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How to study for college math/math theory

  1. Jun 18, 2012 #1
    I'm starting a 7 week calc I course exactly 2 weeks from today. We meet 8 hours a week, so I know I'll need to study a lot, but I'm not sure how to study. In high school, I had a difficult time with calculus, often because I didn't understand it in detail, and sometimes because I froze up on tests. I learned it well enough for AP physics, but that's about it. I actually had to drop AP calculus a few weeks in to the course.
    So, please advise me on how to study so as to
    A: Really understand the material, so I know how to solve more complicated problems, especially proofs.
    B: Not freeze up on tests.

  2. jcsd
  3. Jun 18, 2012 #2
    If it is calculation based and you don't plan on getting into theoretical mathematics it might be easier to not think too much into the problems. Just learn the different ways of calculating the problems and you should be set.

    When I took my first calculus course, I did kind of get confused by the concepts but I ignored them and I did fine. I developed an interest in math afterwards and that is when I started to focus on the theory.
  4. Jun 21, 2012 #3
    I think the best way to learn math is to practice a lot. It's sort of irritating to have to do practice problems on top of your proper homework, but it really does help you prepare for the exams. Also, if you can work with a buddy or visit your professor's office hours, it's also good to talk things through if you don't understand how/why something works.
  5. Jun 21, 2012 #4
    I second finding a study buddy or tutor. Study buddy was better for me, because we were able to meet a lot longer, and my school charges after a certain amount of sessions with the tutors.

    Also, study and do problems a half-hour after class. This will give your brain a chance to breathe and soak in the material, but not enough time to forget everything. I always HATED going back to problems the next day and then thinking, "Wait, how did I forget that already??"

    Do problems EVERY DAY. Don't let your brain rest in such a short course (I took trig as an 8 week course).

    And lastly, after you finish a section or chapter, go back and at least flip through everything before it. So once you finish chapter 2, re-read chapter 1, once you finish chapter 3, read 1-2, etc. You don't have to spend hours re-reading each chapter or doing problems (unless you just don't understand it, obviously), but since this kind of math course builds on everything before it, it's a good refresher to even just remember what was IN the previous chapters. This will help immensely for finals.

    Good luck!
  6. Jun 21, 2012 #5
    Today was the last day of class for my 5 week Calculus 3 summer session and the final will be Tuesday, I'm pretty confident I'm coming out with an A and it has been a really rough road. If you're already iffy on calculus, I would wait for a regular semester and take it during the normal 15/16 week semester because it's a lot of information thrown at you very quickly. If you are not in the position to do that, be prepared to spend every afternoon doing multiple practice problems.

    I left class at 4, got home at 5, worked on homework until about 8 woke up went to statics and then worked on more problems after statics until calc started back at 2. It was literally non stop practicing, and I imagine even a 7 week course will be loaded down with homework/practice problems outside of class.

    As far as tests go, make sure you work through examples that your instructor gives during class and work them inside and out and if you get the hang of those easily, move to something a little harder or that you're less familiar with and do those inside and out until you can do them without hesitation. If you are that comfortable with example problems the teacher gives and homework, then you shouldn't have to think twice during tests. It's a lot of work to get to that point sometimes but it's well worth it when you get back your test and you see you did better than everyone around you.

    Also, someone in my class makes her own study guide before tests, and it really seems to help her out. Plus when she goes to study for the final, she already has all the information compiled that she needs to review. Usually these consist of just concepts and not problems though so reviewing problems is still necessary.

    One last thing, don't miss class or be late. In a short course like this, missing one class in a 7 week course is about the equivalent of missing 2 days in a regular semester. Even coming in late you can miss crucial information. Good luck!
  7. Jun 21, 2012 #6
    IMO, khan academy is the best possible source for learning calculus, short of physical injecting knowledge into your brain. I've personally observed from my friends that calculus is extremely simple if you just learn it from the right view point/explanation. But none of this matters if you don't have the proper college algebra skills...
  8. Jun 22, 2012 #7
    I did this last semester for chemistry and considered suggesting it after I posted my reply. Just MAKING the study guide (a couple of pages of concepts for each chapter) helped me get an A on the final -- I didn't even really have to study it. I also bought a tiny spiral-bound notepad for like $.25 and used that for formulas only so the study guide wouldn't be too long. I have the tendency to think I know everything and then on the exam freak out, thinking something like, "Oh crap, is it PV = nRT or PT = nRV????? AAAAAH!!!"
  9. Jun 22, 2012 #8


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    the best math teacher i ever had said you need to write 3-5 pages for every page you read. that's rewriting material you want to remember, doing problems, making up examples, counterexamples, everything.

    there is no point taking notes in class if you don't go back over them immediately and preferably rewrite them filling in all details and gaps. i also read more than one book on the topic, to see the same thing said in different ways.

    pardon me, as i know the advice is well intentioned, but anyone who ignored the concepts in my class probably failed, at least if i did my job of testing right.

    to understand the subject you have to try to understand it. if you are not going to try to understand it don't waste your time or the instructor's time.

    in other words my class is never purely computational. and freezing up on tests is a sign of tying to memorize material at the last minute and not really understanding it, and not practicing enough. if you are well prepared most questions on a test will look a lot like ones you have done before.

    that does not mean the professor will only ask the same questions he/she has already asked, but that you should have expanded your practice to include many similar situations. one of the most pitiful complaints i ever heard was "but you only did the open top box example in class, and on the test you asked about the closed topped box!"

    The purpose of a test is to check whether you have understood the concepts, and can make the calculations, and can solve problems with them. Sometimes it also includes ability to give theoretical arguments backing up the concepts, i.e. proofs. You should not only know what is true but why it is true, and be able to prove it and to use it.

    I have attached a pdf with some basic advice I used to hand out every semester (in non honors calculus). It seemed as if most students did not even read it however. Once to verify this I added a sentence at the bottom asking everyone to email me immediately so I would have their email. After a week I had heard from only 3 or 4 people.

    I presume you will read this, but of course it is more important to read and act on the advice your own professor hands out or states in class or his/her website, and of course to be there in class to hear it.

    Attached Files:

    Last edited: Jun 22, 2012
  10. Jun 22, 2012 #9
    I whole-heartedly agree. Writing notes as you read, even, is a great way of organizing thoughts and concepts in your head (at least for me). Then, when studying at a later point, re-writing it (perhaps even trying from memory/knowledge) helps even more, I find.

    I'm taking a 6 week Calculus summer course right now and reading an additional book to the course's (Stewart) has helped me tremendously. I am reading and studying Stewart's book and then reading the same material in Apostol, which seems to have helped tremendously. I find Apostol's explanations to be much more clear and thorough. I especially find this true of the proofs. Half of the proofs in Stewart make almost no sense to me. They seem to cut corners and not explain half of their steps/motivations, which confuses me since most people taking a course based on Stewart have little to no working knowledge of proof-based mathematics. The proofs in Apostol have helped me immensely.

    Thanks for the tips, Mathwonk. I wish I had a professor like you!
  11. Jun 23, 2012 #10
    I am a a circuit designer and I have passed the phase of taking courses and learning. I am still learning stuff in math but purely out of interest. After my experience doing research in circuit design, one thing I figured out is that it is not super important to ace every course. If you understand the concepts, you will be able to do well later on as long as you are motivated. However, in this context.......

    I know what it feels like to freeze up. We had a lot of courses in signal processing and in each course, we were given very tough questions to solve that too in the final exams. I had no clue what to do. I used to shiver and my hands literally shook out of fear during the exams. Believe it or not, when I came out the exam hall, it used to take me just 5 minutes to get the solution.

    I am sure calculus is different from signal processing or error control coding. Nevertheless, the professor can give tricky questions. The idea is to give tricky questions to differentiate the smart ones from the "not so smart" ones. Unfortunately, a lot of smart ones miss out simply because of a psyche factor.

    I hope I am not digressing too much from what you need to know. Most of the advices given above cover what you must do in order to do well; however, I am concerned about one thing that you mentioned..... freezing up. I have had a terrible time in the past with the psyche factor. EE department of IIT Madras used to make life of its students a living hell because they wanted to make a clear distinction between the smart ones and the not so smart ones. In the process, they made the papers so tricky that almost 80% of our class had failed in at least one subject. I survived :). We had to take calculus in the math department and it was not easy either; however, the most we got psyched was in subjects related to communication engineering and analog circuit design. I froze like no one else and eventually, managed to get better.

    Apart from working hard, please do work on becoming more confident. The exams are just one hour/3 hours in duration. The examiner might give you trick questions. Imagine that you are facing that moment when in your head, you will be thinking more about your grades than about solving the problem if you get psyched. My suggestion is to get AS MANY older papers as possible and solve them in a time constrained environment. Do your exercises and all that but on top of that, make sure that you solve as many older papers as possible.

    THE ONLY THING THAT WILL HELP YOU IN SUCH SITUATIONS IS CONFIDENCE. I had to fight a huge battle with my diffidence especially when the seniors were telling me "this professor likes to fail the students" while what the prof was really doing is that he was trying to maintain a high standard in his class and maybe, slightly differentiate the smart ones from the others.

    The bottom line is to work hard and understand concepts by solving a lot of problems. On top of that, I feel that the most important thing is to FEEL CONFIDENT. One of the things to acquire that is to solve yester years' question papers in a time constrained environment. Before you go to the exams, you must have known already deep within that "yes, I can solve a reasonably difficult paper in a time constrained environment even if it is a very tricky question paper". MAKE SURE THAT YOU ACQUIRE THAT CONFIDENCE.

    Writing proofs is also an art. There are direct techniques but there are also proofs by contradiction or contrapositive methods. An advanced mathematician is quite familiar with those methods. The prof might just give a question requiring one such method and half the class will simply shake during the exams thinking "boy.. you must be a genius to solve this problem." If you have solved a variety of problems before and know these proof structures, then you might be able to do better knowing which method to apply under different situations. While solving problems, please give SPECIAL ATTENTION to proof structures. It will help you in other areas of math as well.

    Long story short, just practice solving as many problems as possible. If you are reading a book like Spivak, get the solution manual. Refer to it after you have tried the problem for like an hour or so. Make sure you completely understand the structure of proofs along with (obviously) the technical content.

    Finally, solve problems from earlier years. Without the CONFIDENCE to go to an examination hall and sit there for just an hour or so to solve the easy ones as well as the tricky ones, you may not be able to give your best. The most unfortunate thing is that sensitive people (like me) tend to get bogged down simply because the situation seems to be really demanding. It is true that the situation is demanding. The key is to KNOW and completely REALIZE that you can crack the dang paper completely.
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