0. You assume there’s one optimal solution, and that it has already been found, i.e. no room for improvement, and that nothing changes that would affect the solution.
1. You can change the math problem. (Coming up with or finding novel new problems is also useful, but with work, an old problem can be extended. You’ve heard of the Monty Hall problem, but how does it change if you have more doors, and more time with doors being opened? If another person was playing a similar game, given the option, would you gain in expected value if switched with them?)
2. You may find your (old) solution is wrong. (A ‘solution’ you find online can be wrong.)
3. You can try to solve a problem using a different technique. (Try to find 1-the probability first. Exact solution instead of approximate, or the other. Numeric solution, functional solution (in terms of parameters or variables). Often, the set of parameters can be extended.)
4.
Therefore you don’t get the benefit from repeating the same exercises again and again.
If you’ve forgotten something, then you can learn it again. Likewise, review (though with decreasing frequency) allows retention.
Varying the problem helps, as does varying your approach to the problem. Studying math generally involves many years of working progressively complex problems. But this is different from a “kata”, which is a set of moves rigorously repeated in a specific order and invariant manner*.
Psychologically speaking, a kata functions to take a set of moves that the student consciously understands and build muscle memories that can execute the moves effectively at the sub-second timescale of a fight. Reasoning uses different cognitive systems, although once a behavior is consciously understood it is often useful to practice until mental subroutines are developed that enable quick, unreflective execution of the behavior.
*There will be some variation as the student goes from not being good at the kata to being very good at the kata, but that variation is unwanted. The better you are at the kata the less variation there is.
0. You assume there’s one optimal solution, and that it has already been found, i.e. no room for improvement, and that nothing changes that would affect the solution.
1. You can change the math problem. (Coming up with or finding novel new problems is also useful, but with work, an old problem can be extended. You’ve heard of the Monty Hall problem, but how does it change if you have more doors, and more time with doors being opened? If another person was playing a similar game, given the option, would you gain in expected value if switched with them?)
2. You may find your (old) solution is wrong. (A ‘solution’ you find online can be wrong.)
3. You can try to solve a problem using a different technique. (Try to find 1-the probability first. Exact solution instead of approximate, or the other. Numeric solution, functional solution (in terms of parameters or variables). Often, the set of parameters can be extended.)
4.
If you’ve forgotten something, then you can learn it again. Likewise, review (though with decreasing frequency) allows retention.
5. (See below.)
Varying the problem helps, as does varying your approach to the problem. Studying math generally involves many years of working progressively complex problems. But this is different from a “kata”, which is a set of moves rigorously repeated in a specific order and invariant manner*.
Psychologically speaking, a kata functions to take a set of moves that the student consciously understands and build muscle memories that can execute the moves effectively at the sub-second timescale of a fight. Reasoning uses different cognitive systems, although once a behavior is consciously understood it is often useful to practice until mental subroutines are developed that enable quick, unreflective execution of the behavior.
*There will be some variation as the student goes from not being good at the kata to being very good at the kata, but that variation is unwanted. The better you are at the kata the less variation there is.