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100% agree. What I typically tell people is "your mathematical potential has a limit but it's likely higher than you think."

Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation.

(My comment here is basically the intro to a detailed article I wrote on the topic: https://www.justinmath.com/your-mathematical-potential-has-a...)

That's an excellent essay. I especially liked this part:

" Active learning and deliberate practice will be covered in more depth in later posts, but below are some key points:

- Effective learning is active, not passive. It is not effective to attempt to learn by passively watching videos, attending lectures, reading books, or re-reading notes.

- Deliberate practice requires repeatedly practicing skills that are beyond one's repertoire. However, this tends to be more effortful and less enjoyable, which can mislead non-experts to practice within their level of comfort.

- Classroom activities that are enjoyable, collaborative, and non-repetitive (such as group discussions and freeform/unstructured project-based or discovery learning) can sometimes be useful for increasing student motivation and softening the discomfort associated with deliberate practice -- but they are only supplements, not substitutes, for deliberate practice.

- Deliberate practice must be a part of a consistent routine. The power of deliberate practice comes from compounding of incremental improvements over a longer period of time. It is not a "quick fix" like cramming before an exam. "

Thanks! Yeah, I guess I should probably link to some of those later posts about active learning and deliberate practice in the article. If you want to read more about that part you liked, here's the main one I'd follow up with: https://www.justinmath.com/deliberate-practice-the-most-effe...

And when they're knocked off course, they often develop math anxiety. It's a very real sense of dread that's been conditioned over time from test taking pressure, missing foundations, and underperforming.

Then they're not just lacking motivation, they're motivated to avoid math, which makes remediation more difficult. So sad.

Totally. It's a vicious cycle. Once you get knocked off course, you fall into this current that's pulling you further off course. And the further off course you go, the stronger that current is.

I've had similar experiences helping my SO, my sister and a good friend with post-high school math in various forms.

My SO had a teacher at school who'd determined she couldn't do math, and had the worst passing grade as a result. She wanted to go to engineering college and lacking the prerequisites she had to take their pre-course, ie all the math and physics required compressed in a year. She struggled hard from the get-go, and I had to go back to elementary algebra and build up. Yet after a few hard weeks the efforts started paying off, and in the end she nearly aced the pre-course.

My sister had never been into math, and had taken a vocational route working in a kitchen. After some time she wanted to go to college doing something else, and that involved taking college level math. While not as strong as my SO, again similar story where persistence and working the foundations helped a lot, and she aced her somewhat easier college math course on time.

My friend was a bit different, in that he'd never been interested in math but had to take it to get the required points to get into some uni program he wanted. He's fairly smart but struggled with motivation. So for him it required finding the right way of forming the questions so he got some motivation to solve it.

These are my most direct experiences, though I've also helped others here and there. It led me to believe most people could do reasonably well at entry-level college math (ie basic calculus, statistics etc). For some it might require quite a lot of effort to get there, but still doable for someone with motivation.

Why do you think there is a limit at all? What is it about higher level math that is intrinsically incomprehensible to a subset of people?

I suspect that the limit is actually in research and discovery, not comprehension. Calculus took some brilliant minds to develop but now it can be taught to most high schoolers.

As detailed in the article, my conclusion of there being a limit does not rest on the assumption that higher math is intrinsically incomprehensible to a subset of people (though, unrelatedly, I would expect that to be true in some cases).

In the article, the key underlying assumption is that the further you go in math, the more energy it requires to learn the next level up -- and everyone's "energy vs level of abstraction" curve is shifted based on their cognitive ability and degree of motivation/interest.

Here is a quote from the article that gets at the main argument:

"As Hofstadter describes, the abstraction ceiling is not a “hard” threshold, a level at which one is suddenly incapable of learning math, but rather a “soft” threshold, a level at which the amount of time and effort required to learn math begins to skyrocket until learning more advanced math is effectively no longer a productive use of one’s time. That level is different for everyone. For Hofstadter, it was graduate-level math; for another person, it might be earlier or later (but almost certainly earlier)."

https://www.justinmath.com/your-mathematical-potential-has-a...

I always thought I was bad at math. Then I decided to learn it again from the ground up when studying for the GMAT. I hired a tutor who completely re-taught me the basics and got me excited about it for the first time. I was amazed by how quickly I became comfortable with concepts as an adult, topics I assumed I was innately "bad" at. It made me realize how many things I could one day learn, given enough time and interest. Glad there are good tutors out there!

Just wanted to chime in that Nic is an amazing tutor, and if you're someone who wishes they had studied math more rigorously, you should reach out! You'd be amazed how much you can learn in an hour every week or two that's focused entirely on your interests/strengths/weaknesses.

> adults who want to learn proof-based math

What is their usual motivation for this? Do they find they are running into regular work or life situations that require it?

I think about all the math I took in high school and undergrad, and in my adult life I have not used anything more advanced than basic middle school algebra and occasionally some simple trigonometry. I don't even remember most of what I learned, other than very high-level concepts.

Motivations vary a bit, but most of them are just in it for personal enrichment, and the people who are in it for personal enrichment tend to be the most likely to stick with it. There are definitely jobs that require more math than the things you listed, but even if you have one of them the way I teach is usually more optimized for curiosity than professional goals.

> What is their usual motivation for this? Do they find they are running into regular work or life situations that require it?

I have a chip on my shoulder. In university I was depressed and didn't even bother attending lectures let alone doing the work in the first couple of years, couple that with professors who when contested were off by a 20-40% bc they cba to care for a secondary course in another department...

When looking for thesis advisors, I found one interested in the things I was. They made a comment asking whether I had an issue with mathematics. Over the year I learned enough mathematics to get to what I was interested in and understand the bleeding-edge literature (calc, linalg, vec calc, prob theory, etc). I corrected some of his proofs in his classes by the end of the thesis.

Still, my early grades haunt me, and parts of me wants to get a math degree just to prove that it is not a skill (read, intellect) issue.

>I'm a tutor, mainly working with adults who want to learn proof-based math

are you calling college students adults? because otherwise, what adults are trying to learn proof based math?

Good question! No, I don't usually work with college students. Most of my students are actual grown-ups with jobs. I've found quite a few people who just wish they'd been able to study this stuff in college but didn't get the chance for whatever reason and still have some unresolved curiosity about it. It's a very fun group to work with; they're very motivated!