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Some background: I have no mathematical maturity. Last year I completed my schooling and the only time I picked up a math/science book was when exams were due, needless to say I haven't actually given much attention to mathematics for a very long time.

I've got a passing level of familiarity with mathematics which I studied in the past few years, but I recently decided to actually learn mathematics properly. Several sources online seemed to suggest that Spivak was a good first choice for the book.

But it seems that the book is, suffice it to say, very challenging. I've actually enjoyed the book very much since for several of it's problems I was stumped for a long time only to finally figure out a completely new idea. But it seems that right now, practically every new exercise question uses a new idea/identity which I'm not familiar with. I also struggle with exactly when I should actually keep thinking about the problem or ask for help.

Given these circumstances, should I continue trying to solve Spivak? Or would it be a better idea to go through some other book before coming back to Spivak?

Edit: While going through Spivak, I also noticed that I ran into two major errors. One is that I very often fail to use the information that was proved in earlier exercises. Another one is that I often fail to fully conceptualize what a particular thing implies. For example, I often get stumped and look up the solution, only for them to use something implied by a particular statement. Eg $(x-1)(y+1)=0$ implies that one of the two or both are zero. I often don't make the connection and hence get stumped in proofs, usually when the proof is drifting in the direction of contradiction.

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  • $\begingroup$ I'd say try a bunch of different books until you find one that really clicks with you. $\endgroup$ – littleO Jan 17 '16 at 18:28
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    $\begingroup$ Spivak's calculus is an extremely challenging book which should be studied after having a decent amount of mathematical maturity (i.e.,if you want to really learn single-variable calculus and not just a set of "tricks" to solve problems). $\endgroup$ – Mr. Y Jan 17 '16 at 18:39
  • $\begingroup$ @Mr.Y we used it for a first year course at the first college I went to, so I guess they disagree. $\endgroup$ – Matt Samuel Jan 17 '16 at 19:06
  • $\begingroup$ To get inspired, I recommend reading some books on Maths history. Try The Calculus Gallery by William Dunham. Another thing, in maths you don't learn by learning one thing completely and then moving to the next topic. Don't be afraid to skip topics and read ahead and then returning to the difficult problems. $\endgroup$ – user230452 Jan 21 '16 at 16:31
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If you were really mainly interested in using math for other things, like physics or computer science, then I might suggest using an easier calculus book. However, you say you've decided to "learn mathematics properly," which means you need to wrestle with challenging problems like those in Spivak's book, sooner or later.

The real question is whether the number of new methods and ideas used in the problems is simply too high for you at the moment. The way you describe things, I would say the answer is yes. Also, many of these are not necessarily ideas from calculus, but ones from algebra that you might have learned before if your high school education had included a focus on careful proofs and difficult problems (something that is unfortunately very rare).

One possible solution may be to study some precalculus topics from books with that kind of focus before returning to Spivak later. Here are some suggested titles, all of which are short books with lots of problems for bright high school students. (Most of them have hints or solutions.) If I had to pick just one as preparation for Spivak, I would recommend the book on inequalities. But all of them contain interesting math, so you'll never be wasting your time reading these.

  • Algebra. Gelfand, Shen.
  • The Method of Coordinates. Gelfand, Glagoleva, Kirillov.
  • Functions and Graphs. Gelfand, Glagoleva, Shnol.
  • Invitation to Number Theory. Ore.
  • Introduction to Inequalities. Beckenbach, Bellman.
  • Numbers: Rational and Irrational. Niven.
  • Mathematics of Choice: How to Count Without Counting. Niven.
  • Sequences, Combinations, Limits. Gelfand et al.

I'm not a big believer in the value of "proof methods" books. You learn math by doing it. The best "transition to higher mathematics" books are the ones that contain real, interesting math, rather than long descriptions of "modus ponens" and so on. For those who know non-rigorous calculus, Rotman's Journey Into Mathematics isn't bad.

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  • $\begingroup$ From what I've seen, some of the "proof methods" books include sections with interesting sections as well, although most of them use basic set theory and introductory number theory (first chapter of a topology and algebra/number theory book, respectively) as the primary source material. Often times, there are some good problems, but the emphasis just different. The proofs are less terse, and there are some superfluous sections dedicated to a particular idea in logic+method. $\endgroup$ – Andres Mejia Jan 18 '16 at 22:39
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In my experience, if you want to really understand mathematics, hard questions and problems are essential. My recommendation is that if explanations for any book are not clear to you, then get a secondary resource. But struggling is part of math, and solving a hard question will do more good than reading 10 different articles on a particular subject. Perhaps pick up an easier book for a second resource (it will likely provide more details), but continue chugging away, and I think you will see a lot of pay off that way!

In regards to your edit: perhaps pick up one of these new "proof technique" books that create the bridge between simple exercises and real mathematics. I have tutored students who have very good things to say about Ethan Bloch's "proofs and fundamentals." Maybe if you got a handle on the general methodology (there is a chapter dedicated to contradiction and contrapositive) it will be easier to tackle calculus. The pedagogical hierarchy of mathematics is a myth. Sometimes you have to synthesize disparate concepts to really get what is going on.

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    $\begingroup$ Your last sentence is very true. $\endgroup$ – user230452 Jan 21 '16 at 16:29

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