Teaching is one of those things that can be very hard to do well. The first step to teaching is to know what you are talking about. There are people who are "experts" on a particular topic, but they've never organized their thoughts in a way that would make sense to a newbie.
Teaching is story-telling. It's telling a story that makes sense to you and hopefully makes sense to your listeners. Experts sometimes reach a level of unconscious competence. That is, they know what to do, but they can't explain it. Teaching is about moving that expertise into conscious competence, that is, to explaining what you know in a clear, cohesive way.
OK, but that's not all of it. If you're an expert at something, then you will make assumptions about your audience. This is very important. Without assumptions, you will most likely assume they know as much as you do, and so they may struggle with an explanation. For example, if you know statistics backwards and forwards, you might assume that everyone knows the basics really well, which may not be correct.
Some people try to explain ideas using analogies. That only works if the analogy makes sense to the audience. Some people won't get the analogy you are making. Once upon a time, people figured you knew baseball or boxing and they'd make an analogy figuring, for sure, you had to know this.
This isn't to say analogies aren't useful, but only that you have to be careful about it. Analogies are often good at making something concrete that seemed very abstract otherwise. It takes something that lacks familiarity and makes it more familiar.
The key to effective teaching is to understand the worldview of the people you teach. Because students are individuals, you may have a hard time conveying information that will work for everyone. However, since most students generally have similar backgrounds heading into a class, there are some commonalities.
To discover where students are coming from, you need to talk to them. It's surprising how one-sided communication between teachers and students are. Teachers spouting out information, but not engaged in a dialog where the teachers tries to understand why students get confused. To me, this is still the single best way to learn how to teach. Too often, teachers imagine themselves to be something like TV performers with students as the audience kept at arm's length away.
Indeed, one might argue that the lecture format is not the best way to teach students. At the very least, most of us learn by doing.
To sum up, good teaching requires understand the material, being able to tell the material as a story so it makes sense to yourself and to others, to be adaptive to what the audience needs, adjusting the level of difficulty as needed, and to interact with students to better understand how they view what is going on.
Three opinions on theorems
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1. Think of theorem statements like an API. Some people feel intimidated by
the prospect of putting a “theorem” into their papers. They feel that their
res...
5 years ago
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