Becoming a mathematician - how important is IQ?

[arildno]

By seeing,relative to other kids your age, what you are competent at, not the least what you've managed on your own.
THAT's the indicator for level of maturity.

And teachers have certainly an "instinct" for that, an instinct distilled from many years of experience, so that they know what they normally can expect from a 16-year old and what is extremely uncommon.

However, it might be that the most that you can hope for is a few conversations with professionals who can give you good,relevant reading tips.

That should be a realistic goal, and quite attractive in itself, agreed?

I believe that lectures at Danish universities are public and, in principle, open for all to listen to.
But it doesn't follow that your school will allow you attendance in school time, or that it is possible for the university to regard you as a regular student (for example, grading papers, allowing you to sit exam and so on).

But there is no harm in exploring such possibilities, is there?

PS:
Ask your Mom and Dad as well. Moms and Dads want to be involved in such decisions, that's their nature...

 

[Levis2]

By seeing,relative to other kids your age, what you are competent at, not the least what you've managed on your own.
THAT's the indicator for level of maturity.

And teachers have certainly an "instinct" for that, an instinct distilled from many years of experience, so that they know what they normally can expect from a 16-year old and what is extremely uncommon.

However, it might be that the most that you can hope for is a few conversations with professionals who can give you good,relevant reading tips.

That should be a realistic goal, and quite attractive in itself, agreed?

I believe that lectures at Danish universities are public and, in principle, open for all to listen to.
But it doesn't follow that your school will allow you attendance in school time, or that it is possible for the university to regard you as a regular student (for example, grading papers, allowing you to sit exam and so on).

But there is no harm in exploring such possibilities, is there?

PS:
Ask your Mom and Dad as well. Moms and Dads want to be involved in such decisions, that's their nature...

In my case it might just be the majority of the other students in my class that share a mathematical disability lol .. i don't know, but compared to them, I'm 10 years ahead lol :)

And reading tips would definately be of imminemt value! And you are correct, that the university lectures are open to the public, but i don't think my school would tolerate the absence :) My chemistry teacher thought that it was ridicolous for my to stay on 1st year chemistry, so he wanted to move me to 2nd year. This was not possible, because i would then be registered for 1 year absence in my regular chemistry clases lol .. And that will get me expelled :)

I just think it would be hard only to attend the lectures at a university - i really can't say since i haven't been to one, but i could imagine that the exercises they do as well will make the information stick. Might be hard to get a good understanding without them, especially for a 16-year old !:) But if i am allowed by my school, i will definately take advantage of the opportunity.

 

[arildno]

It seems that your science teachers know exactly how outstanding you are.

Now, school rules are to be followed. But in following them, they might on occasion..bend a little bit..

In particular if someone with scientific authority outside the local school environment (say, a university professor) can confirm, through his observations what the level of your abilities are right now.
That would carry some punch with the head-master, in connection with your regular science teachers' assessments.

But, anyhow, I think your first step is to have a good, long talk with your regular maths teacher, and explore realistic options for self-study and getting contact with university people.
I'm sure he already has a bad conscience about you; his job is to help ALL his students, but within the strictures of the normal teaching duties, he is at a loss for helping you on YOUR level at the same time as well.

You're quite right that self-study has its limitations.
But it will be much you CAN study on your own profitably, and a professional can help you sort out which themes you ought to begin with (i.e, suitable for self-studying)

 

[arildno]

And, Levis2:
Reproducing theorems of Archimedes, setting up proper diff.eqs and developing a correct formula for arc lengths are NOT common among 16-year olds, it is quite unique, IMO.

 

[Levis2]

And, Levis2:
Reproducing theorems of Archimedes, setting up proper diff.eqs and developing a correct formula for arc lengths are NOT common among 16-year olds, it is quite unique, IMO.

It probably isn't .. most of the guys in my class are fighting adding two vectors with each other, so probably not :)

 

[arildno]

So, you DO know a little bit of vectors already!

I'm not so sure you would find that part of the university maths particularly hard to follow.

 

[Levis2]

So, you DO know a little bit of vectors already!

I'm not so sure you would find that part of the university maths particularly hard to follow.

yea you know, add, substract, multiply with a number, how vectors are defined by a*vector-i including the proofs (lol worlds simplest haha:) etc .. Orthogonal vectors and so on :) Not enough for college though.

 

[arildno]

One more thing to remember:
When YOU prove some theorem that already has been proven before, it doesn't follow that the guy making the first proof was smarter/more professional than you.

Thus, when doing proofs or devoloping formulae, remember that you are not just training to become a mathematician, you are being one as well!

 

[annelies 123]

Hi!
I think that I'm really understanding your situation.. I'm also 16 years old, but I'm already tested when I was young and I'm having an IQ that is more than 140, my parents don't want to tell the exact results of the test. So I skipped 2 classes and now I'm at University and I'm studying maths. But even I'm sometimes thinking will I be good enough to reach something with maths.. I still don't know but I'll hope so.. For now I'm still before my classmates they have to study for it every day and for me just paying attentions in the lessons is already good enough.. The reason for me why I couldn't be much betteris because languages and things like that couldn't interest me so I negliged it in primary and secondary school but after some time everyone has to work a little bit so I didn't have the time to study math at myself.. If you can I'll surerly say do that! Competing in national and international matholympiads is also great I think.. Try it, I can't compete anymore cause you can't be participating and already be at high school..

I can also tell you a story from someone else, he's only 16 too but he's just in the normal class at school. At home he learns lots about maths I think if he should make examens of the first year at university that he would have enough points.. He's learning so much.. Last year he went to the IMO but didn't got a medal but that's kind of logic he was only sixteen and other people were already 18 or sometimes 19..
If you have some questions I'd love to answer cause I think I'm in the samen situation..

 

[Brandon_R]

Don't worry about your scores on the IQ Test. As Mark Twain puts it, there are three kinds of lies in this world - lies, dammed lies and statistics. If you have the passion to learn mathematics, then by all means follow it.

 

[annelies 123]

Don't worry about your scores on the IQ Test. As Mark Twain puts it, there are three kinds of lies in this world - lies, dammed lies and statistics. If you have the passion to learn mathematics, then by all means follow it.

I'm following it ;) but you even have kind of reliable test on the internet http://www.mensa.org/workout" is the best site I've found.. it says not your IQ but wether you could be highintelligent. To be sure you have to do some real test and so on..

 

[micromass]

I'm sorry, but there is no internet IQ-test that is reliable. Not even the mensa-test. IQ-testing is a very standardized procedure, and anything not following that standard, can not be reliable.
I have serious doubts about the tests validity and unbiasedness...

 

[annelies 123]

I'm sorry, but there is no internet IQ-test that is reliable. Not even the mensa-test. IQ-testing is a very standardized procedure, and anything not following that standard, can not be reliable.
I have serious doubts about the tests validity and unbiasedness...

okay I haven't done research on he reliabilty but if i say you that I've recomended this test a couply of times to peught they might by hyperintellingen. the test has always been right in the cases i know. after that most of them did a qualified test and it gave the same results ;)

 

[elfboy]

Hello - I am a 16 year old danish boy. I'm in what is equivalent in denmark to the 10th grade in the US, and i simply love math. It's funny though, since before i attended 10th grade, i dreaded math due to it being so boring - but i think that was due to the simple arithmetics we did in my previous school. Once i encountered a more pure math in 10th grade, i was sold!

My number 1 goal in this world - the thing that matters most to me - is becoming a mathematician. I want to take a phd in math, and teach at a university, and if I am lucky, end up making a useful contribution. That's what matters most to me of all things atm.

But there's a problem - I am not a child prodigy. I can't do topology or real analysis, and my iq is only 130 ! Ever since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when I'm not that intelligent.

Funny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, let's say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools, but i presume he hasn't met any real good mathematicians lol .. I have also invented a formula by myself for calculating the area of a triangle if one only knows its sides. It looks this this;

A=1/2*c*squareroot(a^2-(c-(b^2+c^2-a^2)/2c)^2)

Where c has to be the biggest side in the triangle. The order of a and b doesn't matter :) All of this is easy stuff though ... nothing worthy a true mathematician :(

Now my question is, can i take a phd in math and become a mathematician, even though I'm not that intelligent? And if I'm barely able to do my phd, will i then be a garbagety and lousy matehmatician ?

it's a thought that takes up a lot of space in my head atm .. I'm so worried that i won't be able to take a degree or contribute to the art of mathematics :(

Help!

you have along way to go

to succeed in math you must work very hard

 

[Waxbear]

Hey Levis. I am a dane too, and i am currently on my first year of a bachelors degree in physics. I just took my exam in Linear Algebra today (i noticed your remark on how simple vector operations are :) ) and i feel that i can safely tell you, that if you are already this proficient with mathematics (just the fact that you know enough calculus to be constructing differential equations) and if you continue to challenge yourself like this until you reach your first year of university, you are going to have a pretty easy time here.

by the way, I am studying at KU. What gymnasium are you at?

 

[Levis2]

Hey Levis. I am a dane too, and i am currently on my first year of a bachelors degree in physics. I just took my exam in Linear Algebra today (i noticed your remark on how simple vector operations are :) ) and i feel that i can safely tell you, that if you are already this proficient with mathematics (just the fact that you know enough calculus to be constructing differential equations) and if you continue to challenge yourself like this until you reach your first year of university, you are going to have a pretty easy time here.

by the way, I am studying at KU. What gymnasium are you at?

Im at HTX Slotshaven EUC in holbæk :)

Just to give you guys an update; I contacted my teacher, and he was feeling the same way you guys were. He has asked the advisors at the university of Copenhagen, whether i would be able to start even though i haven't got a high school exam. Usually you can, if you can show the same qualifications as one with a high school exam - this is not a problem in math, but I'm not that good in danish class... So i can't be admitted this way. The advisors are currently working on getting me admitted through somekind of loop hole in the university regulations. If that's not possible they will attempt to get me a dispensation.

If all this works out - and I'm afraid it's not possible though - i will most likely be able to start university after the summer break :) If so, i will by then have read all the mathematics for high school, so I'm able to follow and complete any kind of admission tests, they might give me :)

 

[Einherjer]

Okay, first of all. Your IQ is so profoundly irrelevant that it is hard to even get started. The IQ only measures one thing, and that is your ability to solve IQ quizes. A skill like any other. If you train consistently on IQ quizes anybody can get an 300 IQ score.

You can't do statistics with only one data point.

 

[qspeechc]

Hi Levis2.
I have some book suggestions if you are interested. Have you done Euclidean geometry with proofs? This is really where mathematics starts. If not may I suggest Harold Jacobs' Geometry:
https://www.amazon.com/dp/071671745X/?tag=pfamazon01-20
(better than his Geometry: Seeing , Doing, Understanding.). If you think that is too simple then try Moise's Elementary Geometry from an Advanced Standpoint:
https://www.amazon.com/dp/0201508672/?tag=pfamazon01-20

Some other books to read before you start calculus:
Principles of Mathematics by Allendoerfer and Oakley (2nd edition or later). Covers high school math up to the point where you can start studying calculus, but written for bright young students like yourself.
What is Mathematics? by Courant (you don't need the updated version by Stewart). Written by one of the greatest teachers ever. Will start you thinking like a mathematician.

Good luck and work hard.

 

[inknit]

Kid, calm down. If math is your passion, by all means pursue it, but don't make your life goal becoming the next Einstein.


Btw, if you're really intelligent, why not study theoretical physics?

 

[Shackleford]

Kid, calm down. If math is your passion, by all means pursue it, but don't make your life goal becoming the next Einstein.


Btw, if you're really intelligent, why not study theoretical physics?

Yeah, maybe he can fix the Riemann integral.

 

[elfboy]

i recommend watching the big bang theory for some career ideas for high iq people like yourself

 

[mal4mac]

In high school, Feynman's IQ was determined to be 125, according to his biographer Gleick. I listened to a debate about this on the radio recently where some psychologists said if anyone has an IQ greater than about 120/130 they can do any job - mathematician, physicist, even psychologist :) More than that doesn't matter, what really matters is *hard work*, finding the right opportunities, interacting well with teachers and peers - so keep posting here, and keep in touch with that nice teacher. Ask him the same questions you ask here, try and find out the best universities to go to, what classes to take and so on... Read Feynman's biography, or those of any other great scientists, note how they always had great mentors, friends, and studied/worked in the right places. I never did. I should have tried harder to find the right people & places! At worst you might fail to become a research mathematician, but you will easily find a job as a maths teacher, quant or IT support person - none of which are bad options (ask your teacher!) And, from experience, I can say that programming/IT support can be a good fall back...

 

[Clairut]

Kid, calm down. If math is your passion, by all means pursue it, but don't make your life goal becoming the next Einstein.


Btw, if you're really intelligent, why not study theoretical physics?

That's the most intelligent comment I've heard...Sometimes we just forget about enjoying ourselves..instead we try to be the best..Levis..if you're trying to find a reliable test try Strict Logic Sequence Examination by Jonathan Wai...I was tested when i was 15..scored around 167(sd=15)...But i will never reach my goal...Living in my own world of fantasy..Without any real passion for anything..

But reading about you..i keep asking myself..What am i doing with my life?
I wish i was you...

 

[aneesh.mulye]

Hello - I am a 16 year old danish boy. I'm in what is equivalent in denmark to the 10th grade in the US, and i simply love math. It's funny though, since before i attended 10th grade, i dreaded math due to it being so boring - but i think that was due to the simple arithmetics we did in my previous school. Once i encountered a more pure math in 10th grade, i was sold!

My number 1 goal in this world - the thing that matters most to me - is becoming a mathematician. I want to take a phd in math, and teach at a university, and if I am lucky, end up making a useful contribution. That's what matters most to me of all things atm.

But there's a problem - I am not a child prodigy. I can't do topology or real analysis, and my iq is only 130 ! Ever since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when I'm not that intelligent.

Funny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, let's say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools, but i presume he hasn't met any real good mathematicians lol .. I have also invented a formula by myself for calculating the area of a triangle if one only knows its sides. It looks this this;

A=1/2*c*squareroot(a^2-(c-(b^2+c^2-a^2)/2c)^2)

Where c has to be the biggest side in the triangle. The order of a and b doesn't matter :) All of this is easy stuff though ... nothing worthy a true mathematician :(

Now my question is, can i take a phd in math and become a mathematician, even though I'm not that intelligent? And if I'm barely able to do my phd, will i then be a garbagety and lousy matehmatician ?

it's a thought that takes up a lot of space in my head atm .. I'm so worried that i won't be able to take a degree or contribute to the art of mathematics :(

Help!

A few observations:

1) IQ is important for mathematical success, no doubt.
2) However, IQ is not stable until the ages of roughly 18 to 21. So the IQ at your age is pretty meaningless, though at 130, highly, highly encouraging - look instead at your results, which seem pretty impressive to me. And there are things you can do to improve it in the short term. (Like exercise, the trick is to keep doing them consistently, so the `short term effect' never effectively ends.)
3) An IQ of 130 is two stddevs above the mean; combined with hard work, this should more than suffice to allow you to get a doctoral degree and teach at the university level. (Original contributions are inherently unpredictable, so I would not suggest worrying about them right now.)
4) Domain knowledge plays a large part in success, even more so in academia. With experience, you will find your mathematical ability increasing. Given how high it is now, that is a VERY good thing.
4) Finally, knowing that you're this far ahead of your age range, and if you're willing to work with dedication, I can say with some confidence that a) your mathematical IQ WILL improve with time, and will perhaps easily reach into 145+ range, and b) you will not have a problem with your further education AS LONG AS you work consistently.



In sum: go for it!

 

[nickadams]

THis is like the 4th time I have seen this thread posted.

 

[elfboy]

Okay, first of all. Your IQ is so profoundly irrelevant that it is hard to even get started. The IQ only measures one thing, and that is your ability to solve IQ quizes. A skill like any other. If you train consistently on IQ quizes anybody can get an 300 IQ score.

You can't do statistics with only one data point.

not true at all. You can't train for a professionally administered IQ test, you can only become familiar with the type of questions.
IQ is a deterministic factor for one's ability to grasp and synthesize abstract concepts.

 

[deRham]

This comment might get overlooked, but I hope not: while IQ seems decidedly NOT the right measure, I believe there are other traits which are also rare. Being a mathematician takes not IQ but ... Surprise surprise, mathematical inclination and talent, as well as hard work.

By mathematician, if you mean a researcher, well it depends if you like the career path, which comes with a lot more than just loving and doing mathematics!

Having a knack for mathematics as a career mathematician is not the same as solving hard puzzles, being very quick, or being able to ace math competitions. That does
not mean all there is to it is working hard even if you love math!

You should learn from an early age that eventually you have to stop using measures like these tests to assess yourself. Rather, ask yourself at all successes and failures what SPECIFIC TO YOU is holding you back or pushing you ahead. You can dazzle someone else with promises that you have a promising future with a high score on X or Y test, but if you have not made honest attempts at figuring out if your specific talents and inclinations are a fit for that future, then you won't fool yourself and will remain anxious.

 

[mal4mac]

So how did Feynman do it with an IQ if 125?

Suggestions:

"... an early aptitude for mathematics, a preference for "hands-on" learning, a passion for tinkering with modern technology (radios in Feynman's day) and a strong indifference to reading (even as a university professor, Feynman hardly read more widely than the "Physics Review" journal)."

http://anne.julienne.org/feynman.html

 

[deRham]

an early aptitude for mathematics

Right there, it's clear he had a skill which is not immediately explained by IQ. It's not that everyone has the same level of skill of course, just that the skill is different from what IQ immediately measures.

I mean, as a different example: not every professional mathematician is a star at the game chess. Both require intellect, practice, skill, etc. But they're simply different things.

So it's not right to ask "Do I have the IQ to become..." rather, do I have the ability. Hard work and smart work go a long way, but a bunch of factors simply give some people a leg up. The key is to learn your specific strengths, so you can bring them to achieving in the fullest way.

 

[nickadams]

The key is to learn your specific strengths, so you can bring them to achieving in the fullest way.

how can you do this?

 

[deRham]

At some level, it happens automatically if you put in a ton of effort into something. You get a lot of ideas as you persevere, make dumb mistakes, and observe what you did. Being an active observer helps quicken the pace of getting better, though.

 

[mal4mac]

... it's clear he [Feynman] had a skill which is not immediately explained by IQ. It's not that everyone has the same level of skill of course, just that the skill is different from what IQ immediately measures.

I agree, so the question is "how do you develop this skill?" Saw Wilckjek in his front room talking to an interviewer recently, and he seemed to be more proud of his large bookshelf full of puzzle books than his Nobel prize medal!

Rather than rushing to get into university I think the OP would be better reading a lot of puzzle books for laymen, or school level mathematics books with lots of puzzling, interesting problems.

University mathematics involves a lot of learning, and quite a bit of "plug and chug", perhaps at the expense of developing mental flexibility.

Maybe doing a lot of puzzles, or fixing old TV sets, develops that kind of flexibility in a young mind? Then when it comes time to "plug and chug" you can go to it with added flexibility.

I remember being impressed by an account by Feynman of winning a school level Mathematics contest through an incredibly flexible approach to a problem - how do you develop that kind of flexibility? Are you just born with it? Is there something beyond IQ - the AQ, the Amazing Quotient, that only people like Feynman and Einstein have? All we can say is that science hasn't found this yet, if it exists. So, as with string theory, we shouldn't take it too seriously. Those with Feynman's IQ should assume, for now, that they can do it! Just do as he did...

 

[Mépris]

I don't think Feynman was trying to model himself after somebody else. While it could be interesting to see how Feynman ended up being Feynman, I think it would be better if one (anyone, really) developed his own kind of talent if that makes sense.

Think of it this way, the world's already heard The Beatles and four of them is enough, while trying to emulate them or present another take of their work (see: Beatallica!) can indeed prove to be interesting or a good learning experience, I find that musicians are better off trying some new. I wouldn't be surprised if the same applied for science in general.

 

[mal4mac]

Read this very carefully:

http://www-history.mcs.st-and.ac.uk/Biographies/Feynman.html

"Richard, [Feynman] learned a great deal of science from Encyclopaedia Britannica and taught himself elementary mathematics before he encountered it at school. He also set up a laboratory in his room at home where he experimented with electricity. In particular he wired circuits with light bulbs, he invented a burglar alarm, and he took radios apart to repair damaged circuits. When he entered Far Rockaway High School his interests were almost entirely mathematics and science."

So instead of going to University early why not get a set of Encyclopedia Britannica, read through maths & physics articles (only!), get an expensive electronics hobby kit to build all these kinds of things, and experiment with fixing your friend's broken down electronic equipment?

I've been bouncing around John Gribbin's "Q is for Quantum" recently - an encyclopedia of quantum physics - and I can see how doing that might develop flexibility of mind.

"At school Feynman approached mathematics in a highly unconventional way. Basically he enjoyed recreational mathematics from which he derived a large amount of pleasure. He studied a lot of mathematics in his own time including trigonometry, differential and integral calculus, and complex numbers long before he met these topics in his formal education."

Note the stress on *recreational* mathematics - same as Wilczek.

"He entered MIT in 1935 and, after four years study, obtained his B.Sc. in 1939. He went there to study mathematics but, although he found the courses easy, he became increasingly worried by the abstraction and lack of applications which characterised the course at this time. He read Eddington's Mathematical Theory of Relativity while in his first year of studies and felt that this was what he wanted from mathematics. His mathematics lecturers presented him with the view that one did mathematics for its own sake so Feynman changed courses, taking electrical engineering. Very quickly he changed again, this time moving into physics."

I just read a biography of Dirac and Eddington's book was very important to him as well!
So put that on the "must buy" list...

"He took Introduction to Theoretical Physics, a class intended for graduate students, in his second year. There was no course on quantum mechanics, a topic that Feynman was very keen to study, so together with a fellow undergraduate, T A Welton, he began to read the available texts in the spring of 1936. Returning to their respective homes in the summer of 1936 the two exchanged a series of remarkable letters as they tried to develop a version of space-time..."

Here the important points to take are:

(i) he really stretched himself by taking a tougher course. Watson (the DNA guy) said he did exactly this as well, taking tough Math classes that biologists were not expected to take...)

(ii) He found a friends to discuss these things with at the highest level - I always felt jealous of Einstein's coffee bar lifestyle where he would go and chat with friends who later became almost as famous as him! If only I had had famous friends! ... But I didn't get chatting to the brightest guys in my class, so it's partly my fault - SO *push yourself forward socially* make friends with the bright guys and talk about important stuff.

(iii) Don't moan that your school doesn't have a class in string theory/LQG/whatever , plough through the tough books, with your like-minded pals, in the breaks! And try and have your own ideas, push them to the limit...

"By 1937 Feynman was reading Dirac's The principles of quantum mechanics and seeing how his highly original ideas fitted into Dirac's approach. In fact Dirac became the scientist who Feynman most respected throughout his life."

Another must read book! (Not too early though ... you need a couple of years of the toughest theoretical physics classes in Uni. before thinking of this...)

"He had the best grades in physics and mathematics that anyone had seen, but on the other hand he was close to the bottom in history, literature and fine arts."

Lesson: Focus on what's most important! UK citizens don't have this problem so much, they don't have to do "history, literature and fine arts." at Uni. They just do physics. So you might want to consider Manchester, like Bohr :), or Lancaster or even Cambridge (wish I had!)

"His doctoral work at Princeton was supervised by John Wheeler, and after finding the first problem that Wheeler gave him rather intractable, he went back to ideas he had thought about while at MIT. The first seminar that he gave at Princeton was to an audience which included Einstein, Pauli and von Neumann. Pauli said at the end..."

Lessons: Find the best mentors, listen to them, use them, hang out with them, but when it comes to the crunch go with your own ideas...

"At twenty-three ... there was no physicist on Earth who could match his exuberant command over the native materials of theoretical science. It was not just a facility at mathematics (though it had become clear ... that the mathematical machinery emerging from the Wheeler-Feynman collaboration was beyond Wheeler's own ability). Feynman seemed to possesses a frightening ease with the substance behind the equations, like Einstein at the same age, like the Soviet physicist Lev Landau - but few others."

But as his "lucky" schooling shows it wasn't (necessarily) some "divine spark" that gave him this ability - it was the things he did, the books he read, the courses he took, the people he talked to. You can't reproduce a "divine spark" (if it exists) but there is much in his life worth pondering on and emulating.

And for you educators out there: Why don't you encourage a culture similar to that encountered by Einstein and Feynman? Why not:

(i) Invite the bright students to the coffee bar for long bull sessions...

(ii) Create reading groups to get reading Eddington's book over the second summer break and Dirac's over the third.. ask them to come up with their own ideas for a theory of space time or Quantum Gravity.

(iii) Act like Wheeler (see "Geons" for another great biography...)

 

[nickadams]

At some level, it happens automatically if you put in a ton of effort into something. You get a lot of ideas as you persevere, make dumb mistakes, and observe what you did. Being an active observer helps quicken the pace of getting better, though.

no i mean how can you learn your specific strengths