Can anyone learn advanced maths? (Researches)

[Drakkith]

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

If this were a perfect world I would I agree with this. If some people cannot learn subject A at skill level X, then it would be beneficial for them not to be forced to try to learn subject A up to skill level X. I don't see how anyone could argue against this. The problem is that we can't identify these people ahead of time. We just can't know who will be incapable of learning a subject. That's why we have to push people along and make them try as hard as we can get them to try to see just how far they can go in a subject. So in our real, imperfect world that is not the logical conclusion to make.

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.

I don't think any is, or realistically could, argue against this. That's not what I'm arguing, nor anyone else as far as I can tell. The only thing being argued is that, in addition to the problems you've given, there are at least a small percentage of people who just cannot learn high-level math. If we include absolutely everyone then this is just a given, as we've already talked about people with severe mental handicaps. So the answer to the OP's question, if we consider absolutely everyone, is a firm "No. Not everyone can learn advanced math."

What I have been saying is that there are people out there who are not considered to be mentally handicapped that simply cannot learn advanced math, regardless of how they were raised and educated. I'm not arguing that most people are like this, I'm not even arguing that the percentage is large. Frankly I believe that with enough time and given enough effort, most people could learn math up to the undergrad level.

 

[Pleonasm]

Hello guys,

I often ask myself if anyone can learn maths to an advanced level? And get really good on it.
I think that every healthy person can get very good at maths. The only condition is that the person is interested in math.
Of course, to get on the level of a Field-Medal winner you have to be blessed a little bit. But I think you can reach and understand a lot just by working out hard.

But are there researches which proof the current state of science in relation to how much the genetic predisposition affects the learning of math?
When you are healthy our neural system should work nearly the same as the neural system of a high-level mathematician or nah? What do you guys think? Can anyone learn maths to a high level?
Or is it important to be 'blessed'? Or do you think it is pretty irrelevant and only relevant for the level of Field-Medal member?

I am really sorry for the grammatic issues. I am still improving my English!

Math yes, physics no.

 

[SchroedingersLion]

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.

I never said raw talent is the only ingredient. But it is a necessity.

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).

Some people are incapable of learning high level mathematics, because it requires a certain amount of intelligence they don't have. This is neither false, nor is stating it arrogant or elitist in any way. Are you trying to attack me because you realize your naive view of the world doesn't hold up to logic and everyday experience?
"Anyone can learn anything, if he tries hard enough or gets the right tutorship" is a nice political correct statement that might even lead to a better education system. But it does not describe reality since talent is real. You can deny it all you want, it doesn't change the fact. How much raw brainpower a certain individuum has depends on a) genetics, b) early childhood education. Psychology is pretty clear on that. And the amount of brainpower determines a certain level of intellectual ability and knowledge you will not cross in the finite lifetime you have.

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.

None of this matters. This 14 years old girl I met is SOMEONE. This SOMEONE will probably never be able to learn high level mathematics.
This answers the question of this thread, wether anyone could learn anything at a high level.
If you could go back in time to her earlier years, you could have improved her current condition. Sure.
But even then: Genetics play a role in that as well so even if we changed the thread's question to "Could anyone learn anything to a high level, if he got the perfect learning environment from the very first minute of his life?", the answer "yes" would become much more realistic, but still not certain.

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?

It's not absurd at all. In fact, it is absurd to assume anyone could learn anything to a high level.

 

[StatGuy2000]

I never said raw talent is the only ingredient. But it is a necessity.Some people are incapable of learning high level mathematics, because it requires a certain amount of intelligence they don't have. This is neither false, nor is stating it arrogant or elitist in any way. Are you trying to attack me because you realize your naive view of the world doesn't hold up to logic and everyday experience?
"Anyone can learn anything, if he tries hard enough or gets the right tutorship" is a nice political correct statement that might even lead to a better education system. But it does not describe reality since talent is real. You can deny it all you want, it doesn't change the fact. How much raw brainpower a certain individuum has depends on a) genetics, b) early childhood education. Psychology is pretty clear on that. And the amount of brainpower determines a certain level of intellectual ability and knowledge you will not cross in the finite lifetime you have.

Again, your contention above assumes that "intelligence" (however way you define it) is a fixed quantity that remains impervious to change over the development of an individual. That is neither an obvious and far from trivial assumption you are making, and has been widely criticized and debated by psychologists for years.

As for the statement "talent is real" -- again, if someone works extremely hard on a certain task and becomes sufficiently skilled at that task, how much of this is dependent on genetics and how much is based on the individual effort? You seem to imply that this is primarily a genetic attribute, whereas I'm proposing that environmental influences (including the very actions of the individual in question) could play just as important a factor.

I should further argue that you repeatedly mention genetics here. Can you, or anyone else, actually pinpoint to research that indicate which genes (or which collective interactions of genes) can be definitively linked to mental or intellectual abilities in humans? As far as I'm aware, I cannot think of any geneticists, neuroscientists, psychologists, etc. who can provide even tentative evidence of such. Of course, if you can link me to published research in this area, I would be more than happy to read this.

 

[gmax137]

In addition to "genetics," repeated use of the word "logic" is inaccurate. This is not a question to be settled by "logic." So far we have four pages of "opinion."

Existence of a gradient in "ability" does not allow us to "logically" deduce that some individuals have zero ability, or even that some have an ability below any given threshold. I've seen no evidence for the slope of this gradient. Maybe it is more shallow than some believe. Maybe it isn't. That's the question, right? How steep is the "ability" gradient among individuals?

Opinions vary. Most driven by anecdotal experiences.

 

[mhl47]

It is really astounding how this discussion polarizes. Unfortunately, I was really hoping that someone would have taken a deeper look in the research of this topic since I have been too lazy ever since to dig into this area. But to be honest, even if there would be empirical studies on this, they would have to face immense issues with the setup, since you can not really isolate all the factors here.

So some interesting questions regarding this topic might be:

.) How much in the brain is really "hard-wired" based on genetics (assuming that nothing is really "hard-wired" but that there is some information already there in the biological structures)
.) Can you circumvent this existing structures in the brain or is there a chance they or their function changes in your lifetime (neuroplasticity)
.) Are you really bound to this preexisting structures whilst building a model of your world? (in AI commonly referred to as world model)
.) What does it actually mean to have brainpower?

One would really have to investigate how "easy" it is for a given state of a brain, starting at birth, to be shaped such that it has a model powerful enough to make a conclusion in mathematics (e.g. linking different fields). At this level of complexity, probably neither neuroscience nor research in artificial intelligence is close to answering such question. But I would be happy to be wrong.

Having said that I would also like to throw my personal opinion and experiences in. From my point of view, anyone can learn anything but it will take longer for some people because the information provided to them might not fit well in their current model of the world. Especially motivation seems to be the key to build knowledge in a way that it can be used to solve problems. Why on Earth would you restructure everything you believe, that kept you alive till now, if you do not have a strong motivation to do so (not something superficial like grades)? If a person does not think, that understanding mathematics can be really beneficial for their life, they will have a hard time learning it. Convincing people that they can benefit from understanding mathematics, and do awesome things with that knowledge, is for me the most important part of teaching. Since one tends to judge that also based on the person that teaches (do I really want to be like my teacher?) it does not exactly help that a lot of math teachers are a bit introverted.

In all the years at university or whilst tutoring, I never had the feeling that a persons struggle with a concept was based on their inability to learn it. Never have I experienced a student with a deep and real interest (working hard on the problems, gathering additional information through books etc., trying to apply the knowledge to real-world situations) who hit a line he/she could not cross.

In a way, it is a really comfortable world-view to say that the ability to learn maths is limited. For the people who are good at it, feeling special and secure in their position, and for the people who seem to be bad at math, convincing them self that they can not learn it anyway. I do not see any real evidence supporting that view.

 

[Drakkith]

Again, your contention above assumes that "intelligence" (however way you define it) is a fixed quantity that remains impervious to change over the development of an individual.

I don't agree with your conclusion here. A 'variable intelligence' could still mean that certain people are simply not going to be able to learn something, as the amount that their intelligence can vary may be too low.

I should further argue that you repeatedly mention genetics here. Can you, or anyone else, actually pinpoint to research that indicate which genes (or which collective interactions of genes) can be definitively linked to mental or intellectual abilities in humans?

There are plenty of genetic diseases that negatively affect mental ability, so there is obviously some link between genetics and mental ability. While there is no conclusive proof that certain specific genes give a person above average abilities, it seems reasonable to conclude that it is possible, perhaps likely. But any link is not obvious and it may not result in a particularly large difference.

Existence of a gradient in "ability" does not allow us to "logically" deduce that some individuals have zero ability, or even that some have an ability below any given threshold.

We aren't looking purely at a mathematical gradient. We have real life people we can look at. The spectrum of mental ability (however you might define that or break it down) runs from 0 (brain dead) through those with severe mental handicaps, continuing on up through people with less severe handicaps, average/near-average, above average, and so on.

In all the years at university or whilst tutoring, I never had the feeling that a persons struggle with a concept was based on their inability to learn it. Never have I experienced a student with a deep and real interest (working hard on the problems, gathering additional information through books etc., trying to apply the knowledge to real-world situations) who hit a line he/she could not cross.

Perhaps the reason that you haven't seen anyone with an interest in a subject hit a hard wall is because the people who have severe difficulty with math (or another subject) don't want to do math because of how difficult it is for them. People rarely enjoy things that are immensely difficult and taxing for them. Also realize that the more difficulty someone has with math and other subjects, the less likely they are to go to college or pursue non-required education. So you aren't as likely to encounter them as you might think.

 

[atyy]

What I have been saying is that there are people out there who are not considered to be mentally handicapped that simply cannot learn advanced math, regardless of how they were raised and educated. I'm not arguing that most people are like this, I'm not even arguing that the percentage is large. Frankly I believe that with enough time and given enough effort, most people could learn math up to the undergrad level.

Can anyone (with "normal intelligence") learn to count? There an interesting case about this: https://en.wikipedia.org/wiki/Pirahã_language.

 

[mathwonk]

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.

 

[Drakkith]

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 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?

 

[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.