Edward Stoddard - Speed Mathematics Simplified

BIBLIOGRAPHY

AMUSEMENTS IN MATHEMATICS

H. E. Dudeney; Dover Publications, N.Y.

ARITHMETICAL EXCURSIONS

Henry Bowers & Joan E. Bowers; Dover Publications, N.Y.

HIGH-SPEED MATH

Lester Meyers; D. Van Nostrand Company, Inc., N.Y.

HOW TO CALCULATE QUICKLY

Henry Sticker; Dover Publications, N.Y.

THE JAPANESE ABACUS: ITS USE AND THEORY

Takashi Kojima; Charles E. Tuttle Company, Rutland, Vt., and Tokyo

MAGIC WITH FIGURES

A. Frederick Collins; Surrey House, N.Y.

MAGIC HOUSE OF NUMBERS

Irving Adler; The John Day Company, Inc., N.Y.

MATHEMATICS MADE SIMPLE

Abraham Sperling and Monroe Stuart; Doubleday & Co., Inc., Garden City

MATHEMATICIAN'S DELIGHT

W. W. Sawyer; Penguin Books, Ltd., Harmondsworth, England

101 PUZZLES IN THOUGHT AND LOGIC

C. R. Wylie Jr.; Dover Publications, N.Y.

RAPID CALCULATIONS

A. H. Russell; Emerson Books, Inc., N.Y.

SHORT-CUT MATHEMATICS

B. A. Slade, Editor; Nelson-Hall Company, Chicago

THE TRACHTENBERG SPEED SYSTEM OF BASIC MATHEMATICS

Ann Cutler and Rudolph McShane; Doubleday & Co., Inc., Garden City

NUMBER SENSE

N UMBER sense is our name for a feel for figuresan ability to sense relationships and to visualize completely and clearly that numbers only symbolize real situations. They have no life of their own, except as a game.

Almost all of us disliked arithmetic in school. Most of us still find it a chore.

There are two main reasons for this. One is that we were usually taught the hardest, slowest way to do problems because it was the easiest way to teach. The other is that numbers often seem utterly cold, impersonal, and foreign.

W. W. Sawyer expresses it this way in his book Mathematician's Delight: The fear of mathematics is a tradition handed down from days when the majority of teachers knew little about human nature, and nothing at all about the nature of mathematics itself. What they did teach was an imitation.

By imitation, Mr. Sawyer means the parrot repetition of rules, the memorizing of addition tables or multiplication tables without understanding of the simple truths behind them.

Actually, of course, in real life we are never faced with an abstract number four. We always deal with four tomatoes, or four cats, or four dollars. It is only in order to learn how to deal conveniently with the tomatoes or the cats or the dollars that we practice with an abstract four.

In recent years, teachers of mathematics have begun to express concern about popular understanding of numbers. Some advances have been made, especially in the teaching of fractions by diagrams and by colored bars of different lengths to help students visualize the relationships.

About the problem-solving methods, however, very little has been done. Most teaching is of methods directly contrary to speed and ease with numbers.

When I coached my son in his multiplication tables a year ago, for instance, I was horrified at the way he had been instructed to recite them. I had made up some flash cards and was trying to train him to see only the answera basic technique in speed mathematics explained in the next few pages. He hesitated, obviously ill at ease. Finally he blurted out the trouble:

They don't let me do it that way in school, Daddy, he said. I'm not allowed to look at 6 x 7 and just say 42. I have to say six times seven is forty-two.

It is to be hoped that this will change soonno fewer than three separate professional groups of mathematics teachers are re-examining current teaching methodsbut meanwhile, we who went through this method of learning have to start from where we are.

Relationships

Even though arithmetic is basically useful only to serve us in dealing with solid objects, be they stocks, cows, column inches, or kilowatts, the fact that the same basic number system applies to all these things makes it possible to isolate number from thing.

This is both the beauty andto schoolboys, at leastthe terror of arithmetic. In order fully to grasp its entire application, we study it as a thing apart.

For practice purposes, at least, we forget about the tomatoes and think of the abstract concept 4 as if it had a real existence of its own. It exists at all, of course, only in the method of thinking about the tools we call numbers that we have slowly and painstakingly built up through thousands of years.

There is space here only to touch briefly on the intriguing results of the fact that we were born with ten fingers, and therefore use ten as a base number for our entire counting system. Other systems have been and still are used, from the binary system based on two required by digital electronic computers to the duo-decimal (dozens) base still in use in buying eggs, products by the gross, English money, inches to the foot, and hours to the day.

Our counting system is based on 10, because we have 10 fingers. As refined and perfected over the centuries, it is a wonderful system.

Everything you ever need to do in arithmetic, whether it happens to be calculating the concrete to go into a dam or making sure you aren't overcharged on a three-and-a-half pound chicken at 49 a pound, can and will be done within the framework of ten.

A surprisingly helpful exercise in feeling relationships of the numbers that go into ten is to spend a few moments with the following little example.

First, look at these three dots:

Nothing very exciting yet. But now we add three more dots, right below them:

How many dots are there? Six, of course. But how did it come about that there are now six? We added three dots to the first three. Then what is three plus three?

Of course you know the answer, and of course this seems pedestrian. But there is a moral.

Did we also double the first number of dots? There were three, and we added the same number. Now there are six. So what is three plus three, again? And what is two times three?

You know the answer, but sit back for a moment and try to visualize the six dots. They are both three plus three, and two times three. The better emotional grasp of this you can get now, the more firmly you can feel as well as understand this relationship, the faster and easier the rest of the book will go.

Now we add three more dots:

How many dots?

What is three times three? Can you feel it? What is six plus three? Pause as you answer to let it sink in.

What is one-third of nine?

Play with these dots a bit. Try to see as many relationships as you can. Notice that three-ninths is equal to one-third. Why? What is six-ninths in simpler numbers?

Oddly enough, all of our arithmeticeven into the millionsis based on the number of dots you now have in front of you. Add one to nine and you have tenwhich is the base of our counting system. We express it with a new one moved over to mean one ten and a zero to mean nothingnothing more than ten.

If we really have a feel for all the relationships within the number nine, we are a long way toward feeling at home with numbers.

Stop for a bit here and, on your pad, set up ten dots. Amuse yourself by setting them up in two rows of five each. See what happens if you try to make any other number of rows with the same number of dots in each row come out to ten. Look back at the two rows of five each and see if you can feel the reason why we can express one-fifth and one-half of ten (or one) with a single-digit decimal, but not one-third or one-fourth.