Sometimes, teaching feels like an Aaron Sorkin film/television show. There’s a lot of walking and talking, and the day goes by lightning fast.

This makes it extra important to slow down and reflect every now and then. One of my geometry classes blew me away today (and it happened to be while my principal was observing :)). We were discussing bisecting line segments. I asked students a what I thought would be a straightforward question: can a line be bisected? Straightforward answer, I thought. A line is infinite, so it cannot be split in half.

Then, came the first nice mathematical insight from a student, who does not believe she is good at math. “Well, wouldn’t 0 be the midpoint of the number line? If you fold the number line at 0 all of the positive numbers would lie exactly on the negative numbers…so it would be like folding it in half.”

What a nice insight! But, I was able to counter: “Well, if you folded it at -1 instead, you would still have an infinite number of points to the right and to the left. So, you would also have to say that -1 is a midpoint of the number line.”

Then the insight of the day came from a different student: “Well, couldn’t you say then that there are an infinite number of midpoints along a line.”

Damn! Teachers live for these insights…insights that cause their own thinking to stop in their tracks.

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