<p>your Dad is wrong.</p>
<p>I can’t do harder linear algebra/analysis type problems, and I have no trouble with calculus or basic probability theory. The average person cannot do calculus or basic probability theory. It has a lot to do with spatial ability and types of working memory, i.e. can you rotate a 3 dimensional object in your head.</p>
<p>Do you remember those who struggles with middle school maths? Do anyone here think that those persons on average could handle a maths degree?</p>
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I would argue that it is physics and not maths, most I know have an easier time with maths than physics.</p>
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<p>Well, I think the main thing is the passion, but yes there’s intelligence; I’ve not met you, but I’d take a wild guess that if you had the passion, you’d be able to do it. </p>
<p>The thing is, generally ability at math and passion feed each other. Trust me, I read math for fun as a high schooler, and frankly if I hadn’t been naturally able to do it with hard work, and didn’t see the fruits coming, I’d not have continued. Part of what drives a math major is the thrill at every step of understanding + realizing how something fits into the puzzle! </p>
<p>The dad is basically wrong, because I would say most people are incapable of getting passionate enough about math, and that in and of itself keeps one from becoming “smart enough” at math. If you have a ton of energy and are focused at it, well math is basically like a huge puzzle where you figure out what is true and what isn’t, and patterns become clearer with time and effort. But you have to like the process enough to do it a LOT. </p>
<p><em>Realistically</em> the average college student at a good university even can’t handle college level math. Let alone get a Ph.D. in it.</p>
<p>As another little note – people do often confuse passion with intelligence. I know lots of people used to call me smart at math for reading relatively advanced college texts in high school, but it was more about energy than ability. Part of that energy carries one to the point of being able to digest the really abstract concepts. It’s kind of a matter of paying attention and rendering abstract thought processes natural to one, and that happens only with passion.</p>
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<p>Hmm, maybe. I think people with natural talent tend to do things that match their abilities, and very quickly, we reach a stage where it’s not natural talent but a very high level of intellectual investment + hard work that really make the difference between success and failure. </p>
<p>Studying math is not a simple matter of reading and writing proofs. It’s about studying the <em>right</em> things, training yourself to figure out what’s important, and training yourself to think in the right way, so that matters become clearer. A master of this art will invariably be more efficient, and even have more success than someone who is a better raw problem-solver.</p>
<p>mathboy is right,
a lot of math is about giving a crap
like you have to be deeply troubled
if you use a theorem
and you’re not sure you’re allowed to use the theorem</p>
<p>and if you don’t know if something’s true or not
it’s not just a -1 on your paper,
you have to prove it or find a counterexample
you can’t be content with a gray area,
everything is either black or white, known or unknown
and if not it screws with your view of the world</p>
<p>my friend who is an IMO gold medalist
was doing a basic multivariable calculus problem today,
about finding minimums by looking at places where the gradient is 0
but the part where he actually found the minimum
took him like five minutes
what bothered him was whether the minimum exists
never mind that we were supposed to assume it existed,
and the teacher probably didn’t care
if you use a theorem
you have to know why you’re allowed to</p>
<p>
I am a natural at this, for some reason I always know which concept is important and then it somehow automatically comes into a nice pattern were everything in maths and sciences in general is woven together neatly. I never do homework or such but things like abstract algebra, partial differential equations, complex analysis and variational calculus aren’t that strange concepts when you think about it so it isn’t really necessary.</p>
<p>I think a problem many have is that they do have no idea of what they do not know. So instead most are wasting a ton of time relearning things they already do know while skipping important stuff in between because of a skewed understanding of themselves. In my opinion the most important thing is to know which parts you haven’t grasped yet.</p>
<p>Usually the things which are most annoying to study are also the things you know the worst, which is probably the reason there is so much truth to my statement above.</p>
<p>The fundamental reason I do not do exercises is because if you could solve them you didn’t need the exercise and if you couldn’t solve it then you just learned that you didn’t know how to solve it. The actual learning part comes when you read the book to understand how to solve the problem, so if you instead know what you don’t know and thus can just read those pages from the start you save a huge amount of time.</p>
<p>That’s quite ironic Klockan3… You’re saying that we suffer from what we don’t know, and then you say that you don’t do problems because you know how to do them. Aren’t you catching yourself in your own statement?</p>
<p>Nope, because I know what I do not know and therefore I do not need to seek for it via doing problem sets etc.</p>
<p>Well if you’ve never done any of you sets how would you know that you could have finished every single problem on the set without any hitch-ups?</p>
<p>Because I know that I didn’t miss any of the theory, and when I know I missed something I just read up on it till I know it? Maths is not about solving a ton of problems, maths is about understanding a few concepts per course, as long as you understand those the tests are trivial.</p>
<p>I was reading The History of Mathematics by W.W.R. Ball this winter on my 13 hour flight. The most powerful insight I got from that book was that math has its roots in philosophy and religion - that one should approach mathematics as a philosopher exploring the realm of mathematical thought. Personally, I must say that this is one of my favorite aspects of mathematics. </p>
<p>However, I feel that since not everyone can be a philosopher the same trend applies to mathematics. That is not to say that not being good at math makes one less intelligent, but that their manner of thought is unsuited for math. My younger sister, for example, is much, much more clever and charming than I am yet when I try and teach her a mathematical concept she has a hard time internalizing the concept. She can apply the concept when needed but she really does not understand why she is doing what she is doing. I would blame the school system except I had the same exact math teacher she currently does and I learned most of this on my own when I was her age. It is my dearest wish to teach her to love and understand math as I do, but it is so hard with the way the schools teach math.</p>
<p>Yeah, I feel the same dill, most who are good at maths could derive almost everything taught in the courses from grade 1-9 without any outside help.</p>
<p>Even though those concepts are quite far away from the concepts taught at higher levels it still gives you a positive feeling towards the subject compared to someone who struggled more with it and not the least it gives you a much better foundation to stand on when you hit the not so easy courses.</p>
<p>I remember how naive I was when I was little and thought that formulas was useless since you could solve every problem using sound logic anyway, kinda like calculating stacking interest or mean values. But when I grew older I realized what formulas really was for, they are short cuts your mind can take for something you have already derived, saving a lot of time.</p>
<p>I think a problem with most is that they do not derive their own formulas but instead just try to learn to think in the way the formulas taught to them do. I would say that it is bad to do it in that way, every person thinks in different ways. Maths is a very open subject, if you do not like a formula you can make your own! If you do not like a proof you can make your own! Everything in maths is what it is defined to be, no more and no less.</p>
<p>Gee, maths is so simple that it is ridiculous 
A single man in an isolated cave could theoretically derive all the maths himself!</p>
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<p>Hm, can’t tell if you’re being serious, but I can tell you that I half got infected with this idea earlier, and would like to give a friendly warning against it. Doing exercises is truly a must – but you should be careful about which ones you do. The reason being, ideally, you’d ask your own questions as exercises and answer them yourself, but in the interest of time, generally one trusts the masters of a field to reveal what’s important in the exercises. </p>
<p>I think this is much like your “Why formulas if you can derive them?” logic – doing some deep problems is really the only way you get a <em>working knowledge</em> of stuff.</p>
<p>In research mathematics, nobody is going to care if you can take a test, really. You have all the books at your disposal, so to speak, and you read and reference them as time comes. Why take classes then? It is to get a working knowledge. Doing certain exercises is a matter of tuning your head to think in a certain fashion, and tuning it without wasting too much time, because there’s an ocean of stuff one probably doesn’t know and has little time to learn before one can actually do research!</p>
<p>Of course, one shouldn’t do exercises that look easily doable. A few that look deep enough are necessary, though.</p>
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Yeah, probably. But I am 100% lazy and if I am getting a phd in anything it will probably be physics, but I am thinking about doing a double masters in physics and maths.</p>
<p>Anyhow, I think that by walking my own path by figuring out roughly everything myself, just following the general guidelines, I can make myself into a much more creative person than the average math-bot the colleges churns out on a regular basis, and without being able to think for yourself you will never achieve anything at all, in the end that is the most important aspect to learn. </p>
<p>It could be a brilliant move, it could also be that I am just really really lazy and just trying to talk myself out of this situation. In the end if I actually did those things I could study roughly twice as fast as I do now. But it is so easy to not care…</p>
<p>Can you give me a reason to try? I mean, it wouldn’t give me more friends or anything like that… Understanding maths is awesome, I can agree with that, but in the end only if you got friends which also thinks that it is awesome.</p>
<p>Respect, it is another thing, I do not want to pull ahead too much from my peers either, I am enough of an alien as it is.</p>
<p>Now, just to clear things up, I never lived in the US, I live in Sweden and study here, I just like to discuss things about college education and thought that this forum was ok. It is a lot easier to get into masters studies here and usually it is expected of you and it is required before you start on your phd.</p>
<p>Edit: Oh, by the way, I learned that when you teach in a subject you learn it a lot better than when you just study it, so I have taught in the lesser maths courses like linalg or calc1-2-3 which is kinda nice. I also often help my peers with maths, giving them tips on how to think etc.</p>
<p>Depends on what school you go to and what track you take. I’m honestly not convinced that the standard math track anywhere requires too much natural mathematical ability (assuming however some basic competence) whereas Math 55 at Harvard is said to be the single most diffcult undergrad course in the country.</p>
<p>What is Math 55? I’m going to assume that is real analysis. Real analysis is the hardest undergraduate course, period.</p>
<p>AS DESCRIBED by Harvard math</p>
<p>Math 55:
"This is probably the most difficult undergraduate math class in the country; a variety of advanced topics in mathematics are covered, and problem sets ask students to prove many fundamental theorems of analysis and linear algebra. Class meets three hours per week, plus one hour of section, and problem sets can take anywhere from 24 to 60 hours to complete. This class is usually small and taught by a well-established and prominent member of the faculty whose teaching ability can vary from year to year. A thorough knowledge of multivariable calculus and linear algebra is almost absolutely required, and any other prior knowledge can only help. Students who benefit the most from this class have taken substantial amounts of advanced mathematics and are fairly fluent in the writing of proofs. Due to the necessity of working in groups and the extensive amount of time spent working together, students usually meet some of their best friends in this class. The difficulty of this class varies with the professor, but the class often contains former members of the International Math Olympiad teams, and in any event, it is designed for people with some years of university level mathematical experience. In order to challenge all students in the class, the professor can opt to make the class very, very difficult. "</p>
<p>havahd math 55
[Harvard</a> Mathematics Department : 21, 23, 25, or 55?](<a href=“http://www.math.harvard.edu/pamphlets/freshmenguide.html]Harvard”>http://www.math.harvard.edu/pamphlets/freshmenguide.html)</p>
<p>This is intended to be the FIRST math course a math major student takes at harvard, assuming they have an illustrious background in mathematics. (of course as you can see from that website there are easier alternatives…math 55 is not required.)</p>
<p>A undergraduate math degree does not have to be easy I guess.</p>
<p>It is actually pretty funny for math 23 (the 2nd one) it describes it by stating “want a proof-based class where you can still see the connection to the real world.” …opposed to math 25, and math 55 where you will most likely go insane and lose all connection to the real world i guess.</p>
<p>I am a math major and I must admit that I am not smart enough for Math 55.</p>
<p>Sucks to be a third-rate math student.</p>
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<p>well seriously, who the hell is</p>
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<p>My best reason is that there’s just a seemingly infinite amount of prerequisite math to learn before you get to the coolest stuff, and I find it’s incredibly motivating to learn prerequisite subjects quickly and efficiently. </p>
<p>This goes to my philosophy that easy to medium exercises are better than hard ones in general for a class’s purposes if the goal of the class is to LEARN. Math 55 at Harvard is not, in my own opinion, the ideal environment to <em>learn</em> for even most of the best mathematicians; it’s the type of environment where you douse your head in ice-cold water to snap out of complacency, and challenge yourself to the limit + bond with peers. In short, there are many courses in Harvard and across the country covering more advanced mathematics than Math 55, but few which have comparable workload and where the focus is on <em>making</em> the class as hard as possible. Generally, in advanced coursework, problems you solve tend to be hard to match the sophistication of the material, rather than being intrinsically debilitating. </p>
<p>So my short answer on why one should do standard math exercises is that they help one learn basic (“basic” can be not so basic!!!) material efficiently, so you get to cool stuff. You’ll HAVE to exercise creative talent to do research anyway, but it’s hard to ever get to that stage of researching if you don’t master the requisite material at a reasonably rapid pace.</p>