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

On the computational side, we answer: What problems should not require a calculator -- or even pencil and paper? What problems should an educated person be able to do mentally? 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 should not require a calculator.

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

On the purely educational side, arithmetic is the first science. Arithmetic studies numbers. Therefore we can now take the time to look and see why Nine 12's are equal to Twelve 9's; that is, why the order in which we multiply does not matter. We can look at what it means to multiply a fraction by a whole number, and therefore why we simply multiply the numerator:

On the computational side, 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. Since the gradual replacement of Roman numerals by Arabic numerals in the sixteenth century, that natural faculty has been taken over by written methods: clever techniques that give answers ("write 6, carry 3") but do not require understanding. The very names -- 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,

But the calculator has changed all that. Therefore we can now do more than teach some written method. Those methods will be found here, but my purpose is to rescue arithmetic from much of their crippling effect.

(To find the difference between 75 and 102, what number must we add to 75 to get 102? 75 plus 25 is 100, plus 2 is 102. 25 plus 2 is 27.)

I say in fact that we're supposed to do arithmetic mentally. The foundation for that is knowing the addition and multiplication tables, and how to multiply and divide by powers of 10. We may use a calculator, electronic or written, only when mental calculation is too difficult. But the teaching of arithmetic can now invite 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 of whole numbers -- half of a number, a third, two thirds -- before the lesson on fractions. To understand that 5 people are the fourth part, or one quarter, of 20 people, has absolutely nothing to do with the fraction ¼. Fractions are numbers we need for measuring rather than counting. 2¼ miles. And what is

taught with fractional symbols

But parts are correctly taught verbally, and they are the best preparation for fractions.

For, to understand that the fraction we write as 1 over 4 is one quarter of 1,

where 1 is now some unit of measure, the student must first understand the meaning of one quarter of anything; e. g, one quarter of 20 people. 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 understood mentally. 5 people are what percent of 20 people? Since 5 is one quarter of 20, then 5 is 25% of 20. Why does 25% mean 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" -- as if algebra were the pinnacle of mathematics education. Algebra is a non-verbal, mechanical skill that will be useful only to students found to 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. It is a scientific one, also.

The inspiration for many of these pages, in addition to Euclid's Seventh Book, has been J. E. Rozán, Aritmética y Geometría, México, D. F., Progreso, 1947.