Question

Do math courses get less detailed/rigorous as the courses become advanced?

Answer 1

I recently finished a master's degree in mathematical logic, which is expected to be rigorous. I would say that 80% of the lecture time was pure, unfiltered handwavyness. However, filling gaps and formulating actual rigorous statements was 100% expected of the students. I personally think as far as lectures go, that the higher the math, the more important it is to convey ideas in a fluent, often informal way. Then, if you are mature enough, you will be able to "clean it up" in your study time.

There are however, some who say that lectures are an absolute waste, and you should always skip them and go straight to books/written sources. I don't share this view, but I understand it.

Answer 2

For calculus 3, this makes sense--in the US, at least, calculus 3 is usually taught not for math majors, but for general STEM majors, and little is proven. For group theory this surprises me--what in an introductory group theory class did you need to take on faith?!! What possibly could have been too hard for your professor to prove?!?

But there's a second issue you describe. "The proof is the same as you've seen in previous courses" is a way to save time. Nobody can do any interesting math by going back to first principles for every single argument; as much as people clown on professors for saying "x is trivial," oftentimes they say that to mean "you should be able to figure this out on your own at this point in your education!" This is not an issue of rigor, it is an issue of the professors assuming you have the competence to fill in details. Maybe you can think they're unreasonable in what they want you to fill in, but that's a different issue.

Answer 3

Less detailed may be true, in the sense that there is only a limited amount of time to cover material so "trivial" things are left out. These things should be obvious in the sense that if you were asked to write a proof, it shouldn't take too long to come up from the definitions.

I suppose technically speaking, handwaving a proof *is* less rigorous, but typically it tends to happen only when the proof itself should be simple enough to produce on one's own.

Answer 4

you are probably expected to fill in the missing details yourself.

Answer 5

They get more rigorous.

Answer 6

Same thing goes for engineering classes. My 100-200 level classes had great teachers who could show that they understood the material and wanted to share their knowledge. In 300-400 level, you're completely on your own and the "teachers" are really just researchers who are required to stand in front of you and talk for a few hours at a time. It's unbelievable that we pay so much for this, especially when all of the same information is available for free on YouTube, where it's also explained more clearly than in any class I've ever paid for.

Answer 7

Many things will be proven in class, but I’ve found that even in rigorous courses where many things are proven, there are some proofs that are simply not suited to presentation in a lecture—they’re so long or messy or technical or filled with awkward notation that the only way to really follow them is to read them at your own pace—and plenty of professors will not waste time on these sorts of proofs in class, even in courses where hand-waving is kept to a minimum otherwise.

Answer 8

Can you give some examples of some group theory material which wasn't proven? In my group theory course every single fact was proven, with exception of some proofs which were left as homework.

Answer 9

The answer is yes. In many graduate level courses, often the proof of a theorem is a sentence "proved by ...... in 2018". In situations where an argument is presented, the easy details are often omitted or left as an exercise and only the key steps were presented.

You don't seem to be anywhere near this level though. I have no idea what "calculus 3" is. Group theory is like the first thing you learn as an undergrad (at least in my undergrad university, Group Theory is a compulsory course for all year 1 math students in the first semester). It makes no sense to handwave through such a fundamental course unless the course is not designed for math majors. It's important for the fundamentals to be as rigorous as possible, even at the cost of teaching speed.

I know that physics and economics undergrads need to learn some group theory in year 2. When I was an undergrad there were some econ friends asking me to explain homomorphism to help with their econ-major course. It's reasonable for such group theory contents to be handwavy because non-math students don't need to know proofs.

Answer 10

As you grow in mathematical maturity, you are expected to look up proofs of interest to you on your own. There is no point in tediously going over every single proof throughout the course.

This is especially true when the proof in question follows a standard structure or uses a technique students are supposed to be familiar with. This is why your world renowned professor taught that way. What's the point in going over a proof if it's basically the same thing you've seen multiple times before? You should be able to see how the proof goes and execute the proof yourself. That's what you're being educated to do, and your professors trust you to be able to do it.

The lectures should focus on proofs which demonstrate novel concepts, ideas, and techniques which the students have not seen before. This way more time can be spent discussing and analyzing new material, instead of wasting time going over the well-known things again.