Can anyone learn advanced maths? (Researches)

[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.
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[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.

 

[FactChecker]

Many people who have contributed comments here, including me, have reached their own limits in spite of years of effort. I know people who easily went beyond my abilities. I have attributed that to a difference in raw intelligence. When someone says that it would just take me more time I have to remind them that, at my age, time will eventually run out.

I have known for a long time that I have always had a weaker memory than most people. I often have to derive or be reminded of facts that others can remember without help. That translates into a serious handicap in learning. I believe that this fact is undeniable in spite of what anyone else thinks.

 

[pinball1970]

Many people who have contributed comments here, including me, have reached their own limits in spite of years of effort. I know people who easily went beyond my abilities. I have attributed that to a difference in raw intelligence. When someone says that it would just take me more time I have to remind them that, at my age, time will eventually run out.

I have known for a long time that I have a weaker memory than most people. I often have to derive or be reminded of facts that others can remember without help. That translates into a serious handicap in learning. I believe that this fact is undeniable in spite of what anyone else thinks.

That is an important point.Memory is essential in building on concepts, if you can’t remember the details of concept one you have to back track before you can proceed to concept two

why not take it further?

What about special needs kids? Kids with learning difficulties? This has been touched on, what did they do wrong? What did their parents do wrong?

Did the kids not apply themselves? Did they need to go on a time management course?Some of those kids will leave school at 16 with a basic understanding of maths and English and nothing much else. Flawed as it is the IQ bell curve tells a story, genes don’t give us calculus but they do build brains.No one is saying the average Joe cannot do well in life (and I am speaking from a position of lofty mediocrity) or assimilate some scientific mathematical knowledge but the OP asked,

“Can anyone get good at maths to an advanced level”(University)That was part of the main question.He may well of asked can anyone play football to high level (spherical ones – I am British)Can anyone make the premiership with proper encouragement, training, diet, mentoring, best facilities, nurturing, loving parents, a stable home life and plenty of time?The answer to that question is not everyone can play football well enough to play for the pub team let alone be a professional or get to the premiership.Why should maths be different?

 

[Dragon27]

Memory is a strange thing. Not always can I understand why certain facts and ideas stick to my mind immediately after the first reading and endure for a long time, while others keep eluding me after many repetitions.
But I'm sure that memory management (if not some kind of innate "raw memory") can be learned and improved as well. To some unknown extent.
And, by the way,

When a person does math beyond their interest level that is when they feel they are working too hard

interests change as well. History was one of the least appealing subjects to me at school, and a history textbook before bed was a sure-fire method for me to quickly go to sleep. But I'm starting to pick up my interest in history now, as a grown-up, and finding myself being fascinated by subject, while being able to memorize historical facts (and even the dreaded "dates") better.

 

[FactChecker]

IMHO, people who try to blame a single factor for a lack of mathematical ability are just trying to oversimplify a complicated subject. They are doing it without scientific proof and they are ignoring other evidence. This thread is very unscientific.

 

[pinball1970]

IMHO, people who try to blame a single factor for a lack of mathematical ability are just trying to oversimplify a complicated subject. They are doing it without scientific proof and they are ignoring other evidence. This thread is very unscientific.

No one is looking at single factors we are putting forward our own experience of learning compared to others and some tutors have given their experience also. It is not a scientific paper but it is evidence.

 

[FactChecker]

No one is looking at single factors we are putting forward our own experience of learning compared to others and some tutors have given their experience also. It is not a scientific paper but it is evidence.

Yes. I was too harsh.

 

[pinball1970]

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?

This is also important. Same topic say calculus same class same teacher. Some kids 'get it' others don't get it straight away others don't get it. All things being equal the only other variable is ability and you find that at every level. Some kids would take too long to get it you would need twice the amount of time or x3 or more. Same applies to trig algebra long division addition with plastic money and pretend shop. Some kids get it.

 

[StatGuy2000]

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.

I had not intended on continuing this thread, but let me make myself clear.

My statements on this thread -- specifically, that any student who do not have a specific mental or learning disability can learn the foundations of advanced mathematics (which I define as pre-calculus, algebra, and trigonometry) provided that they receive a strong foundation of math education at an early age -- is based on my own learning, the learning experiences of students I've witnessed, and my experiences in both teaching and tutoring.

What I have argued against is the notion (expressed by some in this thread) that the capability to learn mathematics is a hereditary or genetic trait that only some people possess, and no amount of education will enable such a student to learn mathematics.