One thing I've learned a lot about in my Discrete Math class, somewhat inadvertently, is how my fellow students learn and the ways they struggle. There are only twelve of us in the class, and our prof typically does only a little bit of lecturing before having us work on problems and lecture each other. Quite frequently we get sent to the board with markers to write out solutions and, more importantly and unusually, explain things to each other. He will often have several of us explain the same concept for the benefit of the class.
There is, of course, a range of ability within the class, and each person's ability also changes from task to task. Sometimes I have seen other people pick things up more quickly than I do, and at other times I've been stunned as the rest of the class has seemed to turn into morons.
One of the topics my class has really struggled with has been probability. I also had some probability this semester in my Prob & Stats class (as you'd expect), and people seemed to struggle with it there too, though it's not as obvious in a large and non-interactive classroom. But I've heard from some people that, when they took that class, they had no problem with the statistics part, but did badly with probability.
I am no expert at probability and I have learned new things in both of these classes, but probability has always come somewhat easily to me. There seem to be a very small number of principles, and then you just have to figure out how to apply them to the situations in the problems. The principles themselves seem to make logical sense. Because there is a not a lot of fiddly algebra and everything makes logical sense (rather than seeming arbitrary), I do very well with it.
I have guesses about why other people struggle with it. I think one problem is that all the problems are word problems. A couple of students in my prob & stats class were talking once about not being able to figure out what the questions were asking - questions like "What is the probability that no more than 4 valves fail?" And in Discrete, even my professor has struggled to understand the questions sometimes, though he's obviously quite smart.
I think the other problem in probability is that, because it's about applying a few rules to diverse situations, it's not a field of math where you are given extremely specific question types and shown how to solve them. In calculus, for instance, you typically have a section of very similar questions and you are shown methods that always yield the correct results. In probability you really have to analyze the situation and think of equivalences between disparate types of events.
These are just guesses. There have been times when I've really wanted to tear my hair out at the struggles my classmates were having with things that seemed very straightforward to me, and I've had to remind myself that they haven't seemed dumb or underprepared on all topics, but that there is apparently something specifically difficult about probability.
What have your experiences been like?