These are two really good points, but #2 seems so obvious yet has never occurred to me.

I believe a property of the "good at math" persona is that the individual can just "magically" solve a math problem _quickly_ in there heads, or more precisely, this individual already possesses the answer, but just needs to recall it, akin to recalling a history fact.

Yes, by practicing maths, you will eventually develop some "muscle memory" for certain types of problems, but understanding or solving a math equation does not inherently have time constraints associated with it, yet we somehow believe that to be part of what it means to be proficient in maths.

This false belief has held me back from believing I could achieve more in mathematics - the idea that if I can't find the solution to a problem in under 10 seconds, I obviously don't know or can't figure out the answer.

Thank you for making this point that is quite obvious, but has enlightened me.

This is how I felt about the Math GRE. On the practice test, when I gave myself plenty of time to think, I was able to answer almost every question correctly. But on the real timed test, I scored very poorly. They gave you so little time to solve the problem there was no "thinking" involved, it was all rote algorithmic manipulation.

I remember that in elementary school a lot of the early math education was especially focused on speed. The same quizzes were basically given to us repeatedly until our speed improved for the entire year. Later on, they intended for this to serve as a sort of placement test for math. I was pretty much always the slowest one, because I would get more stressed when a time limit was given (something around 2 minutes for 60 questions) and I didn't approach it as memorization like many of the other students did. My teachers were convinced that I was unable to learn math because my speeds at answering those questions were not improving as much as those of other students and most of the time I was not able to finish the quizzes. But as soon as my parents requested for the school to give me a normal test they found that none of those problems even existed for me. Afterwards, they ended up moving me to a class that was a lot less focused on just doing the work quickly and that put me ahead of where many of my peers were. I think that if everybody else had been taught in an environment like that were time was not a factor in your grade they would have a much better understanding of the concepts of math and would also find it easier to learn how to do the actual math than how to memorize the answers to repetitive problems.

I think STEM students should be writing more papers across the board. You shouldn't just solve problems and write down the solutions, you should be writing exposition on the methods you use, why they work, and what other options are available.

My writing ability developed in high school and college due to a few specific causes. 1: Being forced to write for my AP European History course in HS (the regular assignments were just lists of questions/prompts to be answered with a few sentences or a paragraph or so, it's amazing how valuable that experience was). 2: Writing proofs in advanced math courses (elementary analysis, abstract algebra, etc.) 3: Participating in usenet newsgroups (by far the biggest contributor).

P.S. I meant to point out the irony that English classes contributed comparatively little. Actually writing about something and putting in the effort to string together something coherent is what really exercises and builds writing abilities. It's nice to have some of the groundwork laid, but practice is by far the most important component.

I got the impression that the article was about K-12 education, where the kids aren't yet completely separated into STEM and non-STEM tracks. (Some kids get onto an accelerated math track). If the schools that my kids attend are typical, they do extensive amounts of writing in all of their subjects. The kind of writing that you describe was pretty common, through 5th grade.

In 6th grade and beyond, the school went to a more traditional math curriculum, I suspect due to milestones such as getting through algebra in 8th grade, etc. For the kids to do more writing in math class requires a choice of what they will spend less time on. Do they learn fewer math topics, get fewer assignments in Social Studies, or get less sleep?

Maybe I'm a freak, but as a student I greatly enjoyed the "pure" nature of math. In other subjects, I found that I could get good grades by basically filling pages with drivel. On the other hand I didn't just write down solutions. I was expected to show a derivation, and in many cases, a proof. I have a copy of my high school pre-calc textbook, and a large fraction of the chapter problems are proofs, which I loved. That's what motivated me to become a math major in college. Today, the proofs have disappeared.

Test anxiety is a multiplier for stereotype threat. It is just the worst for performance. I am a good test-taker because I can put it aside, but it is not fair.

>Timed exams test for speed, which is not something that matters in real mathematics (if you can prove a theorem in one week vs. two it doesn't really matter),

Moreover, some conjectures don't get proved as theorems for decades or centuries, and even after such long periods of time, the conjecture might still sometimes be refuted or proved independent of the axiom framework.

Saying timed exams in math test for speed is a simplification. Perhaps at the underclassmen level, but I've found for a lot of upper level courses the tests were more about making sure you have certain concepts internalized and can make certain inferences without a secondary source. Oral examinations would work just as well, of course, but those are usually considered more stressful.

I do not think 1 is necessarily true. You can group kids by skill, and still develop a culture where you can struggle and feel smart at the same time. In contrast, not segregating classes risks making the not gifted at math students feel not smart, and risks making the gifted students feel that they do not need to work at it.