Some people can spend a long period of intense focus on a math problem while others can't. Those people who can't would usually say something like "I'm not that good because I haven't studied hard enough so it's just a matter of studying lol". The thing is in order to "study hard" mathematics you actually need certain level of intelligence. People who can't "study hard" completely natural things like equivalent classes or groups or even logic(these are basic math first year/late high school students should know) more often than not it's because their brain can't keep track of the structure of the topic, not because they "just haven't tried".
I do realize that there is an important need for everyone to feel smart. But it's simply delusional to think people like Isaac Newton (or any competent professional mathematician) simply "study hard", I mean Newton did study hard, but he could "study hard" concepts like fluxion and binomial series in 17th century, it's because he's a genius...
Simply put, Newton could think intensely about a math problem in weeks or years, the average person could think intensely about a math problem in about 30 seconds before losing track. Newton "studied harder" than the average person, yes, but that's not because he's more hardworking (lol)
Mathematics is about following rigid rules the same way writing novel is about writing grammatically correct sentences. "It's just following rigid rules lol" is a rather dumb remark commonly made by people who never did serious high school math and still believe mathematics is about adding numbers together.
1. "Rigid rules" lead to insight and clarity. To quote Hans Halvorson
“How is it that mathematicians have a firm grip on concepts such as ‘infinity’ and
‘continuous function,’ while speculative philosophers continue talking in circles?”
Analytic philosophy was born out of mathematics, it was an attempt to fix the vast amount of nonsense philosophers of previous era came up with: by making philosophy look like math. There are many heroes in that revolution but I guess it's a story for another time.
2. The "rigid rules" of mathematics led to the creation of formal systems, which led to the idea of the computer which I believe is more creative than any creation of any other art field. Imagine you have to describe concepts like the internet, social network, or simply "software" (all can be reduced to a bunch of arithmetic rules being carried out repeatedly in a tiny piece of metal/plastic) to any artist of the 19th century.
So is competency in math more a result of hard studying or raw intelligence?
A few of my professors are Field medalists, most are ordinary mathematicians, and from what I've seen, there is a threshold.
Below the threshold (e. g the level of group/ring/field/Galois theory, real/complex analysis, differential geometry, topology, homotopy type etc.) there should be two paths to do well in math: the path of hard work, and the path of talent. Either path can get you pass all undergrad math courses and a 300k/year job at PwC, Goldman Sachs or Jane Street.
Above the threshold (grad school level e. g Étale cohomology, algebraic stacks, Langlands program etc) the two paths merge into one and it's difficult to tell where hard work ends and where raw talent begins.
I think there is no possibility for asking this question as "or-question". For people with the kind of intelligence especially suitable for math the amount of "hard" studying is negligible because the things others struggle with the most they just digest without any problems. So their "hard" work is mostly the routine calculations or operations (in abstract formal systems or on lower levels) because they are boring and just have to be written down. Or it's figuring out some really complicated pattern, but again, it isn't thought of by them like of a hard work, isn't perceived that way, it's a challenge, a creative task because when they begin to do the steps one way or another they see them quickly enough.
By the people with the kind of intelligence originally more suitable for other things than math the hard work will mean something completely different. It could be problems to understand some logical pattern just because the brain functions in a different way that doesn't allow to see that pattern directly as it is, to see it's meaning in the frame of the system, it's connections with other elements. It could be problems, if we are talking about real math, to find the way to prove something because the way they think just doesn't give ideas of the right kind suitable for formalization and application in the frame of the current mathematical system (current mathematical system could be abstract algebra, calculus, higher geometry etc.) and at the same time leading to proving the final claim. To just study hard in this case is waste of time because it'd be immensely hard and frustrative to study and even if you manage to force yourself to do it regularly and for hours everything someone from this category will learn will evaporate from their minds quickly. In their case the only way to become succesfull in maths is to change the ways of their minds or not to change but to learn new ones additionally. Some who finally manage to break the wall separating them from people with intelligence tuned for math do it unintentionally and unknowingly, some, probably the least, figure out on purpose what exactly it is in their thinking that hinders them and how it can be changed. That will be the hard work part by these people. Afterwards they fit in the first category more or less completely. In their case the hard work is to figure out HOW to be a mathematician, but of course it means practising and attempting like mad as well - but that's just the form their efforts to figure out/embrace the general way take.
So the two kinds of hard work are completely different. Obviously the larger amount of mathematical society is the people with intrinsically good intelligency math skills, but it's not exactly because of the skills, it's because the skills give them the possbility to live and exist as mathematician without difficulties to maintain inner comfort compared to the other group.
And what about the top (where exactly to draw the line I'm uncertain for my take) I completely agree with IlyaTheImpaler who climb on there need both things already developed - one or another way, the two paths merge into one.
I don't really understand how it works tbh. When I was a kid, in primary school, math was like chinese for me. I couldn't understand it but somehow I was able to solve secondary level school problems and always had straight A's in it. But then tbh, at that time I was never learning at all. I'd sleep in class and wouldn't even touch my books at home.
In secondary school, I had better teachers and was genuinely interested in the subject. It sat very naturally with me. When I was doing my O levels and A levels, my private tutor would give us what he called 'challenges'. They were supposed to be extremely tough problems which we had to solve in class. He'd only give those to a few students. I'd always manage to solve them and a lot of times, I was the only one to be able to do that. I never revised or anything. I didn't have to.
After secondary school, I started learning further maths just for fun, all by myself. A few months later, when I dropped the subject, my teacher asked to see my work. He was impressed. He said that I learnt very fast. He gave me a test and I passed it and he kinda looked surprised lol.
I tutored some students for a while. I haven't done any math for about a year now so I think i'm very rusty. But yeah, I never really understood this sort of knack for mathematics.
I've never put a tremendous amount of effort into math and I don't know much about my iq. However, my mom says that it's higher than average. I was learning things and developing at a faster pace than my siblings so they got me tested.
I'm in no way a genius or whatsoever. If you ask me, I'm kinda dumb lol. I'm gullible and that's not a sign of a highly functioning brain for me lol
The MAIN thing I love about math is that it's not about memorization. You can memorize a few basic formulas and that's common sense, but then calculus will kick your ass. It's being able to figure out a formula yourself and see the relationships between quantities and variations, which is definitely intelligence. That's what I love about math. Anybody can memorize something like history.
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I'm crap at math but that's mostly due to lack of interest in the subject. However I've noticed a lot of people who are good can't actually explain what their doing. They say things like "oh you move that number there and that letter there..." but they can't explain why, it's just doing as they were taught. I've also noticed that people good at maths often struggle badly in other areas. The rigid rules of maths that you can learn seem comforting to them.
Incidentally I always use a calculator even for extremely simple stuff because I have a nagging that says I've got to check 1 1 still equals 2
Studying. I was awful in math, yet I was great at every other subject and would breeze through them. I think the reason for this is that the way math was taught made it unappealing and didn't mesh with the way I thought and learned. So because of that I was less inclined to do it on my own unlike all the other subjects that I readily studied on my own. So I don't buy into the popular belief of "your just good at it", but rather its a matter of enjoying it which facilitates practice and thus increased understanding and aptitude for that subject.
I don't know, math has usually been straight forward to me, it only became hard at master level when you have to do like four pages to come to the answer and there is no way to know if the answer is correct because you can't ball park and see if the number seem reasonable. It becomes greek you can't see what your calculating anymore and if it made sense so spotting a faulty number in all the step was ridiculous. I can't understand that you can fail driving theory, yet people do and that is easier then math.
A combination of both, I think. People with raw intelligence have to put forth less effort in learning math because they can more easily recognize the patterns and memorize the equations. People who are not fortunate enough to have been born with that raw intelligence require significantly more effort in order to grasp the material. Study, and a tremendous amount of repetition. The more difficult the material, the more time and effort required. Eventually we all, regardless of our intelligence, hit a wall. A point in which the math becomes so complex that we struggle to grasp it and don't have the required time to grasp it available to us. For intelligent people, that wall is further off, but even they hit that wall eventually.
Competency and passing an exam or a class are not necessarily synonyous- I would define competency as being able to actually accurately apply math to solving real world problems or teaching it. From my experience tutoring, I think people of at least average intelligence who don't have applicable learning disabilities can pass exams in math through calculus, linear algebra and differential equations if they put enough time into it and the explanations are clear and concrete. Once you hit really abstract material like Real and Complex Analysis, I think raw intelligence sets a ceiling.
I think it's a mixture. There's a level of intelligence and logical understanding. Also math builds on earlier concepts so if you miss or don't understand an earlier step then everything above that one step becomes much more difficult. So a level of study is also involved. But even more so, conceptual understanding, getting the core concepts of mathematics. Such as understanding addition and subtraction is key to understanding number lines, also key to understanding multiplication. Without the understanding of multiplication division becomes almost impossible. Without an understanding of number lines graphing becomes nearly impossible. So someone who takes the same math class probably missed a key concept earlier in their studies.
I want to say studying because I’m all about nurture vs nature but I have studied law for 5 years and I can’t name a single act. I began programming (which requires maths and ‘that type’ of logical thinking) and I’m finding it easier than walking. Let’s say 70% nature 30% nurture? But really, who knows?
Combination of both. If you are blessed with an intuition for math, then it will be easier for you in the beginning. At some point, though, your god given talents won't help you as much as you think and you will have to start working on the subject more seriously. Back in undergrad, we had a few "genius type" people, they took part in math olympiads and did really well with the entry level subjects, because they had been training for years during school.
Later, though, when we had something like measure theory or complex analysis, they realised they couldn't keep up with the coursework without effort.
We also had a guy doing probability theory for.. I think it was 4th time in a row and counting. Whether or not he was studying his butt off isn't important. If it was his 4th time, he was studying math in a wrong way. Cramming and trying to memorise proofs is not efficient in math.
A lot of it is raw intelligence. I’ve always hated math, and I never really put any effort into studying it. But my math score was still in the 88th percentile on the SAT. If I even put half effort into working on math problems, I figure it out pretty easily. One of my friends studied and worked hard on math, but their math score was only the 63rd percentile, and they made poorer grades in math than o did.
It’s really a combination of both? So I wouldn’t really pick one over the other. Everybody has the ability to be proficient in math but some are naturally better at it than others. The ones that are usually bad at it often have to study more to make up for the misunderstanding.
Studying can improve results for anyone, but some require less. I am gifted at math, but have slacked a fair amount. Not everyone has the same abilities and there are different types of math. Math is my wheelhouse, but I am a very slow reader. I loathe reading, because I have to put in 3x the effort. I think you might benefit from the wheel of intelligence, I know from your namesake you also are math inclined, but it describes other abilities as well. I am studying the brain and find neurological abilities go beyond IQ and link better to the wheel. There is of course crossover, but some abilities don't have written test. Math does and ones ability to process it determines competency.
NEITHER!! A GOOD TEACHER!! A teacher that explains WHY, and HOW, rather than the mundane 1950's style rigid lectures that could put a Coke addict to sleep!!
People learn different ways, and a GREAT TEACHER, explaining, in the way that the students understand it just a ROCK STAR!!
Math isn't hard, or intimidating, if you have the right teacher, explaining, and encouraging!
The next step is challenging, and engaging them to want to DISCOVER!!
I think a lot of it has to do with IQ up to a point. Beyond that threshold interest, passion, devotion, hard work, and obsession take over. Personally I can see the practical use in a lot of mathematics, but I have no real interest in advancing the science. It just doesn’t do anything for me. That said my raw intelligence carried me through many advanced math courses with little to no effort with A grades.
Been teaching myself slowly some math/light physics. I’m trying to go against the normal ideas in order to find answers I want to find. Got started because I got seriously fed up with scientists like Neil Tyson saying things weren’t possible, and he such a bully in his answers. Then the president announces unlocking the secrets of the universe, holy shit I was pumped, pushed me even farther.
its a combination of studying and intelligence. Also I would add that people can think clearly, but some can’t think deeply. Vice versa. Has to be a good combo of both I suppose. Have you ever read ‘theory of natural philosophy by Roger Boscovich’ it’s pretty deep, sometimes I feel like having aspirin around.
Here’s a link to the Latin/English version...
https://archive.org/details/theoryofnaturalp00boscrich
I think there's some mindset to it... some familiarity, what you're learning with math is not as much math and theorums themselves but problem solving. You need to know the rules but the rules is not exactly the 'objective'... I get the guys you are talking about... it's also about the actual amount of time spent on it. 'its not that I'm so smart , it's just that I stay with problems longer..."...
This is why I say that math isn't for everyone. You can't teach everyone everything. God didn't say "I'm going to make everyone exactly the same". He made us different for a reason.
A lot of people, no matter how much they try, are not good at math and that's ok.
Math is mostly recognizing a problem and applying repetitive steps to solve the problem. It's when you use math as a language to describe other disciplines in science such as Physics is when it gets more challenging. As such, math like other languages require studying.
There’s a lot of logic in Maths.. so someone who’s more intelligent will probably be better at it than someone who tries really hard but isn’t exactly an analytical thinker.
However, they can surely improve.
You could maybe be born with some well-suited DNA, or live a healthy lifestyle, which would leave you predisposed to handling math problems more easily, but familiarity with those types of problems (math problems in this case), or strong enthusiasm for the subject goes a long way too.
I don't like to use the word "intelligent" because it implies that the person is somehow inherently smarter or wiser than someone else. I believe that's subjective, and probably not even a thing.
I think people are born with some sort of math/numbers competency. It can be improved, taught, learned, but honestly, with some people really struggle, even with huge amounts of effort.
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