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Studying Am I taking too much time?

  1. Jun 20, 2017 #1
    I studied calculus (equivalent to calc 2 perhaps) back in school last year but it was just procedure, patters and tricks. I wanted to do more so I picked up spivaks calculus and it's taking me far too long. It's been a week or two with some 5 hours almost everyday and I'm only in chapter 4. And they say that's the easy part. I would really like to stick to this because because I find it fun but I fear I'm too slow and wasting too much time. Do 'slow' learners get faster with time? Should I give up?
  2. jcsd
  3. Jun 20, 2017 #2


    Staff: Mentor

    Its natural to take longer when you study a subject for yourself. Teachers focus your studies and skip things in a chapter that you are in fact studying. Your knowledge will be deeper.

    One thing that may help your speed is to checkout the mathispower4u.com collection of videos for Calculus I, II, and III. Each video is very short at about 10 mins each doing a one or two related problems. This may improve your speed and understanding.
  4. Jun 20, 2017 #3
    Of course you should not give up! Especially if you find it fun; what else would you do if you gave up on what you found fun??

    Anyway, "learning" is not a binary thing, it's like a spectrum. It may appear that some have "learned faster," but they have likely not learned the material as deeply as you think. (Some students even, unfortunately, learn only how to pass tests.)

    Also... if the book you are reading is actually the right balance of challenging-but-not-out-of-reach, and you are actually contemplating the material deeply, then 4 chapters in 2 weeks is a fair pace!
  5. Jun 20, 2017 #4


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    Staff: Mentor

    I'm in the third week of a 5 week long summer class for Differential Equations right now. I've been putting in at least this much time just about everyday and we've only covered the equivalent of perhaps 2 chapters. As long as you aren't taking time away from other schoolwork, just keep at it.
  6. Jun 21, 2017 #5
    It is normal. I remember Mathwork stating, " I have never finished an entire math book in my eye..." He says he still picks up Courant's Calculus and learns new things from it. Forgive me if I am wrong Mathwonk.

    From my experience, it takes me maybe 7-9 months, to read a math book going over proofs and problems. Sometimes, I may skip a section, if another book of myne covers that section better. Ie, when I was reading Anton: Linear Algebra, to learn the more computational side of LA. I ditched it when he started talking about Basis for Serge Lang. Serge Lang explained this topic in higher detail, shorter pages, and filled the gaps in my knowledge.
  7. Jun 21, 2017 #6
    Are you proving every theorem, doing every major problem, without looking at solutions? Or are you skipping the majority of material? 2 weeks and you covered 4 chapters in Spivak is really fast! I would caution against going that fast.
  8. Jun 21, 2017 #7

    Stephen Tashi

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    Science Advisor

    One learns calculus over a lifetime. It's usual to begin by learning routine procedures and pick up the rigorous content later. But the rigorous content can't be completely appreciated by reading one book because the rigorous content of one level of calculus is usually just a special case of more general mathematics for higher levels of calculus - e.g. material in the first course of calculus is generalized to tackle "calculus on manifolds".

    To a great extent, learning the rigorous content of mathematics is a matter of acclimating to particular atmospheres of abstractions. For example, a calculus student can eventually become accustomed to doing complicated epsilon-delta reasoning, but when that student takes a first course in abstract algebra, its a different ball game to deal with homomorphisms, quotient groups etc. Each new atmosphere of abstractions causes a certain level of discomfort until one gets used to it.

    Each branch and each level of mathematics also has its own bag of mechanic procedures and tricks.

    How thoroughly one should study math and science texts is an interesting question. For example, since the big hurdle is acclimating yourself to new atmospheres of abstractions and bags of tricks, you could conceivable study a mathematical text in spotty manner, skipping over some things you don't understand, and acclimate yourself to the new type of thinking that is required. If you do this, then you might be as well off as someone studied the text thoroughly a few years ago and has forgotten some of the material.

    People who thoroughly "master" the material in a text book sometimes become so attached to the ideas presented in the textbook that they increase their difficulty in understanding generalizations of the material or different approaches to it. For example, someone who begins to think that the continuity of a function is exclusively presented by the treatment of the continuity of a real valued function of a real variable may find it hard to acclimate to the concept of the continuity of a function in a metric space or the continuity of a function in a topological space.

    (An amusing example I encounter is talking to people who have concentrated on the theory of electronic signal processing. Every real life problem gets translated to some scenario that involves filters, spectrum, and bandwith.)

    If your main interest in in math puzzles and math olympiad-type problems, you obviously want to master the techniques useful in solving those types of problems. I don't know how much of the material in Spivak's books is relevant to that task.
  9. Jun 23, 2017 #8

    My first rigorous calculus book was Courant. I made far slower progress than you have with Spivak. Most people find their first go at rigorous mathematics to be very difficult and slow-going. If you're solving the majority of problems successfully and understanding the material then you are doing just fine. Math is always a struggle.
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