Math pole (which class was the most difficult for you?)

[LTME]

I am curious which math classes gave people the most trouble during their time in college. I am not looking for a "this class is harder than that class" kind of ruling as I know it differs from student to student and school to school. I am simply curious to see if there will be a pattern in the collective personal experiences.
So for YOU the reader which class was the most difficult? Did they just get harder as you went along or did one stand out in the middle? Also just incase it proves to be different, which math class involved the most homework?
Thanks for your response.

 

[micromass]

For me, the most difficult class was "probability and statistics". However, this was also one of the classes that I enjoyed the most. It was fairly difficult, but I learned a lot.

 

[nonequilibrium]

Abstract Algebra, without a doubt.

I had had an introductory course into group theory before, but that was very light. This was the first course in abstract algebra (groups, rings, fields) and the high degree of abstraction of it made it totally unintelligible at first. It was only nearing the exam period that I got into it, and eventually I got quite comfortable at it! But it was the first time that I realized I could totally not-understand something to such a degree :) Interesting experience!

(and I never get homework, so that part of the question is irrelevant to me)

 

[Robert1986]

Well, there are two.

The first is sort of in the intersection of math and CS. It was Languages And Computation. The class was, essentially, about encoding problems into some language. By language, I don't mean German or English or something. There is a precise definition. Anyway, the whole course was hard, but it culminated in having to write an interpreter (like a compiler) for a language called "minRE" which was a scripting language similar to AWK (except for the minor fact that mine didn't work.) So, our intepreter had to be able to take a regular expression and search for it in a text document. Now, this was a group project, and the three people in my group were all math majors. The other two people were lazy which means I had (looking back on it, I'd say "I got") to do it all myself (except for some minor documentation, and one part of the coding portion.) I began the Wednesday before thanksgiving. I worked about 4 hours on thanksgiving, about 14 hours each day the friday, saturday and sunday after t-giving, and about 9 hours on monday (I even called in sick at work because I realized late sunday night that I was VERY close to being finished.) It was greulling, but the project actually sort of worked, and it was very rewarding. The other one would probably have to be Combinatorial Analysis. Interestingly, this has also been one of my favourite classes.

 

[OldEngr63]

Without a doubt it was differential equations. I took the course twice, once in the "pure" math department and once in the applied math department.

In the pure math department, the course was taught by a world famous full prof. We spent the entire semester proving properties of the solution of y'' + k^2 y = 0, but we never did actually find the solution, although by the end of the semester, most of us knew what it was. But we knew all sorts of properties of the solution! I thought to myself, "Well, that was worthless! Try again." I got a B in the course, although the one time I opened my mouth, I got cut off at the knees.

So I went to the applied math department and signed up for Diff Eq. The second time around, we did talk about a wide variety of differential equations, but since this course was taught by a grad student, mostly we heard about his research on the Wronskian. I asked a fair number of questions, did all the work, and got a C. I decided I had to move on and learn this material on my on, outside the math department which I ultimately did.

The following semester, I was in an EE class, and everyone else in there knew Laplace transforms, which I had never seen. I got busy learning transform calculus real quick.

 

[Poopsilon]

An introductory class in commutative algebra, geez that stuff is dry!

 

[20Tauri]

I suspect I will have harder courses in the future, as I'm not done with math yet. So far, though, linear algebra was the hardest and mathematical statistics was the most work. Linear was intense because it was the first time I was exposed to proofs or any meaningful level of abstraction. Abstract vector spaces were kind of beastly at first. I ended up learning some pretty cool stuff, though. Math stats was just a lot of homework. It was my first advanced sequence course, so it took up a lot of my time.

 

[homeomorphic]

Functional Analysis, Graduate PDE, Riemannian Geometry, Intermediate Algebraic Topology.

The functional prof is notorious for coming up with very hard problems. Also, his style was too abstract and unmotivated for me.

In PDE, the prof was obsessed with ugly calculations and hated intuition, so my eyes glazed over with boredom. It was more that I didn't have the stomach for it than that it was hard, although it was also fairly hard, too. Ditto with Riemannian Geometry, but combined with a break-neck pace and massive amounts of homework, which left me little time to think about anything besides getting a solution to as many problems as I could.

Algebraic topology was also taught in a fairly heavy-handed manner, similar to the functional analysis class. The definition of a fiber bundle came at us in all its glory, with very little motivation. That is a classic example of teaching for people who already know the subject.

The biggest difficulty for me is when I don't agree with the way the class is being taught.

 

[jbunniii]

I found algebraic topology to be very hard to wrap my brain around. I didn't like the material, nor the teacher's and textbook's (Hatcher's) style, which favored proof sketches instead of rigorous proofs as one finds in algebra or analysis, or point set topology for that matter. It left me with a distaste for topology that continues to this day.

 

[R.P.F.]

Functional Analysis, Graduate PDE, Riemannian Geometry, Intermediate Algebraic Topology.

The functional prof is notorious for coming up with very hard problems. Also, his style was too abstract and unmotivated for me.

In PDE, the prof was obsessed with ugly calculations and hated intuition, so my eyes glazed over with boredom.

Maybe it's because analysis is ugly?

 

[nonequilibrium]

Maybe it's because analysis is ugly?

Analysis is beautiful!

Actually, I once audited a functional analysis class. It was insanely hard (the rich structure of analysis, combined with the abstraction of topology and algebra) but utterly intriguing!

If I had actually taken that class instead of just auditing it, I would probably list that as my hardest so far (really good professor though! He pushed you to the extreme)

 

[homeomorphic]

I found algebraic topology to be very hard to wrap my brain around. I didn't like the material, nor the teacher's and textbook's (Hatcher's) style, which favored proof sketches instead of rigorous proofs as one finds in algebra or analysis, or point set topology for that matter. It left me with a distaste for topology that continues to this day.

Hatcher is pretty good, for the most part. But you might try Munkres Algebraic Topology book. More rigorous. Still, mostly pretty good book.

Anyway, sounds like you haven't had enough experiences that would show you the value of the way Hatcher does things. Sometimes, intuitive arguments can be unsatisfying if they are too far from being rigorous because when you have to try to write proofs, you don't have enough technique or whatever. You don't know how to bridge the gap between intuition and proof. So, I understand that frustration. But my priority is always to have some insight into why things work and what's going on behind the scenes, and Hatcher is very good for that, most of the time.

 

[R.P.F.]

Analysis is beautiful!

Algebra is (more) beautiful!

The hardest ones I had were a graduate topic course related to algebraic number theory and the abstract algebra sequence. I worked 30+ hours per week for algebraic number theory. They are also my favorite classes. Hopefully I can go into algebra in grad school!

 

[Robert1986]

Maybe it's because analysis is ugly?

Say what?!

 

[micromass]

Algebra is (more) beautiful!

Death to the algebrists!

 

[QuarkCharmer]

Any class that is essentially plug and chug. For me that has really only been Statistics so far. I just don't like it at all, and many of the concepts go unproven. I struggled with that class so much, because I just didn't want to do any of the work. Granted, I have only taken up to multivariable calc, DE, and intro linear algebra.

 

[homeomorphic]

Maybe it's because analysis is ugly?

Usually, when something is ugly, it's someone's fault. Hardy said, "There is no permanent place for ugly mathematics."

Functional analysis isn't ugly. I haven't quite figured out the right way to think about it yet, but a good start is the last couple of chapters of Mathematical Physics by Geroch. That's what I call beautiful.

And PDE is ugly in some ways, but not always. To see that, the best place to look is Vladimir Arnold's book on PDE. Beautiful. I love how I know that it's the prof's fault that it was ugly. That allows me a very sweet revenge when I finally understand the subject properly. When I took PDE, I thought, this sucks, I'm going to buy Arnold's book. And it was the perfect cure. Actually, it's a good example of a non-rigorous book that's hard to learn how to learn how to do proofs in the subject from. But still, the extra insight is more than welcome after seeing someone present the subject as if it were all about ugly calculations with not even the possibility of having any physical insight. What a contrast.

 

[homeomorphic]

Any class that is essentially plug and chug. For me that has really only been Statistics so far. I just don't like it at all, and many of the concepts go unproven. I struggled with that class so much, because I just didn't want to do any of the work.

That's why I left electrical engineering. Two classes that were particularly bad in that regard in one semester. I couldn't stand it, so I left and changed my major to math.

 

[Jorriss]

Graduate mathematical methods course in the physics department. I took it before complex analysis, pde's, etc and it was ridiculous to jump into residues on like day two. That being said, if you took complex analysis BEFORE hand it would of been cake... so yeah...

Analysis is also pretty hard but I am only referring to undergraduate. I have not taken any graduate math courses in a math department.

 

[lisab]

You're looking for just math classes? I took a class called Advanced Calculus, which curiously was never offered after I took it (the whole department re-organized). It was all uniform convergence, absolute convergence, blah-blah-blah soooooo ddddrrrryyyyy, please-shoot-me kind of stuff.

 

[jbunniii]

Anyway, sounds like you haven't had enough experiences that would show you the value of the way Hatcher does things. Sometimes, intuitive arguments can be unsatisfying if they are too far from being rigorous because when you have to try to write proofs, you don't have enough technique or whatever. You don't know how to bridge the gap between intuition and proof.

Yes, I think this is accurate. I often had to struggle to convince myself that the "proofs" being presented were actually valid, as they omitted specifying exactly what mapping would achieve what was being claimed. And I didn't find the material compelling enough to invest the effort to gain any real intuition for it. Hatcher was pretty well regarded among the more topologically-minded students I knew, so I have no reason to doubt that it's a good book. It just didn't line up with my (admittedly analysis-leaning) mathematical mindset.

 

[RicciFlow]

The hardest for me are split in two really. The first proof course I ever took was a higher geometry class, just straight up axiomatic euclidean and non-euclidean geometry. There wasn't TOO much work, but as my very first time doing rigorous proofs, it certainly hurt. I also had to write a 10 page paper for that course as well. That was actually rather refreshing. The second was my advanced Linear course. It wasn't particularly difficult, but all the homework problems were a bunch of very long iff proofs. I think my professor just liked to choose the longest ones or something...

My favorite courses though have definitely been my Diff Eq course, and my differential Geometry study. Diff Eq came really naturally to me, and I really enjoyed studying the behavior of the solutions and some of the connections with other areas of math. And Diff Geom. has been just awesome. I have covered two versions of the course really, everything from just geometry of curves and surfaces and the Gauss Bonnet theorem, but also to Manifolds, analysis on Manifolds, and Riemannian geometry. But it's also very daunting, since there's SOOO MUCH diff. geom. stuff. The directions one can go with the applications to other areas of math and even physics is tremendous.