Can anyone learn advanced maths? (Researches)

[mathwonk]

I failed out of college also, but I went back and tried again. My appeal for reinstatement was pretty much that of richard gere's character in Officer and a Gentleman: : "Don't you kick me out. I got nowhere else to go!".

 

[symbolipoint]

I failed out of college, despite my best effort. So where does that put me? Did I just not try hard enough? Or not put in enough effort?

You are the only person who may know that. Maybe you misjudged how hard your chosen major field would be. Maybe you just needed longer time to learn. Maybe you needed to study more hours per week than up to the time you failed-out. Maybe you were not mature enough and needed more time for your brain to develop.

 

[Drakkith]

You are the only person who may know that. Maybe you misjudged how hard your chosen major field would be. Maybe you just needed longer time to learn. Maybe you needed to study more hours per week than up to the time you failed-out. Maybe you were not mature enough and needed more time for your brain to develop.

Sorry, I should have written my post differently. I wasn't really asking a question, but using myself as a counterexample. I can assure you that I did the best I could and that just putting more time and effort wouldn't have helped much. Also, this only happened 6 months ago, so I would hope I was mature enough considering I was 33 at the time.

I'd prefer not to give out any more details in this thread, so feel free to PM me if you have any questions.

 

[symbolipoint]

Sorry, I should have written my post differently. I wasn't really asking a question, but using myself as a counterexample. I can assure you that I did the best I could and that just putting more time and effort wouldn't have helped much. Also, this only happened 6 months ago, so I would hope I was mature enough considering I was 33 at the time.

I'd prefer not to give out any more details in this thread, so feel free to PM me if you have any questions.

Maybe I will but have not yet decided.

As I remember, you recently earned your degree (Bachelors?) in Mathematics, so you were either ready, or you figured out how to succeed.

 

[Drakkith]

As I remember, you recently earned your degree (Bachelors?) in Mathematics, so you were either ready, or you figured out how to succeed.

Lord no, I don't have a Bachelors in anything. I was working on a Bachelors in Optical Engineering up until 6 months ago.

 

[symbolipoint]

Lord no, I don't have a Bachelors in anything. I was working on a Bachelors in Optical Engineering up until 6 months ago.

I confused you with another member. Now I do not remember exactly which other member.

 

[pinball1970]

Here is a quote from Heisuke Hironaka, Fields medalist in mathematics: "I like basic things. Very clever people tend to jump to the new techniques: something is developing very fast, and you want to be on top of it; and if you are smart, you can be a top runner. But I am not so smart, so it is better that I start something where there are no techniques for the problem, and then I can just build step by step."

http://www.ams.org/notices/200509/fea-hironaka.pdf page 1013, line -9.

So even a not so smart person can be a Fields medalist.

Or to quote Adam Sandler: "I am not particularly smart or good looking or talented, and yet I am a multi millionaire".

To include myself, most people think I am as dumb as a bag of rocks, and yet I have a PhD in math, which I obtained when my university said they would fire me if I did not get one, and I was a young father. So I am an advocate of the hard work and serious motivation school of thought.

I rest my case.

He sounds like a humble guy and those kind of special often are.
Einstein said (para) “I am not smarter than everyone else I just stay with a problem longer.”
Feynman described himself as having “limited intelligence” perhaps because his Q was 125? Just shows what the IQ test does and does not measure??
I would actually guess that Heisuke Hironaka’s Intelligence is on the high side as most students don’t have the ability to get into Harvard let alone a PhD in maths from there.Back to the OP can anyone get good at maths (BSc, PhD)? Answer, No.

 

[Dr. Courtney]

My experience has been that persistence works better for me on the applied math side than on the pure math side. I don't know if I was just too lazy in high school (I was really lazy and avoided math like the plague) and missed the developmental opportunity when my brain was still growing and pliable, or perhaps I killed too many brain cells in college (drinking), or perhaps got hit on the head too many times on the playground in elementary school. The math required to be a passable experimental physicist was very hard, but attainable for me with effort. I also managed a number of theory papers in physics, but these were more computational than abstract theory.

Having been a teacher, I think the vast majority of students with normal or better IQs can master math through the normal high school sequence (Algebra 1, Geometry, Algebra 2, Precalc) and intro college Calc and Statistics. But it will take a lot of effort for most that are closer to average. Their failure is more a matter of effort than ability. But original research in pure math is several steps above that - big quantum steps.

One of my working hypotheses is that tremendous amounts of effort can get people performing 1-2 standard deviations above their innate abilities in most things - math, physics, music, sports, etc. But to be a pro at really hard things like sports, music, math,and physics, one needs to be 3-4 standard deviations above the average. The real standouts in most fields are the rare folks who begin life 2-3 standard deviations above the average with their natural gifting and then go 2-3 standard deviations above their natural gifting by working harder than just about everyone else.

 

[mathwonk]

Maybe this will encourage someone. My opinion is that since no one knows the answer to this, in fact unanswerable, question, the only way to find out whether someone can learn advanced math is to keep trying.

https://www.nytimes.com/2018/10/08/...er-too-late.html?login=email&auth=login-email

 

[mhl47]

Maybe this will encourage someone. My opinion is that since no one knows the answer to this, in fact unanswerable, question, the only way to find out whether someone can learn advanced math is to keep trying.

https://www.nytimes.com/2018/10/08/...er-too-late.html?login=email&auth=login-email

Why should this be a fundamentally unanswerable question? One day we might have an understanding of the human brain, the way it is prestructured at birth and the way we learn such that we can address this question.

I know that ANNs (artifical neural networks) do not come close to the complexity of a brain. But for them you could make an argument why a particular setup of nodes, connections or the number of layers might not be suited to create a strong enough model to solve a problem. In a way this should also be possible for an arbitrarily complex structure...unless the human brain is not complex enough to grasp it's own complexity. So maybe one day an AI will explain to us what limitations we have and to which level an individual can learn math :D.

 

[IjustlikeMaths]

So maybe one day an AI will explain to us what limitations we have and to which level an individual can learn math :D

That is why I love AI. It will offer us so much value in the future. :D

Well, first of all, I want to thank everybody in this thread. I wouldn't have expected that this thread would 'explode' like this :D
And you all have my respect for being so friendly even when you have different opinions!

Since nobody got any research, maybe because there isn't any research which could answer my question I still believe, that when a person is interested in maths it can achieve a doctor or professor level.
As I said the only conditions for this are that the person is :

1. interested in math.
2. willing to work on it and put an effort into learning math.
3. healthy. I mean without any mental disability or anything else. Just an average healthy person.
But I also think, that there are levels which you won't reach like Field-Medal level. even tho I think anyone can learn math to a high level there will be an area where you need to be blessed or your brain has to 'work a little bit different'.
But till professor or doctor level I think it is possible for everyone who fulfills the conditions I mentioned.

 

[mathwonk]

unfortunately as several of us can testify from our own experience, just making an effort, even by someone interested capable and healthy, does not always lead to success. The effort has to be directed correctly, and it takes time to learn how to do this. As a college professor for some 40 years, I almost never had a student who I thought "could not" have succeeded in my classes, if they had behaved in the way needed for success. I.e. only a handful out of thousands of students seemed ill advised to be there. However the actual success rate was extremely low, e.g. passing rates in calculus were often below 50%, even with generous grade inflation.

The number of students who simply attended class regularly (i.e. basically always), read the book, handed in assignments, asked questions in class and came to office hours, were usually numbered on one or two fingers out of a class of dozens. Students who actually entered class knowing the stated prerequisites were almost non existent, and usually limited entirely to foreign students. All my students apparently thought they were trying hard, or as hard as should be expected, to pass.

A (should be) famous study by Uri Treisman at Berkeley, followed a group of racial minorities who for some unexplained reason were failing miserably out of calculus, even though they did many of these things I mention, and it was found they lacked other more subtle study skills, like working in groups, challenging each other with the hardest problems, and (I would recommend) reworking tests they had already taken to be prepared for the same questions on the final. When Treisman taught them these skills and organized a study group for them, the same group of minorities became the stars of the class.

http://www.utdanacenter.org/about-us/staff/p-uri-treisman/

The same change of study habits worked for me, from a first failing experience in college to a later honors level one. When I attempted grad school, I also found that working together with others was extremely valuable, and I eventually succeeded in my goal, long delayed, of getting a PhD, although that was the hardest thing I ever did, and my advisor was a huge help.

In my opinion, most people who are not clearly unqualified, can succeed in undergrad and even grad school by employing the right techniques. It is not so clear to me that anyone knows how to predict just who can produce interesting thesis research at the doctoral level or beyond. No doubt there are also techniques that work to assist in research too, such as reading the work of top researchers in the field and trying to prove the results oneself, or generalize them. One is often confident that ones brightest students will succeed in obtaining a doctorate, but even then, the apparently smartest ones do not always produce the most interesting research. Imagination and originality are somewhat hard to measure reliably, and they do differ from technical power, although without that it is hard to finish a project, no matter how well conceived.

Come to think of it, Uri Treisman refined his program to apply also to doctoral level studies and went on to produce a large number of successful PhD's.

But as far as just learning to appreciate advanced math, this is a project anyone can enjoy, by beginning at the bottom, and trying to understand elementary, but significant math, such as Euclid's geometry, or number theory.

 

[Terrell]

any mathematician could win the Fields medal, if he only worked hard enough.

All talent-genetics-based arguments aside. This is impossible due to the pigeonhole principle. We factor in an average of 20 years working in the field of mathematics and the fact that the fields medal is awarded only every 4 years and the number of recipients every awarding year.

 

[PeroK]

There is an argument put forward here that goes something like this. Let's say I decide to try to go back and get a PhD in maths.

Work hard at maths, get back to undergraduate level at 1st class honours.
Start PhD, make slow progress. Work harder. Make more progress. Work harder and harder. Spend every waking hour doing maths. Get PhD.

And, the line of reasoning continues to get better and better and more and more successful you just need to work harder and harder and longer and longer hours.

But, what about the alternative:

Work harder and harder, have mental breakdown. Recover, come back, work even harder, commit suicide.

To me, the idea that you can just go on putting in more and more hours and never break down is absurd.

You have this in music, sport as well. Overwork and overtraining eventually lead to physical and/or mental breakdown. It happens all the time.

Not everyone who puts in the maximum effort becomes a top tennis player, concert pianist or gains a PhD

I worked in IT and on one particular project two people had nervous breakdowns - the second person had logged, I think, 150 hours work one week. You can't try harder than that! But it led to disaster not success.

It may be that everyone, if forced to, could gain a degree in maths, say. But, you are going to have to exclude those who break down or kill themselves trying. They, if no one else, are going to spoil your 100% success rates.

 

[pinball1970]

All talent-genetics-based arguments aside. This is impossible due to the pigeonhole principle. We factor in an average of 20 years working in the field of mathematics and the fact that the fields medal is awarded only every 4 years and the number of recipients every awarding year.

You have to be less than 40 too.

Gives you about 5 chances but realistically 3 or 4

If you make a ground breaking discovery at 40 you are not eligible.

 

[Terrell]

To me, the idea that you can just go on putting in more and more hours and never break down is absurd.

I have a theory that people's interest in mathematics lies in a spectrum. Then people like tao, ramanujan, grothendieck, etc... have their interest levels at the far extremely interested level. While people with Phds is in the extremely interested level, with BSc's in the moderately interested level, and so on and so forth. When a person does math beyond their interest level that is when they feel they are working too hard. They're force feeding their brain with mathematics more than their mathematical appetite. Then the analogy becomes complete when they start throwing up; i.e. committing suicide, mental breakdowns, etc... I think it all start for having the wrong reasons of doing math such as recognition, accolades, etc...

 

[symbolipoint]

Terrell's posting #86 means, "Why are you studying Mathematics"? or "Why do you want to study Mathematics"? The answers would then be either for the right reasons, or the wrong reasons, or some kinds of in-between reasons.

 

[pinball1970]

I have a theory that people's interest in mathematics lies in a spectrum. Then people like tao, ramanujan, grothendieck, etc... have their interest levels at the far extremely interested level. While people with Phds is in the extremely interested level, with BSc's in the moderately interested level, and so on and so forth. When a person does math beyond their interest level that is when they feel they are working too hard. They're force feeding their brain with mathematics more than their mathematical appetite. Then the analogy becomes complete when they start throwing up; i.e. committing suicide, mental breakdowns, etc... I think it all start for having the wrong reasons of doing math such as recognition, accolades, etc...

Substitute ''interest' with ''ability'' and I am with you.

 

[Dr. Courtney]

I hated math for many years. Studied it out of necessity (to learn Physics, which I loved.)

Is that a right reason, or a wrong reason?

For me, it was a right reason - since the laws of nature are written in the language of mathematics, I had to learn the language first.

(Likewise, many who need English for other disciplines may never like English, but need to become proficient enough at it to study their real passion - perhaps law or history.) Come to think of it, I also hated my required high school courses in English - too much literature that I found boring. I did appreciate grammar, logic, and elective courses in creative writing and composition.

Eventually, I learned to like math, but more for its power than its beauty.

 

[Terrell]

Substitute ''interest' with ''ability'' and I am with you.

I feel like it is a good mixture of both.

 

[Dragon27]

Start PhD, make slow progress. Work harder. Make more progress. Work harder and harder. Spend every waking hour doing maths. Get PhD.

Work harder and harder, have mental breakdown. Recover, come back, work even harder, commit suicide.

I have a theory that people's interest in mathematics lies in a spectrum.
…
When a person does math beyond their interest level that is when they feel they are working too hard. They're force feeding their brain with mathematics more than their mathematical appetite.

Indeed, working harder to get more progress is often a bad idea. Relaxing, going easier on yourself, not working harder than your brain and stress tolerance allows may lead to better results and maybe even more progress in the long run. People usually try to strain themselves when they feel time-constrained (like in the university). It is not necessary to run fast to finish a marathon, on the contrary, it is inadvisable.
The word "advanced" in the "advanced math" doesn't necessarily mean that it requires more effort to learn it. Abstract concepts often feel easier (and more like "cheating") than low-level manipulation with some more basic stuff. They just require more time and "unpacking" to get them. But not necessarily more ability.
​

 

[pinball1970]

There is an argument put forward here that goes something like this. Let's say I decide to try to go back and get a PhD in maths.

Work hard at maths, get back to undergraduate level at 1st class honours.
Start PhD, make slow progress. Work harder. Make more progress. Work harder and harder. Spend every waking hour doing maths. Get PhD.

And, the line of reasoning continues to get better and better and more and more successful you just need to work harder and harder and longer and longer hours.

But, what about the alternative:

Work harder and harder, have mental breakdown. Recover, come back, work even harder, commit suicide.

To me, the idea that you can just go on putting in more and more hours and never break down is absurd.

You have this in music, sport as well. Overwork and overtraining eventually lead to physical and/or mental breakdown. It happens all the time.

Not everyone who puts in the maximum effort becomes a top tennis player, concert pianist or gains a PhD

I worked in IT and on one particular project two people had nervous breakdowns - the second person had logged, I think, 150 hours work one week. You can't try harder than that! But it led to disaster not success.

It may be that everyone, if forced to, could gain a degree in maths, say. But, you are going to have to exclude those who break down or kill themselves trying. They, if no one else, are going to spoil your 100% success rates.

There are two camps on this with a middle ground seeing both sides. I am with perok. One can improve but you cannot train to have ability, you may as well train to
Be an inch taller. The inch taller guy will always have a better reach. All the interest in the world is no substitute for raw intelect. All the smart Guys are being modest imo

 

[StatGuy2000]

There are two camps on this with a middle ground seeing both sides. I am with perok. One can improve but you cannot train to have ability, you may as well train to
Be an inch taller. The inch taller guy will always have a better reach. All the interest in the world is no substitute for raw intelect. All the smart Guys are being modest imo

I believe both you and @PeroK are guilty of placing too much emphasis on so-called "raw intellect" (rather than the "smart" guys being modest). Furthermore, I believe you are incorrect that you cannot train to have ability, because you are implicitly implying that "ability" is a fixed genetic trait. In actual fact, a person's ability is in large part a combination of persistent training and dedication, combined with whatever "raw intellect" people have, and the educational and social resources made available to the individual.

 

[StatGuy2000]

There is an argument put forward here that goes something like this. Let's say I decide to try to go back and get a PhD in maths.

Work hard at maths, get back to undergraduate level at 1st class honours.
Start PhD, make slow progress. Work harder. Make more progress. Work harder and harder. Spend every waking hour doing maths. Get PhD.

And, the line of reasoning continues to get better and better and more and more successful you just need to work harder and harder and longer and longer hours.

But, what about the alternative:

Work harder and harder, have mental breakdown. Recover, come back, work even harder, commit suicide.

To me, the idea that you can just go on putting in more and more hours and never break down is absurd.

You have this in music, sport as well. Overwork and overtraining eventually lead to physical and/or mental breakdown. It happens all the time.

Not everyone who puts in the maximum effort becomes a top tennis player, concert pianist or gains a PhD

I worked in IT and on one particular project two people had nervous breakdowns - the second person had logged, I think, 150 hours work one week. You can't try harder than that! But it led to disaster not success.

It may be that everyone, if forced to, could gain a degree in maths, say. But, you are going to have to exclude those who break down or kill themselves trying. They, if no one else, are going to spoil your 100% success rates.

You are conflating two distinct things, hard work on the hand and excessive overwork and the stress associated with this. You are also conflating effective ways to manage time and learn material from simply logging in more and more hours without necessarily being productive in the tasks at hand -- and this is something that has nothing to do with innate ability or training.

To use your example in IT -- if a project requires someone to log in 150 hours of work in one week, that should be an immediate red flag of either poor goals and organization on the management, over-commitment to unrealistic timelines or goals, poor resourcing, or any combination of these. Of course such a situation would end up being disastrous. But that situation is not analogous to the process of learning.

 

[FactChecker]

I believe both you and @PeroK are guilty of placing too much emphasis on so-called "raw intellect" (rather than the "smart" guys being modest). Furthermore, I believe you are incorrect that you cannot train to have ability, because you are implicitly implying that "ability" is a fixed genetic trait. In actual fact, a person's ability is in large part a combination of persistent training and dedication, combined with whatever "raw intellect" people have, and the educational and social resources made available to the individual.

You are hearing the same comment from many people here that there is a significant variance in individual raw capabilities, but you seem to ignore it. When you talk about your experience with students, are you sure that you were not just ignoring them too? It's not clear to me that any evidence would change your mind.

 

[StatGuy2000]

You are hearing the same comment from many people here that there is a significant variance in individual raw capabilities, but you seem to ignore it. When you talk about your experience with students, are you sure that you were not just ignoring them too? It's not clear to me that any evidence would change your mind.

On the contrary, I am well aware of the variance in individual capabilities -- but from my observation, these capabilities have more to do with the pace of learning or the manner of learning, rather than some hard limit. I have tutored numerous people throughout my life, and I have never found any student who was ultimately incapable of learning, but certain students took longer to understand certain concepts. What I've also found was that these students, when given good resources (i.e. strong tutoring from qualified instructors, accessible and effective textbooks) and taught (or acquired on their own) effective time-management skills, learned faster and became better in the material. Now would you argue that these people became "smarter"?

Again, what I consistently read and hear in the thread is how deep is the bias among Americans and the British (most commentators on PF being from the US or the UK) about mathematical ability being a "genetic" trait.

 

[clope023]

I believe both you and @PeroK are guilty of placing too much emphasis on so-called "raw intellect" (rather than the "smart" guys being modest). Furthermore, I believe you are incorrect that you cannot train to have ability, because you are implicitly implying that "ability" is a fixed genetic trait. In actual fact, a person's ability is in large part a combination of persistent training and dedication, combined with whatever "raw intellect" people have, and the educational and social resources made available to the individual.

I'm more in your camp than those saying the opposite but raw ability can be a thing that can't be trained for; an example being boxers with a reach that's just several inches beyond that of their opponents, there's ways of getting around that but if a boxer has t-rex arms there's not much that can be directly done in that respect.

 

[Drakkith]

On the contrary, I am well aware of the variance in individual capabilities -- but from my observation, these capabilities have more to do with the pace of learning or the manner of learning, rather than some hard limit. I have tutored numerous people throughout my life, and I have never found any student who was ultimately incapable of learning, but certain students took longer to understand certain concepts. What I've also found was that these students, when given good resources (i.e. strong tutoring from qualified instructors, accessible and effective textbooks) and taught (or acquired on their own) effective time-management skills, learned faster and became better in the material. Now would you argue that these people became "smarter"?

And I've tutored people who simply did not make any appreciable progress in their math skills despite their best effort.

Again, what I consistently read and hear in the thread is how deep is the bias among Americans and the British (most commentators on PF being from the US or the UK) about mathematical ability being a "genetic" trait.

I don't know how you got that impression. What I've seen in this thread is mostly an argument that some portion of ability is genetically based, but a much larger portion is based on education and upbringing.

I still stand by what I wrote earlier in the thread:

I'd also like to say that I don't like this black and white concept of learning math. Instead of asking whether or not anyone could learn some level of math, it seems much more reasonable to ask will some students find learning math so difficult that it would be unreasonable to expect them to do so? I mean, if someone spent ten years learning Calculus, how long would it take them to learn higher level math? If it would take them longer than their own lifespan to learn all the math necessary to get to a certain level then I would say that that means they will never learn math at that level. It doesn't mean that they can't learn math at all, it just means that the difficulty is so large that they cannot be reasonably expected to learn math to that level.

 

[clope023]

And I've tutored people who simply did not make any appreciable progress in their math skills despite their best effort.

Best proported efforts can be deceiving, lots of people make the claim that they can't loose weight despite best efforts for example but that can often be limited by the fact that they don't know what they don't know about calories, macro-nutrients, and such like; so best reported efforts might not actually be their best efforts.

 

[Drakkith]

Best proported efforts can be deceiving, lots of people make the claim that they can't loose weight despite best efforts for example but that can often be limited by the fact that they don't know what they don't know about calories, macro-nutrients, and such like; so best reported efforts might not actually be their best efforts.

The impression I get from them is that they are giving a substantial amount of effort for very little progress. Is that a better way to word it?

 

[clope023]

The impression I get from them is that they are giving a substantial amount of effort for very little progress. Is that a better way to word it?

Perhaps, just saying they were probably doing something wrong is more likely.

 

[Drakkith]

Perhaps, just saying they were probably doing something wrong is more likely.

I'm not sure what you mean by this.

 

[StatGuy2000]

And I've tutored people who simply did not make any appreciable progress in their math skills despite their best effort.

Please don't take this personally, but have you thought about the possibility that some of the students you tutored did not make any appreciable progress because you were not tutoring them effectively?

I raise this point because when we talk about their "best" effort (or as you clarified it, substantial amount of effort), they may well be putting in their efforts in ineffective ways. Also, people can approach the same problem from different vantage points (some learn by repetition, others learn by example, still others learn by visualization, etc.). If you don't teach or tutor them in ways that customize their particular style of learning, they may not always be able to pick up on the material.

 

[Drakkith]

Please don't take this personally, but have you thought about the possibility that some of the students you tutored did not make any appreciable progress because you were not tutoring them effectively? I raise this point because when we talk about their "best" effort (or as you clarified it, substantial amount of effort), they may well be putting in their efforts in an ineffective way. Also, people can approach the same problem from different vantage points (some learn by repetition, others learn by example, still others learn by visualization). If you don't teach or tutor them in ways that customize their particular way of learning, they may not always be able to pick up on the material.

Furthermore, learning isn't necessarily linear. Students can struggle for long periods without making any visible, apparent progress and then suddenly things "click" -- anecdotally I've seen this occur among numerous students.

I'm not talking about people who are stuck on, say, the chain rule in calculus. I'm talking about people that have serious difficulty doing anything with math. People who come in day after day and struggle immensely at even understanding what a simple problem is asking them to do. People who can't even comprehend a simple abstract concept like a variable.

Note that I've tutored students in a sort of 'special needs' program in college, and I've also tutored people who weren't. In both areas I've encountered people like I described above. The difference is that most of the people in the latter group are effectively 'normal' in other areas despite being abysmal at math.

I'd also like to point out that I myself was part of this program in college, as I have serious difficulty writing papers and dealing with things that are heavily language-based. So I can tell you from firsthand experience that if there isn't a hard limit to someone's skill at something, the effort vs progress graph can certainly be logarithmic.

 

[symbolipoint]

Is this topic making any progress yet?

Can anyone learn advanced maths? (Researches)

I have no "researches" to refer to. I'd say "advanced Mathematics" might be anything beyond the range of typical university Calculus 1, 2, 3. Below this range of courses may be Algebra 1 & 2, Geometry, Trigonometry, College Algebra (part of "Pre-Calculus"). I'll take Drakkith's statement on it (Can anyone learn advanced maths?). His statement is based on his own learning, and on teaching or tutoring experience. One should try teaching or tutoring, to be more familiar what that is like.