## Tuesday, September 14, 2010

### Ponderings on Geometry

At a conference this summer I saw an interesting invited talk about teaching geometry. The speaker discussed some of the historical challenges to Euclidean geometry, and discussed how these were eventually overcome. The problem for schools being that while mathematicians like Hilbert managed to fill in all the gaps left by Euclid, the resulting geometry was very complex. It took a great deal of work to get from the axioms up to the interesting Euclidean results. In fact it took too long for a single year high school course to even get to the beginning of Euclidean geometry. As a result, the speaker claimed, most high schools eliminated axiomatic and proof-based geometry from the curriculum, replacing it with practical, computational geometry only. (He went on to detail a new approach, which allows a more rigorous approach to Euclid's geometry in a high school without starting from scratch, and using a book which is designed to be used in an inquiry based class. It looks fascinating, and I'm hoping to teach a class based on this book soon, but that's a digression.)

What caught me off guard was his statement that rigorous axiomatic approaches to geometry using proofs had been all but eliminated from most public high schools due to difficulty dealing with the challenges to Euclid. I had to wonder, was this actually true? I certainly had a proof based and axiomatic geometry in high school, and that was... ok, a little over twenty years ago. (Geez I feel old now.) And while my high school was mostly fine, it wasn't really an elite school. (When I taught an analysis class, I actually used several problems phrased in the form: "My high school calculus teacher said ___. Prove rigorously that he was wrong.") So had schools actually eliminated proofs from geometry since I had taken it? Or had my school been an odd stand out for not eliminating it?

Tonight I was about to start a geometry unit with a class. I asked the students for a show of hands of how many people remembered doing ruler and compass constructions. Less than a third--and maybe less than a quarter--raised their hands. Now I know from experience that some of the students may very well have seen constructions and just have forgotten everything five minutes later. But still, it seems that a number actually didn't do constructions. And I'm inclined to think that if they didn't do any constructions, they probably didn't do any proofs, although I could be wrong on this. (And of course some may have done constructions but not proofs.) So maybe the summer speaker was right, and axiomatic geometry is mostly gone.

But I question his reasoning about why the proofs are gone. He believed it to be because it proved too difficult to fill the gaps found in Euclid. I can't imagine boards of education, state legislatures and such largely care--or even know--about that. My first suspicion was that someone, somewhere decided it was "too hard", or that students just couldn't do it, and replaced it with other things.

But why too hard? Abstract reasoning and logic is hard, but given time and effort, most people can make progress. Ah, but there's the problem: Time. To do a good job with a class about proofs, it takes time spent working on problems, trying ideas, failing, and trying again. Learning to reason is a long and difficult process. And curriculum tends to fill up with all sorts of nonsense as everyone and their dog proposes new lists of things that "everybody" ought to know. Most of those things are lists of facts and formulas. I can almost hear the litany start for a geometry class: "Students must be able to give the formula for the area of a circle, a semicircle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus; Students must be able to find the perimeter of a circle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus..." (And it's worth noting that the only really interesting things in the list I just gave are probably the area of the circle and the rectangle, and perimeter of a circle. The rest should be easily derivable from some good geometric thinking. But the students will instead be given a list of formulas to memorize.)

We add and add to curriculum, and nothing is ever taken out. We add numerical approximations and work with computers. We add calculators, then graphing calculators, then geometry software. We add three dimensional shapes, and trigonometric functions, and whatever else anyone can think of because ... well, because someone else happened to remember it and thought it would show how rigorous we were being if the list of things for students to know was really long. (It should be noted that I am not opposed to any of these topics in high school math classes. But we must recognize that we cannot do every possible topic all at once.)

But a long list of topics is not rigor. That's just memorizing a bunch of stuff. Take any of those topics and spend some time with the students doing a long and careful analysis of some challenging problems, and you'd have a recipe for a great math class. The central questions will always be the following: What do you think is true? How do you know that? Why? Is this like anything we have done before? Can we generalize this result?

All of this of course misses the biggest elephant in the room as to why high school math has dropped most reasoning and replaced it with lists of tasks and formulas: It's very easy to write a statewide multiple-choice test which to see if students can choose the formula for the area of a circle. It's very difficult (or perhaps impossible) to write a state-wide multiple choice test which determines how skilled students are at reasoning and solving complex problems. But I'd much prefer a student who can work out how to find the area of a trapezoid based on what she already knows than one who has only memorized the formula. I regularly have students who have memorized a formula corresponding to a figure, but can't figure out if it's the formula for the area or the perimeter. (When I ask how to find the area of a circle, I can guarantee about half the class will respond "2 pi r.")

And I'd really love to have students who learned enough reasoning to write proofs in their geometry class, but apparently that's rare now. No wonder we have trouble teaching proofs later in college.