Regarding the last quote from your math teacher: I had heard a variant that goes something like, "It's more useful to be wrong right now then to be correct a few weeks later." The idea being that dealing with whatever incorrect conclusions you have directly, as soon as possible, is better in terms of improvement if you can rapidly iterate from that failure.

Obviously, this doesn't work well with many different things (anything with a high cost of failure), but it is a problem-solving method I've not heard promoted on LessWrong that is good to have in your toolbox.