There is a statement that I hear quite frequently that I’m not sure what to do with: “Well, I’m not really an algebra person. Maybe I’ll like geometry better.”
I mean, obviously there is a basic sensibility to it in the same way as a historian who studies the Roman Empire finds something in it more exciting that studying the crusades of Alexander the Great. “Well, I’m not really an Alexander the Great person. Maybe I’ll like the Roman Empire better.” (I don’t know, do history teachers hear stuff like that?)
But like differentiating the Great Alexander and Rome, it is difficult to fully understand one without, at least a mild working understanding of the other. Julius Caesar was said to be incredibly motivated by all that Alexander accomplished. Pompey the Great (one of Julius Caesar’s brothers-in-law turned rival) returned from campaign and wore the cloak of the Great Alexander in Triumph. To assume that you can study Rome while completely ignoring Alexander would be to leave a lot of the story untold.
Likewise the other way around. If someone were to try to consider the greatness of Alexander (“great” in the sense that he was mighty and powerful, probably not because he was a fantastic guy…), to ignore how his conquests contributed to the formation of the ancient world would be to leave a lot of the story untold.
They aren’t the same, but they are connected.
And so it is with Algebra and Geometry.
Yes they’re different. They need to be taught differently, but they require each other. There are times when algebraic methods and thinking fit in really, really well with geometry concepts (similarity, 3-D figures, and circles, for example). There are also time when it doesn’t fit in as well (rigid transformations and triangle congruence, for example). It’s the same with Algebra. There are times when geometry works well to support the learning of algebra (distance formula, numeracy with the number line, slope and rates of change, for example). There are also time when geometry doesn’t fit in as well.
As math teachers, we need to have a discipline to let the content dictate the techniques. My main critique of most geometry textbooks is that they are incredibly heavy in the algebra.
This problem is probably working too hard to try to put Algebra in a place where it isn’t a natural fit. If a student were going to try to prove that triangle NPM is congruent to triangle NPQ, wouldn’t trying to prove a reflection be a much more reasonable choice?
So, I suppose that I am simply recognizing that as we meander through the wonderful course of geometry, it would be wrong to make it algebra-heavy. It would be wrong to make it algebra-free.
Perhaps our goal as teachers whose course is between Algebra I and Algebra II is to show the students what algebra looks like and how well it works when it is used in the appropriate spots.
Or as my father-in-law often says, “proper tool, proper job.”