One of the lessons I learned in my first year of teaching - fortunately before the end of the year - was that it's not enough for students to learn; they have to see clearly that they are learning; and for this purpose, it is not enough to give them ways to apply the new knowledge & skills; you have to really spell out for them here is what you have learned.
This morning, I saw an interesting illustration of the power of feeling that you do know something, in Oliver Sacks' account of one of my favorite stories of scientific discovery.
Dmitri Mendeleev had been playing with arranging and rearranging cards representing the various known elements, trying to discern the underlying pattern; one night he fell asleep and dreamed the Periodic Table, more or less as we know it.
Here is Sacks' footnote to the story:
This, at least, is the accepted myth, and one that was later promulgated by Mendeleev himself, somewhat as Kekule was to describe his own discovery of the benzene ring years later, as the result of a dream of snakes biting their own tails. But if one looks at the actual table that Mendeleev sketched, one can see that it is full of transpositions, crossings-out, and calculations in the margins. It shows, in the most graphic way, the creative struggle for understanding which was going on in his mind. Mendeleev did not wake from his dream with all the answers in place but, more interestingly, perhaps, woke with a sense of revelation, so that within hours he was able to solve many of the questions that had occupied him for years.
(Oliver Sacks, Uncle Tungsten. New York: Random House, Inc, 2001. p. 198)
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