“Its not whether they pass or fail, but how much they suffer along the way.”
I make a point of telling my students that making mistakes is not failure, and reassure them that making mistakes is all part of the process to learning mathematics. There is no way to get around it, studying math takes time and effort.
The best students learn ahead of the rest of the class, and spend a LOT of time making mistakes, and fumbling along, and feeling lost. Average students wait for an introduction from the instructor, and will copy what the instructor does until it starts to make sense… both of these types of student will spend time feeling frustrated, and have the uncomfortable sensation that they’re not getting something. It is uncomfortable for them up to the point when the clue-by-four smacks them in the head and they get it. Sitting with your own ignorance is part of the process, and when that cognitive-dissonance finally goes away it it both a relief and a pleasure.
Bad students… the ones that make me clamp down on my own eyeballs to keep them rolling audibly… bad students are the ones who either (a) say “I can do that” and go back to updating their Twitter status until test time or (b) think that you should explain things over and over so they don’t have to do all that uncomfortable thinking. “I don’t know” is no excuse to not try. But it is a quick ticket to an angry mathematician.
Grrrrrr… Oh, wait. I fail those students!
Yes, that means when I tell my students that math teachers don’t really engage in shadenfreude, I’m lying. Just a little.
So seriously students – copy the process. Take one step at a time: even if understanding doesn’t happen all at once, it will happen by individual steps.