Math problems are like “katas” for rationality. The difference is that, once you’ve solved a problem once with rationality, you can solve it again much more easily from memory without engaging your rational facilities again. Therefore you don’t get the benefit from repeating the same exercises again and again.
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.
I wonder, though—maybe there are some rational skills that do benefit from repetitive practice? Overcoming bias comes to mind—even after you recognize the bias, sometimes it still takes mental energy to resist its temptation. Maybe katas could help there?
I can think of a few skills that, while not “rationality” in themselves, make it much easier to reason effectively. Numeracy is one. The innumerate can’t really see the difference between a million, a billion, a trillion, and a godzillion.
It helps to have, in memory, a set of references to compare to. For example, there are about a third of a billion people in the United States. Therefore a billion dollars is roughly $3 each, a trillion dollars is roughly $3,000 each, and a million dollars is roughly nothing (.3 cents) each.
A working knowledge of history is also helpful, as is a rough understanding of manufacturing.
I do sudokus. These are computer-generated, and of consistent difficulty. so I can’t solve them from memory. Perhaps something similar could be done for math or logic problems, or story problems where cognitive biases work against the solutions.
Perhaps, but it would surprise me if you don’t have hundreds of common sudoku patterns in your memory. Not entire puzzles, but heuristics for solving limited parts of the puzzle. That’s how humans learn. We do pattern recognition whenever possible and fall back on reason when we’re stumped. “Learning” substantially consists of developing the heuristics that allow you to perform without reason (which is slow and error-prone).
Right, and if doing computer-generated sudokus is a kata for developing the heuristics for doing sudokus, then perhaps solving computer-generated logic problems could be a kata for developing the heuristics for rationality.
I think we need to distinguish between some related things here:
Rote learning is the stuff of katas, multiplication tables, etc. It’s not rationality in itself, but reason works best if you have a lot of reliable premises.
Developing heuristics is the stuff of everyday education. Most people get years of this stuff, and it’s what makes most people as rational as they are.
Crystallized intelligence it the ability to reason by applying heuristics. Most people aren’t very good at it, which is the main limitation on education. AFAIK, we don’t know how to give people more.
Fluid intelligence is the ability to reason creatively without heuristics. It’s the closest to what I mean by “rationality”, but also the hardest to train.
Interesting. I think of heuristics as being almost the same as cognitive biases. If it helps System 1, it’s a heuristic. If it gets in the way of System 2, it’s a cognitive bias.
Not a disagreement, just an observation that we are using language differently.
I basically agree. A heuristic lets System 1 function without invoking (the much slower) System 2. We need heuristics to get through the day; we couldn’t function if we had to reason out every single behavior we implement. A bias is a heuristic when it’s dysfunctional, resulting in a poorly-chosen System 1 behavior when System 2 could give a significantly better outcome.
One barrier to rationality is that updating one’s heuristics is effortful and often kind of annoying, so we always have some outdated heuristics. The quicker things change, the worse it gets. Too much trust in one’s heuristics risks biased behavior; too little yields indecisiveness.
Math problems are like “katas” for rationality. The difference is that, once you’ve solved a problem once with rationality, you can solve it again much more easily from memory without engaging your rational facilities again. Therefore you don’t get the benefit from repeating the same exercises again and again.
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.
I wonder, though—maybe there are some rational skills that do benefit from repetitive practice? Overcoming bias comes to mind—even after you recognize the bias, sometimes it still takes mental energy to resist its temptation. Maybe katas could help there?
It’s also helpful to practice when you’re first learning something.
I can think of a few skills that, while not “rationality” in themselves, make it much easier to reason effectively. Numeracy is one. The innumerate can’t really see the difference between a million, a billion, a trillion, and a godzillion.
It helps to have, in memory, a set of references to compare to. For example, there are about a third of a billion people in the United States. Therefore a billion dollars is roughly $3 each, a trillion dollars is roughly $3,000 each, and a million dollars is roughly nothing (.3 cents) each.
A working knowledge of history is also helpful, as is a rough understanding of manufacturing.
I do sudokus. These are computer-generated, and of consistent difficulty. so I can’t solve them from memory. Perhaps something similar could be done for math or logic problems, or story problems where cognitive biases work against the solutions.
Perhaps, but it would surprise me if you don’t have hundreds of common sudoku patterns in your memory. Not entire puzzles, but heuristics for solving limited parts of the puzzle. That’s how humans learn. We do pattern recognition whenever possible and fall back on reason when we’re stumped. “Learning” substantially consists of developing the heuristics that allow you to perform without reason (which is slow and error-prone).
Right, and if doing computer-generated sudokus is a kata for developing the heuristics for doing sudokus, then perhaps solving computer-generated logic problems could be a kata for developing the heuristics for rationality.
The problem is that to be good at rationality you need to be good at interacting with the real-world with all it’s uncertainty.
I think we need to distinguish between some related things here:
Rote learning is the stuff of katas, multiplication tables, etc. It’s not rationality in itself, but reason works best if you have a lot of reliable premises.
Developing heuristics is the stuff of everyday education. Most people get years of this stuff, and it’s what makes most people as rational as they are.
Crystallized intelligence it the ability to reason by applying heuristics. Most people aren’t very good at it, which is the main limitation on education. AFAIK, we don’t know how to give people more.
Fluid intelligence is the ability to reason creatively without heuristics. It’s the closest to what I mean by “rationality”, but also the hardest to train.
Executive function includes some basic cognitive processes that govern people’s behavior. Unfortunately, it is almost entirely (86-92%) genetic.
Interesting. I think of heuristics as being almost the same as cognitive biases. If it helps System 1, it’s a heuristic. If it gets in the way of System 2, it’s a cognitive bias.
Not a disagreement, just an observation that we are using language differently.
I basically agree. A heuristic lets System 1 function without invoking (the much slower) System 2. We need heuristics to get through the day; we couldn’t function if we had to reason out every single behavior we implement. A bias is a heuristic when it’s dysfunctional, resulting in a poorly-chosen System 1 behavior when System 2 could give a significantly better outcome.
One barrier to rationality is that updating one’s heuristics is effortful and often kind of annoying, so we always have some outdated heuristics. The quicker things change, the worse it gets. Too much trust in one’s heuristics risks biased behavior; too little yields indecisiveness.