Introduction - A complete course in arithmetic

Have you noticed, too, how people with a talent for calculation are naturally quick at learning almost any other subject; and how training in it makes a slow mind quicker, even if it does no other good.

I have.

Also, it would not be easy to find many branches of study that require more effort from the learner. For all these reasons we cannot do without this form of training.

I agree.

Plato, The Republic, Book VII

OUNTING is a natural faculty of the mind. Yet does anyone doubt that, once out of the classroom, the student will not use a calculator? What can we possibly teach, then, that a student will actually find useful? What deserves to be called educational?

Our response is: What problems should not require a calculator? What problems should an educated person not even have to write down

We are speaking of something as simple as not writing 15 + 6. We are speaking of understanding that Half, or 50%, of 308 is Half of 300 plus Half of 8. This should not be a written problem. And it certainly should not require a calculator.

The calculator has in fact freed arithmetic to resume its true nature, which is the art of counting.

The Internet, moreover, can now bring something of a balance to what has become an all-too-powerful central government, as it were, of mathematics textbooks.

To begin with, most of us have grown up thinking we're supposed to do arithmetic with pencil and paper -- which is itself a calculator. But arithmetic is something we do naturally in our heads. For more than four hundred years, though, that natural faculty has been taken over by written methods: clever techniques that give answers ("write 6, carry 3") but do not stimulate understanding. The very names of the operations -- addition, subtraction, multiplication, division -- have become names of written methods. To "subtract" 75 from 102 has come to mean: Write 75 under 102, draw a line, and

But the calculator has changed all that -- we need to do more than just teach those methods. Written methods will be found here, yet my purpose is to rescue arithmetic from much of their crippling effect.

(To find the difference between 75 and 102, add 25 to 75 to get 100, then add 2. 27.)

Since we are no longer completely dependent then on methods, we can give our attention to understanding -- which can only be expressed verbally. For it is only with our normal, spoken language that we show that we understand anything.

Understanding that in DIVISON, we must find how many times one number is contained in another.

And understanding PARTS. For a unique feature of these pages is the verbal introduction of parts -- half of a number, a third, two thirds -- before the lesson on fractions. To understand that 5 is the fourth part, or one quarter, of 20, has absolutely nothing to do with the fraction ¼. Fractions are numbers we need for measuring rather

1 out of 4" -- is actually parts taught with fractional symbols

But parts are correctly taught verbally, and "1 out of 4" has nothing to do with measuring.

In fact, in order to define the fraction we write as 1 over 4 as one quarter of 1,

the student must first understand the meaning of one quarter of anything. For example, one quarter of 20. Understanding fractions thus depends on first understanding parts. Confusion arises because the English names of the fractions are the same as the names of the parts: half, a third, a quarter, three quarters.

But most important: Many problems traditionally taught with fractions can now be done verbally, which is to say, mentally and with understanding. 6 is what percent of 24? Since 6 is one quarter of 24, then 6 is 25% of 24. Why does 25% means one quarter? Because 25 is one quarter of 100.

These pages, then, present arithmetic as its own science -- not as a stepping-stone to algebra. This is not "pre-algebra." The explanations in these pages will be free of the odor of algebra. For if there is one thing apart from written methods that has caused a child's natural sense of number to atrophy, it is the early introduction of algebra. The vast majority of students will find it of no interest and of no value. Algebra is not educational. It is a non-verbal, mechanical skill that will be useful only to students who have the talent for going on in mathematics, economics or science.

Arithmetic, on the other hand, when properly taught, is the most educational subject. When the algebra teacher gives the rule for dividing negative numbers, or the chemistry teacher asserts that a hydrogen atom has one proton, the student must accept it on authority. ("Yes, teacher. Thank you.") But the arithmetic student can see a fact itself -- One plus One is Two. Whoever understands the meaning of those words can decide directly whether or not that is true. It is not a question of authority or belief. That is an educational experience.