How to Study Mathematics (2017)

Intuition vs rigor

• Several comments contrast “school-level” intuitive math with university-level abstraction.
• Common view: intuition is crucial but not sufficient at higher levels; it must be rebuilt on top of formal experience.
• Intuition itself changes: from visual/geometry-based in school to tool/structure-based in advanced math.
• Experience and exposure are framed as prerequisites for useful intuition; early university often feels like an “intuition vacuum” until that experience accumulates.

Definitions, theorems, and proofs

• Strong emphasis on memorizing exact definitions; precision is needed to check proofs and avoid subtle misconceptions.
• Some argue “internalize, not memorize,” but others reply that beginners usually need literal memorization first, then internalization follows.
• Multiple people want textbooks to include a short “reason” or “motivation” line explaining why a theorem is true, in addition to a formal proof.
• Mixed views on proofs: some recommend memorizing only outlines; others argue that deeper proof recall is essential if you want to prove new results.

Problem-solving and practice load

• Many advocate solving lots of problems (even all exercises in a text) and filling in every “obvious” proof step.
• Others note practical limits: some books (e.g., dense analysis or statistics texts) have hundreds of proof-style problems and can’t realistically be exhausted during a course.
• Concerns about textbooks without solutions: students may get stuck or be unable to verify their work, hurting motivation.
• Debate over “brute-force” exposure: some see it as key to real understanding; others think more conceptual, analytic resources are needed as well.

Study strategies and the university transition

• Techniques mentioned: spaced repetition of definitions, reflective study diaries, intense peer study groups, and active participation in TA sessions.
• Several describe the shock of moving from “can do everything” in high school to feeling completely lost in university; this confusion is framed as normal and even necessary for learning.
• Warnings about falling behind in foundational courses, where gaps quickly compound.

Enjoyment, motivation, and effort

• One thread stresses that enjoying math is a major predictor of persistence and success; another counters that confidence and early small wins matter more for many learners.
• “Sitzfleisch” (ability to sit with hard problems for long periods) is praised as a key trait, though there are anecdotes of highly imaginative people relying more on collaborators’ persistence.
• Some see fear of failure with exercises as a major obstacle.

Teaching quality and making math engaging

• Several commenters blame poor teaching, “just trust me” attitudes, and lack of motivation/intuition in lectures for students’ struggles.
• Suggestions to rekindle curiosity include off-syllabus explorations: map coloring, infinite series paradoxes, spherical triangles, and “pathological” curves—used more as playful exploration than fully rigorous study.

Tools and resources

• A few mention external outlines and modern adaptive platforms as helpful for self-study and spaced practice, with positive personal experiences of relearning or advancing in math later in life.