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Slightly tangential, but every time I work with someone on basic kinematics and they say, "Oh, well, if I just flip the sign from positive to negative, I get the right answer! I'm done!" a little piece of me dies.

It's like... you're not even trying to understand the concepts, are you?

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I think this is hard because it's not immediately obvious that direction is such an important concept in kinematics. It's probably the most important thing to get across when teaching the subject, but you have to battle against prior knowledge:

Math students (that is, everyone) earlier learned that this sign-flipping strategy is ok and perhaps an encouraged shortcut when e.g. subtracting.

(see also that Khan Academy video - "first do the multiplication, then think about the sign")

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