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I just realised the "=" sign means the OPPOSITE of what many students think it means. Many students think it's an indicator that something was done, but when you write an "=" sign, the point it is making is that actually you haven't actually done ANYTHING because it's THE SAME.
Of course I haven’t *just* realised that students think the “=“ means “here is the result of what was done”. I definitely knew that already! What was a new idea for me was how this is the exact opposite of its actual meaning.
Ooh I’ve just had some thoughts flowing on from this...
I think this might be one reason why completing the square is so difficult for a lot of students. In completing the square you are very explicitly not actually doing anything, and there is no “answer” at the end. Bamboozling if “=“ means “the result of doing something” for you.
And all those students who write no “=“ signs at all when they are just doing algebraic rearranging. They are trying to *tell* me that they understand nothing is being done! The *lack* of “=“ signs for them is saying “nothing really happened”!
When you do a long calculation to get an answer, you’re not really doing anything, just trying to find the identity of the expression. It’s not all that different from completing the square in the sense you are trying to write the expression in a certain format: a single number.
Going back to the original realisation, I realise it works the other way too. The main reason you are able to declare two things to be equal using an "=" sign, especially when doing algebra, is precisely that NOTHING HAS CHANGED https://twitter.com/DavidKButlerUoA/status/1277905699876397057?s=20
The idea of things connected by “=“ signs looking different but acruallly being the same is reminding me of this scene from Mystery Men.
