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Courses Struggle in Graduate Analysis course despite 'doing the right things'

  1. Oct 27, 2011 #1
    Hi,

    I just joined this community today after browsing the forums here for a few weeks. I was hoping that perhaps someone could help me out in my situation. I am a graduate student in Mathematics going for an MS in Mathematics. I attended classes last Fall, but left the program on Medical Leave half-way through the Semester and I just returned this Fall.

    I have a very tough Analysis course with a great professor who expects perfect rigor in the proofs. I have been struggling enormously to obtain any semblance of a decent grade. My problem set scores are usually 12/20 or so and my first Exam was a 23%. The problem is, however, that I put in tons of time to the material. And I don't use the time inefficiently. I see the professor often and get tips on how to best study the material. The problem lies in trying to come up with proofs to the Theorems/Problems. We use the Rudin book by the way.

    It seems like each proof requires at least one major step that seems to come out of the thin air. I mean, it is logically sound, but it is something so abstract that I don't know how I would think of it on my own. And even replicating it is hard. I cannot seem to remember the 'tricks' in the proofs for the life of me. So that is why I do poorly on both Problem Sets and Exams. Now, I do have a factor going against me. I suffer from a severe form of Bipolar Disorder and am taking multiple medications for it. One of them, an antipsychotic drug Zyprexa (Olanzapine) is widely considered one of the most sedating agents on earth (literally), and its effects on the memory are devastating. However, I know from having this illness for many years, that Zyprexa is the only drug that truly keeps my symptoms at bay. I have tried at least a dozen other medications, none of which have prevented relapses.

    Strangely, I have an A average in my other class, Vector Spaces (graduate linear algebra). However, that professor is pretty lenient IMO. But most other students in my Analysis class are doing at least B-level work.

    I am scared, because I do not want to be expelled from the Program for doing poorly in this course. This is already my second time taking it (remember I took it last Fall...) so it is not really a matter of not being good at proofs. It is clear to me that it is the medication that is severely limiting my ability to think abstractly and to remember important things.

    What tips would you give me to do better. I am already meeting with the professor very often. The other students either don't want to help me or don't have the time to help me, and I have tried many different studying techniques.

    I have an Exam tomorrow, for which I have been studying for, pretty much non-stop, since Saturday (although I have technically been studying for it since the LAST exam). So let me know if you have been through a similar thing or just have any ideas for me.

    Thanks.
     
  2. jcsd
  3. Oct 27, 2011 #2
    I'm in a similar boat, in an identical class with nearly identical grades, and struggling for the same reasons (not the whole medication thing but rather trouble figuring the proofs out as you described).

    I've asked this question to a lot of people and I keep getting the same answer: Do more math. Do more problems. If you think you've been doing enough, well obviously you're wrong because then you wouldn't be struggling.

    My class just had an exam last week. It's a introductory grad analysis class just like yours (Rudin). Class avg was 18%: There is no magic formula. If your desire to understand the material is strong, you will eventually put in enough work to figure it out. Don't let the seemingly ingenius people fool you. Yes, maybe a few of them figured it out through sheer brainpower. But I'd bet most of them just studied so freaking hard that they know all the tricks and have been doing so many problems that they've seen it all!

    My professor summed it up perfectly, and it really struck home since he's very accomplished at a very, very young age. He basically said, "Look, this your chosen career. This is your job now. This is what it entails, it's very hard work."

    Now I know you're just masters so probably not getting paid to be there. I'm just undergrad, so it's similar. But still, you're there because you want to be. What's the trick? There is no trick. If you think you've worked hard enough, then obviously you're mistaken because you wouldn't have asked this question.
     
    Last edited: Oct 27, 2011
  4. Oct 27, 2011 #3
    hope that didn't sound too callous. the point is that this is the realization I'VE come to when faced with the same issue as you.

    the most efficient use of your time would be to
    1) learn the definitions
    2) do nothing else but problems , problems, problems etc etc etc.......then, yep you guessed it DO MORE PROBLEMS

    understanding the proofs in the book is probably the least important at this early stage of your career. my professor suggested not to worry TOO much about the hard proofs; save that until YOU have to teach the course yourself. even though the proofs seem second to the definitions, that they should be understood before doing problems, i think that's wrong: doing problems will make the proofs seem easier.
     
    Last edited: Oct 27, 2011
  5. Oct 27, 2011 #4
  6. Oct 27, 2011 #5
    Hi. Thanks for your reply. That strategy is generally what I have been doing the last month and a half or so: do as many proofs as I can of all types. I feel it has helped, but clearly not enough.

    I should mention that the medication throws another problem into play with that concept. Since it is so sedating, I end up needing to get at least 11 hours of sleep every day. I cannot even get out of bed if I don't get that much. So my time to do work is significantly shortened especially bringing in work from Vector Spaces and my TA responsibility (I do get paid thank goodness). In addition, I can only work for a few hours at a time without requiring a long break due to how exhausted I get so easily from the sedation.

    So I feel as though I am right around the supremum of the amount of time I can put in realistically without compromising my mental health. I have tried other meds like i mentioned but only this one works. And I have tried lower doses, but that doesn't work either. That is why I think there must be a way to study more efficiently, perhaps?

    I feel as though if I could deeply understand the proofs, then I would not need to spend as much of my study time doing memorization, and I could get to more proofs that way. Do you have any success stories so far in your Analysis course?

    I still am confident though, and best of all, the professor knows that I am putting in the time and he encouraged me to stay in the class after the "withdrawal" date passed. I know that if I get "negative" about the whole thing and start complaining, my mental health will suffer and then I will do even worse!
     
  7. Oct 27, 2011 #6
    Also, funny coincidence, I just bought that book two weeks ago for the same reason! It has been an eye-opener to the "tricks" used in Analysis and has also corrected some of my faulty logical assumptions. For example, I had the hardest time convincing myself that a set in a given Metric Space could be neither closed nor open.
     
  8. Oct 27, 2011 #7
    Yeah that book was a life-saver for me when we had to do a rewrite of our exam. There was a problem that asked us to prove that if (sum from 1 to infinity) {a_n} is a convergent series, then there exists a sequence of positive numbers {c_n} that goes to infinity but that (sum from 1 to infinity) {c_n*a_n} converges.

    In retrospect, this should not have been as hard as I was making it out to be, but a counterexample in that book provided me the insight and ability to be more creative with my answer. It's been a humbling experience trying to get over the fact that I wasn't able to figure it out on my own, but its taught me a valuable lesson in that math is not about pride and being the smartest student. Rather, its about the desire to understand the material, no matter how that understanding comes about.

    ...success stories so far? haha uhhmmm, maybe I'm kinda proud of a couple of my homework solutions, but beyond that a true success story will have to wait until the next midterm/final to be told (hopefully!).

    Another piece of advice I've been getting is to seek problems outside of Rudin. Specifically, seek problems that were on graduate comprehensive exams and practice those. Some will be from Rudin, but many will be new. I'm pretty sure if you go to Berkeley math they have old analysis comps and solutions on their website, and it's probably the same for most universities. My school also has their comps online, no solutions though.

    And something I've been doing is going back to the first Analysis book I ever used (Bartle: Intro to Analysis), and doing the problems from that book that are easier than the problems in Rudin. I think it's helped alot, it builds up my skills more gradually. Rudin has a really steep learning curve and can be very discouraging. So i go back to Bartle when I need to find my groove, and I find that the problems that I used to struggle really hard with in Bartle are now pretty easy, and it helps me gain more "momentum" for the harder problems found on comps and in rudin.

    Beyond that, i don't know, maybe try posting problems and working through them with some of the really helpful people who post on here.

    Good luck on the exam, and remember it's really tough for pretty much everyone.
     
  9. Oct 28, 2011 #8
    Well, the exam is over. I just finished it a few minutes ago. The professor let me have extra time for my health issues (which I didn't mention here originally) and I think that helped a good amount. I was able to scrap ideas for proofs that were going nowhere fast (you know, they weren't converging haha).

    I think my extra studying did pay off quite a bit. I knew the general strategy of all of the questions. I think there may be a few "holes" in two of the five problems given, so I don't think I did great, but I think I probably got a 65/100 - 75/100 which is a big improvement over the 23 percent last time. I am curious to see my results and to share them here.
     
  10. Oct 28, 2011 #9

    micromass

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    Ok, I think you're studying wrong. There is nothing wrong with doing a lot of problems, and it will help. But it's a waste if you don't learn anything from it. Every problem has a little trick, and it is that trick that really matters.

    You say that in some problems, there is always something pulled out of thin air. This is very far from the truth. If you'd really understand it, then you'd see why they do the things they do. Try to think more about "why do they do this" and "how can I generalize this step to other occasions". The more tricks you know, the better you'll do.
     
  11. Oct 28, 2011 #10
    thanks for that post micro.
     
  12. Oct 28, 2011 #11
    Yes. This is what I was trying to convey more or less. I don't realize why those "tricks" are used, and how they were picked. So to me, they seem pulled out of thin air. So I am glad you explained this so that others reading this know exactly what my problem is.

    So, from this knowledge, how do I study more efficiently? I always DO say to myself "why do they do this" and I can always see why. However, I do not know what prompted the author of the proof to employ such a trick. I can see clearly why it works, but I don't know HOW they thought of it. Thus, I am unable to reproduce it because it doesn't make sense "deep down."

    So do you have any ideas of explicit things I can start trying this weekend so that I can do even better on my next Exam? I would really appreciate it and I am so glad that I found this forum. It is nice to talk to other people who genuinely want to help other people and enjoy studying for endless hours.
     
  13. Oct 28, 2011 #12

    micromass

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    It really depends. Some tricks are just what they are tricks. They work because they work. These are the things you should memorize. But most tricks are different, they work for a reason.

    My advice is: whenver you encounter a strange trick: ask about it. Even asking here on the forums should help you.
     
  14. Oct 28, 2011 #13
    Rudin often uses more "tricks" in his proofs than other authors. I (and many others) don't like the book for that reason.
     
  15. Oct 28, 2011 #14
    I'm only an undergraduate, but I think there are a couple things I can contribute.

    Like Micromass said, tricks are tricks; that is just what they are. To see why the author is prompted to employ the trick, try to look for patterns. In the spirit of mathematics, let us consider the most basic scenario where a trick is employed.

    [PLAIN]http://math.ucsd.edu/~wgarner/math20a/prodrule_files/eq0002M.gif [Broken]
    The trick employed was to subtract and add a term.

    [PLAIN]http://math.ucsd.edu/~wgarner/math20a/prodrule_files/eq0004M.gif [Broken]

    Now there is where you would stop and ask yourself.. if I had never seen this proof or trick before how would I have constructed it by myself? In what scenarios would I use it?

    If you spend time reflecting upon the question you should realize that you can use the trick whenever you see that your missing one term to make it look like the definition of the derivative. You might also reason that next time you would look for terms that look close to definitions you have seen before then try to manipulate it accordingly.

    Try to look for a pattern. There is always a scheme used for different type of proofs. Your challenge is to realize what scheme to use and when. When you learn a proof, you must learn to know to expand that knowledge in solving other proofs. It might be helpful to write down the tricks you have seen over time and why they were used.

    I hope it helps.
     
    Last edited by a moderator: May 5, 2017
  16. Oct 28, 2011 #15
    Having been on some pretty intense medications myself, albeit for sleep, I fear you may simply not be able to do this level of math with the medication you're on. Antipsychotics are no joke, and I know for myself a few of the medications I tried caused such apathy and cotton-brain that I simply lacked the mental acuity to do proper mathematics. I know this isn't helpful, but it may very well be the reality of it.
     
  17. Oct 28, 2011 #16
    He said he has an A average in all of his other classes. If you ask me he is doing great and judging by his performance I would have never guessed that he has any thing against his abilities.

    Be very careful before you make excuses for someone, it all too easy of a scapegoat. It seems he is just having a harder time to adjust to proof based classes.. which isn't out of the norm. Granted that medication always has an affect good or bad, I don't think we should use it as a scapegoat just yet given this circumstance.
     
  18. Oct 29, 2011 #17
    I'm in a pretty similar boat. I did really well in my undergraduate analysis class and recently had been having some trouble in my graduate level class. Here are some tips that I found really help me understand stuff better.

    1: Really pay attention during class. For undergrad stuff I could kind of doze off during lecture and robotically take notes and then learn from them later. With the graduate stuff this take way too long and was an incredible waste of time.

    2: On the topic of wasting time, if there is something you just really don't get, instead of trying for 2 hours to understand a theorem, just move on to the next one and get help understanding the first one. Obviously it's not good to skip stuff but I find if you're not making headway after 15 min or so staring at it for another 2 hours isnt gonna help.

    3: But, when you are looking at theorems, make sure you understand them perfectly. Not kinda understand them. You should be able to give me the proof or at least the general idea of the proof. I find what helps me a lot with this is taking notes as I am reading the book. Just see if you can follow along with the proof by writing it down. And it helps to review after every section to see if you can do the proofs after. Even at this level almost all test questions can be done directly from definitions so make sure you really got that down.

    4: finally, someone up there said do lots of problems. I'm gonna have to disagree and say doing problems never ever really helped me that much. THe problems are not hard to do open book because you can just kinda follow what you think is the right chain of thought without deep understanding of the theorems or even definitions. I think the best way to learn this stuff is just to hammer in the definitions and theorems. When you got that I think you'll find it comes a lot easier.

    Just what works for me. Hope that helps.
     
  19. Oct 29, 2011 #18
    While your enthusiasm is great, Nano-Passion, you don't get good marks in Graduate Linear Algebra if you are "just having a harder time to adjust to proof based classes". You don't even get to graduate Analysis without good proof skills, which the OP obviously has.

    Higher-level analysis is a slippery subject. It has some seriously unintuitive aspects that revolve around how infinity behaves when you try to pin it down. I agree with the other posters here that many proofs in Analysis are much harder to internalize than other areas like Abstract Algebra (for me, at least).

    The only suggestion I would have for the OP would be to talk to your department about spreading your courses over a longer period so that you only have to take one at a time. Particularly if you are having difficulty getting enough productive hours in a day. Time can be the best weapon against difficult subjects.
     
  20. Oct 29, 2011 #19

    jk

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    I would say that if you think you understand the theorems but can not apply them todo the problems, you probably have not understood them as well as you think.

    You really need to do both
     
  21. Oct 29, 2011 #20
    Hi eigenvalue I'm a medical school student and I want to help you as your condition is likely a medical condition. You should consult your doctor about your problems as there are many medications for bipolar disorder and many of them are less sedating (Sedation is due to the action of the drug on the histamine receptors in the brain) . The doctor can also try with you other types of psychological therapy and give you less powerful drugs which may improve your condition also he can tell you about an optimal diet with balanced amount of vitamins and minerals (Brain food).Your abilities are likely to improve by time trust me
     
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