I meant "problems" in a broad sense -- I loved disorganized professors who would pause and stare at their lecture notes in silence for a minute, realize their proof or example contained some flaw, and then have to correct it on the fly.
I found those moments really valuable if course-correcting was non-trivial -- the typical Definition-Theorem-Proof-Example format certainly is essential for organizing one's thinking and communicating new math in a way that's digestible to other mathematicians, but it is not how mathematicians actually think about math or solve novel problems
In the grad analysis sequence this "course correcting" mechanic was built into the course, since we were required to regularly solve a challenging problem and then present its proof to the class and withstand intense questioning from both the professor and peers. If you caught an error in someone's proof and could help the presenter arrive at a correct proof, you'd both earn points.
The thrill of surviving an incredulous "Wait a second..." from that particular professor (who later became my research advisor) was hard to beat
Anyway my intent was to analogize math lectures (whatever they might look like) with language courses or immersion in the sense that they are an opportunity to practice speaking and listening, and to immerse yourself in cultural norms. I think it goes a bit deeper than this, in that language is inextricably connected to most thought and vice versa -- we experience this in a very explicit way whenever we find our thinking clarified in the process of formulating a question, but it's always there
That said, pure immersion for language learning is actually easy to beat -- lots of research shows that immersion together with explicit grammar instruction has far better learning outcomes than immersion alone. Immersion alone misses lots of nuance -- and it relies on the speaker being acutely aware of the difference between their output and target forms.
With your verb conjugation example, lots of time can be saved by knowing that there's a thing called the subjunctive and that it is distinct from tense and it shows up in a myriad of places tending to concern hypotheticals
Similarly, I gain a lot from talking to mathematicians and attending conferences. But I also need to spend time alone consulting relevant theory, reading papers, and playing with examples. Both are important, but in math it seems you one get away with less immersion
I found those moments really valuable if course-correcting was non-trivial -- the typical Definition-Theorem-Proof-Example format certainly is essential for organizing one's thinking and communicating new math in a way that's digestible to other mathematicians, but it is not how mathematicians actually think about math or solve novel problems
In the grad analysis sequence this "course correcting" mechanic was built into the course, since we were required to regularly solve a challenging problem and then present its proof to the class and withstand intense questioning from both the professor and peers. If you caught an error in someone's proof and could help the presenter arrive at a correct proof, you'd both earn points.
The thrill of surviving an incredulous "Wait a second..." from that particular professor (who later became my research advisor) was hard to beat
Anyway my intent was to analogize math lectures (whatever they might look like) with language courses or immersion in the sense that they are an opportunity to practice speaking and listening, and to immerse yourself in cultural norms. I think it goes a bit deeper than this, in that language is inextricably connected to most thought and vice versa -- we experience this in a very explicit way whenever we find our thinking clarified in the process of formulating a question, but it's always there
That said, pure immersion for language learning is actually easy to beat -- lots of research shows that immersion together with explicit grammar instruction has far better learning outcomes than immersion alone. Immersion alone misses lots of nuance -- and it relies on the speaker being acutely aware of the difference between their output and target forms.
With your verb conjugation example, lots of time can be saved by knowing that there's a thing called the subjunctive and that it is distinct from tense and it shows up in a myriad of places tending to concern hypotheticals
Similarly, I gain a lot from talking to mathematicians and attending conferences. But I also need to spend time alone consulting relevant theory, reading papers, and playing with examples. Both are important, but in math it seems you one get away with less immersion