Chapter 1 - Number Sense

Number sense is our name for a "feel" for figures—an 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 answer"—a 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 soon—no fewer than three separate professional groups of mathematics teachers are re-examining current teaching methods—but 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 and—to schoolboys, at least—the 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 ringers, 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 ringers. 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 49½¢ 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 arithmetic—even into the millions—is based on the number of dots you now have in front of you. Add one to nine and you have ten—which 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 nothing—nothing 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.

Seeing Only the Answer

Beyond working at a "feel" for number relationships there are certain specific rules of procedure that will speed up your handling of numbers.

The first of these is simply a matter of training. Quite new training for many of us, and one directly contrary to the way arithmetic is often taught, but one that offers an amazing improvement all by itself.

The technique is to see only the answer.

When adding, we learn to "see" the two digits 4 and 3 as 7—not as 4 and 3.

Then, multiplying, we learn to "see" the digits 4 and 3 as 12—not as 4 and 3.

This may seem elementary. You may already be doing something very much like it in your own number handling. Yet some conscious work in this direction will pay handsome dividends.

Try to remember, if you can, how it was when you first learned to read. You spelled out each word letter by letter. It was slow and painful and not really very enjoyable. But now you grasp whole words and phrases at a glance. It's not only faster, it is easier.

This is unfortunately just the opposite to the way most arithmetic is taught, so most of us have to unlearn what was drilled into us in school. But it is well worth the effort, and it is essential to many of the streamlined methods and short cuts later in the book.

Arithmetic has been called the language of business. In many most important senses it really is, and in order to understand income-expense and financial statements you need a good grasp of it. Our insistence on the importance of seeing only the answer—of seeing 6 x 7 as 42—is basic to a vocabulary of the language. The methods and short cuts to come later might be called the grammar, but grammar is useless without vocabulary.

From time to time in this book I will slip in a little casual practice at seeing only the answer. Please do not skip these examples. They are important. They directly affect every other element in the book.

Add these numbers: 8 7 6

Did you see the digits 8, 7, and 6? You were probably taught to add "8 plus 7 is 15; 15 plus 6 is 21."

This is too slow.

Instead, practice looking at the 8 and the 7 and thinking, automatically, "15." Try to do this without saying or thinking either the 8 or the 7. Then, thinking only "15," glance at the 6 and see "21." You don't say or even think "6" at all.

If you have never tried this, the idea may be not only new but rather shocking. You can get used to it very quickly if you try, and it will speed up your number work substantially even without the other techniques. It isn't hard. It takes a bit of practice, and knowing your addition tables so you don't have to cudgel your brains to remember what 8 and 7 add up to. It's just what you do when you look at m and e and think "me" without consciously putting the two letters together.

Try it again: 8 7 6

Now practice reading the following additions by seeing only the answer. Don't say to yourself, and try to avoid even thinking to yourself, the digits you are adding. Do your best to "see" 4 plus 5 as 9—not as 4 plus 5. Read the answers to these additions just as you would read i and t as it, not i and t:

If you found yourself beginning to see only the answers, very good. If not, you might find it helpful to try again.

Work With Numbers, Not Digits

The second step to developing number sense goes even further in aiding a natural and sure speed with figures. This step is far more drastic than seeing only the answer. It violates almost everything we are usually taught about numbers, yet you will quickly see how much sense it makes and how important it can be.

This rule, agreed on by almost every teacher of short-cut mathematics, is to work from left to right—not right to left.
 
This is just opposite to what is taught in school. We are taught to add, subtract, and multiply from right to left. It is easier to teach to children and easier to learn from the "imitation" standpoint of learning by rote, but it is directly contrary to the way we read and think about numbers.

There are at least three important advantages to working from left to right.

First, it is the way we look at everything else on a page. We read from left to right.

Second, it is the way we look at a number right up to the moment we begin doing something to it. For instance, look at the number 164,928. You read it one-hundred sixty-four thousand, nine-hundred and twenty-eight. But when you begin to add or subtract or multiply it, you are taught to tackle it as 8, 2, 9, 4, 6, 1.

It isn't the same number at all. At the very outset we are taught to combine this number with another in a totally foreign, unrecognizable form.

There is still a third reason why it is faster and better to work from left to right. You develop your most important numbers first and work toward the "details."

Suppose you are a salesman who has just sold a $423 order for which you will get a 6% commission. If you work from left to right (you will learn how later), you know by the time you get just one digit that your commission will be twenty-something dollars. You know when you have finished two digits that your commission is $25 and change.

But if you work in the schoolroom, right-to-left way, the first two digits you develop tell you only the change. You know only that you will get something dollars and 38¢. Not until you finish working out the whole commission do you know that your commission will be $25.38.

That 38¢ may be important to a bookkeeper, but its importance in the number itself is relatively a detail. You care a lot more about the $25 than you do about the 38¢.

This is true of every number and every application, whether or not a decimal point happens to break it into dollars and cents. The first digit in a number is ten times as important as the second, a hundred times as important as the third, and so on down the line. If the order we just discussed were a hundred times as large, you would still care a great deal more about the $2,500 part of the commission than you would about the $38 part.

Working from left to right reveals to you, step by step, the most important numbers first. For this reason alone, the new methods for doing this are one of the most valuable quick estimating tools you can have.

The fact that each digit in a number decreases in importance by a factor of ten as it moves one place to the right is the reason why many companies today report their operations and financial position in "round" numbers: rounding off the pennies or, in very large companies, tens, hundreds, and even thousands of dollars. It is the number to the left that is most important. Even the U. S. government now permits each of us to figure our income tax in round numbers, to the nearest dollar for each deduction and part of the calculation. If your income-tax report is at all complicated and you do it yourself and have not tried rounding it off, you will be astonished next time you do it. It saves close to half the time of doing the report.

If any one technique in this entire book is worth more than the price of admission, I would be tempted to put the left-to-right methods of working first on the list. There are other valuable techniques, but the left-to-right methods are utterly unique.

The value of this approach to your number sense can only develop as you learn the methods that make it possible. The point to be made here is simply this: work at it. It is, as you learn to use it, as black-and-white a difference as thinking of the number 462 or approaching it as 2, 6, 4.

Convert to Simpler Forms

Most of us convert some of our figuring problems to simpler forms, when we can and when we notice that we can, without thinking very much about it.
You wouldn't give a second thought to wondering how much you had in terms of dollars if you found three 25¢ pieces in your hand. We call 250 a quarter because that is just what it is—a quarter of a dollar. In fact, if you take one out of your pocket right now you will find that it doesn't even say anything about cents. The official designation is "quarter dollar."

Whether anybody has ever called your attention to it or not, you are thinking now in terms of aliquots. An important chapter comes later on the short cuts that aliquots make possible. The whole concept, once you get used to it, is merely an extension and refinement of your instinctive understanding that 750 is the same as ¾ of a dollar.

This is conversion to a simpler form.

Perhaps, too, you have noticed that you can more easily multiply 692 by 99, by subtracting one 692 from a hundred 692's (69,200 - 692) than by setting up the whole problem with a pencil and paper and going through the classical form, which would look like this:

                                                6 9 2
                                                   9 9
                                            
                                             6 2 2 8
                                          6 2 2 8

                                          6 8 5 0 8

Which is quicker and easier? Yet in doing the first you were merely using a basic and helpful form of the technique we call "round off and adjust." It can apply to many more numbers than 99.

This, too, is conversion to a simpler form.

Or perhaps, in quickly trying to come up with an appropriate tip for a meal check where 15% is standard, you noted that you could mentally take one-tenth of the check and then add one-half of that number to the one-tenth. A five-dollar check, for instance, would call for a 75¢ tip. One tenth of five dollars (50¢) plus one half of 50¢ (25¢), gives 75¢ quickly and easily.

It is obviously more convenient to arrive at 75¢ this way than to try (mentally or on the edge of the check) to multiply in the classic manner:

                                                $5.0 0
                                                    .1 5

                                                2 5 0 0
                                                5 0 0

                                                .7500

Yet in doing this little trick, you are merely practicing a fairly simple form of the short-cut method called "breakdown."

There are other useful forms of conversion, such as factoring and proportionate change. The application of these methods to number sense will become plain as you learn and begin to apply them.

The Four Steps to Number Sense

Here, for quick review, are the four steppingstones to number sense:

Practice seeing relationships

How does 5 relate to 10? 3 to 9?

See only the answer

Read 4 + 3 as 7—not as four plus three.

Work from left to right

27 is 27—not 7, 2.

Convert to simpler forms

25 ¢ is both 25 ¢ and a quarter of a dollar.

99 is 100 minus 1.

15 is 10 plus ½ of 10 (And more conversions to come.)

Before going on to the first real "working" chapter of this book, get in practice for using it as well as reading it by trying to see only the answers to the following multiplications. Remember, 6 x 7 is 42—not six times seven:

                        2X9                 7X3                 8x4
                        5x5                  4X6                 3X2
                        4x7                  5X8                 3X9

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