In first grade, I gave my son money for the book fair, expecting him to bring a cartoon book home. He instead brought home something like “Algebra for Dummies”. My wife and I laughed, and then a few weeks later we were shocked when he was able to solve multiple equations with multiple unknowns. Not exaggerating at all when I say that he literally taught himself algebra at age 7, and his math skills just kept going from there.
He has a math brain, and he was born that way.
except if you are the parent of a kid who is “gifted” or highly gifted or needs an IEP/special adaptations or is twice exceptional.
By definition you are leaving out about 25% of the class behind or lacking material for them. Why should education only teach to the middle?
And if that’s the case, then why do we have top universities where you have to be exceptional to get in the door? That makes no sense. And frankly, I think it divides kids into haves and have nots, those whose parents can get access to what they need or those who can’t. Or, if “luck” has it those who fall right in the middle where some people think they ought to be to get an education that fits. Outside the lines and not worth talking about. hmm.
In the specific case of Einstein, I would say his accomplishments are a product of his personal effort and dedication. I would say the same of most great scientists. Of course, that does not at all invalidate just how far above most people Einstein was in terms of cognitive ability. I’m simply arguing that he got there through exceptional dedication and not innate ability.
What causes some people to have this sort of dedication towards a subject is an entirely different matter.
One of my kids was invited to do accelerated math during his 8th grade math honors class starting his freshmen year. He proudly accepted the invitations. His class covered 2 years of math during their freshmen year. Pre calc sophomore year. Most of the group choice to skip over Calc AB and went directly into Calc BC as juniors and some of the kids took AP stats as juniors concurrently with either AB or BC or took AP stats as a senior. Our public high school paid for my son’s MVC class at local U with college students during his first semester of his senior year and would have paid for linear algebra 2nd semester, but there was a scheduling conflict with his HS classes and the times offered at the U. Half of the kids in the accelerated program took AP physics 1 as soph and AP physics 2 as juniors. The other half were going to take AP physics 1 and 2 combo as juniors during a 0 hour at 6AMish except some of the kids backed out leaving 9 kids. Unfortunately, the class was cancelled. The Physic teacher dropped of a summer self-study program to my son and the other kids for AP physics 2, to allow the kids in the accelerated math program to take Physics C together as seniors. When my son talks about Math, he puts his hand over his heart and say I really love math. Yesterday, we found out he was accepted to Carnegie Mellon. Last night I was reading about CMU and discovered their motto is “my heart is in the work”. Sounds like a match made in heaven.
The speculation is that Einstein was on the autism spectrum. Neuro-atypical. If that is what caused his devotion to his field, then I think we are saying the same thing.
Absolutely. Great story.
I see your point. I guess my argument is that, ignoring those individuals who are at the extremes, everyone else is more or less at the same cognitive level. The bell curve is narrow. I would put myself (someone who gets straight A’s at a top 20 school) in the same group as someone getting just about any grade at any school when it comes to cognitive ability. Any differences that arise in performance are most likely not due to some innate advantage I hold or some innate disadvantage on their part. In that sense, most people attending top schools aren’t exceptional at all the way I understand the word. Rather, I would describe them as people who learned how to learn thanks to both their own efforts and their upbringing.
That’s not how it works with a “Bell curve” (actually I suspect if you could quantify ability on some sort of linear scale you would find its more fat tailed than a Gaussian distribution). There are kids that are 1 in 100, 1 in 1000, 1 in 100,000 and 1 in 10 million. What you do for the top 5 kids in a class of 500 (1 in 100) is different to what you do for the kid you see once in a teaching career (1 in 1000 to 1 in 10000). But all of those situations are ones that need to be considered by a regular high school. They are nothing to do with someone like Tao that occurs perhaps once a decade in an entire country.
All the way along that distribution you find the 1 in 100 kid being massively outdistanced by the 1 in 1000 kid who is massively outdistanced by the 1 in 10000 kid, who in turn is massively outdistanced by the more able kids, all the way up to the Fields Medal winner. It is just very obvious in math, in a way that it wouldn’t be in a bunch of high school English or History essays.
You can read Hardy’s “A mathematician apology” (https://www.math.ualberta.ca/mss/misc/A%20Mathematician's%20Apology.pdf) where he talks about how much better Ramanujan was at maths than him (“Well, I have done one the thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms”).
The abilities of students in math are highly dispersed. Even the best teacher wouldn’t be able to do an adequate job if her class has students with highly disparate skills. To be their best, students also need to be challenged by their academic peers.
It is true that 9th grade track is the “normal” track but an awful lot, perhaps even more kids compress their math and take algebra in 8th. All of the gifted programs (which have much broader admissions than most states) offer algebra as a 7th grader as the high option. And in LAUSD, which has a massive population, the highest math offered is algebra in 6th grade at three highly gifted programs. My daughter is actually taking that track right now.
Ironically, although the charter schools are generally stronger than district schools, many kids leave to seek out more challenging math because the district offers it.
Totally agree that digging in and doing what it takes is a superior skill to extreme brilliance in a singular thing. I’d take the hard worker for most jobs, any day.
I would even go out on a limb and say grit is distributed on a bell curve! (that’s a joke, sort of)
The Steph Currys of the world are the rare people who are on the far end of the curve for both grit and innate ability. Throw in an environment that fosters study skills and time management, and you have yourself a rock star in the chosen field. Hard to have that trifecta, though. Having only one doesn’t get you far at all.
That may be your opinion. But what evidence do you have to support it? Have you looked at the distribution of scores in long tail math tests (Putnam, IMO etc)? I think all you are demonstrating is that most US grading, even at college level, is not oriented towards long tail discrimination of ability (and is heavily weighted to working hard instead).
I’ll give an example of how UK college exams work in math (at Cambridge): you do four 3 hour exams over two days. To pass you need a total of 3 questions correct in the four papers combined, the mid ranking students get 6-10 correct, the top 30% get 12-14 correct and the very top student (senior wrangler) gets 35+ correct. A decent student, who might spend an hour to solve one question, has no ability to compete with the best, who can see the answer immediately and intuitively.
For a comparable US example read about the Feynman lectures and how only “one or two dozen students” (out of I believe 180 of the best students in the country) were able to follow him (The Feynman Lectures on Physics: Feynman’s Preface )
Super cool to read. Thanks for posting it.
Unfortunately, many (most?) American high schools disagree. And most middle schools really disagree. For many, if you get over the threshold requirements, their work is done.
Well I said “considered” because they will encounter those situations. That consideration may result in a conclusion that “we don’t care to provide anything different/extra for those kids”. Then the parents either sue or give up, which for the wealthier ones may involve finding solutions elsewhere, in private school or extracurricular enrichment.
In our area, many parents of “gifted” kids in high SES areas have figured out a work around to get their kids needs met. They apply for an IEP and then use the IEP as a legal cudgel to ensure their kids get what they need from the public school systems.
A report for 9th grade is at https://go.boarddocs.com/ca/sandi/Board.nsf/files/BEFPLW6531B2/$file/SB%20359%20Report%207-30-19.pdf . In 2017-18, it looks like 22% were recommended for accelerated math, and 29% were placed in accelerated math. I believe the totals refer to the 2 “Advanced” Integrated Math paths in the flowchart – AI 1 and AI 2.
There is also a 3rd path to calculus in the flowchart involving dropping down from Advanced Integrated 1 8th grade to Regular Integrated 2 9th grade. I believe this is uncommon since AI 1 → Reg 2 is not an intended switch point and would involve discontinuities in curriculum. Other documents suggest this path is typically recommended to students receiving a D in AI 1 (ABC advance, F repeats, D switches sequence).
My understanding of IB HL math is that it teaches analytic geometry in 3D using vectors and coordinates, not the Euclidean geometry that I was referring to. Nonetheless, it still requires some ability to visualize, just not to the same extent as in Euclidean geometry.
I think quite a few of the accelerated math kids tend to go around the school via AoPS. The school transcript might only show Calc BC but they have also had Number Theory plus Combinatorics/Probability as well as some of the other subjects like Algebra and Pre Calculus at a much deeper level.
That’s a great way for kids who can’t get their needs met in-school to continue with their love of math. Most kids don’t care about credit. But they don’t want to sit in a class where they explain the same concept for the third or fourth time.