THE TEACHING AND LEARNING OF ARITHMETIC
In the United Kingdom young children are encouraged to learn their "times-tables" : 2 x 2 = 4 through 6 x 9 = 54 up to 12 x 12 = 144. Alongside, and after this, they are shown how to multiply larger numbers together using their times-tables for each step.
They are also taught something called long division 156735/273. They use their times-tables to help them with this.
The result of this way of teaching is that there is a very substantial fraction of the adult population that cannot perform either multiplication or division. Within this fraction there are people who are deeply, psychologically, afraid of any sort of calculation. The remaining portion of the population, those who understood multiplication and long division and perhaps went into some field where calculations were an essential skill, never actually used the skills that they were taught : They used hand calculators for small jobs and some sort of computer for larger ones. No grown up person would ever think of doing long division. The teaching of arithmetic in its present form is pretty well a waste of time : Some of the school population end up afraid of sums and never attempt them and the remainder never use the systems they were taught.
Most problems could be avoided if all pupils were given a basic calculator on Day One.
Teachers, as a group, will not buy into this idea.
The teaching profession believes that times-tables and long division give students a feel for numbers and that this is an indispensible requirement.
However, these systems of multiplication and division were devised because there was no other way of getting the answers to problems. The Romans did not have calculators, nor did the Italian pioneers of accountancy. Nevertheless, they were alive to the tedium of repeated similar calculations and were open to new ideas. If God had given the Romans hand held calculators, times-tables and long division and all the other barbarisms would not exist.
Once Napier had explained the idea of logarithms, it became possible to do quite extensive multiplications and divisions by adding or subtracting the logarithms of numbers. It was not yet a machine calculation, but it was much easier to get an approximate answer. Table of logarithms became an essential adjunct to the calculating man
Then the first stage of machine calculation was brought into being, what is called a slide rule or a slip stick. This had the logarithms printed on it and by sliding one part against the other, you could do multiplications and divisions to sufficient accuracy for many industrial purposes. Slide rules were increasingly used from the middle of the seventeenth century until late in the twentieth.
Useful as they were to the savant, logarithms and slide rules were not much used by the general populace in the West. The abacus, widely used in Eastern countries, also seems not to have been popular in the West.
After about 1980 hand held electronic calculators became the tool of choice for small calculations. Even some of the die hard arithmetic haters were persuaded to give them a trial. Teachers in schools either pretended that calculators did not exist, or they threw up their hands in despair because they weren't allowed to teach arithmetic using them.
Teachers have become trapped in an orthodoxy which cannot easily be broken. Senior pedagogues believe in the system, teach their juniors, force a teaching curriculum on schools and generally ensure that the system is not changed. It is a vicious circle.
It is not unknown for a large academic group to maintain an orthodoxy in the face of a mountain of accumulation of contrary fact. Dieticians in the 1950's and 1960's were coerced into a theory of nutrition by a man called Ancel Keys. Keys became so influential that if anyone tried to counter his views they stood a good chance of losing their job. Only now, armed with slightly more freedom of academic speech by virtue of the internet, is this stranglehold being slowly broken. Going further back, the teaching of Latin in schools persisted well into the 1960's as a compulsory subject if you wanted to enter some of the better universities. Latin teachers averred that you needed to know Latin to thoroughly understand English. What they really meant was "We'll lose our jobs if ...."
The teachers' belief that you get "a feel for numbers" by the use of times-tables and the other rigmaroles is probably quite true. What is not true is that you can't get such a feeling for numbers in any other way. I have got a jolly good feel for numbers because I am always dealing with them. I haven't done a long division for more than forty years and I certainly don't do multiplication by using times-tables. You get a feeling for numbers by working with them a lot.
I think that if all pupils were given a calculator on Day One a much greater percentage of them would leave school confident that they could manage simple arithmetic, more of them would do their own house keeping, tax and other domestic manipulations, and overall more people would have this "feel for numbers".
If an optional exam were introduced in which pupils could use calculators, I doubt whether any existing, roughly equivalent, exams would survive.
Still, all power to the teachers : Their methods keep me in business.
D C W Morley January 2016