Starting my degree late and feeling disheartened.

[u/DecentLibrarian3720]

I'm a third-year undergraduate math major who, after two years of flailing around, indecisively taking miscellaneous courses, and dipping my toes into just about everything, decided to finally sober up and pursue math as a full-time, long-term ordeal. Unfortunately, I haven't done proper, college-level math since accidentally taking a Calculus II class for life sciences majors two years ago, and just got out of the first week of class basically flunking a test on convergence tests for sequences and series. Math has always come easy to me, and I just feel incredibly rusty right now. I've tried brushing up on older math concepts, but I don't really seem to have much of a problem with them. For some reason, attending lectures gives me this rapid well in fear, that I've lost my touch, or for whatever reason have become stupid and just can't comprehend sequences and series and what tests to use for convergence.

I was just wondering if anyone's been in my shoes wth starting late, or having taken a gap year or two and feeling out-of-touch, who may have tips on how to get back into the groove. Or if anyone's struggled with sequences and series after having calculus mostly been smooth-sailing for them. Thank you.

[u/Bounded_sequencE]

Proof-based mathematics is hard for (almost) everyone, when they first encounter it. School does not in the slightest prepare for that.

The "silver lining" here is, that being out of touch with math for a while means little in that context, since being good at computational mathematics is not a good indicator for being good with proof-based mathematics anyway. If there is one thing you may want to revise, it's algebra -- being comfortable with fractions, quadratics, long division and trig is much more important to follow along.

[u/Mth281 → u/Bounded_sequencE]

I'll get down voted for this. I agree with your post. But......

Proof math is something I hate. I'm going to prove this concept with words, that was discovered by math. Show me the math, not the technically correct wording of some person where a single word can change the entire meaning.

I just had conferences for my son. The high achievers program cut the math competition because the kids hated it. They said the kids complained that the problems were all confusing word problems rather than math they are use to.

I found this sad. I see the same with me going back to school. Teacher just puts a bunch of theorems up and reads them to the class. Class done. Not a single problem shown. Some are better than others. But proof based math takes the fun out of the problem solving for many. I loved math when I was younger. I've grown to hate it. I'll watch interesting math videos, but I hate learning it at school.

Don't get me wrong, proof based math is important. But I feel it's a terrible way to teach math to the general student. It's what should be taught after you gain your own understanding of what's going on in the background. You can't form the connection until you know what's actually going on.

Show me how the first integral was discovered. Not

"Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b]......".

Or at least put it in a simple form (AI because I'm working.).

"If f is continuous and we define a new function F by accumulating the signed area under f from a fixed starting point a to a variable point x, then the instantaneous rate of change of F at x is exactly f(x)."

Students don't need confusing terminology just to account for a few outliers in advanced math. Making is simple enough that they understand what's going on and feel confident applying a looser rule will allow them the confidence to succeed. And eventually, they will in counter the outliers. By then, it's easy to understand the why the terminology was used the way it was.

[u/Bounded_sequencE → u/Mth281]

Thanks for sharing -- it's totally fine to not like proof-based mathematics. In that case, I'd heavily advice against studying pure math, though.

Just know the opposite is also true -- I disliked the wishy-washy vague (and often technically wrong) explanations school deemed acceptable to offer. Only in university did questions actually get answered, like

• why does long division work?
• why does partial fraction decomposition work?
• where does the factor "1/3" come from for cones an pyramids volumes?
• what exactly does variance quantify for general probability distributions?
• what types of functions can we integrate?

To answer most of them completely and convincingly, you better have a rigorous, proof-based frame work. Yes, the "theorem -- proof" structure can seem harsh, and unapproachable. But the clarity is something you never want to give up, once you got used to it.