1968 issue of The Arithmetic Teacher was most enjoyable.

Alice's Adventures in Wonderland by Lewis Carroll (New York: Random House, 1946) was the source for one of my eighth-grade arithmetic lessons. I used certain portions to show the mathematical significance in them. The Humpty- Dumpty selection was applied in the same manner as Mr. Adkins used it - to illustrate "the use of different terms to name the same ideas."

Another passage was read to demonstrate noncommutativity with words:

"Do you mean that you think you can find out the answer to it?" said the March Hare.

"Exactly so," said Alice. "Then you should say what you mean," the

March Hare went on. "I do," Alice hastily replied; "at least - at

least I mean what I say - thas the same thing, you know."

"Not the same thing a bit!" said the Hatter. "Why, you might just as well say that 1 see what I eat' is the same thing as *I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!"

"You might just as well say," added the Dor- mouse, which seemed to be talking in its sleep, "that 1 breathe when I sleep' is the same thing as *I sleep when I breathe'!"

A third selection was introduced merely to have fun with mathematics:

"I couldn't afford to learn it," said the Mock Turtle with a sigh. "I only took the regular course."

"What was that?" inquired Alice. "Reeling and Writhing, of course, to begin

with," the Mock Turtle replied; "and then the

different branches of Arithmetic - Ambition, Dis- traction, Uglification, and Derision."

"I never heard of 'Uglification'," Alice ven- tured to say. "What is it?"

The Gryphon lifted up both its paws in sur- prise. "Never heard of uglifying!" it exclaimed. "You know what to beautify is, I suppose?"

"Yes," said Alice doubtfully; "it means - to - make - anything prettier."

"Well, then," the Gryphon went on, "if you don't know what to uglify is, you are a simple- ton."

The students seemed to enjoy this change of pace. In addition to utilizing mathematics concepts, good literature was exhibited in the classroom. - Carolyn R. Wood, Knoxville Junior High School, Pittsburgh, Pennsylvania.

Dear Editor:

The articles "Physical Representation for Signed-Number Operations" by Ashlock and West and "Multiplication of Integers" by Loye Y. Hollis in the November 1967 issue of The Arithmetic Teacher suggested several procedures for rationalizing the operations with integers. However, an analogy ceases to be useful when it becomes more difficult to understand than the principle it illustrates. I doubt very much that a student of twelve or thirteen can easily visualize a "film running backwards of an automobile in reverse." If he can, it is doubt- ful that an intelligent youngster will see this as irrefutable evidence establishing the law that the product of two negative integers is a positive integer.

Since mathematics is at least partially a study of patterns and structures, a more positive teaching approach to the discovery of the multiplication law for integers is given below. 1. Complete the upper right-hand portion of the

[Continued on p. 506]




Article Contentsp. 493p. 494p. 495p. 496p. 497p. 498

Issue Table of ContentsThe Arithmetic Teacher, Vol. 15, No. 6 (OCTOBER 1968), pp. 490-584Front MatterEditorial commentAs we read [pp. 490-491]

Mathematical activity [pp. 493-498]Letters to the editor [pp. 498, 506, 528, 544, 550, 563]Learning laboratories in elementary schools in Winnetka [pp. 501-503]Making and using graphs in the kindergarten mathematics program [pp. 504-506]An annotated bibliography of suggested manipulative devices [pp. 509-512, 514-524]A physical model for teaching multiplication of integers [pp. 525-528]Focus on researchResearch on mathematics education, Grades K-8, for 1967 [pp. 531-538, 540-544]

Forum on teacher preparationA mathematics laboratory for prospective elementary school teachers [pp. 547-550]

In the classroom"Stock-market" unit [pp. 552-556]Discovering centigrade and Fahrenheit relationships [pp. 556-559]Semipermanent chalk: a valuable aid in the mathematics classroom [pp. 559-560]Patterns of intersection [pp. 560-562]Writing equations for "story problems" [pp. 562-563]

ReviewsBooks and materialsReview: untitled [pp. 564-565]Review: untitled [pp. 565-567]Review: untitled [pp. 567-568]

Books received [pp. 568-570]

President's report: The state of the Council, 1967/68 [pp. 571-575]Minutes of the annual business meeting [pp. 575-576]Officers, directors, committees, projects, and representatives (1968/69) [pp. 578-582]Your professional dates [pp. 583-584]

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Mathematical activity