Some people believe that children shouldn’t be allowed to use calculators in mathematics and science classes in schools – I disagree.
Scientists use calculators for many of the routine and tedious calculations that they frequently have to perform. So, if schools are teaching children how to do science, they should teach them to use calculators intelligently. It’s very easy to press the wrong buttons when using a calculator, so it’s important to check that the results you get are a sensible result for the intended calculation.
For example, suppose you want to calculate 987 multiplied by 1 213. The answer must be about 1 000 000 (because the first number is a bit less than 1 000 and the second is a bit more) and must have 1 as the last digit (because 7 × 3 = 21). When we perform tasks like this, we need to think mathematically. So intelligent use of a calculator doesn’t stop children from understanding mathematics.
However, arithmetic is often taught in a way that prevents children from thinking mathematically. Many people can multiply 987 by 1 213 to get the right answer without thinking. They simply follow instructions that they were taught at school. They may not understand why the method they have used gives them the right answer. Mathematics is about not about following instructions without thought – it is about logical thinking.
Until the latter part of the twentieth century, calculators were not widely available. So, people had to perform tedious calculations without them. As a result, children were taught methods for multiplying and dividing large numbers. This enabled them to work as bank clerks, accountants etc. I don’t believe that the purpose was to help them to understand mathematics. If you believe that mathematics and arithmetic are the same (as many people of my age seem to do) – see appendix 1 of post 16.50.
Now banks and accountants use computers and you probably have a calculator on your mobile phone. The world has changed, and education should change with it. Children still need to be able to use numbers, but they don’t need to learn methods for performing very complicated calculations. Instead they should learn how to perform simple calculations, make approximations and evaluate numerical results sensibly. Indeed, evaluating numerical information is especially important in many countries of the world where we are constantly being led astray by journalists and politicians who seem to be incapable of using numbers sensibly.
When I was at school, we were taught to use tables of logarithms (post 18.3) to perform simple calculations. These tables gave the logarithms of numbers between 1 and 10 with up to four decimal places. They would tell you that log10(9.870) = 0.9943 and that log10(1.213) = 0.0839. Since 987 = 9.87 × 102, (see post 18.2 if you’re not sure what this means)
log10(9.870) = log10(9.870) + log10(102) = 0.9943 + 2 = 2.9943
(see post 18.3). In the same way, you could find that log10(1 213) = 3.0839. So
log10(987 × 1 123) = log10(9.870) + log10(1 213) = 2.9943 + 3.0839 = 6.0782.
The previous step, and the next, are explained in post 18.3. It follows that
987 × 1 123 = 106.0782.
We were also given tables of what we were taught to call “antilogarithms” – they were tables of values of 10x, where x was a number between 0 and 10. They would tell us that 100.0782 = 0.1972, so that 106.0782 = 1.1972 × 106. The result is equal to 987 × 1 123 to four decimal places – the number of decimal places in the logarithms we have used, limits the precision of the result (see post 16.7).
You might think that we had to understand logarithms very well to use them for these calculations. But we didn’t! We were simply taught routine methods that enabled us to use them without thinking – very similar to the way some people use a calculator.
So, using a calculator doesn’t stop us understanding mathematics. And much of the mathematics teaching that people received, before calculators were available, didn’t help them to understand mathematics either!