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I am teaching a mid-level calculus course, and I see my students making the freshman's dream mistake of thinking that every function is a homomorphism. In particular, they think that exponents can be distributed, so $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ and $(a + b)^2 = a^2 + b^2$ in their world.

We have discussed this in class, and I've demonstrated how this will lead to contradictions like $\sqrt{2} = 2$. When I see this mistake, I struggle with grading it because I have the opinion that properties of exponents should have been learned long ago. I came here to ask how I can gently get my students to unlearn this mistake as soon as possible. Am I being unreasonable in thinking that this is a big problem at this stage of learning mathematics?

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    $\begingroup$ I would say 90% reinforcement (drill, punishment, whatever you want to call it) and 10% explanation. For one thing, the idea that these kids have never had it explained to them is silly, silly, silly. You don't break a bad habit by talk. You break it by repetitive reinforcement to the opposite. $\endgroup$ – guest Sep 28 '18 at 22:24
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    $\begingroup$ One small trick that I have seen is to label the error with a cute name. "freshman's dream" is a good one. Then when the kids are at the board doing recitation, you can hoot an point out the "freshman's dream" error. That will make them take notice. We are social monkeys. Not computers. $\endgroup$ – guest Sep 28 '18 at 22:27

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