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Can someone truly love math, and master and remember discovered calculations, counterexamples, proofs; but still fail to invent anything new (e.g. incapacity to prove anything unseen, calculate something that demands new tricks, or discover counterexamples)? I ask not about dyscalculia that appears irrelevant, as I'm not asking about arithmetic.

I bolded the relevant parts from the Reddit Post beneath that's anecdotal: is there any evidence?

Although I am usually not a full-time teacher, I've taught hundreds of students over the last 35 years.

Students fall into five categories:

  1. A tiny number of brilliant students who will teach you something.

  2. A moderate number of pretty smart students who would probably do an OK job if you just handed them the textbook and left.

  3. A large number of average students for whom your technical class is serious work, but will get through with help from you and study.

  4. A moderate number of people who don't like or care about math and/or aren't particularly talented who might pass if they put in the work but probably won't.

  5. A small number of people who will never understand the material, no matter what.

When I was young I refused to admit that that last category existed. I put a lot of time into a few people who worked really hard but tried to memorize and fake their way through, or just couldn't get it.

And then something happened to convince me otherwise.

I had a friend who started Math in University at the same time that I did, but he was about five years older. He'd already had two successful careers - he had been a journalist and then quit that to run a campaign for a politician who won his election - but he'd always loved math.

He did perfectly well in first year - not exceptionally but fine. But in second year, the trouble started. He could not create new proofs - no, not at all. He understood the material quite well, with some gaps, but writing proofs was a bridge he could not see how to cross.

I spent hours with him, but he started to do worse and worse.

Eventually he vanished. I finally got in touch with his brother who said, "I really appreciate your having contacted us, but I'm not going to tell him you called. He had a nervous breakdown and his doctor said he should be kept away from any memory of what happened."

Now don't get me wrong. My guess is that at least half the people who fail math do so because they just aren't interested enough to put the time in, and if they really wanted to, they would succeed.

But there's a core of people, often otherwise smart people - people who are even really interested - who simply don't think that way, and all the coaching in the world won't fix this problem.

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    $\begingroup$ I think this is off-topic for the site; it's not a question about teaching math. $\endgroup$ Jan 14 '18 at 18:41
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    $\begingroup$ I think the question is acceptable for the site. Whether there are people who are simply incapable of learning what a proof is, despite otherwise reasonable success, is relevant research for teaching mathematics. $\endgroup$
    – Tommi
    Jan 14 '18 at 20:12
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    $\begingroup$ I think it is WAY ON TOPIC. Pedagogy versus audience is RIGHT IN THE SWEET SPOT. Love the story and the five categories Think the discussion of ability versus effort versus strategy needs to proceed. $\endgroup$
    – guest
    Jan 14 '18 at 20:14
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    $\begingroup$ I also think this is on-topic. Although I have lots of anecdotal evidence after decades of teaching and witnessing teaching from freshman undergrad to advanced grad-level, it is rarely entirely clear to me the source of "troubles"... Actual research (which would probably be too invasive in a usual teacher-student relationship) would be enlightening. $\endgroup$ Jan 14 '18 at 20:37
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    $\begingroup$ @guest: Note that "discussion" is inherently not what the site is for. "Avoid questions that are primarily opinion-based, or that are likely to generate discussion rather than answers." (emphasis in original) matheducators.stackexchange.com/tour $\endgroup$ Jan 14 '18 at 22:11
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I feel like the question you are interested in is whether there is a "kind of person that can produce mathematics" and a "kind of person that can't." I don't know of any conclusive research on this subject and lot of the interest in it is heavily politicized. That said, it would seem to me that the evidence is overwhelmingly that there are not two kinds of people.

The main reason I offer is that we as humans are shockingly good at education. Every time we move the goal posts a new generation of students stands up and dutifully learns how to do what we've required of them. For example, we now have nursing students diligently passing calculus courses because colleges have decided to require it for premed. Twenty years ago that would be unthinkable and people would have smugly said "of course the type of person who becomes a nurse couldn't do calculus, there are some people that can do math and some people cant."

So then, why do some people struggle with proofs? Well, the current goal of mathematics education isn't (in general) to produce students who can write proofs. The goal instead is to produce students who can apply theorems and perform calculations. If we revised this goal, I am confident that all the students would follow us into real analysis, just as they've followed us into calculus.

In addition to my beliefs above, there are a few things I want to point out about the text quoted in the question.

First, I really chafe against the equivocation of a student not being in a position to pass a course right now and somehow being not the kind of person who can do math. There will always be students that struggle, sometimes it's their way of thinking about the material, sometimes it's their professor, sometimes (very often) it's external stressors. Sure, every semester I will have a student that can't seem to "get" the material no matter how long we work at it together. Sometimes these students try everything and fail, but sometimes that student gets a tutor or makes new friends, stops showing up to office hours, and passes the class. It's very cocky to draw the conclusion "this student will never have any capacity to understand math" from "I can't teach this student this subject in math right now."

The second thing I want to point out is that the conclusion of the anecdote seems horribly flawed to me. Student stress and its effects on mental health are well documented and one of the symptoms is lower cognitive function. I have unfortunately watched mental breakdowns happen to my peers in graduate school, they are debilitating, life destroying events that usually have a long run up to them. They are exacerbated by the fact that the stress they cause make it monumentally harder to do cognitive work. Going from "my friend had a mental breakdown the caused him to be removed from the college he was attending" to "gee, he must be really bad at math!" makes much less sense to me than "my friend must have been going through some serious shit to have a mental breakdown, I wonder if stress was part of why he was having so much trouble doing math."

I think the phenomena that are being described here are much more accurately accounted for by thinking about what factors might make a person struggle in a particular situation, rather than just positing that "some people can't do math." It seems to me that every time we assume that people as a whole can do something and proceed to try to educate them to do so we are proven correct.

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There are really several distinct questions tangled up in this, so let me try to tease them apart.

  1. The first question is, Are there people who are intrinsically incapable of creative work in mathematics?
  2. The second question is, Are there people who eventually become incapable of creative work in mathematics?
  3. The third question is, What evidence exists for the answers to the above questions?

The reason I separate out the first and second question is because the anecdotal case mentioned in the OP was not a blank slate, but rather someone who had already gone through a full sequence of primary and secondary education before the OP met him. These initial years of education play a major role in shaping not only what mathematics people know, but also how they approach the subject. There is an abundance of research that demonstrates that primary and secondary education tends to emphasize memorization and speed, and pays little attention (if any) to opportunities to reason creatively in novel contexts. It seems extremely plausible to me that a student who is successful in such a context may find that whatever innate mathematical creative potential they may have once had becomes calcified, or atrophies (or whatever metaphor you think is best).

To take a somewhat clichéd analogy, it is widely acknowledged that it is very difficult to become a proficient musician if you do not start taking lessons until adulthood. It's not impossible, of course, and with hard work one can learn to play with some passable level of skill -- but to become a truly great musician may be beyond the reach of someone who starts too late. Does that mean that such a person could never have become a great musician? Not necessarily -- but it does suggest that there is a window of opportunity that eventually closes.

Now mathematics is not music, and mathematical creativity is not the same as musical talent, but I think there is something fruitful in the analogy nevertheless. The story in the OP was about someone who, in adulthood, found that he was not able to do creative work in math. It's noteworthy that this individual was "about 5 years older" than the narrator of the story, and had been away from mathematics instruction for that period of time. It seems quite reasonable to me that whatever potential for creative work in mathematics this person once possessed may have simply dried up by the time he returned to university study.

Anecdotes are not data (as the OP knows, which is why he asks for evidence), so my own anecdotal experience is not worth much here, but I don't think very many people will be surprised to learn that students who return to university study after a number of years in the workforce tend to struggle more so than their younger peers. They have a lot of stresses on them; they are more likely to have spouses and children, they are less likely to have direct financial and emotional support from their parents, and so on. The fact that the person in the OP's story had a nervous breakdown suggests, to me, that there was probably a lot going on that had nothing to do with his mathematical coursework.

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