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> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2

...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.

A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.

I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.



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