A Plea in Defense of Euclidean Geometry
While I grew up in snow country, I can’t tell my kids that I trudged miles through snow to get to school. But I can tell them I learned proofs in high school geometry, which could become as much a part of a vanished virtuous past.
One of the pleasures of being on the faculty at Caltech is interacting with our bright undergraduates. For the past two years, I’ve asked the incoming freshmen in my calculus/probability class whether they had proofs in their high school geometry course. About 40% have not, and more than half of the remainder had at best a cursory few weeks. So less than one-third have had the kind of rigorous theorem/proof course I had back in James Madison High School in Brooklyn more than 30 years ago.
Why do I mourn this loss of what was a core part of education for centuries? After all, we no longer require Greek and Latin in high school and Euclid was just one of those Greeks, wasn’t he? While the geometric intuition that comes from the classical high school geometry course is significant, what is really important is the exposure to clear and rigorous arguments.
Modern mathematicians don’t use the two-column proofs so beloved by my high school geometry teachers, and real life rarely needs the precise rigor of mathematicians, but those who have survived those darned dual columns understand something about argumentation and logic. They can more readily see through the faulty reasoning so often presented in the media and by politicians.
It is not merely a question of good citizenship. In the global economy, our young people will be in competition with young people the world over. If I talk about American high school education with scientific visitors from abroad, they are either aghast or amused. Immigrants I know from the former Soviet Union tell of fifth-grade Russian mathematics texts at a higher level than what we teach juniors in high school. For a large number of jobs in our technologically based world, a solid scientific and mathematical training is essential and our foreign competitors are beating us there.
The trend away from theorem/proof geometry seems to be based on the following reasoning by the educational establishment: Some high school students are just unable to get this theorem/proof stuff. If we place them in a separate track, they’ll wind up with a low self-image. So to prevent some students from having a low self-image, we won’t ask anyone to understand it. This summary is admittedly a caricature but it is the core of the reason that Euclidean geometry is disappearing.
I’m not particularly worried about the lack of Euclidean geometry among my Caltech freshmen. While my colleagues also have noted a marked decrease in the quality of preparation of our students over the past 15 years, we see that these students are just as smart and motivated as Caltech students ever were. They are forced by us to absorb notions of careful proof at a less leisurely pace than they would experience in high school and the process can be painful, for them and their teachers.
But I am concerned about the country as a whole. The dumbing down of high school education in the United States, especially in mathematics and science, is a crime that must be laid at the doorstep of the educational establishment. We must demand that the level of high school science and mathematics being taught be improved, starting, of course, with Euclidean geometry.