Frustration Studying Mathematics
As the title suggests, I’m feeling pretty stuck and frustrated with my self-study of mathematics.
My background is a BS in math education and an MS in math ed from a smaller state school, both completed about 20 years ago. I’ve spent most of my career teaching developmental math, college algebra, and calculus, so I’m very comfortable computationally and pedagogically. But I’ve realized there’s a major gap between being able to teach and compute mathematics versus being genuinely fluent in higher-level abstract mathematics.
I’m in my 40s now, and once my kids are out of the house I’d like to pursue mathematics seriously again, possibly even toward a PhD someday. Not because I’m chasing prestige or the PhD, but because I genuinely love the subject and want a deeper structural understanding of it. When done, I'll probably resume teaching because I enjoy sharing math with others.
Anyway...
The problem is that I feel trapped between levels. Introductory material often feels too shallow, but most advanced books assume a level of mathematical maturity, proof fluency, and abstraction that I simply never developed formally. I can follow ideas when they’re explained carefully, but I struggle to build intuition from dense theorem-proof exposition alone.
Lately I’ve been wondering if the right approach is to stop trying to “jump ahead” and instead work slowly through foundational texts in areas like algebra, linear algebra, number theory, discrete math, and proof writing—focusing heavily on examples, constructions, computations, and writing mathematics by hand until the abstraction starts to feel natural.
Has anyone here rebuilt their mathematical foundation later in life in a similar way? If so, what worked for you?