While I was at university, there was a professor in the Computer Science college who was known as incredibly difficult. Dr. D was unquestionably bright, having two doctorates, one in Mathematics and one in Computer Science. He knew his material, and while his Chinese accent was at times difficult to understand, it wasn't a major hindrance to learning the material.

I had Dr. D for 2 classes: Discrete Mathematics, and Computer Science II. I had never worked so hard for a B in my life. Dr. D was not as some may imagine lacking in the spirit of teaching. Quite the contrary, he loved his job, and he cared and encouraged his students. He often complained that not enough students came to his office hours. Sometimes when our class needed a respite from the non-stop barrage of ideas that Dr. D issued forth, we would (and often did) get him talking about his research on number theory and the rest of the class period would be lost to tangential discussion.

What made Dr. D so difficult was the way he taught us. He taught all his undergraduate classes as if we were graduate students. Instead of giving us concrete examples, we got lots of theory. Maybe I'm just not smart enough, but the bridge from theory to practice has always been difficult for me. The concept of mathematical induction seems easy enough, but when applying it to a problem, unless I've seen something similar, it becomes a major struggle. I understand the means behind the methods: Learn how to learn. That's all fine and dandy, but before I can learn to learn, I need to understand.

Mark Guzdial wrote an excellent article for the ACM discussing just this issue. He brings to light a 1985 paper which is very telling, and coincides with my own personal experiences:

The original 1985 Sweller and Cooper paper on worked examples had five studies with similar set-ups. There are two groups of students, each of which is shown two worked-out algebra problems. Our experimental group then gets eight more algebra problems, completely worked out. Our control group solves those eight more problems. As you might imagine, the control group takes five times as long to complete the eight problems than the experiment group takes to simply read them. Both groups then get new problems to solve. The experimental group solves the problems in half the time and with fewer errors than the control group. Not problem-solving leads to better problem-solving skills than those doing problem-solving. That's when Educational Psychologists began to question the idea that we should best teach problem-solving by having students solve problems.

Why then, if there is such a significant statistical difference, are teachers still insisting on teaching by theory alone? If seeing worked out problems has a better chance of teaching students how and why things work, then let's push that. I know that this certainly works for me. The more I see of a problem worked out, the more I begin to understand the mechanics of it, so when I do come across a problem that is slightly different, my understanding is greater, and my chances of solving are greater as well. This isn't to say that theory should never be taught, of course it should. Guzdial simply states, and I agree, that theory and problems without samples shouldn't be the way to start learning a subject.