The biggest and most important part of my job is to convince my students that they know math.
I can go and see how they have attempted the problem, and show them how most of it is ok, and where the error is, if they haven't found it yet. Just like with cd and the furnace.
One problem is that many students want to memorize "correct recepies" for each case. Math is not like that. We learn tools, and we are allowed to use any of our tools on any problem. Similar looking problems may require completely different tools. For example I had problem with integration in university exam. I was trying to find correct substitute function, but I stumbled upon solution instead, so I declared first law of integral calculus on my guess, showed that the derivative matched initial function and got full points.
My computer science degree is from last millenium, there are no course books to base lectures on, and computers keep distracting the students from all the boring things they are supposed to learn.