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?

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A trifecta on classical education

Gene Veith, one of my favorite religious writers and the proprietor of the terrific Cranach blog (and provost at Patrick Henry College), has three quick posts today on classical education. He touches briefly on teaching content rather than process, and how classical education teaches bothl; on critical thinking; and on learning styles and the teaching of “meaning”. Some clips: 

The key factor in learning is grasping meaning, a concept that evades any of these sensory approaches. (While cultivation of meaning is what classical education is all about.)

and:

More substantive scholars say that being able to think critically requires (again, see below) CONTENT. You have to think ABOUT SOMETHING. Whereas much of the critical thinking curriculum is all process, trying to provoke content-free thinking. (The classical solution: DIALECTIC, featuring questions AND answers, as in that great model of classical education, the catechism, which, properly used, helps the student answer the question, “what does this mean?”)

I am pretty sure that Prof. Veith has this overall definition of “classical education” in mind, but I am not sure exactly how he defines it. And I wonder if all of what he says still works if you replace “classical” with the more generic “liberal arts”.