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Studying Looking for some basic feedback on math book to learn next

  1. Jul 8, 2016 #1

    Radic S

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    I started learning math at 29, I went through the high school curriculum in Ontario taking the advanced math courses route. The last book I studied was "nelson calculus and vectors"

    (Table of Contents)


    ( Curriculum Correlation) http://math.nelson.com/calculusandvectors/Resources/Curriculum Correlation Calculus.pdf

    I asked my brother who is studying CS to recommend a book I can start after completing that math course. He told me to start "Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by https://www.amazon.com/s/ref=dp_byl...ield-author=Tom+M.+Apostol&sort=relevancerank "

    I've started reading and I find this book a bit difficult to go through on my own and I'm not sure if I'm prepared for studying this book at a reasonable rate. Would anyone have any alternatives to this book that would help me progress in my understanding of math? What I find the most striking is the completely different approach to presenting the material. Previously I was exposed to a concept and it was applied in a few examples, then you had variations of it. The language now being used is more general and assumptive? (I’m not sure if that’s the best way to describe it) Is there a book that has a more gradual approach to beginning to learn in this style? Or a better method? I’m willing to hire a tutor or take some courses which are within reasonable proximity.

    Extra info:
    I don't have a large background in math, this was the 3rd math class I've taken. I'll be applying for fall 2017 for the math/physics specialist program at UofT and a variation of it at WaterLoo. When I looked up the class that i'm required to take first year I got


    http://www.math.toronto.edu/murty/MAT 157Y Syllabus.pdf
    The book they use is a "Calculus by M. Spivak (4th Edition) (Publish or Perish)" which my brother has and I've looked over it. I'm rather confused because it says that the prerequisite for this class is the calculus I took and mentioned above. Meanwhile when I go through the book it seems to be leaps and bounds different from anything I've been exposed to. Now I'm not sure how I'm suppose to prepare for a book/courses of that sort when it's prerequisite is clearly some sort of joke.

    The other course is an

    Algebra I



    The book they use is “Linear Algebra with Applications

    1st Edition (Hardcover), by J. Holt”

    course which I heard is taught from scratch, so I'm not too concerned about it’s prerequisites. I do have the available time I would like to study it on my own and move ahead, but this calculus/analysis class has me stumped at what to do.
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Jul 8, 2016 #2
    Can you tell us a bit what your difficulties with Apostol are exactly?
  4. Jul 8, 2016 #3
    Our math curriculum here in Alberta isn't too far off from Ontario, so I know what you're talking about. I took all four math classes - from algebra to calculus - and did not encounter one single proof. I guess they wanted to make math more accessible to everyone by drilling you with examples and removing any sort of abstraction. Everything is spoonfed to you. So now I've met people in their first year of engineering who can tell you that x3+x2+x+1 is a polynomial, but cannot decipher anxn+an-1xn-1+ . . . +a2x2+a1x+a0.

    I'm not familiar with Apostol. I personally used Serge Lang's A First Course in Calculus. I find it to have the right balance of rigor and intuition for someone with a highschool level of math. My only suggestion would be to find a secondary source on limits, particularly if your going to be using Spivak.
  5. Jul 8, 2016 #4


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    Hey Radic S.

    Is it just the terminology and/or the symbols/vocabulary throwing you off? Perhaps you could explain precisely what is throwing you off so we can assist you.

    The ideas and conceptual understanding will remain the same despite the terminology and rigor and being able to map these things to the specific form that the textbook uses will be important and not so much relying only on the language used to convey those ideas/information.
  6. Jul 11, 2016 #5

    Radic S

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    Sorry for the delayed response. I used these intro pages as an example of what I’m having issues with.

    After the first paragraph on page 5, the author introduces k as “k= 0, 1, 2, 3… n” but then they show k^2 in the example, if he’s representing k = n, then why wouldn’t they use n in the example instead of k? When in figure 1.5 we see (kb)/n and eventually we get b=(nb)/n. Why are we switching between the two letters throughout the whole example? If I’m not mistaken the variable k used in this context is referred to as the (a in ax^2+bx+c) showing weather the parabola is being stretched or compressed.

    The next step 1.3 the author makes a claim that “there is an interesting identity which makes it possible to evaluate this sum in a simpler way” without mentioning why this identity is simpler, where it’s from, why we want to make it simpler etc. The following page he whips out a formula from thin air and rewrites it, again no explanation for why or where, just “hey here’s something we can use, then rewrite it and we have our [3k^2+3k+1 = (k+1)^3 -k3].

    Having earlier defined “k =0, 1, 2, 3… n”, we now get “k= 1,2, … n-1”, so is k= first example, second example or interchangeable for both? Using the same letter twice just makes a mess for me without a clear reason for doing so.

    From there we have [3(n-1)^2+3(n-1)+1=n^3-(n-1)^3] which I understand how that was arrived at, but the step after, where the author “adds these formulas” obtains some giant one line formula that was arrived at by adding which formulas? The last 3? 4? Or a different combination of them.

    On page 7 the author says “to prove that A=(b^3)/3….” they get [ 1^2+2^2+…+n^2<(n^3/3+n^2), but if we expand (n-1)^2 we get (n^2-2n+1), so how does adding n^2 cancel out the (n-1)^2 in the leftmost inequality in 1.5?

    Then for the exercises on page 8, the author whips out some more overly complicated wording. “Check through the principal steps in the foregoing sections and find what effects this has on the calculation of the area” Which were the principal steps all of them? Some of them? Does the exercise want me to carry out the actual computations that are required in the case like they did?

    It really seems that the language used in these books is over the top, which is really annoying. I’ve noticed this underlying tone in mathematics, where simple ideas are presented in overly complicated wording without any real justification for it. I’m not sure if this is intentional or just a bad habit that has been passed along mathematicians over the years.

    These are the primary reason why I’m trying to find a more efficient method for studying/preparing for those math classes, it took me a good few hours to go through most of those 3 pages understanding what was going on. At this rate it will take me way too long to go through this book, which is why I’m not sure if it’s too ‘advanced’ for me or if I require a tutor to help guide me through the book faster.

    That's exactly how I feel, it's not like being thrown in the ocean and told to swim after you've completed a swimming course in the shallow end. Some ideas I can see connected from what I've learned before, but it's completely different.

    I'll try to find that book to see maybe it's a better fit.
    Last edited: Jul 11, 2016
  7. Jul 11, 2016 #6
    I hope you know that this will remain true throughout the degree. The math pages in Apostol are pretty easy compared to the texts you'll need to digest later.
  8. Jul 11, 2016 #7

    Radic S

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    I hope so! I've always wanted to learn the language, that's why at my age I've started from scratch and i'm moving forward. I have 14 months to prepare for that course, I came on the forms seeking for some advice and guidance on what my best route is. I can study it myself and post questions on the forms, but i'm not sure if it's the best approach to get the most out of it. What would you Micromass suggest I do to prepare myself? Should I get a tutor? Take an online course? Start a different book?
  9. Jul 11, 2016 #8


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    You are correct in that (often) simple ideas are made convoluted by authors and unfortunately there are many reasons why that's the case.

    I have seen this in mathematics all the time and unfortunately there is so much "baggage" linguistically that it can take a while to translate this to what it really is.

    I think you will be better off being able to use the simplest explanation and possibly have some sort of notepad where you can translate all the jargon into something useful.

    As a result of the above, my advice for preparation is that you learn the jargon and translate it to something easier to understand.
  10. Jul 12, 2016 #9

    Radic S

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    chiro and micromass, I get a sense with your responses that I haven't showed enough of where I've been left with the 1 and only calculus math course I've taken that is a prerequisite for spivak book, so I added pictures of one section of derivatives to help give a better idea of how different the two are.


    I've read about a international baccalaureate exam for math or Euclid Mathematics Contest (University of Waterloo), thought maybe preparing for those would be better for me maybe? Not really sure if they hold any gems for development.

    chiro, do you know of anyone online that does mentoring for the math goals i'm seeking or where i could find a person with the knowledge to help better guide me? I was considering going to one of the top local universities and looking for TA's that can help me.
  11. Jul 12, 2016 #10


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    I live in a different country (Australia) so I won't be able to help you with those specifics.

    I think you should either seek a professional university for your education and continue to ask questions on forums like this.

    Just note that it's best to stick with specific questions (and one at a time) rather than going too general.

    The progression for learning topics is pretty standardized across all universities and that will give you a guide on what to do and the order in which to do it.

    If you are self-motivated then all you really need are some textbooks, stationary and computer access (including internet) and time. If not (like the majority of people), you will need to attend a university or a community college and have instructed learning.

    Also - if you seek a TA or tutor be prepared to potentially spend a lot of money on an hourly rate as mathematics is a specialized subject - particularly at university level.
  12. Jul 12, 2016 #11
    Hey Radic,
    I am not far along in my studies, so take anything I say with that in mind, but I used Apostol's book to study single variable calculus and plan to continue with his second volume. That being said, I can relate to many of the issues you've been facing, as this was the first textbook I used for post high school math. The best I can say is that, with time, the language begins to clear up and you will not feel as lost in the methods employed to prove or explain things. It personally took me about month to properly work through the introduction, given the jump in expectations. I also found some of the introduction material, specifically the induction formulas, to be poorly motivated, but this was never the point of the book. I find that the rest of the chapters provide very thorough and motivated approaches to the material, although it is still easy to feel lost at first. It helped me to look through lectures on MIT OCW as well as other free online resources when a certain theorem or concept seemed too difficult as presented in the book. At other times, you may just need to take a concept and play with it for a few days, or revisit it after a while, to grasp it. If you decide to continue with Apostol or an equivalent text, expect progress to be slow, especially at the beginning. Good luck!
  13. Jul 18, 2016 #12

    Radic S

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    Hey Chiro,

    thanks for the advice! I'll progress with the book on my own for now while searching for a TA. You mentioned I need some textbooks, what other books would I require besides the one i'm using?

    One more thing, do you know what background from a tutor I should be looking for? I.e classes taken, papers written, TA experience etc..

    I'm doing this book to prepare for the "Calculus by M. Spivak (4th Edition) (Publish or Perish)" analysis course I'm required to take first year in fall 2017.

    Hey Akorys,

    I've looked at the MIT OCW classes myself and listened to the lectures, I don't find them difficult at all. Rather I found the single variable calculus course pretty straightforward. I do know that many of these universities don't post their specialist courses online, just their general first year courses. As an example MIT has a 1 year physics class with Professor Lewin, but they also have a more advanced first year class which you can only find some videos recorded by students. I don't think this apostal book is a general first year calculus course class which is why it's a bit more challenging.

    Thank you for the advice! I was hoping there's a smoother transition to that book. I'd rather not spend 3 hours on a simple problem, if I can do an easier book to prepare me for it. Time's very precious and I hate to waste it.
  14. Jul 18, 2016 #13


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    Often a good tutor is someone who has done the course you are being tutored on plus say a year or two of coursework within the relevant fields on top of that.

    The reason for this is that the extra stuff makes the earlier stuff a lot easier and a lot more understandable meaning it can be better taught.

    For textbooks you will need to tell us exactly what courses you are studying but if you are near a university I'd recommend you go and see if you can purchase lecture notes for subjects in mathematics since you might get lucky and buy the notes for very cheap (far cheaper than the textbooks) and that can serve as a pretty good guide. Books are not as customized or tailored to student needs as many lecture notes are - particularly if they have been updated over a long period of time and reflect a lot of student feedback (not saying that books don't necessarily have this though).

    If you are getting tutored, try and get a sense of how they explain things in your own intuitive language or "plain english" (substitute other language of choice here as well if necessary).

    Tutoring is not about papers written or prestigious awards or anything like that - it's about the ability to explain difficult non-intuitive things in an intuitive way and that requires more than just aptitude for a subject.

    I'd recommend looking at some really hard concepts before getting a tutor and seeing how well they can break things down and explain them to you in the simplest possible language. If they can't do that then find a new one and don't spend any more money.
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