Once upon a time I was pretty good at math but either I just stopped liking it or the series of dismal school teachers I had turned me off of it. I ended up taking the social studies/humanities rout and somewhat regretting it. I’ve studied some foundations of mathematics stuff, symbolic logic and really basic set theory and usually find that I can learn pretty rapidly if I have a good explanation in front of me. What is the best way to teach myself math? I stopped with statistics (High school, advanced placement) and never got to calculus. I don’t expect to become a math wiz or anything, I’d just like to understand the science I read better. Anyone have good advice?
I’m currently trying to teach myself mathematics from the ground up, so I’m in a similar situation as you. The biggest issue, as I see it, is attempting to forget everything I already “know” about math. Math curriculum at both the public high school and the state university I attended was generally bad; the focus was more on memorizing formulas and methods of solving prototypical problems than on honing one’s deductive reasoning skills, which if I’m not mistaken is the core of math as a field of inquiry.
So obviously textbooks are good place to start, but which ones don’t suck? Well, I can’t help you there, as I’m trying to figure this out myself, but I use a combination of recommendations from this page and looking at ratings on Amazon.
Here are the books I am currently reading, have read portions of, or are on my immediate to-read list, but take this with a huge grain of salt as I’m not a mathematician, only an aspiring student:
How to Prove It: A Structured Approach by Vellemen—Elementary proof strategies, is a good reference if you find yourself routinely unable to follow proofs
How to Solve It by Polya—Haven’t read it yet but it’s supposedly quite good.
Mathematics and Plausible Reasoning, Vol. I & II by Polya—Ditto.
Topics in Algebra by Herstein—I’m not very far into this, but it’s fairly cogent so far
Linear Algebra Done Right by Axler—Intuitive, determinant-free approach to linear algebra
Linear Algebra by Shilov—Rigorous, determinant-based approach to linear algebra. Virtually the opposite of Axler’s book, so I figure between these two books I’ll have a fairly good understanding once I finish.
Calculus by Spivak—Widely lauded. I’m only 6 chapters in, but I immensely enjoy this book so far. I took three semesters of calculus in college, but I didn’t intuitively understand the definition of a limit until I read this book.
I’ve learned a lot of equations from Wikipedia, but I’ve not really learned a lot of real math—that’s really come from doing homework problems and thinking about them later.
I’ve definitely learned a lot of math from Wikipedia. I don’t generally do the proofs myself, so I don’t really have any of the elusive “mathematical maturity”, but I definitely have learned a lot of abstract algebra, category theory and mathematical logic just by reading the definitions of various things on Wikipedia and trying to understand them.
On the other hand, I am pretty motivated to learn these things because I actively enjoy them. Other branches of math, I am much less interested in and so I don’t learn that much. But it is possible!
I don’t understand why/how anyone would learn equations without understanding them.
I agree that wikipedia is not a good substitute for textbooks in general, neither does it replace actual practice by problem solving. You can still learn a lot of math (even complete proofs) from it: get good first impressions on whole areas. It even contains high quality introductory material on certain important topics and facts.
However I completely agree with you: the most important thing in math is to think about problems. Undergraduate Springer books (yellow series) typically contain a lot of problems alongside actual text. My method is the following:
1) Read one chapter and write up the statement of every theorem.
2) Go through all statements and reproduce the proof without rereading the material
3) Iterate 1)-2) if your are stuck with any of the proofs
4) Proceed with the problem section and try to solve all problems. Omit problems only if they are marked as hard and if you are stuck after an hour of thinking.
The most topics books to start with are linear algebra and calculus. Working through the undergraduate material in the above way takes a long time, but you will build a firm base for further studies.
I’ve always found that memorizing proofs or actually doing the exercises (as opposed to taking time to understand the structure of the solutions to some of them, if the main text doesn’t already cover the representative propositions) hits diminishing returns, in most cases anyway, when you are learning for yourself. The details get forgotten too quickly to justify the effort, the useful thing is to get good hold of the concepts (which by the way can be glossed over even with all the proofs and exercises, by relying on brittle algorithm-like technique instead of deeper intuition).
I’ve always found that memorizing proofs or actually doing the exercises (as opposed to taking time to understand the structure of the solutions to some of them, if the main text doesn’t already cover the representative propositions) hits diminishing returns, in most cases anyway, when you are learning for yourself. The details get forgotten too quickly to justify the effort, the useful thing is to get good hold of the concepts (which by the way can be glossed over even with all the proofs and exercises, by relying on brittle algorithm-like technique instead of deeper intuition).
I don’t vote for blind memorization either. However, I think that if one can not reconstruct a proof than it is not understood either. Trying to reconstruct thought processes by heart will highlight the parts with incomplete understanding.
Of course in order to fully understand things one should look at additional consequences, solve problems, look at analogues, understand motivation etc. Still, the reconstruction of proofs is a very good starting point, IMO.
Sure. I’m pointing to the difference between making sure that you can do proofs (not necessarily reconstruct the particular ones from the textbook) and exercises, and actually reconstructing the proofs and doing the exercises. Getting to the point of correctly ruling the former can easily take 10 times less time than the latter. You won’t be as fast at performing the proofs in the coming weeks if need be, but a couple of years pass and you’d be as bad both ways (but you’d still have the concepts!).
Perhaps I should have said “looked up” instead of “learned.” That is, I understand the Laplace transform, and have done many homework problems that involved deriving common transform pairs. However, when I need one, I don’t try to re-derive it or rely on memory; I go look it up at Wikipedia and use it.
Which textbook is good on a given topic depends on the student’s current level, and more importantly on what exactly do you want to learn. “Math”?.. A couple of random suggestions that appealed to me aesthetically, but YMMV:
F. W. Lawvere & S. H. Schanuel (1991). Conceptual mathematics: a first introduction to categories. Buffalo Workshop Press, Buffalo, NY, USA.
S. Mac Lane & G. Birkhoff (1999). Algebra. American Mathematical Society, 3 edn.
If the Simple Math of Everything were a real text book, I’d read that. But I’ve gathered calculus is the right place to start. Probability theory would be next, I guess.
Once upon a time I was pretty good at math but either I just stopped liking it or the series of dismal school teachers I had turned me off of it. I ended up taking the social studies/humanities rout and somewhat regretting it. I’ve studied some foundations of mathematics stuff, symbolic logic and really basic set theory and usually find that I can learn pretty rapidly if I have a good explanation in front of me. What is the best way to teach myself math? I stopped with statistics (High school, advanced placement) and never got to calculus. I don’t expect to become a math wiz or anything, I’d just like to understand the science I read better. Anyone have good advice?
I’m currently trying to teach myself mathematics from the ground up, so I’m in a similar situation as you. The biggest issue, as I see it, is attempting to forget everything I already “know” about math. Math curriculum at both the public high school and the state university I attended was generally bad; the focus was more on memorizing formulas and methods of solving prototypical problems than on honing one’s deductive reasoning skills, which if I’m not mistaken is the core of math as a field of inquiry.
So obviously textbooks are good place to start, but which ones don’t suck? Well, I can’t help you there, as I’m trying to figure this out myself, but I use a combination of recommendations from this page and looking at ratings on Amazon.
Here are the books I am currently reading, have read portions of, or are on my immediate to-read list, but take this with a huge grain of salt as I’m not a mathematician, only an aspiring student:
How to Prove It: A Structured Approach by Vellemen—Elementary proof strategies, is a good reference if you find yourself routinely unable to follow proofs
How to Solve It by Polya—Haven’t read it yet but it’s supposedly quite good.
Mathematics and Plausible Reasoning, Vol. I & II by Polya—Ditto.
Topics in Algebra by Herstein—I’m not very far into this, but it’s fairly cogent so far
Linear Algebra Done Right by Axler—Intuitive, determinant-free approach to linear algebra
Linear Algebra by Shilov—Rigorous, determinant-based approach to linear algebra. Virtually the opposite of Axler’s book, so I figure between these two books I’ll have a fairly good understanding once I finish.
Calculus by Spivak—Widely lauded. I’m only 6 chapters in, but I immensely enjoy this book so far. I took three semesters of calculus in college, but I didn’t intuitively understand the definition of a limit until I read this book.
I’ve learned an awful lot of maths from Wikipedia.
I’ve learned a lot of equations from Wikipedia, but I’ve not really learned a lot of real math—that’s really come from doing homework problems and thinking about them later.
I’ve definitely learned a lot of math from Wikipedia. I don’t generally do the proofs myself, so I don’t really have any of the elusive “mathematical maturity”, but I definitely have learned a lot of abstract algebra, category theory and mathematical logic just by reading the definitions of various things on Wikipedia and trying to understand them.
On the other hand, I am pretty motivated to learn these things because I actively enjoy them. Other branches of math, I am much less interested in and so I don’t learn that much. But it is possible!
I don’t understand why/how anyone would learn equations without understanding them.
I agree that wikipedia is not a good substitute for textbooks in general, neither does it replace actual practice by problem solving. You can still learn a lot of math (even complete proofs) from it: get good first impressions on whole areas. It even contains high quality introductory material on certain important topics and facts.
However I completely agree with you: the most important thing in math is to think about problems. Undergraduate Springer books (yellow series) typically contain a lot of problems alongside actual text. My method is the following:
1) Read one chapter and write up the statement of every theorem.
2) Go through all statements and reproduce the proof without rereading the material
3) Iterate 1)-2) if your are stuck with any of the proofs
4) Proceed with the problem section and try to solve all problems. Omit problems only if they are marked as hard and if you are stuck after an hour of thinking.
The most topics books to start with are linear algebra and calculus. Working through the undergraduate material in the above way takes a long time, but you will build a firm base for further studies.
I’ve always found that memorizing proofs or actually doing the exercises (as opposed to taking time to understand the structure of the solutions to some of them, if the main text doesn’t already cover the representative propositions) hits diminishing returns, in most cases anyway, when you are learning for yourself. The details get forgotten too quickly to justify the effort, the useful thing is to get good hold of the concepts (which by the way can be glossed over even with all the proofs and exercises, by relying on brittle algorithm-like technique instead of deeper intuition).
I’ve always found that memorizing proofs or actually doing the exercises (as opposed to taking time to understand the structure of the solutions to some of them, if the main text doesn’t already cover the representative propositions) hits diminishing returns, in most cases anyway, when you are learning for yourself. The details get forgotten too quickly to justify the effort, the useful thing is to get good hold of the concepts (which by the way can be glossed over even with all the proofs and exercises, by relying on brittle algorithm-like technique instead of deeper intuition).
I don’t vote for blind memorization either. However, I think that if one can not reconstruct a proof than it is not understood either. Trying to reconstruct thought processes by heart will highlight the parts with incomplete understanding.
Of course in order to fully understand things one should look at additional consequences, solve problems, look at analogues, understand motivation etc. Still, the reconstruction of proofs is a very good starting point, IMO.
Sure. I’m pointing to the difference between making sure that you can do proofs (not necessarily reconstruct the particular ones from the textbook) and exercises, and actually reconstructing the proofs and doing the exercises. Getting to the point of correctly ruling the former can easily take 10 times less time than the latter. You won’t be as fast at performing the proofs in the coming weeks if need be, but a couple of years pass and you’d be as bad both ways (but you’d still have the concepts!).
Perhaps I should have said “looked up” instead of “learned.” That is, I understand the Laplace transform, and have done many homework problems that involved deriving common transform pairs. However, when I need one, I don’t try to re-derive it or rely on memory; I go look it up at Wikipedia and use it.
...reading textbooks?
I’m looking for specific advice. Do you know of good text books?
Which textbook is good on a given topic depends on the student’s current level, and more importantly on what exactly do you want to learn. “Math”?.. A couple of random suggestions that appealed to me aesthetically, but YMMV:
F. W. Lawvere & S. H. Schanuel (1991). Conceptual mathematics: a first introduction to categories. Buffalo Workshop Press, Buffalo, NY, USA.
S. Mac Lane & G. Birkhoff (1999). Algebra. American Mathematical Society, 3 edn.
(Both can be found on Kad.)
What, specifically, do you want to learn?
If the Simple Math of Everything were a real text book, I’d read that. But I’ve gathered calculus is the right place to start. Probability theory would be next, I guess.