Recommend books/resources for amateur to learn mathematics
December 27, 2003 10:37 AM Subscribe
I'd like to learn Math. I'm particularly interested in learning trig and calculus. I'm don't need to learn these disciplines for any purpose. I'm just interested. I'm a reasonably bright guy, with a logical mind (I've worked as a programmer), and I'm a good self-learner. I'm not in a rush (don't mind working at this for a few years). What books/resources would you recommend? I should probably go all the way back to Algebra, which is pretty much where I left off in High School years ago.
Response by poster: Thanks, oissubke. Maybe you (or someone else here) can answer a related question. The worst thing about Math education (when I went to school, anyway) was the hours and hours I had to spend doing problems even after I completely understood the concept. What I'd love to do is to just quit doing a particular set of problems as-soon-as I get the concept they're trying to explain. But I imagine it's a good idea to keep working at the problems until one commits the concept really well to permenant memory.
What's a good self-study rule about how many problems to do and when is it okay to move on to the next concept?
posted by grumblebee at 10:53 AM on December 27, 2003
What's a good self-study rule about how many problems to do and when is it okay to move on to the next concept?
posted by grumblebee at 10:53 AM on December 27, 2003
Spivak's Calculus" is excellent, it starts off with the basics and goes through the principles of calculus in detail, instead of just giving you a bunch of formulas to memorize.
It's a lot more intellectually stimulating than most calculus books. The problems presented are complex but not elaborate, they test understanding rather than making you worry wether you misplaced a minus sign somewhere.
posted by bobo123 at 10:56 AM on December 27, 2003
It's a lot more intellectually stimulating than most calculus books. The problems presented are complex but not elaborate, they test understanding rather than making you worry wether you misplaced a minus sign somewhere.
posted by bobo123 at 10:56 AM on December 27, 2003
i wouldn't do any more problems than you need to get the concept. i think school is like that just so everyone can keep up - at university it tends to be more one question per concept and you're done.
i once helped at a numeracy centre (that taught people in the community maths). however, most customers wanted to learn very basic skills (to manage their money - addition, subtraction). you might try seeing if you have a local numeracy centre, but you could find that you're a bit out of ther normal range (imho there's nothing worse than being taught by someone who's not confident with the subject matter).
i doubt trig and (basic) calculus will take very long to learn if you find the right book and it will open up new areas of programming: i use trig a lot for generating "art"; once you've got calculus sorted look at numerical methods for integration, etc - that's the basis for being a "mathematical programmer" in my experience of industry. in fact you might find the relevant chapters of numerical recipes interesting as you learn (don't be put off if at first they read as gibberish - try them once you've mastered the maths books). good luck.
oh, and finally, i find that as i get (a lot!) older, i learn maths (i was a physicist, so there was a lot of maths that i never learnt at university) in a very different way - i see a lot more connections between things and a lot more practical uses. i guess it's just experience with life. anyway, it makes learning more fun. i hope it's the same for you!
posted by andrew cooke at 11:50 AM on December 27, 2003
i once helped at a numeracy centre (that taught people in the community maths). however, most customers wanted to learn very basic skills (to manage their money - addition, subtraction). you might try seeing if you have a local numeracy centre, but you could find that you're a bit out of ther normal range (imho there's nothing worse than being taught by someone who's not confident with the subject matter).
i doubt trig and (basic) calculus will take very long to learn if you find the right book and it will open up new areas of programming: i use trig a lot for generating "art"; once you've got calculus sorted look at numerical methods for integration, etc - that's the basis for being a "mathematical programmer" in my experience of industry. in fact you might find the relevant chapters of numerical recipes interesting as you learn (don't be put off if at first they read as gibberish - try them once you've mastered the maths books). good luck.
oh, and finally, i find that as i get (a lot!) older, i learn maths (i was a physicist, so there was a lot of maths that i never learnt at university) in a very different way - i see a lot more connections between things and a lot more practical uses. i guess it's just experience with life. anyway, it makes learning more fun. i hope it's the same for you!
posted by andrew cooke at 11:50 AM on December 27, 2003
Saxon Math. They go all the way through calculus. I used the series when I homeschooled and it rocks. The instruction is very incremental , the explanations are exceptionally clear, and the problems are set up so that you have a few to work thru to ensure you get the procedure-then the main problem set contains a mix of problems from the beginning of the book up till where you are at that point-reason being they think repetition is a key to really "getting it" yet not wanting to bore you silly with the review. You wouldn't have to do that section if you didn't wish to.
The books aren't cheap, but if you are pennypinching you can sometimes find used ones from the homeschool crowd.
posted by konolia at 11:56 AM on December 27, 2003 [1 favorite]
The books aren't cheap, but if you are pennypinching you can sometimes find used ones from the homeschool crowd.
posted by konolia at 11:56 AM on December 27, 2003 [1 favorite]
Oh, and I think the program 60 Minutes did a segment on the fellow that came up with the Saxon program. It was rooted in disgust with how poorly kids did with algebra as it was presently taught. He managed to get some school systems to use it to very good effect.
My kids got top-of-the-line math scores in their yearly standardized testing, thanks to the Saxon series.
posted by konolia at 12:01 PM on December 27, 2003
My kids got top-of-the-line math scores in their yearly standardized testing, thanks to the Saxon series.
posted by konolia at 12:01 PM on December 27, 2003
grumblebee -- what kinds of problem domains are you interested in working in? This is an important question to ask yourself up front. In fact, it might be good to tell us (or at least yourself) if there are any specific kinds of problems you'd like to be able to solve.
A "novel" approach: read just about anything by Keith Devlin, but especially his Mathematics: The Science of Patterns for a high-level, easy reading view of mathematical topics. Also, books on Math History can be really good... I think that to some extent, the developments of math history mirror developmental stages individuals go through as far as their ability to understand new abstractions. If you are better at symbolic thinking than geometric thinking, skip past the greeks, though... start in at 7th centurly development of algebra, where a modern number system is starting to reigh. If you are better at geometric thinking, though, don't avoid the greeks. I've got Howard Eves' History of Math book, and it's pretty good. I read Carl Boyer's History of Calculus when I was 18 and got quite a bit out of that, too. George Polya's How To Solve It is a bit old and stiff, but if you want to bone up on general problem solving skills, it's a good book to digest.
My first college calc prof wrote a calc text that I thought was pretty accessible. It's out of print, and therefore available cheaply when available.
I ought to emphasize again that you should find some problems you are interested in and chew on them, without a book if possible. Doing this will increase your problem solving skils and chances are you'll find yourself discovering solutions and techniques before you're taught them. This is especially important because you will forget much of what you learn. I forget integration by parts all the time, but know the basics of the idea well enough I can re-derive the formula when necessary.
Finally, something else I told another MeFite a while back: See if you can approach Math with the spirit of play for a bit, because in something of the way that poets play with words to try to articulate emotion and experience and insight, mathemeticians play with symbolic thinking and logic to articulate pattern, structure, and insight. If you catch that spirit -- and if you're a programmer, you already have half of it, probably -- you're almost set.
posted by weston at 12:04 PM on December 27, 2003 [4 favorites]
A "novel" approach: read just about anything by Keith Devlin, but especially his Mathematics: The Science of Patterns for a high-level, easy reading view of mathematical topics. Also, books on Math History can be really good... I think that to some extent, the developments of math history mirror developmental stages individuals go through as far as their ability to understand new abstractions. If you are better at symbolic thinking than geometric thinking, skip past the greeks, though... start in at 7th centurly development of algebra, where a modern number system is starting to reigh. If you are better at geometric thinking, though, don't avoid the greeks. I've got Howard Eves' History of Math book, and it's pretty good. I read Carl Boyer's History of Calculus when I was 18 and got quite a bit out of that, too. George Polya's How To Solve It is a bit old and stiff, but if you want to bone up on general problem solving skills, it's a good book to digest.
My first college calc prof wrote a calc text that I thought was pretty accessible. It's out of print, and therefore available cheaply when available.
I ought to emphasize again that you should find some problems you are interested in and chew on them, without a book if possible. Doing this will increase your problem solving skils and chances are you'll find yourself discovering solutions and techniques before you're taught them. This is especially important because you will forget much of what you learn. I forget integration by parts all the time, but know the basics of the idea well enough I can re-derive the formula when necessary.
Finally, something else I told another MeFite a while back: See if you can approach Math with the spirit of play for a bit, because in something of the way that poets play with words to try to articulate emotion and experience and insight, mathemeticians play with symbolic thinking and logic to articulate pattern, structure, and insight. If you catch that spirit -- and if you're a programmer, you already have half of it, probably -- you're almost set.
posted by weston at 12:04 PM on December 27, 2003 [4 favorites]
Response by poster: Weston, that's a great question. I don't have a very clear answer except to say that I keep reading that calculas is one of the greatest human achievements. If this is true then spending my life without exploring it is like spending my life without ever seeing the pyramids or without ever listening to Beethoven's 9th Symphony.
Mathematicians also talk a lot about the beauty of numbers. I want to experience that.
The closest I get to anthing practical is that I mess around with Actionscript programming a lot, and I really love graphics programming. Trig seems really useful for that. I already use a lot of trig when I program, but I don't have a deep understanding of it. I just plug in other people's formulas.
posted by grumblebee at 1:06 PM on December 27, 2003
Mathematicians also talk a lot about the beauty of numbers. I want to experience that.
The closest I get to anthing practical is that I mess around with Actionscript programming a lot, and I really love graphics programming. Trig seems really useful for that. I already use a lot of trig when I program, but I don't have a deep understanding of it. I just plug in other people's formulas.
posted by grumblebee at 1:06 PM on December 27, 2003
If this is true then spending my life without exploring it is like spending my life without ever seeing the pyramids or without ever listening to Beethoven's 9th Symphony.
I'm going to tell this to my friends sometime. This is the best analogy I've heard since I read Zen and the Art of Motorcycle Maintenance and the narrator talked about the "high country of the mind" being like mountain climbing -- it's hard to get there, it's sometimes inhospitable, but unless you learn to go there, you're stuck in certain valleys your whole life.
Keith Devlin, the author I mentioned before, is really good about the beauty thing.
The closest I get to anthing practical is that I mess around with Actionscript programming a lot, and I really love graphics programming. Trig seems really useful for that.
Indeed it is. :) Trig all comes from understanding ratios and correspondences between sides lengths and angle sizes of right triangles. Once you have that down, the rest usually falls into place (except manipulating complicated trig identities, which is a highly heuristic art that just takes lots of practice, I think).
You might also be interested in transformational geometry and basic linear (vector) algebra. Both are useful in 2D and 3D graphics programming. Lots of math related to rotating, moving, or stretching/distorting an object. It takes vector calc to do light/shading and a few other things on 3D models.
A college-level geometry textbook might well contain all these topics, except the linear algebra & vector calc. I haven't found one I really like, though.
posted by weston at 2:00 PM on December 27, 2003 [1 favorite]
I'm going to tell this to my friends sometime. This is the best analogy I've heard since I read Zen and the Art of Motorcycle Maintenance and the narrator talked about the "high country of the mind" being like mountain climbing -- it's hard to get there, it's sometimes inhospitable, but unless you learn to go there, you're stuck in certain valleys your whole life.
Keith Devlin, the author I mentioned before, is really good about the beauty thing.
The closest I get to anthing practical is that I mess around with Actionscript programming a lot, and I really love graphics programming. Trig seems really useful for that.
Indeed it is. :) Trig all comes from understanding ratios and correspondences between sides lengths and angle sizes of right triangles. Once you have that down, the rest usually falls into place (except manipulating complicated trig identities, which is a highly heuristic art that just takes lots of practice, I think).
You might also be interested in transformational geometry and basic linear (vector) algebra. Both are useful in 2D and 3D graphics programming. Lots of math related to rotating, moving, or stretching/distorting an object. It takes vector calc to do light/shading and a few other things on 3D models.
A college-level geometry textbook might well contain all these topics, except the linear algebra & vector calc. I haven't found one I really like, though.
posted by weston at 2:00 PM on December 27, 2003 [1 favorite]
"i wouldn't do any more problems than you need to get the concept. ... at university it tends to be more one question per concept and you're done"
The problem with that advice, in my opinion, is that if you plan on taking your studies to any depth, you're going to run into a lot of concepts. Those concepts will build on each other progressively. My experience is that if you do "one question per concept" for concept A, you'll feel like you've gotten it. Then 2 months and 15 concepts later you'll run into concept P, which you can't grasp without thoroughly knowing concept A, which you've now half-forgotten & mixed up with concepts B, C & D. My advice is to do as many problems as you need to get the concept, then do a few more to work it into your long-term memory.
Another piece of advice, for calculus at least. Give yourself enough time to properly absorb each chunk before moving on to the next. For years I wanted to teach myself calculus. I'd pick up a book, zoom along for a few chapters, feeling like I had a handle on it, then just crash into a dead-end. This past year, I went back to school and just finished up 4 quarters of calculus. I realized the problem previously was that I was trying to read the calculus textbook the way I would a book on history or evolutionary biology - just zoom straight on through. When I got to class, I realized that it's ok to spend a couple of hours just absorbing a few pages - and a full year to work through a single textbook. Math information is just more concentrated that way.
posted by tdismukes at 2:52 PM on December 27, 2003 [1 favorite]
The problem with that advice, in my opinion, is that if you plan on taking your studies to any depth, you're going to run into a lot of concepts. Those concepts will build on each other progressively. My experience is that if you do "one question per concept" for concept A, you'll feel like you've gotten it. Then 2 months and 15 concepts later you'll run into concept P, which you can't grasp without thoroughly knowing concept A, which you've now half-forgotten & mixed up with concepts B, C & D. My advice is to do as many problems as you need to get the concept, then do a few more to work it into your long-term memory.
Another piece of advice, for calculus at least. Give yourself enough time to properly absorb each chunk before moving on to the next. For years I wanted to teach myself calculus. I'd pick up a book, zoom along for a few chapters, feeling like I had a handle on it, then just crash into a dead-end. This past year, I went back to school and just finished up 4 quarters of calculus. I realized the problem previously was that I was trying to read the calculus textbook the way I would a book on history or evolutionary biology - just zoom straight on through. When I got to class, I realized that it's ok to spend a couple of hours just absorbing a few pages - and a full year to work through a single textbook. Math information is just more concentrated that way.
posted by tdismukes at 2:52 PM on December 27, 2003 [1 favorite]
Well I'm on your same boat, more or less ; I have to learn calculus (or at least some rudiments of) for a univ exam.
My univ-approved textbook basically sucks your will to learn
and manages to ruin the little I do know about maths, so I'm actively seeking for better books, expecially E-Books (ok expecially "free" ebooks cause I really don't have the money to buy stacks of them) ; so far I can recommend you the following tomes:
Calculus-Concept and Contexts. Brilliant book with practical problems like "given the surface of a box and the price of box paper per square feet, understand how you can estimate the cost of a box by its surface" and other problems. Yet it's definitely not a "for dummies" vaporbook.
Another excellent test, but more formal is Calculus, One-Variable Calculus with an Introduction to Linear Algebra a damn expensive (+-$140) book , definitely university-level one, but worth every penny imho. This book also provides demonstration for many axioms and theorems you'll probably never find in "dummies" kind of books.
In my limited experience I haven't yet found a calculus book that I can understand almost immediately , consistently providing me the "unveiling" sensation I crave when reading books. So far I have only found snippets of that sensation by reading these two books I recommended you.
One good free site about math and calculus is SosMath. Not as formal, accurate and rich as the books previously mentioned , but may just work for you.
If anybody happen to have links to good free ebook on calculus (expecially applied to economics) please let me know !
posted by elpapacito at 3:15 PM on December 27, 2003
My univ-approved textbook basically sucks your will to learn
and manages to ruin the little I do know about maths, so I'm actively seeking for better books, expecially E-Books (ok expecially "free" ebooks cause I really don't have the money to buy stacks of them) ; so far I can recommend you the following tomes:
Calculus-Concept and Contexts. Brilliant book with practical problems like "given the surface of a box and the price of box paper per square feet, understand how you can estimate the cost of a box by its surface" and other problems. Yet it's definitely not a "for dummies" vaporbook.
Another excellent test, but more formal is Calculus, One-Variable Calculus with an Introduction to Linear Algebra a damn expensive (+-$140) book , definitely university-level one, but worth every penny imho. This book also provides demonstration for many axioms and theorems you'll probably never find in "dummies" kind of books.
In my limited experience I haven't yet found a calculus book that I can understand almost immediately , consistently providing me the "unveiling" sensation I crave when reading books. So far I have only found snippets of that sensation by reading these two books I recommended you.
One good free site about math and calculus is SosMath. Not as formal, accurate and rich as the books previously mentioned , but may just work for you.
If anybody happen to have links to good free ebook on calculus (expecially applied to economics) please let me know !
posted by elpapacito at 3:15 PM on December 27, 2003
I've read of professional mathematicians who were originally inspired by What is Mathematics? by Courant and Robins. It's a deep book. And I will second the recommendation for Spivak's Calculus. Courant's two volume Introduction to Calculus and Analysis is also a challenging and classic text.
MathWorld isn't a text, but it's an amazing reference.
Finally, the math section at MIT's Open Courseware project (previously discussed) should be a terrific resource, though I haven't really investigated very much, yet.
posted by stuart_s at 6:59 PM on December 27, 2003
MathWorld isn't a text, but it's an amazing reference.
Finally, the math section at MIT's Open Courseware project (previously discussed) should be a terrific resource, though I haven't really investigated very much, yet.
posted by stuart_s at 6:59 PM on December 27, 2003
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posted by oissubke at 10:41 AM on December 27, 2003 [4 favorites]