[iopq]

Some students are taught this way:

x + 5 = 10 the equals sign is a magical mirror so when you take operations across it it changes them to the opposite of what they were

so adding five becomes subtracting five, multiplying by two becomes dividing by two, etc.

x = 10 - 5 by way of magical mirror

[jiaweihli]

This is how I learned it - in hindsight, I feel this is easier to reason about, as it forces you to apply operations synchronously. (As opposed to, say, subtract 5 from both sides, then add 8 => which we can reduce to add 3. But this reduction might just be noise to your brain.)

[iopq]

This way you get "proofs" like: http://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml

If you have an intuitive understanding like "both sides represent a number, as long as I manipulate it the same way the equation stays true" then you wouldn't have problems with logic when dividing by zero

[jiaweihli]

Yes, I understand the concept that you quoted. Theoretically, it's more sound.

However, I want to point out that the purpose of the 'magic mirror' method is to ease mental math. It frees your brain from having to spend time / space on the extra step of applying the same operation to both sides.

Putting this in the context of 'showing your work':

Normal: x / 3 = 4 => 3 * x / 3 = 4 * 3 => x = 12

Mirror: x / 3 = 4 => x = 4 * 3 = 12

It's a bit complex to describe - I feel how you do basic math such as this is hardwired into your brain at a very young age.

[R_Edward]

I'm not sure we do our students any favors by suggesting that math is "magical."