Introduction to arithmetic - A complete course

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

Our answer is to emphasize problems that should not require a calculator -- or even pencil and paper. What problems should an educated person be able to do mentally? For arithmetic is, and always has been, a spoken skill, based on knowing the addition and multiplication tables. To add something as simple as 6 + 4 + 3, you have to say, "6 plus 4 is 10, plus 3 is 13."

50%, or Half, of 308? "Half of 300 is 150. Half of 8 is 4. Therefore Half of 308 is 154."

That 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 however have grown up thinking we're supposed to do arithmetic with pencil and paper -- which is itself a calculator

The natural faculty for counting has been undermined by those 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 the 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 those routines. They 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 by speaking -- mentally or aloud. It is that which distinghishes arithmetic from algebra, which is a written skill. We may use a calculator, electronic or written, when mental calculation is too difficult. But the teaching of arithmetic can now invite number-sense, which can only be expressed in words. It is only with our spoken language that we show that we understand anything.

Understanding that MULTIPLICATION by a whole number is repeated addition -- even multiplication of a fraction.

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 aliquot parts: half of something, a third, a fourth, a fifth -- before they become names of fractions. With the exception of the word half, the ordinal numbers express division into equal parts. Those words belong to language itself; mathematics does not own them. In the sentences "5 is half of 10," "A quarter of the class got B," the words half and quarter are not the fractions ½ or ¼. The fraction ¼ is a number we

need for measuring, and is one quarter or the fourth part of 1, on the number line. Obviously then the student much first understand the meaning of "one quarter." Confusion arises because in English (not in every language), the proper fractions have the same names as the parts. In fact, what is usually taught as fractions -- "¼ means 1 out of 4" -- is actually parts taught with fractional symbols.

But parts are more effectively taught verbally; they are the best preparation for fractions and percent. In fact, many problems traditionally taught with fractions can now be understood directly. 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." Algebra, which will be useful only to certain students, is the written manipulation of symbols. But arithmetic transcends its symbols, for the words have meaning and therefore truth. ("5 is one more than 4, but two less than 7.") Properly taught, arithmetic 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. But the arithmetic student can see a fact itself -- 5 is the third part of 15. Whoever understands the meaning of those words can decide directly whether or not it is true. It is not a question of authority or belief. That is an educational experience. It is a scientific one, also.

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