# What is a classical education approach to mathematics?

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction:

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

(I took out the mini-rant against the gosh-awful Saxon method.)

Any thoughts from the audience here?

Filed under Education, Liberal arts, Math, Teaching

### 9 responses to “What is a classical education approach to mathematics?”

1. I’d like to see the anti-saxon rant. I have three elementary school kids taking saxon math at public school right now. The younger two, who have never know any thing else, are doing ok, but the older one who started on a different method, is swimming. They never master anything before they move on. He is in a constant state of confusion. My wife is a product of saxon math. She’s bright, talented and an excellent manager of business. But she still struggles with fractions. Not exactly what I’d wish on my kids.

I’d like to see a well thought out argument against saxon that I could use to help get them to let at least my oldest kid use a different method.

2. @Jay: You can actually read the mini-rant if you go to the comment I left at Veith’s site. Upshot: Saxon kids learn how to solve any problem, so long as it looks EXACTLY like a problem from the Saxon book with NO DEVIATIONS. But if you get them into a problem they haven’t seen before, the transfer of skills which are drilled in the exercises to tools that need to be employed in the new problem is basically zero. Saxon trains students not in math, but in the exercise sets found in their books, and call me crazy but that’s not learning math.

3. Thanks, I’ll go over and read the rant. Your assessment is exactly what I’m seeing in my kids. Not what I want for them. As an adult educator who often teaches things involving applied math, I see wide variation in people’s math skills. In fact, just last week I taught a class that involved calibrating a center pivot irrigation system to apply chemicals along with the water. You haven’t lived until you try to teach pi r squared and 2 pi r to 60 year old men who just want to get their permit. 🙂

4. What do you think about the http://www.mathematicallycorrect.com site?

You mentioned that students who had used Saxon do poorly in your classes. What textbook did the students who do well in your classes cut their teeth on?

While we have used the Saxon materials in the early grades with great success, I find that I’ve had to supplement my more advanced students with things like Harold Jacobs’ texts, Martin Gardener’s puzzles, and the like.

To weigh in on the Saxon method — I see it as useful for elementary math — “grammer” level up to high school algebra. It needs to be supplemented and teacher input is required to connect the little nuggets presented in each lesson. I might draw an analogy to a basketball player shooting lots of foul shots — eventually the shot is smooth and the player doesn’t spend any conscious energy on the mechanics of the shot. This can free up mental energy as concepts are introduced…

I hate to see my intro university physics students get hung on simple algebra — handling fractions or systems of linear equations — when they are trying to apply Maxwell’s equations.

As for Classical Mathematics education — Just a comment on elementary-high school level instruction for now. I think much of the “grammar” could be greatly enhanced and made more interesting — definitions and facts can be layered with historical context; some significant time spent in Euclid’s elements — working the constructions during middle school geometry can be easily made accessible to middle school students, for example. So can Euclid’s geometric proof of the pythagorean thm. …include early introduction to logic and proofs/derivations — for example, a simple proof that the sqrt(2) is irrational …

6. Oh-my-god. We have not seen the likes of Keister since Morris Kline. He even besmirched Baby Rudin!

7. I found a used copy of Zwier and Nyhoff online and bought it. The 1969 copyright date was its real selling point.

“It’s out of print but…” seems to be the way I begin way too many discussions these days.

8. Editorial: For context on Adrian’s comment above, go to Veith’s original post and scroll down to comment #22. (It’s a little confusing to have both the pro- and antagonist in that comment sub-thread be named “Adrian”. though.

@Myrtle: Yes, 1965-1970 were good years for math books. I’ve blogged my praise for those books before.

9. It’s a little confusing to have both the pro- and antagonist in that comment sub-thread be named “Adrian”. though.