So you failed, then took your time and practiced more to succeed. I don't see how that contradicts anything I said. Circumstances matters, effort matters. Talent being a huge factor doesn't contradict that.
You asked..
> how do you explain the very large number of people who spends tons of time and energy trying to get good at maths but fails to even grasp simple concepts like calculus?
One answer, as vector_spaces said, is that the learning environment can completely change the outcome of this effort. You agreed:
> Circumstances matter
It's true that talent being a factor doesn't contradict that, but given how homogenous early-life learning environments are, it's -extremely- difficult to tell whether a person "failed" because of lack of talent, vs. lack of the right environment to experiment and learn.
Basically, because a "wrong environment" fail looks like a "lack of talent" fail, I believe a lot of talented people don't find out that they have the talent in the first place, as they believe that an attempt in the wrong environment is as good as any attempt, and never try again - particularly if they've been chastised and humiliated -during- that learning process, as vector_spaces was.
In any case, the reason I wanted to jump in was because of your original statement:
>I don't think this is because people told them anything
I would just ask that you reconsider that position, based especially on your agreement that "circumstances matter." Thanks for reading.
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So basically, just because someone managed to improve their grades and pass classes with hard work doesn't really say anything about my points. There are still people who pass easily without any of that. In short, my view is that hard work and environment can help improve your results, but unless you are completely prohibited from attending school you wont need that if you are smart enough.
For example, I skipped a lot of classes, was abused, was suicidal, never studied for any tests and didn't do any homework assignments in high school. I still got perfect math grades. Most people have better environments than that and spend more time on maths than that yet still do worse. I spent a lot of time doing 1:1 teaching in college, explaining things in intuitive ways etc helps, but ultimately I had to conclude that I can't make others understand things the way I do. They simply lacked something.
I appreciate you sharing your story - I'm sorry you had to deal with that experience.
That is not to say that some people didn't have to put as much effort as others - learned faster. But when I come across people, mostly online, brushing off math at or above high school level as easy have either received subpar/simplified math education (happens almost exclusively in the US and some shitty EU countries), or have too much confidence from their positive experience with low-difficulty exams.
> I had to conclude that I can't make others understand things the way I do.
If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
> If you've had this happen too often you're probably just bad at explaining things, or even worse you haven't fully understood them.
I understood the subjects well enough too solve problems on exams I hadn't seen before. That was the main way I managed to get good grades without doing practice problems or studying. I didn't manage to get others to understand things that well. I don't think that failing to get people to that level means I am bad at explaining as there is nobody in the entire world who can get average college students to that level.
For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades. I'm not sure if that was hard or not, but to me it didn't seem hard. I didn't continue doing maths as I didn't really have enough passion for the subject to continue doing research, it is pretty boring. After that I realized I needed to earn money, so I learned programming, started getting top 100 spots and sometimes top 10 spots in world wide programming competitions and joined Google. That didn't seem particularly hard to me either, as it took about half a year just doing things on my own.
Why aren't I famous etc? Because I just don't care that much about things. I get bored and stop doing them long before I have worked enough hours on things to get any notable results. I aren't bragging here, I know my value to society thus isn't that high, I am just giving my perspective on things. There are tons of people online arguing that I can't exist, I don't like that.
No offense but that's exactly what I felt you were doing on your first comment when you were talking about people having trouble understanding math as easily as you can. We are on the Internet so it doesn't sound as condescending as it would IRL, but it still makes you come off as trying to convince us that you're superior. I'm certain you're not doing it intentionally, I'm just trying to explain why as you said, are there tons of people online arguing that you can't exist. People with academic achievements akin to yours tend to remain humble when similar topics about intelligence and genius are brought up. I don't think they act this way because they legitimately do not consider themselves smart, they just know it's quite socially inappropriate to boast about it. On the contrary, again on a systematic basis, those with negligible life achievements almost always try to inform everyone in the room that they feel superior in terms of mental capability. Based on your academic biography above you seem to be an exception to what I just described and have observed both online and IRL.
> For my masters thesis I solved a problem that had stumped one of the more renowned mathematicians in Sweden for decades.
Sounds impressive but at the same time many would be quite skeptical about a really challenging problem being solved in the short, in terms of time, course of a Masters degree. Again, not attempting to undermine your achievement, but you usually hear about hard unsolved problems being a topic of a PhD thesis which almost always includes a number of publications being done during its course as part of it. I'd appreciate a few more details here but I understand you may hesitate stating more because of potential privacy concerns.
It could be, but it doesn't really matter. There are people who can think circles around me (and probably you). And I don't think that they ought to have some flaws just to not be totally superior out of some universal law of fairness.
Obsessing about it is obviously not healthy.
The latter assumes the former, since if such a thing exists, you can be sure that external factors would limit such a person's ability to succeed in traditional settings.
In either case, I don't understand why such a notion would be useful to anyone except as a justification for gatekeeping and maintaining a culture of elitism in math, which is very much not needed. I'm genuinely curious as to why you find the concept useful
Is this useful information for how to treat typical students? No. However, talent having such a profound impact on peoples ability to learn, understand and apply mathematics is something most people are aware of and hence it isn't strange that people who studied maths thinks that genius requires inborn talent. The author of this piece instead takes a jab at it and tries to say that they just believe that since they got told by their parents that they were smart, I don't think that is true. Most people studying maths doesn't view themselves as good at maths, so what is more likely is that they take their own struggles despite hard work and passion as evidence that genius requires inborn talent.
The trick was mostly that most things just clicked for me, and I could use that to figure the rest out and fill in the gaps during exams to solve the problems they didn't explain in any classes.