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I'm finding that I am getting to the point of being hopelessly behind in one of my courses (Fourier Analysis).

What is the best thing to do when it feels impossible to get caught up in the literal sense.

Being "caught up" to me means having read through all lecture notes up to date, and understanding all the proofs/definitions/theorems covered therein.

The place in the course notes that I am "caught up" to is probably about 2 weeks behind where we are in the course currently. I try to catch up outside the class but the pace at which I can go through the proofs is slow at best. I hit certain steps in the proofs that are incomprehensible to me and take 1-2 days turn around time between me and the instructor to get resolved. There is no textbook to fall back on because this is a grad course. There are recommended references but the books are usually just as advanced and are written in different notation and would involve extensive re-reading of material I already understand just to get to the point where I am caught up with notational differences; so I feel like this would not be helpful. The worst part is we're not even 1/3 of the way into the semester and haven't even had our first midterm yet.

I feel like I have only two options:

1) try to rigorously understand the material and continue slogging forward slowly and ever getting more behind

2) barrel ahead reading only the definitions and results until I am at the place we are in class and then try to keep up from there.

The second option seems like trying to build a stone house upon a sand foundation. If I can't follow the proofs of the theorems I have no hope of completing the assignments without someone holding my hand. And the first option seems doomed to get me an assured failure in the course (we are supposed to maintain an A- average to stay in the program). I can't be the only one who's found themselves in this situation?

I feel like I need to phone a math crisis center hotline or something!

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My advice to you is to start drinking heavily. OK, but seriously if you're not making any progress try a different tack. Ask questions here if you're stuck on a proof step. Work through given examples. Try a different textbook. –  TheMathemagician Jan 10 '13 at 17:20

2 Answers 2

up vote 5 down vote accepted

In my experience, skipping forward is not the path to success - It may be possible to get a good grade in the course, but you definitely will not have a solid understanding of the material.

Personally, I would spend several hours working backwards from the current point in class to uncover all the dependencies. While this takes some work, it's not as difficult as slogging forward & attempting a rigorous understanding of the proofs of every theorem.

Essentially, pinpoint the material you don't understand - finding out what you don't know is easier than learning everything you don't know (especially because you'll miss things you don't know you don't know). Then, go to whatever help outside class is available - office hours, TAs, tutors, etc with your list of material and ask. Ask ask ask ask. Don't be afraid to ask the stupid questions. If you can get your professor to spare you an hour, prepare a list of the most critical things to ask him in person & get reexplained. If you're at a large university this may be difficult to do, but do you best to get some face-time with the professor. If nothing else, s/he'll know you're trying.

I cannot emphasize enough the importance of reaching out for help - be it here, or to the resources available at your university. It may be possible to catch up on your own, but it will be much easier and far less frustrating to do it with help.

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Skipping forward always felt like a superficial way of learning material (or prentending to) anyway. And I've always advised others against this at all costs; so it would be silly of me to do it now. I will try to break things down as best I can and identify the specific problems but right now I am overwhelmed so that feels like simply everything. I certainly will be asking a lot of questions here in the near future (I just discovered this site). I'm not trying to have a pity party here - so I apologize if it comes across that way. –  Kyle Schlitt Oct 2 '11 at 19:42
    
It may seem like everything now, but if the course is well constructed, more recent proofs should build on previous ones. Assume referenced theorems are true and see if the proofs make sense given that truth. Then, when you step back to the earlier theorems later proofs rely on and prove those, you already have a rigorous understanding of later proofs. As I said, the idea is that what seems like everything now is hopefully only a few key concepts - identify those and you're golden. –  Drew Christianson Oct 2 '11 at 19:45
    
Well it seems that the further into the course I go, the briefer the proofs get. Since its not only the results from previous proofs that are taken for granted, but even the techniques used within them which are now not as detailed. For example the theorems are not usually referenced because it's probably supposed to be obvious to me by that point which theorem is being used. –  Kyle Schlitt Oct 2 '11 at 19:52
    
Ah. That makes things tougher, but I would argue that working backwards still helps. Given that brevity of proofs is that much of a problem, I would also look into getting different & more detailed reference material - I'm sure a reference request here or to your professor would generate useful responses. –  Drew Christianson Oct 2 '11 at 19:58
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I think you should directly tell your professor he is going too fast for you. It is possible that everyone is feeling equally lost, or that the only people who aren't are those who have seen the material before. Your profile suggests that this is a graduate course, so it is probably small and probably being taught by someone who, though he knows the material very well, hasn't had much experience teaching it. Feedback could be invaluable for him. In any case, he asked how he was doing; he shouldn't find it rude if you answer. –  David Speyer Oct 2 '11 at 20:10

I was in a similar situation when I took Fourier Analysis and PDEs my first year. I think a good way to check whether you understand the material or not is to read through the important theorems and then be able to prove them yourself. That is the best (and in my opinion...only) way to to learn. As Halmos said, for the passive reader, the proof enters one ear as easily as it leaves the other.

The reason why I think this method of learning is helpful is because it teaches you important techniques (especially so in Fourier Analysis) and approaches when trying to prove essential statements. If you are unable to prove a theorem by yourself, you will look back in the book, and be reminded of the essential technique you missed out on in your initial approach. Then the technique will stick with you better than if you had passively read the proof.

The point I am trying to make is that you really need to do mathematics to learn it. There is no other way out of it, but I sympathize with your situation. I was in a similar situation, and I think one of the factors that tends to make you feel like you will never be able to catch up is anxiety (which is a result of falling so far behind). However, if you keep pushing yourself in actively learning the material, as I have mentioned to you above, the anxiety factor will fade away. You will feel more optimistic as you will realize you are actually learning the material and not just bullshitting the assignments by piecing together different theorems you don't understand (all too common in mathematics courses these days unfortunately...)

And most importantly, you will remember most of the specifics of fourier analysis even two or three years later. If you superficially learn the material, you will remember it only up to the final exam. So, go ahead, and start reviewing the material and don't agonize too much about behind behind as long as you feel like you're fully learning the material. Of course, use good judgment and be practical too. If you have an assignment due the next day, make sure to do the necessary reading (even if it is superficial) to finish the assignment.

Best wishes!

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Also, this approach tends to make the exercises a bit easier, atleast in my experience. –  r.g. Oct 2 '11 at 20:01

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