[wenc]

So I've taught undergrad courses, and my sense is, the average US college engineering sophomore with linalg and say Calculus II (but no Real Analysis) might struggle with this material somewhat. They may know the material for linalg and calculus (and may have gotten As), but my feeling is that many would not have reached the mathematical maturity to truly internalize concepts.

I would place this course maybe at the senior level (with graduate level cross-registration)...400-500 level elective.

What is your sense?

--

Side note: it's interesting in that in other countries, e.g. say France, the math curriculum is so darned advanced. In undergrad Year 1 at École Polytechnique, real analysis and variational methods are already covered in common courses.

https://programmes.polytechnique.edu/en/ingenieur-polytechni...

Functional analysis in Year 2.

https://programmes.polytechnique.edu/en/ingenieur-polytechni...

Then again the top French schools filter out non-math folks via classes prépas and exams.

[nightski]

I'm in the demographic you describe, yet I've had a hard time finding resources to develop that "mathematical maturity" short of going back to graduate school. Which I'd love to do, but am at a point in my career where that would be devastatingly expensive to my future since these are the prime earning and wealth building years.

I wish there was more of a self directed way to achieve this.

[ghufran_syed]

I just finished an MS in math and statistics a long time after doing a non-mathematical undergrad degree. I feel a lot of what is called mathematical maturity is actually getting comfortable doing proofs, which I think is hard to learn while also learning more advanced math. I would recommend working through "Mathematical proofs" by Polimeni/Chartrand/Zhang. Unlike math at an earlier level, you can't just check your answers against the official ones to see if you made a mistake - writing proofs is more like writing essays, the grammar is the easy bit, it's the process of putting the arguments together in the right detail and the right order that's important and hard to do without feedback. So you also need to get feedback from mathematicians on your proofs if possible. The best way to do that if you don't have a buddy who happens to be a mathematician is to learn to use LaTeX and ask questions on math.stackexchange.com.

An alternative is to do the proof and abstract algebra courses via (asymmetric) distance learning at https://westcottcourses.com/courses.html

I know someone who took these courses and felt like they got good feedback on their homeworks from the profs running the course.

Feel free to get in touch if you want to chat, I spent a long time trying to self-learn this stuff before starting my math MS, so happy to help in any way I can!

[marai2]

Since it's seems like you were already motivated and interested in learning Math on your own, how would you describe what your learnings were before you enrolled in formal studies? In other words if you could travel back in time to talk to yourself before you made the decision to enroll in a Master's program, what would the younger you have asked the older you and what would be the response? For example I'm thinking a reply might be like "well you're going to miss out on opportunities xyz by commiting to a Master's program, but because I know you and know you wouldn't be happy without satisfying your desire to learn Math in a more formal study the trade of is worth it. And you don't know this yet but when you start learning about P,Q,R you'll really get a kick out of it" :-)

[nightski]

Thank you for the resources, it is greatly appreciated!

[wenc]

Just curious, the topic above aside: what would you say is the main reason behind why you'd like to strive for "mathematical maturity"?

(from what you wrote, I gather it's not for reasons of vocation?)

[int3]

I'm currently doing MITx's Fundamentals of Statistics MOOC, which seems fairly similar to this one. The course material doesn't require too much understanding of real analysis, though the instructor does make cursory acknowledgements of the "technical details" he's glossing over, presumably for the more advanced students. The only analysis concepts we've used are continuity, differentiability, and convergence. It isn't a proof-based course: the instructor does prove some theorems in class, but the psets are all just computation. I do agree that it'll be harder to internalize some of the concepts w/o a background in analysis, but I think you can get a reasonable amount out of the course regardless.

That said, I did study analysis (but not measure theory) in college, and I don't entirely remember what I learned from calc vs analysis classes, so I may be a bit off here.

[doukdouk]

The "undergrad Year 1 at École Polytechnique" is really the junior year, since the freshman/sophomore years of university education would have be done in prépas. It is undergrad, and it would be their first year at the school, but it is quite misleading to call it "undergrad Year 1". Given that undergrad is three years in France, "undergrad Year 1 at École Polytechnique" means "last year of undergrad".

[wenc]

Ah what you say is true... upon examination, this curriculum is for the 4-year ingénieur polytechnicien program, which culminates in a Master's degree (diplôme d'ingénieur).

[doukdouk]

Note that this is unusual in that it lasts four years, not three.

Standard curriculum is three years of bachelor's, then two years of master's.

In the prépa - engineering school track, it is two years of prépa, then three years of engineering school that gives out a master's in engineering.

Thus the first year of engineering school maps to a (third) last year of undergrad, the second to a first year of a master's and the last to the last year of a master's.