My son can do basic algebra in his head and he isn't 6 yet.... How do I best hone his natural skill at math being that he is at such a young age?
You have a very curious son, and other people, not me, would say "boy is he talented", but I don't say this, because I remember vaguely what it was like to be in his shoes. The thing he is doing is impressing YOU with social tricks, by learning whatever rote button-pushing he needs to do to make you happy about him. This is not actual knowledge, because it is very easy to impress most adults by a simple act of original thinking, since most adults have forgotten how to do it, and how easy it is to do it. This type of creativity and talent, while statistically exceptional, is not particularly exceptional for CHILDREN, they are wired for doing this.
Most children are in principle able to do what he does, as it is a simple formal structure. This is something that children are able to internalize by that age, since they are certainly able to internalize their native language grammar, to the point of understanding embedded sentences, at close to this age. The mathematical structure of language grammar is somewhat more elaborate than this level of algebra (as is learning to read).
It is a waste of his time to learn "algebra" the way it is taught in school, as these formal rules are not real mathematical knowledge, or rather, they are extremely trivial mathematical knowledge which is taught by rote. Something that might get him excited is doing multiplications and discovering the laws of prime numbers, or doing multiplications base 2 or base 16, these are interesting to children at this age, or learning geometric proofs, something Gauss could handle.
So you can draw a triangle, and imagine a man walking around the triangle, and counting how much he turns. He walks along one edge, and then turns by an angle, then along another edge, and turns by an angle, then along another edge, and turns by an angle, and he is back to where he started. Since he got back to where he started, and he is pointing the same way, the angles he turned by add up to 360 degrees, which, after some of the algebra he's already learned, implies that the angles on the interior add up to 180 degrees. This is real mathematics. It can be extended to find the sum of angles of any polygon, and then he will notice that the turning number (the sum of exterior angles) come in integer clumps of 360 (obviously, you always come back to where you started), but this integer is also equal to the number of self intersections of the curve. Sort of. You have to count the intersections in a certain signed way, because sometimes there are pairs of intersections that sort of cancel out. This leads to the fruitful 19th century concept of winding number and turning number of a curve, which are appropriate geometrical concepts for children who have no formal mathematics.
Another thing you can show him is that if you are "inside" a polygon and shoot out a line, the number of intersections of the line with the polygon is always an odd number. While if you are "outside" it's always an even number. Almost, you need to count the self-intersections in this signed way to make it work, but this introduces the concept of "generic" and "exceptional" lines.
You can hone his mathematical skills by simply teaching him more formal things, like doing careful column decimal addition, and calculating a number like sqrt(2) to a certain distance by hand. This is a very difficult calculation at his age, but he might do it, and it will give him the satisfaction of having worked a long time for a tangible answer. Long decimal calculations can be done at his age, so long as the algorithm is understood and spelled out. Unfortunately, most algorithms require Taylor series to understand why they work, and some require continued fractions. These are both excellent topics for later, but this is probably too early.
The puzzles in puzzle books are very useful here, and also the puzzles from IQ tests, and things like this. If he can master these, he will learn some new concepts and principles. From this point on, obviously, his IQ will be stratospheric, but this doesn't mean he knows anything about anything except IQ tests. But these are useful skills they are testing for, and you shouldn't deprive him of them just because they are misused by classist bigots.
Other proofs from Euclid are also appropriate, the pythagorean theorem can be proved the chinese way, or Euclid's way. But doing it Euclid's way, you should introduce Cavalieri's principle, that the act of identifying equal areas is through sliding infinitesimal segments (Euclid doesn't do this, but it's easy to imagine if you look at the proof Euclid gives). A copy of Euclid would go a long way here, and you can find it online for free, I am sure. Cavalieri's ideas are explained on Wikipedia, and are a good motivation for calculus.
The calculus of finite differences is good for children, as it is a half-way house to infinitesimal calculus.
Once he understands the Euclid theorems (and he might not, I didn't get Euclid until I was relatively old, pooh-poohing it because I knew coordinate geometry from my exposure to computer graphics), you can move on to more sophisticated geometry, and real infinitesimal calculus. For calculus, it is important to explain the concept of infinitesimal clearly and cogently, and this is not done in modern rigorous books. But you can do it in a few minutes yourself. Then the rigorous epsilon-delta proofs can come, perhaps at around age 12-13, maybe later, they require familiarity with the notion of a formal proof.
Another thing is to install a distribution of GNU/Linux on one of your home computers, and plonk him down in front of a terminal, and show him a simple language, like python. Children can easily and quickly absorb computer languages, since they are abstracted from natural language. If you go to another country, he will pick up a second language quickly, and this is equivalent mental training. This is not realistic for most people, of course, but exposing him to another language at this age can be done more easily than transplanting your family. Learning to use a computer will produce an infinite number of the most interesting kinds of mathematical problems in his head almost automatically, the moment he wants to write some nifty program for himself, they appear like cockroaches in your head, you can't avoid it except using the bug-spray they seem to have handy at school.
None of this is intended as a knock on your son. He might be the world's greatest mathematician someday. But this has nothing to do with the abilities displayed on this video, which say more about that you are a good parent, and notice and hone his mathematical skills. The ability to produce great mathematics can only be demonstrated by producing great mathematics (or first mediocre mathematics, which is all I have done personally, by the way). Great mathematicians, aside from producing great mathematics, are usually ordinary people, not superheroes. They just spend a hell of a lot of time on mathematics in the correct exploratory way.
Gauss was a child prodigy, and also a great mathematician. Galois was a prodigy and also a great mathematician. Penrose was the opposite of a prodigy and a great mathematician. Einstein was not a prodigy at all, and was a great scientist. 99% of all children showing this kind of talent do not do anything with it. Again, not a knock on your son, but the mathematics is not a magic property of a person, it's something you painstakingly evolve in your head over years and decades.
It also helps at some point to talk about the dangers of marijuana, and how it will prevent him from further growth in mathematics. Mathematics is the best anti-drug there is, since the act of doing it will reveal the mental confusion of drugs immediately and clearly.
The most important thing, after the basics are laid down, is to expose him to mathematical resources which are written by actual researchers. Arxiv is very good for this, as you can find research papers which can be read at any level, if you know calculus, algebra, some geometry, and search dilligently. But it is best to read this stuff after learning the basics yourself, so that you have a good immune system for the political nonsense in the literature, and learn to identify the pure original work.
It is also important to make sure he develops in freedom, so if he wants to be a novelist, or a professional soccer player, or a musician, or a plumber, or a stock-market analyst, you don't impose. Mathematics is so entrancing, that it doesn't need pushing, in fact, it is so addictive, that society needs to put negative motivators in place to prevent everyone from learning it.
Here's something no one else has said that I think is way more important.
For exceptionally bright children, going to school with normally developing peers is a nightmare.
I'm fairly convinced that the main point of the education system is to socialize children. From personal experience I can tell you right now that if your kid wants to talk about Shakespeare/Homer/politics/etc. with peers who are reading Dr. Seuss then he is going to be in for a very bad time.
So I'd say it's far more important to make sure that your child grows up in a supportive environment. Most bright people are also exceptionally curious. They'll chase things on their own. It's connecting with others that is hard.
Your mileage may vary but this was my personal experience. I hope he doesn't have the same.
Constant Boredom: Because the teacher has to move at the pace of the slowest student, I spent about 90% of my time not doing anything related to class. The classes where my teacher got mad at me for pulling a book out I had nothing to do which made me bored. Children do not handle boredom well. I got in trouble quite a bit.
Isolation in the classroom: Here's what happens. You finish first way ahead of everyone else. When the reasonably smart kids finish they don't want or need your help. When the other kids finish they don't want your help because you're not their friend. You're not their friend because you have nothing in common. The reasonably smart kids help them instead.
Isolation outside: My sample size was small so this may be wrong. But a lot of professors look at you oddly when you're ten or twelve and you say you want to discuss their research.
Generally being difficult and tiring people: It must feel horrible to talk to someone a third of your age and then find that they know that they know more than you. And that they're disappointed. Also the whole talking discussing a thesis one minute and then throwing a temper tantrum thing is kind of bizarre.
I'm not sure how to solve this. Skipping many grades is bad as Jessica pointed out. Magnet schools seem decent from what I've seen. Maybe starting college early (I wish my parents didn't say we don't have money to send you when I was in fifth grade, or that I had known to look around for scholarships).
The other thing I'd like to mention is to be very careful of forcing your child to do things. Or even worse making your love conditional on progress.
My best friend is a chess prodigy. He was US champion a few years ago. He doesn't play anymore. He hates that his parents forced him to practice forty hours a week. He hates that they were disappointed with any result less than first place. He dislikes his parents immensely, even to this day.
Be very careful of trying to vicariously live some sort of success through your child. Actually don't do it it's evil.
Would you like to promote this answer as well? I'm fairly new and not particularly popular, yet.
I think it boils down to mass education and "college prep" is completely useless for kids who are reading thesises and able to absorb new information in real time.
Honestly more than anything else it depends on the parents. I knew people whose parents forced them to be mathletes and that was basically their identity. And then they went to school and couldn't even be in the same league as that kid who could do square roots or cube roots of large numbers in his head. And they reacted poorly and felt worthless. Which is both unreasonable and unhealthy.
Or the kid who is modestly talented but is now at the bottom despite working his ass off. Because most of his peers have never needed anything to be explained twice. And they don't study at all so they can pass and still have lives.
There's also anti intellectualism in American culture. In other parts of the world it is cool to be intelligent.
It is easy enough to teach any child to do square roots and cube roots, and any other simple calculation. in their head: you have them find the nearest integer square/cube root, and correct with one or two terms of Taylor expansion, so for example
you think "let's see, 4^2=16, 40^2 = 1600, 1503=1600-97, so sqrt(1600-97) = 40 sqrt( 1 - 97/1600) ~ 40 sqrt( 1 - 1/16) = 40 sqrt(1 - .125/2) = 40 sqrt( 1 - .0625) ~ 40 (1 - 1/2 .0625) = 40 (1 - .03125) = 40 - 4*.3125 = 40 - 1.2520 = 38.758 good enough to 3-4 places, 38.76 done", it's a few seconds in the head.
Actually, seing it now, you see 39^2 is (40-1)^2 = 1600 - 2*40 + 1 = 1521, so you have 1521-18, and a better answer is 39 - 1/2 18/39, 39 - 9/39, 39 - 3/13, 39 - 3/(12 -1) = 39 - 3/12 + 3/144 = 39 - .25 + 1/48 = 39 - .25 + 1/(50-2) = 39 - .25 + 1/50 + 2/2500 = 39 - .25 + .02 + 1/1250 = 38.77, better.
Then you say 38.77. I did it without paper, showing the steps in the head, usually I do it in the head using these steps, jotting out a few intermediate steps. You can then go further, by adding back the correction left out, etc, etc. It's easy if you know Taylor series and have a little experience with calculation.
This is not how most prodigies do it, they learn "mathlete" tricks, which are useless. Feynman describes the proper way in his popular book "Surely Youre Joking Mr. Feynman", embedded in a funny anecdote about him and Tukey in grad school at MIT.
The kid at the bottom is usually suffering from some other problem, like not knowing how to read, or having divorced parents, or living in a bad neighborhood, or smoking marijuana. Other times, there is a spiritual, or social block, where the school system makes it impossible to learn certain things, for example, if he is in large part Native American, and the school forces him to describe the genocide of his people as a great happy event, then it is difficult for him to pass history. If mathematics class doesn't explain properly, and there are no matheamatically competent adults around, he might not get it, and have no recourse.
Whenever a normal human child is not as good as you at everything, and actually, tons better, it's a complete waste of their talent. Children should run circles around adults intellectually, that's the way it's supposed to be.
I don't think this answer should be promoted, I think you should get off your high horse and accept that you are a human being just like everybody else, with no special brain.
You are wrong: the kids who want to talk about "Shakespeare and Homer" are usually either parrotting what adults want to hear, in order to label them prodigies, or else they are doing it because this is socially accepted thing that society says makes them "smart". They have no exceptional talent. They are going through knowledge by absorbing everything uncritically, something which is easy to do when young, simply by reading tons. The kids reading Dr. Seuss are reading literature of extremely high quality, Dr. Seuss is a fantastic writer of tight verse. The measure of intelligence is originality, and that is something which comes with time, and it is lacking in the prodigy. The other children around you are just as bright as you, but don't sell out to the adult social structure by doing adult tasks, rather they are faithful to the child social structure, and this involves spiting the adult social structure and learning child things. A prodigy should realize that quickly.
Your peers were just as bored as you. School is boring for everyone.
It is ridiculous to assume that you were any smarter than the reasonably smart kids. You are just using this as a propaganda point. I know the illusion of superiority, it is ridiculous. I went to school with such prodigies, and it is easy to outperform them in mathematics, because they think that mathematics is something that just comes without effort.
Professors look at you oddly when you discuss their research without having a clue about it, but pretend that you do. This is what most child prodigies do. It is annoying as all heck, because it is impolite to tell a child that he doesn't know what he's talking about. I also had such interactions with professors of physics as a 10-12 year old child, and I remember in disputes, them being more often right than I was.
It is not horrible to talk to someone who is a third of your age who knows more, it's a welcome relief! It just never happens.
There is no solution, because there is no problem. You should just go through school with everybody else, it's a joke for everyone else too. If you are born poor, rushing through school is just a hidden form of child labor, because you end up delivering money earlier.
BTW, what's your field?
+1 for the perspective, though I think the truth is somewhere between these two extremes.