There’s a big difference between being smart and being gifted in mathematics or in any other science. A student who is very good at math understand the procedure and can follow up on any theory. He can solve any problem within the limit of his math book. On the other hand, a gifted math student may give an alternative procedure that’s out of the limit of his book. The teacher might answer him by saying “yes, but you will study that after three years”. So, you see that a gifted person is quite creative and usually has visions and intuition to spot the answer before knowing the preliminary procedure.
To give you a clear picture, if you ask each student what is the sum of all the integers from 1 to 100. Assume neither know there’s a formula.
The student who is good at math adds up all the numbers and, within a few seconds, comes back with the answer (5050).
The gifted student realises you don’t have to add up all the numbers as 1+100=101, 2+99=101 and all the pairs add up to 101. So with 50 pairs of numbers x 101 each gives 5050.
The world needs both – the gifted student for when there is an easier way, and the good student for when there isn’t. The gifted person is most likely to become creative. He might invent new math theories, or new apparatus. The gifted one will always display ideas outside the limit of the curriculum. In some cases, he might come up with a mind-boggling solution that nobody ever expected.