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 counting. "One, two, three, four," and so on.  An educated person should be able to add 45 + 6 simply by saying, "45 plus 5 is 50, plus 1 is 51."

That should not be a written problem ("Bring down the 1, carry the 1"). 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,

and do the method.

But the calculator has changed all that.  Therefore we can now do more than teach those written 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?  Say, "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, whether mentally or aloud.  The foundation for that is knowing the addition and multiplication tables, and taking advantage of positional numeration to easily 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 number-sense, which can only be expressed verbally.  For it is only with our normal, spoken language that we show we understand anything.

Understanding that in SUBTRACTION, we must find what number to add.

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.

Understanding that PERCENT -- per centum -- means how many
for each 100.

As for DECIMALS, an innovation of these pages is to introduce them immediately after whole numbers, not after fractions.  Decimals continue the positional numeration of whole numbers, and there is no logical justification for teaching fractions first. We do most practical calculations with decimals -- not with fractions.  Operations with decimals can then be taught in the same lessons as whole numbers. For we can do arithmetic only with whole numbers, and then correctly place the decimal point.

The only topic logically prior to both decimals and fractions is the meaning of the aliquot PARTS:  half of a number, a third, a fourth, a fifth -- before they are assigned as names of fractions.  For to understand that the

number we write as 1 over 4 is one quarter of 1, and correctly place it on the number line, it is necessary to first understand the meaning of one quarter of anything; e. g., 5 people are one quarter of 20 people.  That statement has absolutely nothing to do with the fraction Œ, which is a number we need for measuring.  Confusion arises because the English names of the fractions are the same as the names of the parts. In fact, what is usually taught as fractions -- "Œ means 1 out of 4" -- is actually parts  taught with fractional symbols.

But parts are correctly taught verbally, and they are the best preparation for fractions, and percent.  In fact, many problems traditionally taught with fractions can now be understood directly, which is to say, by speaking. 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.

No fractions

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 crown of mathematics education.  Algebra, which will be useful only to certain students, is the written manipulation of symbols.  But arithmetic, which relies on speech, transcends its symbols and has meaning. 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.