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.
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.