A Problem With High School Math Education

When I studied math in high-school, I absolutely hated geometry. It pretty much made me give up on math entirely, until I renewed my fascination in university. Geometry and I had a hate/disgust relationship since. I am putting it out there so if you want to blame me for being biased, hey, I just made your day easier.

But it made me question: why are we even teaching geometry? Unlike the ancient Egyptians, few of us will need to collect land taxes on rapidly changing land sizes. The usual answer is that geometry is used to teach the art of mathematical proofs. There are lemmas-upon-lemmas building up to interesting results. They show how mathematical rigor is achieved, and what it is useful for.

This is bullshit.

It is bullshit because there is an algorithm one can run to manufacture a proof for every true theorem, or a counter-example for every non-theorem. True, understanding how the algorithm works takes about two years of math undergrad studies. The hand-waving proof is here: first, build up the proof that real number pairs as points and equations of lines as lines obey the Euclidean axioms. Then show that the theory of real numbers is complete. Then show the theory of real numbers is quantifier-free. QED.

When we are teaching kids geometry, we are teaching them tricks to solve something, when an algorithm would do it at well. We are teaching them to be slightly-worse than computers.

The proof above is actually incorrect: Euclidean geometry is missing an axiom for it to work. (The axiom is “there is no line and a triangle such that the line intersects the triangle in exactly one edge”.) Euclid missed it, as have generations after him, because geometry is visually seductive: it is hard to verify proofs formally when the drawings look “correct”. Teaching kids rigor through geometry is teaching them something not even Euclid achieved!

Is there an alternative? I think there is. The theory of finite sets is a beautiful little gem of mathematics. The axioms are the classic ZF axioms, with the infinity axiom inverted (“every one-to-one function from a set to itself is onto”). One can start building the theory the same as building the ZF theory: proving that ordinals exist, building up simple sets and the like — and then introduce the infinity axiom and its inversion, show a tiny taste of the infinity side of the theory, and then introduce the “finiteness axiom”, and show how you can build the natural numbers on top of it.

For the coup-de-grace, finite set theory gives a more natural language to talk about Godel’s incompleteness theorem, showing that we cannot have an algorithm for deciding questions about finite sets — or the natural numbers.

Achieving rigor is much easier: one uses Venn diagrams as intuitions, but a formal derivation as proof. It’s beautiful, it’s nice, and it is the actual basis of all math.

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