Curtailing the calculator

Will Miles

Will is a maths teacher at Merchant Taylors' Boys' School, Crosby, Liverpool.

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Almost everyone who studied maths at school will at some point have bumped into BIDMAS or its twin, BODMAS.

BIDMAS stands for Brackets, Indices (powers), Division and Multiplication, Addition and Subtraction. BODMAS is identical, except “indices” are called “Order”. Both acronyms tell us the same thing: the order in which to tackle the operators in an equation.

For example, in the sum 1 + 22 - (3 + 2) x 4, we must first evaluate the contents of the brackets. Indices (powers) would be next, then division and multiplication, then finally addition and subtraction.

This order of precedence is crucial in all arithmetic. To do things in a different order (to start solving the above sum by doing 1 + 2, for example) would often lead to a different and incorrect answer.

At this point in the explanation, perhaps after a couple of examples have been covered on the board, a pupil will cautiously raise his hand.

“Sir. Can’t I just type the question into my calculator and press equals?”

It’s a fair question. After all, isn’t that what the calculator is for? If the pupil inputs the question accurately, it will provide the correct answer. Given that one can just use the calculator, why learn about BIDMAS at all, let alone memorise or practise using it?

I’m afraid that in order to see why, we shall have to consider another maths problem. Worse still, this one will involve algebra.

Let’s keep it really simple: evaluate the expression a + b2 where a = 2 and b = -3.

It seems, I hope, straightforward. One substitutes the values of a and b into the equation and then performs the necessary calculations.

Yet with this sort of question, pupils who routinely do arithmetic without calculators, and therefore have BIDMAS firmly in mind, are at an advantage. They think of b = -3, and square -3 to get 9. Indices before addition. Then it's very simple: 2 + 9 = 11. Correct!

Pupils who rely on calculators are more likely to slip up. Not thinking of BIDMAS, they don't realise that when inputting -3 into the calculator, they need to surround it with brackets in order that the minus sign be applied to the 3 before the power is calculated. They simply press the buttons in what seems the obvious order: 2,+, -, 3, x2, =. The calculator, dutifully applying BIDMAS, first works out that 32 is 9. It then calculates that 2 + – 9 = -7. Wrong!

More interesting than this mistake is the way pupils react when told that they have the wrong answer. Being unfamiliar or at best unpractised with BIDMAS, they struggle to understand why they needed to enclose the -3 in brackets. (To be clear, it’s so that -3 is squared, rather than 3; remember, b = -3 and we are trying to calculate b2.) Their incomprehension often manifests in the confused response: “The calculator gives me a different answer”. It is as if the pupils think that using the calculator makes it impossible for them to get the wrong answer.

This reaction suggests a deeper problem: pupils often see the calculator not as a tool to help them do maths, but as a device that does maths for them, that relieves them of the need to understand mathematical concepts.

Perhaps the reason the calculator has this effect has something to do with its equals button. On the calculator, equals is the button that gives the answer. In real maths, equals is a sign that has a different and far more profound meaning. It states that what is on its left is equal to what is on its right. From this fact, it follows that if you change what is on one side of the equals sign, the other side must be changed by the same amount.

This concept is a cornerstone of mathematics. Without understanding it, one will struggle to grasp more advanced maths, as well as the meaning of the equals sign in disciplines such as physics and computer science.

Calculators are useful and often necessary. Pupils should be taught how and when to use them: as a tool for dealing with large or complex numbers; not as a semi-magical answer-producing device that obviates their need to truly understand the principles of mathematics. For it is through learning and practising these principles that one develops and sharpens the mighty logical and reasoning capabilities of the mind - a calculator far more powerful and profound than any contraption of wire and electronics.

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