29

Certainly someone your age (or even much older) can learn Calculus, even get a degree in mathematics. And get a good job afterward. That "young man's game' quote refers to doing mathematical research at the highest level. There is some validity to it, but there are also well-known counterexamples. (And even most young men never make it there.)


25

For a little historical perspective, which at least explains why older books on non-elementary mathematics rarely have "solutions": Imagine first that there's no internet, that long-distance phone calls are prohibitively expensive, and that photocopying is not only expensive but mostly inaccessible to students. Imagine also that people use the same homework ...


19

Unguided self-study of mathematics is difficult, and harder for someone with little experience at it. It is normal to take time to advance. One should think in terms of months not hours. A typical one semester class in linear algebra does not cover every detail in the typical linear algebra textbook, nor require a student to work every exercise in that book. ...


15

You asked for anecdotal evidence. I was a "gifted student". The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade. I never formally learned algebra 1, but I understood it. They gave me a book for 11th grade math and I learned. I don't think it did me any harm - but I do think I understood the concepts and didn't just ...


15

I have a bit of anecdotal evidence. I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I was not seen as a particularly bright student. Due to bullying I looked for alternatives to the local high schools, and ended up applying to university early ...


15

To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some point, mathematical sentences where written without symbols; the introduction of letters to denote mathematical object improved quite a lot the depth and ...


14

Some suggestions Michael Spivak's Calculus This is really an honors level calculus text, but it might be useful to have around. It's pretty expensive, however. Stephen Abbott's Understanding Analysis Back in January 2003, in sci.math, I wrote: I think it's the best written introductory analysis book that's appeared in the past couple of decades. ...


13

You may have been able to remember a lot of math for a short time, but you ma have lacked understanding. When you talk to people who excelled in school mathematics, they might be perplexed by your plight, or they may think you just didn't stick with it long enough. They might mistakenly tell you that you're just not good at math. However, traditional ...


13

By abstraction, we mean that we step back from concrete objects to consider a number of objects with identical properties simultaneously. For instance, consider the following three objects: The set of functions $A,B,C$ defined on the set $\{1,2,3\}$ by $A(1)=1,\;A(2)=2,\;A(3)=3$, $B(1)=2,\;B(2)=3,\;B(3)=1$, $C(1)=3,\;C(2)=1,\;C(3)=2$, The set of complex ...


13

I tend not to answer the questions that I have no professional knowledge about or the questions that are too general to be answered here. But, your question reminded me of my own undergraduate years when I had the exact same strategy as you! I am writing this to confess that it was not a right strategy. Let me explain, using two examples. Suppose you have ...


13

An proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so that the other mathematician is convinced that they could (in principle) give a fully rigorous argument. Even that isn't quite accurate because no one (short ...


12

As you have noticed, mathematical text is often quite concise and it can be difficult to squeeze it into tighter space or write in your own words in a nontrivial way. Therefore it is easy to end up copying things verbatim if you want to take notes. Taking useful notes in mathematics is quite different from many other subjects, and you are not alone with your ...


12

If you solve a mathematical problem the wrong way, you get a wrong result, which can be checked. If you "solve" a philosophical problem there is no way to check the result in any decent timeframe. May be you made no sense, may be you invented a new branch of philosophy. In the hard (read: checkable) parts of philosophic thought, the tools are mathematical ...


11

I think it's very commendable to try proving things yourself first; even a failed attempt has value. However, it's also important to learn from others' proofs, so don't be afraid to sneak a peek at the full proof, if only the first line or two to give you a hint. You may also find it worthwhile to get hold of a book on proof techniques, such as Velleman's ...


11

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a while because there's no one to talk in it with. There are two types of mentalities about intelligence and success in life. Studies have shown that children who'...


11

There is evidence that group studying in college is negatively correlated with improvements in critical thinking, as measured by a test called the CLA (Collegiate Learning Assessment). This is described in Arum and Roksa, Academically Adrift: Limited Learning on College Campuses, p. 100.


10

It's great to hear about your enthusiasm to learn math! No fewer than three answers have suggested Khan Academy. Khan Academy sucks. (In my not-so-humble opinion.) The Khan Academy videos basically crank through calculations, without encouraging any deeper understanding. It sounds like you have bits and pieces of algebra. Next, you probably want to check ...


10

My first suggestion is not to prepare for your exams alone. Prepare for your exams with some of your colleagues, but make sure each of you reflects on your own progress toward readiness independently; the role of you and your colleagues is to provide feedback for each other. My next suggestion is to identify areas of concern and focus your efforts on these ...


10

Don't denigrate "pre-calculus mathematics and calculus." Many of the problems of students is a lack of a solid foundation in the lower mathematics, especially algebra. Make sure you can do the algebra reliably and quickly. I would say the next most important knowledge is that of basic logic--not necessarily mathematical logic, which will be taught in ...


10

To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-bloc education, for a variety of reasons which are irrelevant to the question. But, either way, there are some implicit hypotheses that are (in my opinion) worth ...


10

You will retain something as long as you practice it. It just so happens that for many, many theorems, it's the statement of the theorem that matters more than the proof. I think a good example is the Fundamental Theorem of Algebra. The proofs for it are obnoxiously technical compared to how easy it is to state it. (You could explain the theorem to a ...


10

In addition to other insightful answers/comments/remarks, apart from the issue of "degree" and "what kind of genius", I think a large part of the problem is exactly that mathematics is mostly portrayed as a school subject, in which a large part of "success" is measured by a certain conformity to "the teacher"'s expectations, and conformity with the worldview ...


9

This question makes me think of the James Gleick quote on two kinds of genius: "There are two kinds of geniuses: the 'ordinary' and the 'magicians'. An ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what they've done, we feel certain ...


9

Taking a break from a convoluted computation... To me the key limiting factor is space. I can spread out several sheets over the table and have various bits and pieces directly visible. I cannot do this to the same extent on my tablet or my computer. Therefore I also use loose sheets not something that is bound together. One might argue I could set up a ...


9

I left a comment a while back, but I have been mulling over this (off and on) since then, and feel that I should expand on it. As a first-term master's student, I made some off-hand comment about not liking applied mathematics. My advisor cracked down on me pretty hard at the time, and essentially explained that I lacked the mathematical maturity to have ...


8

Following points have always worked for me: Learn the concepts, not the formulas. Practice. If you really want something to stick in your brain, you have to convince the brain, you need it. The clue is not how many times you repeat something, but how many times you try to recall it. Only do one manageable theoretical step at a time and let yourself time to ...


8

In my experience whenever answers are provided a huge majority of students will rush for the answers before even giving it a try. Unfortunately the best you learn from an exercise, especially a non routine one, sits in the gloomy time during which you're trying to figure out what to do and have no clues... I think giving immediately all solutions to ...


8

I think this question, with editing, has a lot to do with mathematics education, in that it points to the problem of learning to read and interpret mathematical writing. I would instead ask something like: "How does one glean the ideas in a mathematical paper without reading it line by line?" Now for a somewhat rambling answer to this question: Note that ...


8

This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to this question on Math Overflow. The opening of the quotation I posted there is particularly telling: "Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then ...


8

I guess that (by the question's metric) the most valuable things to learn are visual and conceptual, instead of the ability to perform detailed computations. Some concepts are very easy to learn, and have substantial value, so the ratio is very high. For example: Some of the most common logical fallacies. Some of the most dangerous logical fallacies. How ...


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