User:IssaRice/Tiers of learning in mathematics

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Name Description People at this level
Computation-based The material is taught, but proofs of the main results are not covered. Example: high school/freshman calculus courses, in most cases. High school students, lower-level undergraduate students
Proof-based, but focusing on exercises The material is taught using proofs, but where the student is not expected to prove the main results themselves. Instead, the student is expected to be able to solve (often artificial) problems (including proof-based problems) that make use of the main results. Example: math 13X and math 33X at UW. Upper-level undergraduate students
Building the theory from scratch The material is taught using proofs, and the focus is on being able to prove the results oneself, where the exercises are the proofs of the main results. Example: Tao's Analysis; reading a textbook that gives proofs of main results, but where one tries to prove the theorems oneself first. Graduate students, most mathematicians (?). My current feeling is that most mathematicians stop here for subfields that they do not specialize in (but I don't talk to enough mathematicians to know).
Building the theory from scratch in parallel The material is taught using proofs and the focus is on being able to prove the results oneself in multiple ways. The difference with "building the theory from scratch" can also be stated in another way: instead of being given an ordered list of theorems to prove (where the order gives clues about how to prove theorems), one is given an unordered list of theorems to prove, and asked to derive a "plan of attack" by combining "earlier" theorems to prove "later" ones. Example: working through multiple analysis books in parallel and keeping a dependency graph of theorems in one's mind as one does this. Or in linear algebra, instead of being given definitions for things like linear independence, span, basis, and so on, one knows what one wants to show (e.g. "I know intuitively what a basis should look like") and then comes up with definitions and theorems to prove this. Textbook authors (?). My current feeling is that textbook authors get to this level, and also people who specialize in a field. However, even people at this level might only do this in an ad hoc manner/intuitively, rather than systematically drawing dependency graphs and checking each possible edge and so forth. Even with textbook authors, most of them choose only to write one path through "theorem space" in the end; they must have alternative paths in mind (some might put these in exercises), but they never seem to talk about how they made their choice of which path to write about and why it's better than other paths. (Note: again, I don't talk to enough mathematicians to know.)
Reverse mathematics Mathematics from a "logical strength" point of view, where one is building the theory from scratch in parallel but also tracking the strengths of various results. I think people at this level might also have a strong historical understanding of math, and be able to visualize how the results came to be proved, but also have "alternate histories" in their mind. John Stillwell :D

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