Years ago, a colleague was convinced one of his students understood more than his written work suggested.  So he took the proverbial bull by the horns.  After school one day he took an essay the student had written and typed it out on a computer, breaking the essay into individual sentences.  Then he asked the student what you needed to know in order to move from each sentence to the next.  Here was the right way to go about filling in the blanks.  By the end of the little session, the student had not only shown the teacher how much he knew and understood, he had learned a lot about what he needed to do to make himself understood.

This was a lovely thing to do.  Simple, direct, right on target.

Here’s another example.  Suppose you are teaching geometry and you want students to consider parallel lines.  This is not a vocabulary lesson.  You can take it that everyone knows what a dictionary would say about the word “parallel”.  It is, instead, an opportunity to come to appreciate issues and ideas surrounding parallel lines.  One of the things you might do is tell your class that two lines each perpendicular to a given line are parallel.  Or you might tell them that if you draw a line that crosses two other lines, then, if the alternative interior angles are equal, the two lines are parallel.

But here’s something else you could do.

Take out two lines, say, two meter sticks, and lay them on the floor.  Now ask the kids to tell you if they are parallel.

Assuming you set them on the floor so that they didn’t cross one another, it is possible they are parallel.  But how would we know whether they would cross one another if we extended them?  Is that important?  Suppose you line them up pretty well and they look like they wouldn’t intersect.  Then you shift one a little.  It still looks like they wouldn’t intersect.  Can two different lines that pass through a given spot be parallel to the same line?  That’s a neat question, too.

What would you need to establish that the meter sticks are parallel?

This is a good question.  And with a little guidance from a wise teacher, and a protractor to measure angles, that class is going to figure out that two lines mutually perpendicular to a given line are parallel.  And they are going to see that when two lines are parallel, the alternate interior angles of a transverse line are going to be equal.  And because they figured it out, they have also learned some powerful habits of mind surrounding proofs and making sense of things.

There’s a lovely line from Goethe that I ran across in a piece by Kurt Vonnegut.  “Whatever it is that you have inherited from your father, you are going to have to earn it if it is to really belong to you.” Seems right here, and Euclid is our father.  (See “Roots” in Vonnegut’s Palm Sunday.)