The Hardest Lesson

[stoutfellow]

Someone on Facebook posted a quote from Stephen Hawking on the nature of intelligence, and it brought to mind something I tell my math students, prior to the first test in each class. I tell them that I test for three things: factual knowledge (in this context, the statements of definitions and theorems - and I tell them which they need to know), skills (and I tell them which skills I expect them to display), and understanding. Then I tell them that I can't tell them what I expect them to understand, because to display understanding is to deal with situations you haven't seen before. That generally involves word problems (at lower levels) and proofs (at higher).

Understanding is the hardest thing to teach. It may not be teachable at all.