A very interesting article in the NYT Magazine, Building a Better Teacher, by Elizabeth Green came out earlier this week. On page 8 of the article, Green writes:
Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less.And these are not even the most interesting observations that the article makes.