Ozzie wrote:
No. This is not always true. I've been a math teacher for 15 years, mostly at the high school level. I have worked with some highly intelligent students who are not very good at what I am teaching. Some of those students are lazy, like the kid who runs 5:05 in the 1600m for 4 years, but does so without practicing. As a freshman that is an alright time and got him by, but by the time he is a senior and running the same time, he's way behind. For a lot of students, they missed something along the way.
I taught a girl who got a B- or C+ in my algebra 2 class that was taught at half speed for those who struggle. She ended up going on to a JC, which is pretty typical of a student from that grade range. What is not typical is that after 3 years, yes 3 years instead of 2 at the JC, she transferred to Berkeley and ended up finishing with a degree in mathematics. Last I knew, she took that to law school. She should be done by now, but we haven't kept up. Back when she transferred to Berkeley and decided to major in math, I asked her what happened. She said that at Christmas Break of her second year of JC she realized that she understood the concepts that the professor was talking about, but couldn't keep up with her calculations in the notes. She spent time every day of the break going over her basic multiplication tables...the ones we learn in 2nd grade. She got that down in a way that she never had and after that, the degree was easy. That's a true story and it blew my mind when she told it. It gave me a whole new perspective.
There's a widespread misconception that math is a natural capability like perfect pitch instead of a subject to be learned where working on later topics depend on knowing the earlier ones. So many who struggled with calculus in college had the intelligence to do well but didn't have the foundation in trigonometry and algebra. Maybe in high school they were able to solve problems in these subjects, calculus too, and even do well but didn't really bother to learn it.
Those type of stories are so great. It reminds me of stories about WW2 soldiers from small towns going to school on the GI bill and blossoming. It's not even about being humble. Just living life seriously and taking it as it comes. This person is an example of someone who's lost in the crowd. Good enough to get passing grades so there really wasn't a problem. She just needed the right experience and time to subconciously percolate and figure out what was really going on. Such people usually first look for answers on resources like https://plainmath.net/, and then they help the same seekers there. I doubt her talent could have been brought out deliberately.