I consider the value in math lectures to come from the speaker’s explanation of why to expect certain things. Is this an obvious fact in another context, rewritten for this application? Is this a surprise? What reasons besides the rigorous argument are there for believing the theory?
A lot of the theorems I learned in school weren’t particularly amenable to intuitive explanations like that.
For example, take Galois theory. The fact that a polynomial’s solvability by radicals depends on the solvability of its Galois group is surprising and not intuitive at all. The fundamental theorem of Galois theory is a very technical result utilizing purpose-built mathematical structures that were developed specifically to study the solutions of polynomial equations.
