Lately I’ve had the experience of trying to drag someone across the Pons Asinorum. It has put me in mind of teaching and the nature of child development.
Pons Asinorum (Bridge of Asses) is an ancient, colloquial name for Euclid’s second triangle proposition. There are some disagreements about why it is called this, but I don’t really see why. Elements I.5 is the first proposition that requires the student to really see some things for the first time. You can’t drag someone through it anymore than you can drag a recalcitrant donkey across a bridge. Either give them the answer to memorize or wait for them to get older.
So having gone through the proposition once without success, I have to take stock of what to do next. The “answer” to I.5 is quite simple: in an isosceles triangle, the angles opposite the equal sides are themselves equal. Should I hand that over as a definition and move on?
But I.5 has some really important geometry skills in use for the first time. It involves overlapping triangles and seeing things with shared angles or sides. The mental dexterity of shifting from one triangle to the other is necessary for future success. How long should we put that off? I’ve known 10th-graders that can’t do it, and I’m working with a 5th-grader.
There’s also the issue of angle subtraction. This is where Euclid first uses it, and it’s not the last time we’ll see it. So if I skip I.5, I at least have to make sure we all get the idea of angle subtraction–that if I have two big angles that are equal to each other, I can “subtract” smaller equal angles from each and end up with equal angles.
My general educational outlook is that current methods foolishly try to force the “abstractive moment” in students. It’s remarkable how much time my kids spend on explanation and universalizing instead of learning math facts. There are certainly kids who can and do leap from strength to strength this way, but for someone who is not ready for that leap it’s a buzzsaw of discouragement.
So we’ll play a few games with angles and triangles from I.4, then circle back around to the Pons Asinorum to test the waters. And if it’s still too much, we’ll decompose to key elements and move on. This is one of the reasons I’m working through the Elements with a young student: to get a handle on what is and is not within reach at that age.
Teaching Elements, by the way, has been really interesting for me as well. I’m pretty good at geometry but I’ve never done the Elements itself. Things that I thought were definitions (like this one, about isosceles triangles) are actually proven. I like knowing where things come from!
I’ve also started dreaming (ish) about geometry. I’m either remembering things long forgotten or figuring out some things on my own based on the things we’ve done so far. This morning I realized how to prove that opposite angles are equal wherever two lines intersect. It’s so simple! And again, something that I learned by rote in school. Much more satisfying this way.
But would I have thought so 30 years ago?