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 education from the University of Wisconsin–Oshkosh, both completed about 20 years ago. Looking back, I don’t feel the programs prepared me well for advanced mathematics. The emphasis was heavily on pedagogy rather than mathematical maturity or proof-based thinking. That was fine for most of my career as a community college educator, but now that my kids are getting older and moving out of the house, I want to study mathematics more seriously (possibly even pursue a PhD). Not because of the title, but because I genuinely love the subject and want deeper fluency and structural understanding.
The problem is that I feel trapped between levels. Introductory material often feels shallow, while more advanced material is overwhelming. For example, I recently spent a good part of an afternoon working through the proof of the division algorithm. The statement itself was immediately clear (as it probably should be), but the proof left me thinking: “I understand the argument and can reproduce it, but how would someone come up with this on their own?” This happens to me often and I know I'm missing a transitional piece that I need to be successful.
Lately I’ve been wondering whether the right approach is to stop trying to “jump ahead” and instead work slowly through foundational proof-writing texts (maybe Velleman or Hammack) alongside a focused subject like elementary number theory, spending substantial time on examples until abstraction starts to feel more natural. Right now, I often feel like I’m spinning my wheels more than actually progressing.
Has anyone here rebuilt their mathematical foundation later in life in a similar way? If so, what approaches, habits, or resources helped most?