Why I teach the identity symbol
Two years ago I had a decision to make. Should I teach kids the identity symbol? I discussed it with my head of department, who wasn't really in favour of the idea, though was happy for me to try. The argument against is that it would introduce an unnecessary layer of complexity. It's a valid argument, and I wasn't sure I was doing the right thing. However, I went ahead, and two years on, I'm glad I did.
First, what exactly is an identity? It took me quite a while to figure it out, and I spent a long time asking fellow teachers and tutors, none of whom had an answer. From A Levels all I was ever told was that it means 'exactly equal to...' but, doesn't the equals sign also mean 'exactly equal to'?!
Here are two pairs of expressions:
Do the expressions in pair 1 have the same value? It should be apparent that they do, since one is the expansion of the other. So, you don't need me to tell you anything more to know that those two are equal; me telling you they're equal adds nothing, that was already obvious.
What about pair 2? We can't know whether or not they have equal values. So, if I now tell you that they're equal, by including an equals sign, I've added additional information. Those two expressions aren't going to be equal for all values of x... in fact, with a little analytic algebra, we can see that they are only equal if x = 6. So, if they are equal, the unknown quantity x must be 6.
By contrast, we have no idea what value the x represents in pair 1; it could represent any value. Here's an activity I use to help show the distinction between equations and identities. Take an identity, such as pair 1 above, and get half the class to try solving it analytically (using the balance method) while the other half try to solve it numerically (using trial and improvement.) The second half can start with whatever number they please, and it won't take long before the triumphant cries of "I got it first time!" kick in. Of course, it doesn't take long into the group discussion for everyone to realise that everyone got it right the first time; that in fact no matter what number you choose, the values on both sides will be equal. By contrast, the analytic bunch should by now be thoroughly baffled, as they've ended up with something like x = x or 0 = 0. Well that's obvious... and unhelpful; it doesn't tell us anything we didn't already know. Which is, of course, the whole point of the distinction between and identity and a solvable equation.
But why bother going through all this? Why teach it? It's additional mathematics which I could argue is important, and interesting. I might even point out that this knowledge is now required as part of the Foundation tier exam GCSE exam specifications:
But none of these explain why I went ahead; I did so for a different, and I think, compelling, reason. Consider these two mathematical processes:
All of the statements are valid, and the manner in which the working is shown is fine... except I feel that by keeping to the equals sign only we have somewhat masked the fact that two very different processes are taking place. With the first process all manipulation takes place on the right hand side, and there is resolutely NO rearrangement happening; there is no crossing the equals sign, whereas in the second process manipulation is happening on the left hand side (though it could be either side) and rearrangement is happening across the equals sign. Consequently, we confuse when we are solving an equation with when we are manipulating an expression. By including the identity symbol, these two distinct processes now look on paper like this:
An alternative that I've seen some teachers choose to adopt is to have children set out working for manipulation like this, doing away with any connecting symbol altogether, and focussing on the way we process each now step onto a new line:
I do think this has some merits, and I even slip into it myself sometimes, especially if space on the board is scarce. However, on balance I believe it misses an opportunity to introduce a new piece of mathematical notation that beautifully demarcates solving equations by rearrangement from simplifying expressions by manipulation. It also states that there is a connection between what is written on the left, and what is written on the right. For these reasons, I teach the identity symbol.