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gjayb comments on Two More Things to Unlearn from School - Less Wrong
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So there's an argument for going into the classroom (as a teacher) completely unprepared, stumble through the material, reason things out in fron of the students, go down hopeless calculations for a while, then say "scratch that", "let's see....hmmmmm", etc....Nothing should be clear, the students would have to make huge efforts just to find out what's on the hw. I know some people like this (not by design). I wonder if their students learn some important life-skills though.
In my own experience, this can work well in a small group with engaged students. I had an excellent optics class where we would try to derive a known result as a group: the professor would explain the experiment, draw a picture, and then ask us to help. If we got him going, he would take a few steps, then ask again. Now, I remember next to nothing of equations for optics, but I have a very good idea of how to go about figuring out the outcomes for various experiments theoretically.
On the other hand, I've had professors stop referring to notes partway through a derivation or proof, get dreadfully confused, and simply frustrate themselves and their students. So this may be an all-or-nothing: for a given day or proof or class, either do a group derivation or present the material on a platter.
I will say that I also had a high school English teacher who would use the wrong word or give a ridiculous interpretation in the hopes that a student would correct him and learn to not always trust authority. I liked the theory, but in practice it meant that the attentive students had to do work that was frequently repetitive and irritating, such as correcting word choice or grammar (as these were students who were already thinking) while those who could learn most from such a lesson never noticed it.
I had a teacher somewhat similar to that my freshman year in high school, except she was a last-minute replacement and was not really an English teacher. Her grammar was atrocious, and I ended up getting detention for correcting her too often (interrupting class or lack of respect or some such was the reason given on the detention). It was probably my first real experience with an authority figure being so utterly and obviously wrong, and I wasn't sorry at all for the detention. It was well worth it.
Here's my bad teacher story:
When I was 13 or 14, my physical science teacher was talking to the class about space probes with trajectories that take them outside the solar system. He said that such probes get faster and faster as they go. Thinking he either had misspoken or was intentionally being wrong to see who would catch his error, I corrected him. To my surprise, he said he had not misspoken and that he was correct. We argued about it a bit then he told me to write down a defense of my position.
Later that day, kids came up to me and said, "Why are you arguing with Mr. S? You know he's right!".
I wrote a weak attempt at a defense of the law of inertia (using a reductio ad absurdum argument if I remember correctly). When I gave it to him the next day, he praised it and conceded the argument -- but only privately. He never admitted he was wrong in front of my classmates.
Yeesh, that's terrible. It kind of figures that he'd rather mislead a class full of students about the way physics works than own up to his mistake.
It reminds me of an error I had been taught about the way airfoils work that wasn't corrected until I read a flippin comic strip on the subject almost a decade after I graduated high school.
I was stunned, and spent the rest of the afternoon learning how airfoils really work. What makes this particular example so tragic is it leverages another principle of physics that you won't realize doesn't fit if you are taught to accept everything the teacher says as gospel. What's worse is I'm pretty sure the mistake is still there in the vast majority of textbooks.
I argued publicly with my German teacher about the derivation of 'case' in class. At the beginning of the next lesson, she started with an admission that she'd been wrong and I'd been right. In conceding to a twelve year old on her home ground in front of a class of other children that her job was to control, she taught me an awesome lesson about honesty and humility. I held her in huge respect after that and was her ally ever after. Thank you Ms Eyre.
Evidence-based education does suggest teaching to learn is great for learning