Introduction to algebra

The equals sign in algebra - part 1

Surely the equals sign always means the same thing, doesn’t it? What the equals sign means strictly in mathematical terms is not the same thing necessarily as the way it is interpreted in practice. When doing arithmetic, that is manipulating number, pupils mostly think of the equals sign as an instruction to do something with some numbers, to perform an operation. They see ‘3+ 5 =’ and respond by doing something; adding the 3 and the 5 to get 8. So, given the question, 3 + ? = 5, many younger pupils put 8 in the box; they see the equals sing as an instruction to perform an operation on the numbers in the question, and naturally respond to the ‘+’ sign by adding them up.

Pupils also use the equals sign simply as a device for connecting the calculation they have performed with the result of the calculation. It means simply, ‘This is what I did and this is what I got…’

Given the problem ‘You have €28, earn €5 and spend €8, how much do you have now?’ pupils will quite happily write something like: 28 + 5 = 33 – 8 = 25. This way of recording the calculation is mathematically incorrect, because 28 + 5 does not equal 33 – 8. But this is not what the pupil means, of course. What is written down here represents the pupil’s thinking about the problem, or the buttons he or she has pressed on a calculator to solve it. It simply means something like, ‘I added 28 and 5, and got the answer 33, and then I subtracted 8 and this came to 25’.