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:
A tiny number of brilliant students who will teach you something.
A moderate number of pretty smart students who would probably do an OK job if you just handed them the textbook and left.
A large number of average students for whom your technical class is serious work, but will get through with help from you and study.
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.
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.