Thursday, March 10, 2011

What's in a name?

How do we know what an apple is? Does the abstract concept of "apple" mean anything to you until you've seen it, held it, tasted it?

Generalities are only useful when we have seen specific examples of what the general concepts represent. Then the general term or concept serves as an efficient notational shorthand for the example or idea as a whole. But until we feel what it means in our bones, the general term only serves to mystify and confuse.

I see this happen all too often in math classes and textbooks. Someone will define a concept and then compose that definition with another and another. If you understood what the first definition was saying, you could reasonably follow the compositions. But if you didn't, you'd be stuck trying to build a sandcastle .... without knowing or understanding what sand is.

I've had the same experience this year as I've embarked on my master's research on machine learning. In the first few months, I'd come across terms like "posterior distribution", "forward distribution", "expectation of the log likelihood", "marginalize the joint". And such phrases were absolutely meaningless to me. Now, I use the same phrases without consciously thinking about it and with a very precise image in my mind of what they mean. I'm so immersed in the terminology by now that I can still be surprised when a colleague blinks in confusion at my use of these phrases. What's the matter -- don't you have the same picture in your head that I do in mine?

This problem is even more pronounced when you're teaching. If you're a good teacher, you probably understand the material better than your students do (though, this is not always true ;). But that's not enough. A good teacher must also misunderstand the material as their students do. As a student, I was always delighted when the teacher seemed to get exactly what I was confused about and then clear that up. As a teacher, I now realize how hard it is and am impressed even more by former teachers who were able to do it. It requires getting out of your own head where you know exactly what an apple is and into the head of the student where an apple is just an assortment of five random letters.

As I near the end of my MIT career, I'm focusing even more on the art of effective exposition -- how to explain things really, really well. This semester, I'm teaching two classes and writing my thesis. Both are activities that are really challenging and transforming my understanding of what a good explanation is. I'm finding that it not only helps me understand something better when I explain it to someone else, but it often helps them too! I'm finding that I can't always anticipate what's going in my student or reader's head, but when I can, it makes a tremendous difference. And I'm finding, most of all, that I really enjoy explaining things clearly. I've still got a lot to learn and it's often hard and stressful (especially when I think I explained something poorly), but I'm loving the experience and hope to continuing improving at it.


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