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.

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 intelligence. 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:

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

Written methods will be found here, yet my purpose is to rescue arithmetic from much of their crippling effect.

For, now that we are freed from rule-learning, we can give our attention to understanding -- which can only be expressed verbally. For it is only with our normal, spoken language 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 of 20, has absolutely nothing to do with the fraction ¼. Fractions are numbers that we need for measuring rather than

of 3" -- is actually parts taught with fractional symbols! But parts are correctly taught verbally, and "2 out of 3" has nothing to do with measuring.

2 is two thirds of 3 -- and the fraction itself is defined as being two thirds of 1. Thus understanding fractions depends on first understanding parts. Confusion arises because the fractions, in English, have the same names as the parts: half, a third, a fourth, three fourths. Those names in turn become the names of what are called ratios. And with an understanding of ratio, many problems that are now taught with fractions, especially percent problems, can be done verbally and mentally. For percents are ratios. The lessons on ratio and proportion therefore are the most significant pages.

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 smell 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. Algebra appeals neither to common sense nor to intuition, and the vast majority of students will find it of no value. As for stimulating students to go into math and science, arithmetic and plane geometry are the first sciences. They look at facts themselves. That is more than can be said of modern physics; which has degenerated into algebra.

When properly taught, arithmetic is the most educational subject. When the algebra teacher gives the rule for multiplying fractions, 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 belief. That is an educational experience.