[bpyne]

My biggest problems as a math major at the undergraduate level were proofs. I could muddle my way through abstract algebra proofs but real analysis just didn't click.

The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.

[bitxbitxbitcoin]

I had a similar experience doing my math minor. All the way through Caluclus classes and Diff EQ and up until the first half of Linear Algebra, everything is plug and chug. After the first half of Linear Algebra when proofs started making an appearance, I came to the realization that proofs are another type of mental activity entirely. Survey of Algebra, Basic Real Analysis, and even the dedicated proof writing course I took were all exponentially harder to pass.

[vharuck]

>the dedicated proof writing course

What was yours like? The one I took didn't cover much on the actual writing of proofs. The professor accepted reasonable essays with high-school level notation. Instead, he gave us a toolbox for proving things: pairing terms in a series to find the sum, rewriting recursive equations, etc. It was mostly to show that clever tricks are how mathematicians prove new things. But that might've been because the professor was a guy who reveled in clever solutions.

[bitxbitxbitcoin]

It has been over a decade so I don't exactly remember - I looked it up and the class was Introduction to Mathematical Reasoning.

What I personally recall was following along in class while the professor walked through classic proofs emphasizing what each new notation meant as well as the difference between Direct, indirect, and induction proofs. Tests and assignments were essentially recreating proofs cherry picked to be similar to ones we walked through in class.

I realize that's kind of how all my higher level math classes were run. It's just that once the cherry picking becomes looser, intuition doesn't necessarily catch up :(.

[bpyne]

My university realized the need for a proof writing course a little too late to help me. Students who did well in the more abstract math classes, aside from the outlier "gifted" mathematicians, formed study groups. I wasn't mature enough at the time to realize how valuable those groups were so I went it alone and my grades reflected it.

[bitxbitxbitcoin]

I hear you! I wasn't mature enough to join study groups or go to office hours during undergraduate, either, and my grades also reflected it.

[stephen_greet]

I had a very similar experience during undergrad. I loved most of my classes through Linear Algebra but developing proofs felt like trying to learn a new language. Unfortunately, it never really clicked for me.

I'd love to know how to develop an intuition for writing proofs from scratch.

[v64]

I also strugged with analysis. I always felt like epsilon-delta proofs involved pulling some absolutely strange value for delta out of your ass that happens to work out in the end, and I never developed an intuition for that. Same with integrating by parts, oh it just works out so nicely if you rewrite u in this totally obtuse way.

[vector_spaces]

The tricky thing with analysis that I don't think many professors are good at conveying is that the ordering of statements in the proof isn't the same as the ordering of steps the proof writer performs to come up with the proof.

Basically you sort of write the broad strokes of the proof up front, leaving the right hand sides of statements like "Choose epsilon such that epsilon = __" blank. Then you do a bunch of scratch work to figure out what epsilon needs to be so that your proof works out in the end.

Another challenge with analysis is that inequalities are central, so fluency in their manipulation is absolutely critical. And naturally most students aren't fluent with them by the time they take analysis, so they get bulldozed by Baby Rudin and learn to hate a pretty cool (and useful) branch of math

[bpyne]

Yes! That was my experience exactly.

[kevmo]

Highly recommend the book "How to Prove It" by Velleman.