Sometimes the misconception isn't their fault.
Take the "-" symbol for instance. Are we talking about subtraction? Negative numbers? How about "the opposite"? Or inverse; maybe it's inverse.
I gave students this number line today with the prompts.
1. Tell me everything you can about the number P.
2. Show where -P is on the number line. Tell me everything you know for sure about -P.
They did very will with the first prompt. Lots of responses like:
"P is on the negative side."
"P is a negative number. It's between -2 and -3."
"P is probably about -2.7 because it's closer to -3 than it is to -2."
Ok, I'm loving this. Then they drop the hammer on me.
"-P is negative."What are your first steps when you encounter thing like this?
"-P is also on the negative side."
"-P has a negative sign in front of it so it's also negative."