Can anyone learn advanced maths? (Researches)

  • #26
We can go back to high school. There are pupils who just understand maths immediately, as soon as the teacher presents the material. And there are others, like the one I tutored today, who do not, who even have to spend a lot of money on extra tutorship and still get average grades or worse.

There is a clear innate ability involved here. Call it talent, call it intelligence. There are gaps you cannot cross with hard work alone.

I think it is more complex.
Do they still put effort into math? Do they really learn?
Are they still interested in math?
If the answer is no, then it is no wonder that they struggle.
What if the students had a bad teacher when they were in school?
What if they lack the elementary things of math?
If you lack the basics in math you will obviously struggle with the advanced stuff.

First of all, I never claimed that certain people do not understand math more easily than others (if you want to call that innate ability or talent, sure go right ahead). It's also worth keeping in mind that the understanding of mathematics is cumulative, so those who may not have learned the fundamentals at an early stage will have more difficulty in later years (all arguments for ensuring students receive the highest quality of math instruction and education at the earliest years).

That does not mean that (a) earlier obstacles cannot be overcome, nor (b) not everyone is capable to learn or develop an understanding of mathematics.

Exactly this is what I think.

If by 'good' you mean about the average at an average state university in the US, that's much more tenable, especially if you lax the requirement to 'only' be the basics needed for a major that 'only' uses applied math like engineering or physics.

Of course, I don't mean Field-Medal level since I said they have the talent to get there.
But I think talent only matters in this high-level area.
University level and a little bit above but under Field-Medal level is for everyone possible to reach. That is my hypothesis.
 
  • #27
Defaulting to logic can be faulty, this is an example of 'straight' logic:

George Washington is a dog
All dogs go to heaven when they die
George Washington is dead
Therefore he is in heaven
If you start a logic chain with a wrong fact, it's obvious it's not going to work.
That some people understand maths more easily than others, that there are even drastic differences, is fact.

It depends on what you mean by 'good' at math, if by that you mean Fields Medal, than even most mathematicians will never get there.
There is no qualitative difference between an average mathematician and a Fields candidate. The difference is quantitative: One has more talent than the other.
If some mathematicians are not good enough to become Fields medalists, then it follows that some people are not good enough to be mathematicians.
Saying anyone could become a successful mathematician is like saying any mathematician could win the Fields medal, if he only worked hard enough.
Not going to happen.

If by 'good' you mean about the average at an average state university in the US, that's much more tenable, especially if you lax the requirement to 'only' be the basics needed for a major that 'only' uses applied math like engineering or physics.
I agree with that. That doesn't mean anyone could reach one of these levels.

On a forum like this, I think most people here underestimate how smart they actually are. I know it has been researched that smart people tend to underestimate their own abilities by a lot. There are even IQ measurements by course and guess who scored highest? Mathematicians and physicists. They are FAR above the IQ of an average person and the average person is far above persons with weak intellect (not mentally disabled).
Have you guys ever tried to tutor really weak pupils?
Today, I mentored a 14 years old girl, I saw her for the first time. She did not understand how to calculate -10+1 (one year after she was introduced to negative integers). Now, you can try to explain it with real life examples (you have a temperature of -10°, it gets hotter by 1°, how many degrees do we have, or sth. like this). You could give her the hint, that she could also calculate 1-10 so that it would look more natural (the '-' being in between both numbers). You could show her a number bar of integers and use this to get the results. She just doesn't get the concept of negative numbers. Will she eventually understand it? Probably. After a LOT of extra time (and money). But what next? Her course will have moved on by then. She would have to take MANY extra years just to reach the maths level to ENTER a university. And what if it was a quantitative field like engineering or physics, where the topics are far more advanced and abstract and where the pace is orders of magnitude higher? How many extra yeas would it take someone like this to finish this degree? 10? 20? Now, don't get me started on pure math courses...
 
  • #28
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There certainly are cultural influences that affect the general population's mathematical ability. But there are just as certainly individual differences that affect mathematical ability. A genius can understand mathematical concepts before they are even taught to him. While that is an extreme case, it is the extreme case that proves the less extreme cases in a continuum.
 
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  • #29
Drakkith
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Again, there is a commonly-held view throughout Western countries (particularly by Americans and the British) that somehow mathematical ability is a "genetic" trait that only a certain people are blessed with the capability to understand. In no other subject that I can think of is such a view held -- not in, say, foreign languages, not in geography, history, art, music, etc. This is a view that I categorically reject -- mathematics is no different than any of these subjects.

No one here could reasonably argue that certain populations have a significantly higher 'raw talent' with math than other populations (I use 'populations' instead of 'races' because I want to generalize). I don't know of any evidence supporting the idea. Any significant differences are almost certainly due to differences in culture and schooling.

University level and a little bit above but under Field-Medal level is for everyone possible to reach. That is my hypothesis.

I strongly disagree. I've tutored people who literally cannot add numbers without using a calculator, but who are otherwise capable adults.

Have you guys ever tried to tutor really weak pupils?

Yep. And the difference between these students and even 'average' students is striking. What's interesting to me is how different these students are to each other. Some have severe problems because they have trouble focusing, some because they have memory problems, some because they can't understand abstractions very well, and the list goes on.

One thing to consider is that, if you consider someone who has an obvious mental handicap, what about another person who is slightly better than the first, and then another person slightly better than the second, and so on and so on. It seems obvious to me that at least the first few people in this chain aren't going to perform well enough to get anywhere close to Field's-Medal-Level. So where do you draw the line?

You can believe that anyone can reach the highest levels of math, but given the fact that human abilities tend to be on a continuum I just don't see it.
 
  • #30
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Today, I mentored a 14 years old girl, I saw her for the first time. She did not understand how to calculate -10+1 (one year after she was introduced to negative integers). Now, you can try to explain it with real life examples (you have a temperature of -10°, it gets hotter by 1°, how many degrees do we have, or sth. like this). You could give her the hint, that she could also calculate 1-10 so that it would look more natural (the '-' being in between both numbers). You could show her a number bar of integers and use this to get the results. She just doesn't get the concept of negative numbers. Will she eventually understand it? Probably. After a LOT of extra time (and money). But what next? Her course will have moved on by then. She would have to take MANY extra years just to reach the maths level to ENTER a university. And what if it was a quantitative field like engineering or physics, where the topics are far more advanced and abstract and where the pace is orders of magnitude higher? How many extra yeas would it take someone like this to finish this degree? 10? 20? Now, don't get me started on pure math courses...

I used to teach martial arts and alot of the students at my school were from low income households as well as overly religious households that home schooled their kids without the parents themselves being qualified to do so (ie biblical creationists and such), an example being that the 14 year old boy was asked to write down his new belt color on his certificate after his test and he wrote the word blue 'bloo' (he was not joking). I don't know if he actually had any severe mental illnesses, he was very socially anxious because as far as I could tell he only left the house to come to karate class and maybe go with his mom to do errands, obviously education is not highly supported at a house like this. A severe lack of preparation can make a student appear weaker than they actually are, for this student you're talking about it could be the case that she didn't even have the extremely basic mathematical foundations of play problems at the level of 'if I have 5 apples and I take away 3 how many do I have?' due to lack of opportunity while she was young.
 
  • #31
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Just about what level are we talking about as "advanced maths" here? Are we talking undergraduate? Graduate? Post doc?
 
  • #32
Matterwave
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Just about what level are we talking about as "advanced maths" here? Are we talking undergraduate? Graduate? Post doc?

The OP mentioned in post #6 that by "high level" he means "University level" math. I assume that would be Undergraduate or Graduate level math.
 
  • #33
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The OP mentioned in post #6 that by "high level" he means "University level" math. I assume that would be Undergraduate or Graduate level math.
Sure, but i find it kind of vague. I mean, undergraduate, graduate as well as phd or post doc is basically "university level" maths. That is, maths taken at a university.
Personally i have only taken undergraduate level of maths so far, but i would asume there is quite a difference between undergraduate and graduate, and even more so between graduate and phd or post doc level maths. But since i don't have any direct experience with that level i could very well be wrong.

On a personal note i always viewed myself as a weak student in maths specially, i struggled a lot with high school maths (to be fair, i wasn't very interested in studying anything at that time) and i have had to put a lot of work in on the undergraduate maths. So i sort of figure, if i can do it really anyone can do it. So by that standard i believe anyone can learn at least undergraduate level maths.
 
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  • #34
Drakkith
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Just about what level are we talking about as "advanced maths" here? Are we talking undergraduate? Graduate? Post doc?

The OP originally asked about the math that wins the Fields medal, so quite advanced. Definitely beyond undergrad and possibly beyond graduate.
 
  • #35
gmax137
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OP originally asked about the math that wins the Fields meda
i think you need to reread posts 1 and 6.
 
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  • #36
Drakkith
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i think you need to reread posts 1 and 6.

Hmmm...

On a high level, I mean the level like when you study maths in university. That you need some kind of talent to get a Field-Medal or even compete it is obvious. That is not what I meant with high level. High level is university level for me.

Well there we go then.
 
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  • #37
Just about what level are we talking about as "advanced maths" here? Are we talking undergraduate? Graduate? Post doc?
I would say Doctor Professional level
 
  • #38
The OP originally asked about the math that wins the Fields medal, so quite advanced. Definitely beyond undergrad and possibly beyond graduate.
Nono. I never meant Fiel-Medal level. I just used the Field-Medal as a Benchmark where you need to be blessed! You need kind of talent to get there.

I mean the University level like doctor or professor in math.
 
  • #39
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I would say Doctor Professional level
At the research level, there certainly are individual limitations that would prevent a large percent of people from achieving that ability.
 
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  • #40
StatGuy2000
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There is no qualitative difference between an average mathematician and a Fields candidate. The difference is quantitative: One has more talent than the other.
If some mathematicians are not good enough to become Fields medalists, then it follows that some people are not good enough to be mathematicians.
Saying anyone could become a successful mathematician is like saying any mathematician could win the Fields medal, if he only worked hard enough.
Not going to happen.

First of all, I dispute your characterization between the average mathematician and a Fields candidate -- Fields medals (like Nobel prizes) are awarded based on discoveries, and raw talent is not the only ingredient in making discoveries -- there is a considerable element of random chance involved as well.

But setting that aside, your very premise that some mathematicians cannot become Fields medalists => some people are incapable of learning math (your contention, @SchroedingersLion ) is both false, and is extremely arrogant and elitist on your part (I'm being kind, btw, with my choice of words, due to PF rules).

As for that 14 year girl, how much of her difficulty is based on poor instruction in her earlier years? If she had received better resources earlier in her life (and encouraged to both study and persist in studying math), she very well could have learned math effectively.

Again, what is the difference between studying mathematics and, say, studying say, French as a foreign language? Or geography? Or history? Or biology? I've met plenty of students who have trouble in all of these subjects -- do you then conclude that students are incapable of learning all of these subjects?
How absurd does this sound to you?
 
  • #42
Drakkith
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But setting that aside, your very premise that some mathematicians cannot become Fields medalists => some people are incapable of learning math (your contention, @SchroedingersLion ) is both false, and is extremely arrogant and elitist on your part (I'm being kind, btw, with my choice of words, due to PF rules).

Per wiki:
Elitism is the belief or attitude that individuals who form an elite — a select group of people with a certain ancestry, intrinsic quality, high intellect, wealth, special skills, or experience — are more likely to be constructive to society as a whole, and therefore deserve influence or authority greater than that of others.

Since no one has said anything about high-skill mathematicians (or anyone else) deserving greater influence or authority, I find your accusation simply wrong.
If you cannot discuss this without making such accusations then please leave the thread.

As for that 14 year girl, how much of her difficulty is based on poor instruction in her earlier years? If she had received better resources earlier in her life (and encouraged to both study and persist in studying math), she very well could have learned math effectively.

That's pure speculation. We can't go back in time and try things differently, so there's no way to know.

Again, what is the difference between studying mathematics and, say, studying say, French as a foreign language? Or geography? Or history? Or biology? I've met plenty of students who have trouble in all of these subjects -- do you then conclude that students are incapable of learning all of these subjects?

Yes, absolutely. I completely agree with the idea that some students will never be able to learn a second language, or geography, or history, etc. Are most students who are having difficulty in these areas incapable of learning them? No, probably not. Are some? In my opinion, yes.

How absurd does this sound to you?

It doesn't sound absurd at all. There are many, many students that have significant difficulty in all of the subjects you listed.

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.
 
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  • #43
symbolipoint
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Crazy hot-headed arguing just like would have been expected of this kind of topic. No clear conclusions.
 
  • #44
StoneTemplePython
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At the research level, there certainly are individual limitations that would prevent a large percent of people from achieving that ability.

I don't understand how this statement is controversial.

A lot of mathematics is about clear thinking and I'm seeing a large deficit of that on this thread.
 
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  • #45
gmax137
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If some mathematicians are not good enough to become Fields medalists, then it follows that some people are not good enough to be mathematicians.

That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."
 
  • #46
Drakkith
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That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."

Well that's a bit extreme. A more appropriate statement would be: "There are runners who will never win a marathon. It follows that some people will not be good runners." The implied assumption here is that running is a skill and that various people have different skill levels or ability at it.
 
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  • #47
StatGuy2000
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Yes, absolutely. I completely agree with the idea that some students will never be able to learn a second language, or geography, or history, etc. Are most students who are having difficulty in these areas incapable of learning them? No, probably not. Are some? In my opinion, yes.



It doesn't sound absurd at all. There are many, many students that have significant difficulty in all of the subjects you listed.

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.

My issue with your stance above is that the logical conclusion you would make is the following:

some people are incapable of learning Subject A (math, history, geography, language) => it's a waste of time for some people to study Subject A => we should identify these people and stop them from even learning Subject A

The problem is that educators (whether at the K-12 level, or in post-secondary level) by and large have no idea why their students are struggling with their subjects. What I fear is that educators may well see a student struggling and automatically conclude that these students are hopeless cases, whereas they may well be suffering from poor preparation in their preceding years (due to poor teaching or poor resources).

The other issue is that people do not always learn subjects in the same pace nor do they necessarily learn material in an orderly, linear path. There have been many documented cases where students who have struggled with a subject like math in the early years end up catching up with the material and excelling in the subject at an older age. However, if an educator (or parent) looks at said student from the earlier years, they may be led to believe (erroneously) that the student will never learn math, and thus actively discourage or prevent the student in further studies. To me this is a tragedy.

As educators, the goal should not be to focus their attention on the high achievers, but to be evangelists in their subjects, to bring their passion to the subject accessible to as broad a swathe of students as possible, and to instill the discipline the students will need to learn the subject matter at hand. Concluding beforehand that certain students can never learn a subject is a betrayal of that goal.
 
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  • #48
symbolipoint
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That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."
That analogy does not work.
 
  • #49
This may be a bit tangential, so dear moderators, please feel free to (re) move it.

My niece has no problem with calculus, linear algebra, etc. She enjoys the math part of her syllabus. On the other hand, she is not too comfortable with her programming language courses. She has trouble figuring out how to approach a programming problem, i.e. what kind of loop structures she would probably have to use, etc. etc. (Once I walk her through the solution, she ultimately understands it, but if she doesn't revisit it for a few days, she might find the same problem nearly as impenetrable as before).

I'd like to know if anyone else knows someone who is OK with math but not so OK with programming. Any thoughts about how to nurture the programmer's way of thinking for such a student?

Coming back to the original thread topic, it is conceivable that my niece may, at some point, break through the barrier and begin to progress much faster. It's also possible that her progress will remain slow compared to other students. Extrapolating from this, it is conceivable that anyone may in principle be able to learn a lot of math or programming if they went at it persistently and diligently. But then, if it took them most of their life to achieve college level competence, then it's practically equivalent to a total inability to progress to doctoral level stuff, IMHO. So it's reasonable to say that in a practical sense, some may be incapable of learning doctoral level math in one lifetime, simply due to the sheer number of years they might need. It's not a huge leap to then say the same thing about college level math (e.g. calculus and linear albegra), except that statistically the fraction of people in that category would be less.

Another thought: Consider some hypothetical person, whose manifested math aptitude is just slightly below average. Imagine that somehow they are fortunate (?) enough to live for 300 years with undiminished mental faculties and reasonable physical condition. Can they manage Fields Medal type achievements just by staying around and working really, really hard for those 300 years?
 
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  • #50
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My niece has no problem with calculus, linear algebra, etc. She enjoys the math part of her syllabus. On the other hand, she is not too comfortable with her programming language courses. She has trouble figuring out how to approach a programming problem, i.e. what kind of loop structures she would probably have to use, etc. etc. (Once I walk her through the solution, she ultimately understands it, but if she doesn't revisit it for a few days, she might find the same problem nearly as impenetrable as before).

I'd like to know if anyone else knows someone who is OK with math but not so OK with programming.

Answering this side note, i am similar to your niece. I enjoyed calculus (at least multivariable, vector and tensor) and linear algebra, but i absolutely loathe programming. I cannot make sense of it and it is the first subject i have come across that i actually cannot bring myself to want to learn even a little. I seem to be near immune to the logic and structure of it as well.

This is actually part of the reason i decided to not go for robotics as i originally planned and decided to switch to physics instead.

I don't feel like the two subjects necessarily are interwined when it comes to the actual understanding, sort of like physics and maths are a bit different in that way.
 

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