Gifted Issues Discussion homepage Forums General Discussion Gifted children could learn math much earlier

(As background context, I'm a mathematician, and I've taught gifted children math over a span of ~10 years.)

Something that I've noticed lately is a widespread implicit acceptance of the norm for gifted children to learn math at grade level, or just 1 year above grade level. My experience has been that even moderately gifted children (IQ ~130) can learn algebra in at age ~11, and that highly gifted children (IQ ~145) can learn algebra at age ~8. Moreover, I think that there are strong arguments in favor of this.

Developmental capacity

• It's not uncommon for moderately gifted children to be 2+ years ahead in reading and for highly gifted children to be 5+ years ahead in reading, so one might expect them to have mathematical potential that's 2+ or 5+ years ahead of grade level (respectively).
• IQ was for a time believed to be "mental age divided by chronological age" multiplied by 100. This notion has (rightly) fallen out of favor, but it's sufficiently close to the truth for people to have believed it. Under this assumption, a 10 year old with IQ 130 has mental age 13 and a 10 year old with IQ 145 has mental age 14.5, and these 10 year olds are correspondingly cognitively ready for curricula aimed at people of their mental age.
• I know of people of IQ ~160 who have learned calculus at age 7: this suggests that in some respects IQ understates "mental age."

Desirability

Some people have suggested that it's better for gifted children to learn a broad range of things rather than accelerating, because if they accelerate then they'll be out of sync with their peers. I think that this is true in some contexts. But I don't think that the benefits of being better in sync with one's peers outweigh the benefits of accelerating through the K-12 math curriculum specifically.

• Grade school math is key to understanding the natural sciences, statistics and economics. Remaining at grade level in math substantially delays a gifted child's ability to understand these things.
• Learning math well builds general reasoning ability, which has benefits across domains.
• Many gifted children find math especially enjoyable once they become deeply involved in it.
• Being far ahead in math can build confidence on account of being an unambiguous signal of intellectual ability.

In The Calculus Trap Richard Rusczyk at Art of Problem Solving argued that rushing through the standard curriculum is not the best answer for mathematically talented young people, suggesting that students should instead focus on learning how to solve complex problems. I agree with him that learning how to solve complex problems is more important than acceleration through the standard curriculum. But the two things aren't mutually exclusive: gifted children can both learn how to solve complex problems and accelerate through the standard curriculum.

Learning precalculus and calculus was a transformative experience: it allowed me to understand physics, it gave me a thrill, and it made me better understand myself on account of tapping into my latent mathematical ability. It was when my intellectual development really accelerated. I was 16 at the time. I wish somebody had encouraged me to start earlier. A sizable minority of the most intellectually impressive people who I know I know had similar experiences.

There are large potential returns to gifted children learning more math earlier on.

I completely agree that depth and breadth can be achieved in tandem through child led acceleration. Children have an innate number sense, and for HG+ children, this can mean that a 2 year old intuitively grasps grade 1 or 2 math implicitly through free play, without any instruction. How keeping that child at grade level--effectively sentencing him/her to a 4-5 year brake on learning--could be construed by educators as being in the best interest of the child is beyond me. Innate interest merits, IMO, a supportive approach to foster those latent abilities.

We do a great disservice to anyone when we predetermine an achievement ceiling based on stereotypes, such as age.

I agree, but would only add: while in principle accelerating and doing hard problems aren't mutually exclusive, in practice very few children, including those who are accelerated, seem to get anything like enough experience with hard problems. If I were going to encourage one cultural change, it would be "more hard problems" not "more acceleration".

DH teaches graduate students statistics and often comments that the foreign students (who are mostly from China) are generally much better at math (and harder-working) than U.S. students. I wonder if that's because math is taught differently in China--I did hear something on NPR a little while ago about how students (maybe in Japan??) were taught to keep working on a problem even if it seemed unsolvable, while U.S. students were quicker to give up. But I wonder if that's all of it, or if compared to our math curriculum the schools in other countries are 'accelerated,' and if so how is it different. Too bad there aren't "Chinese Math" schools or something like that in the U.S.--it might prompt other schools to allow more acceleration, too.

Does this fill that quota? http://www.prismsus.org/ I don't know much about it, but seems pretty neat. Princeton, NJ is full of really excellent (albeit expensive) schools.

Having a little math head myself, I certainly have evidence that math can be learned at a young age - DS3 has become a little mental calculator of late, multi-digit addition/subtraction, and even (to a less extent) mult/div IN HIS HEAD. So clearly on paper it's pretty darn easy for him. I swear the kid can mentally calculate faster than I can. The concept of algebra just seems to make sense to him (at least, the beginning stuff. he blew through dragonbox months ago). honestly, all of it just seems to make sense to him. it's eerie really. His brain is clearly hard-wired for calculations. We discovered he could skip-count pretty much every number to 100 so early on...2.5ish maybe? and we still don't know how he 'learned' that. So, for at least some kids, the ability is in there. Honestly, I think if we sat with him and did legitimate math, every day ...I can't even imagine what the little brain could do. (normally we just follow his lead, whenever he WANTS to sit and do math, we will, but otherwise, it's mostly via conservations....especially in the car)

The concepts are just there, as is the interest. It's pretty awesome to watch.

I'd go so far as to say (at least in my own experience with myself, my DH, and our DD-- all of us HG+)-- that such children rarely learn math in any other way.

Without challenge, there is no real engagement, mentally speaking. It's like trying to "learn" from standing in line at the department of motor vehicles otherwise.

I estimate that, had we followed her natural arc developmentally, that she would probably have been well-prepared and intrigued by algebra and geometry at about age 8-9. Instead, school ruined her. Well. I say "ruined" but I have watched this process in action, let's just say, and the results have been striking.

We've joked that we gave her school a curious, engaged 6yo with a two hour attention span and they gave us a 14yo with a twenty second attention span and cynical, perfectionistic, avoidant outlook on life(gallows humor, obviously).

She's "learned" that the "right" answer isn't about a process, and that if at first you don't succeed, it's because there is something Very Wrong and that it is Probably With You, But Could Be With The Problem-- best to give up.


So I do think that this is partly to blame on how schools think that all children MUST learn mathematics. We should have intervened when they forced her to spend 3 y 'learning' stuff that she knew before she was even four years old. She was at one time like Marnie's DS.

Yes, this. I have a nine-year-old daughter in the same position. She and her older brother (11) have suffered significantly under a math teacher who ranks as bad among bad teachers. I'll be spending time this summer fixing what she's done to them.

Jonah, have you ever seen Hung-Hsi Wu's stuff on the Common Core? He's an emeritus professor of mathematics from UC Berkeley who was an author of the standards. He went beyond the call of duty in that regard and has created a lot of materials for teaching mathematics. IMO, his stuff is very well-suited to gifted kids. IMO, there's also a need to create learning materials based on it (the large textbook manufacturers seem to be failing in that regard).

Here are some links that may interest you and others here:

Teaching fractions according to the Common Core standards

Teaching Geometry according to the Common Core standards

Article about the philosophy behind the Common Core


More stuff here

But there are, lot of private 'Chinese' tutoring centers around where I live to choose from. I'm not convinced that the 'Chinese Math' is inherently "better" there system seems to include a lot of drill.

@ ColinsMum — Thanks for sharing your observations. The gifted children who I've worked with are unrepresentative in various ways (I teach for Art of Problem Solving) and it's good to hear what you think the situation is more broadly.

@ HowlerKarma — Thanks for sharing your experience. I'm very sorry to hear this. I think that this sort of thing is very common. Fortunately, it's not too late (even if it would have been better had things gone differently).

@ Val —  I think that broadly, school is best conceptualized as a constraint that one has to work within (assuming that homeschooling isn't an option, etc.) rather than a place to learn, and to think about time outside of school as the time for learning. Accepting this can be liberating. I was familiar with Hung-Hsi Wu, but these PDFs are new to me: thanks for pointing me to them, they'll be helpful.

@ bluemagic — I think that elementary Chinese math instruction is better than American math instruction on average in that the students actually develop computational fluency, but I certainly wouldn't recommend it as a good way for gifted children to learn math.

I gave the school maths loving advanced kid (not hugely maybe 1 to 2 years), they taught him to add using his fingers and haven't challenged him in two years. Worse than that they haven't improved his weaker subjects either. I would homes hook if I could.