No Algebra till Junior year

The number one failure of classical education is teaching mathematics, and since I’m convinced that all non-vocational pre-college education should be classical, it follows that everyone is failing to teach math. I was reminded of this while re-reading an explanation that Brandon gave in 2016 of Whewell’s reasons for basing mathematics instruction on geometrical proofs rather than analytic mathematics (i.e. algebra and geometry).

[Whewell] is not claiming that analytical mathematics — algebra and calculus — are inferior mathematics, or that they are ill-suited to discovery of mathematical truths; in fact, he thinks the opposite is true. He is also not saying that they should not be part of education at all. His argument is rather that they are not good as foundations — education, insofar as it is mathematical, should build up to them, not take them as basic. It’s the perfection of analytical mathematics, its capacity for extremely abstract representation, that makes it poorly suited for getting people used to thinking mathematically and rationally. And this is because in analytical mathematics, as such, you don’t actually think about problems — you think about formulas and abstract relations that can be interpreted in many different ways for many different kinds of problems. If people get started too early on this, it is easy for them to start using these x’s and y’s and formulas concerning them as nothing but a crutch.

Read the whole thing, along with Rob’s development in the comments section. We’ve clearly fallen into exactly the fault that Whewell wanted to avoid, and have dumped the geometrical approach almost entirely while dedicating years to teaching analytic methods.

The fundamental pedagogical mistake is in this approach is that it teaches abstractions before teaching what they are abstracted from. The mistake borders on insanity – it is literally the same thing as teaching kids formal logic without telling them the word equivalents of the symbols they are using. We could certainly run a logic class like this, and treat the process of training the kids to use logical symbols as the same sort of thing as programing computers to manipulate the same symbols. The problem is that the sort of kid who succeeds in such a Kafka-esque curriculum will be more harmed by his education than enlightened by it, since he will be succeed only by suppressing his natural desire to understand what he is doingMath that doesn’t start with the concrete is therefore a sort of anti-education that teaches kids they can never know what they are doing or why what they are doing is true.

As Brandon points out, formal systems are far more powerful, rigorous, and simple than any geometry could hope to be. We’ve known for a hundred years how many things Euclid leaves out or relies on his pictures to prove, i.e. how many things he cannot reduce to an axiom or definition. But we’ve confused the necessity of getting past Euclid with the mistaken idea that we could teach math without him. The power and rigor of the abstraction belongs to it precisely as abstraction, and unless we teach the foundation we will be left with a madhouse game of translating coded-messages into different message codes forever. Most kids won’t get math, we shouldn’t expect them to, and they might well learn more from rejecting the pedagogy than accepting it.


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