A Lot of Advanced Math Is Taught Backwards

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When you’re a math major in undergrad, and you get up into the junior and senior level courses:

• Real Analysis
• Abstract Algebra
• Topology

You start to realize that a lot of this stuff is really taught backwards.

Theorem, proof, theorem, proof.

But you don’t really have a concrete grasp of what’s even going on.

You’re just pushing symbols around.

Maybe you can follow the rules, but you don’t really understand the subject.

That’s because you need the concrete examples first.

You gotta work through computations and then do the abstraction.

Skipping the concrete stage doesn’t make the course rigorous.

It just makes it inefficient.

Further reading: The Necessity of Grinding Through Concrete Examples Before Jumping Up a Level of Abstraction .

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