## Geometry first

Jackie at Continuities is wondering whether the usual path through high school mathematics — Algebra I, then Geometry, then Algebra II, etc. — is out of order, and whether geometry ought to come first:

As far as I can tell the only difference between Alg II and Pre-Calc is that trig is taught during Pre-Calc and Pre-Calc introduces the concept of the limit. Functions are developed a bit more rigorously too.

The first semester of Algebra II is mostly a repeat of Algebra I as they’ve forgotten it with the year “off” during Geometry.

Why not then teach Geometry first? I’m talking about plane and solid geometry with an emphasis on reasoning, and right angle trig. Obviously there would need to be some supplementing needed (work with radicals, solving equations). Most students have “seen” the solving of equations in 8th grade (Have they mastered it? No, of course not).

I completely agree. It seems to me that the reason Geometry gets sandwiched between Algebra I and Algebra II is that people want to use algebra concepts in geometry. But I think that doesn’t necessarily have to be the case. If you look at the source — Euclid’s Elements — you will not find a drop of algebra in it. All the concepts that we, today, would label as being algebra or number theory or what-have-you are just latter-day retrofittings of Euclid’s ideas. Euclid himself phrased everything in terms of geometry, with the algebra and number theory done in terms of commensurable lengths and other geometric terminology. I wouldn’t go so far as to say Euclid knew nothing of algebra or number theory, but if you follow Euclid you don’t need algebra, as we know it, at all in your geometry.

That would leave a geometry course that is mainly about logical reasoning, cogent organization of facts, objective deductions from data, and clear exposition of an argument. One might add to this list the art/craft of forming conjectures from experimentation and then writing an argument in favor of your conjectures, which is astoundingly simple these days thanks to Geometers Sketchpad and other fun, low-cost dynamic geometry software packages. (My students who use Sketchpad in their student teaching report, to a person, that students really turn on when they use Sketchpad and do some very good mathematics, for 8th-9th graders.) This sounds like precisely the kind of foundation, and buffer zone, that students need to acquire before tackling algebra with a view towards understanding how it works rather than just memorizing facts. (Indeed, memorizing facts in algebra is quite hard unless you understand why the facts work.)

Of course, if you ask ten people whether they liked their geometry class in school, eight will probably say “no” and seven of those eight will say it was because of “proofs”. But I wonder what that really means. Perhaps, having gotten a taste of equation solving in algebra and therefore acquiring the “there’s only one right answer and I have 30 seconds to find it” mentality about mathematics, they are spoiled for ever encountering mathematics as it really is (which is something that geometry is a lot closer to than algebra I). Perhaps they had a geometry teacher who was not really good at, trained in, or interested in math at all — or someone who was like so many teachers out there who “just love kids” but who choose not to translate that love into teaching their kids how to think well.

But I think if you put a geometry class like what I described above into the hands of a competent, mathematically astute teacher with a mind to help his/her students become excellent thinkers, a year of that could very well change a generation of kids.