> If you don't teach proof-based mathematics at the undergraduate level then aspiring mathematicians would have to take those courses at the graduate level. Thi

> If you don't teach proof-based mathematics at the undergraduate level then aspiring mathematicians would have to take those courses at the graduate level. This would cause advanced degrees to take considerably longer to obtain. No it wouldn't. Most math PhDs require six or so courses before the research - you can begin with analysis or abstract algebra for proofs. The current system does have unnecessary redundancy. > Like programming, writing proofs is a skill that takes practice to get good at. If you were to suggest that we shouldn't teach programming until graduate-level CS courses, you would be laughed out of the room. Programming is more comparable to the computational mathematics that I've already mentioned should be in undergrad; more to the point, a lot of computer science theory need not be taught in undergrad either. Computer science theory is much closer to proof-base mathematics. > I think the best mathematicians in history started learning how to prove things as children. Approximately no one is one of the best mathematicians in history. That's an attractive idea, but I think it's more realistic to assume they were extraordinarily precocious than that proofs are easier to digest at a younger age. I could point to people who mastered computational mathematics at a young age too (like Einstein, at 14), but that doesn't say anything about which is easier to learn. I'm talking about which is more widely applicable for practical work.