#mathblr go ahead and argue the point

[transientpetersen]

My expectations for a good undergraduate-level education in mathematics:

Good grounding in proof techniques and an introduction to formal proofs

 can be combined with a particular topic, often topology or analysis

 should be very early in your studies to get the most benefit

Introduction to broad areas of mathematics

 topology (nice to have before calculus since it provides another way of thinking about continuity)

 analysis (complex, real, manifolds, etc.)

 algebra (groups and fields at a minimum)

 combinatorics or number theory or graph theory (so you're forced to deal with at least one construct that does not relate to vector spaces)

Study of more specialized kinds of calculation

 calculus and diff-eq (nice to do this after/alongside analysis)

 linear algebra (applied manifolds)

 statistics (if you're lucky you get some measure theory here)

 numerical methods, linear programming (if you feel the siren call of computer science)

The first category, I would expect your typical fresh student to take two semesters with: the first semester to internalize that there is more to mathematics than calculation and the second semester to actually learn how to write proofs. If you're already comfortable with the idea then this can be accelerated to one semester in which you're practicing the forms.

In the second and third categories, each of their four subareas can easily take two semesters of work so you're probably looking at two years (each) unless you want to get particularly aggressive. Typically students will prune more of the advanced topics from the second category if they want an applied degree or prune the third category if they want a more theoretical degree.

For a graduate-level, I'd expect to condense all of the above into three or four semesters with either the less advanced topics dropped (for those who've already studied mathematics) or some subcategories dropped (for those who haven't). Then a trailing semester to perform research.