Yes, like sb1201 said, the cut-off age (according to Doman’s book) for the ability to perceive quantity is actually strongest between the ages of 0 and 30 months. And yes, my son started at age 2 1/2 (at 29.5 months, practically ON the “cut-off” age) and succeeded superbly.
I do not know why this is, exactly, so I can’t give any definitive answers. Perhaps by then most children are becoming so accustomed to counting, or have been introduced to numerals (abstract symbols for quantities, such as 1,2,3 or IV, V, VI), and discontinue to simply look at numbers as wholes and start trying to figure out the individual parts, one by one (counting). Like I said, I can’t give any definitive answers on this.
My best speculation is that by the time they’re 3 most children are counting so much, they fail to see the numbers as wholes and, the area of the brain that is responsible for instantly deciphering quantity (don’t know what part, just know it exists because I have seen it done!) is about gone by lack of use.
It’s a shame the window isn’t open longer, like for reading you can be taught really at any age (although the younger, the easier, but nonetheless even adults can still be taught).
I don’t have much of an authoritative point on the exact age, only what Doman has said in his books, because I’ve never tried it with a 36-month-old, or 40-month-old, or whatever, so I don’t know exactly how long the window may last, but like I said, my son started at 2.5 and did very well. Doman also recommends that with slightly older children, they may learn the numbers 1-20.
As for math methods for teaching older kids who missed this window, I don’t have any personal recommendations, but as the old rule goes is any way you can teach a child in an honest, factual, and joyous way. I know it is possible to still attain superb mathematical results with older children, even beginning as late as four and five years old, at the Institutes kids starting this late are often enjoying trigonometry at the ages of 11 and 12. This was discussed a little bit in other posts.
I have heard that working with an abacus, with practice, can enable calculator-speed calculation, even kids who don’t have the abacus with them manipulate imaginary abacuses and can calculate huge sums instantly, no more of that “subtract the five and carry the seven” sort of slow procedures.
Here is a youtube video of kids doing these sort of calculations:
http://www.youtube.com/watch?v=EueFhYZ4HxI
I personally think that the way we traditionally (at least in the USA) teach kids mathematics gives them a handicap in ever mastering it throughout their lives: we begin by teaching them to count, which has about as much mathematical value as reciting the ABC’s or do-re-me. Then once they’re doing simple arithmetic, then begins the long, slow process of weaning them off of their fingers and getting them to do arithmetic in their heads. Then we spend from pre-K to 5th grade teaching them not a whole lot more than basic arithmetic, most kids don’t graduate from long-division until 5th grade (that’s 7 years to learn how to add, subtract, multiply, and divide!). And, most high school graduates are not very confident adding up their grocery lists in their head. It seems to me that their must be a better way, perhaps people with older children can share their experiences in teaching their preschoolers math and let us know what worked.
I know Domanmom has had great success with her son and he was right at the cut off when he started. Perhaps she will get on and share her experience.