Proof in Geometry
Many years ago, when I was still teaching high school, I added a Teaching Proof page to my website, which included a bit of philosophizing and links to the relevant parts of the site. If you’re looking for ideas and materials on this topic, you should definitely check out that page.
I’ve also addressed proof many times on this blog, both in specific cases and in more general contexts. You can search for “Pythagorean theorem” for some specific examples, or search for “proof” to get a full list. Here are three highlights:
Teaching proof only makes sense if students have some familiarity with the underlying math, as well as the maturity to appreciate the necessary rigor. In the absence of these ingredients, this part of the curriculum turns into a mechanical process of sequencing statements with no understanding. Alas, that is too often what students encounter, especially as geometry courses are taught to younger and younger children.
Proof in 11th and 12th Grade
The above links are mostly about a more meaningful and effective approach to proof in geometry. Today, I’d like to complement those, and argue for the teaching of proof to high school juniors and seniors, in courses beyond Algebra 2. Unfortunately, there is no room for this in many schools’ math programs, and in fact it is not really supported by state standards. Still, I hope you will not stop reading! You may find what I have to say interesting, even if you cannot implement those ideas in the immediate future.
While it is true that the math in Algebra 2 and beyond is a requirement for further work in STEM fields, I think we need to also think about teaching math as one of the humanities — a great achievement of the human spirit. One way I tried do that when I was department chair was to resist the pushing of Calculus into students’ schedules prior to their senior year. Putting all genuinely precalculus content into a one-semester Functions course left room for one-semester electives which were available to Calculus-bound juniors, seniors who were not taking Calculus, and math aficionados who took one of those courses while taking Calculus.
The most popular and accessible such elective was a basic statistics course, which we used to introduce data analysis and sampling, along with a review of algebra and precalculus topics such as logarithms. But I also designed two “pure math” courses: Space and Infinity, which were intended to show students that mathematics covers a much wider range of topics than the ones found on the highway from arithmetic to calculus, including many topics that are intrinsically far more interesting to teenagers. You can find out about these offerings by clicking the above links. In this post, I want to focus on how proof figures in those courses.
Students at that age are more mathematically mature than they were when they took Geometry. This is especially true of students who choose to take a math elective course off the Calculus highway, and we have an opportunity to reach them with powerful, foundational ideas.
Geometry 2
Early in my high school career, a student asked me “Why is there an Algebra 2 course, but no Geometry 2?” I didn’t have a good answer, and I eventually decided to design such a course, which I called Space. One topic in the course was an exploration of polyhedra, which was largely but not exclusively based on building 3-D models with Zometool, using lessons from Zome Geometry, a book I co-authored with mathematician-artist George Hart. Students were guided through proofs of the fact there are only five Platonic solids, as well as Euler’s and Descartes’ formulas about polyhedra. Hands-on work with Zome created a uniquely motivating context for those proofs.
A highlight of the course was the protracted development of the proof that for any two congruent figures in the plane, one is the image of the other in a translation, a rotation, a reflection, or a glide reflection. This was quite a project. First, I had to introduce students to (very basic) abstract algebra, because the proof relied on manipulating the composition of geometric transformations. Then, we needed to prove a number of fundamental theorems. And finally we combined all this in an epic case-by-case study to reach our destination. You can read a detailed overview of this, illustrated with interactive GeoGebra applets, here. A multi-week project like this one is not commonly attempted in high school, but the step-by-step development of the proof and the use of interactive geometry software made it accessible. Arriving at the punch line was quite satisfying for the students, as they got to appreciate that not all proofs fit on half a page, and that one branch of mathematics can throw light on another.
Proof by Contradiction, Proof by Mathematical Induction
I once had a student whose mother tried to talk her out of taking my Infinity class. The reason was that the class was reputed to be difficult, and she worried that her daughter (who was not a math superstar) might end up with a B. The student was not dissuaded. She took the class, got a B, and did not regret it. The reason is that the class touched on ideas that are thrilling to teenagers. (Of all the classes I taught, that was the only one that students discussed with their parents at the dinner table!) One key idea was that unlike the rational numbers, the real numbers are not countable. That, of course, can only be understood with the help of Cantor’s diagonal proof — a proof by contradiction. In earlier classes, we tell students that √2 is irrational, and that there are an infinite number of primes. But why should they take our word for it, when the classic proofs of these results —proofs by contradiction— are accessible to them once they are high school juniors?
Up to 10th or 11th grade, we encourage students to look for numerical patterns and generalize them using an algebraic representation. That is an important skill. However, some patterns do not generalize! After seeing examples of that, students appreciate the need to prove that specific patterns do generalize, which is where mathematical induction comes in. The fact that this kind of proof is related to recursive vs. explicit formulas (a topic that is familiar to students) and to computer algorithms helps to make this accessible and meaningful.
Conclusion
I hope I convinced you that teaching proof in high school can and should go beyond the Geometry course. I support NCTM’s recommendation that there should be a range of math electives in 11th and 12th grade. If it does get implemented, some such courses can include “pure math” content, including a variety of approaches to proof. The Common Core State Standards (and thus most state standards) fail to include proof by contradiction and proof by mathematical induction, or in fact any mention of proof outside of the Geometry course. That’s too bad, because while those concepts are not essential in any practical way, they should be part of a well-rounded education — just because they are interesting.