If you are a waiter carrying a platter full of food at a fancy restaurant, the small action of releasing your grip can cause a huge mess, a lot of wasted food, and some angry customers. Small error → large consequences.
Likewise, if you are thinking about a complex problem, a small error in your chain of reasoning can lead to massively mistaken conclusions. Many math students have experienced how a sign error in a lengthy calculation can lead to a clearly wrong answer. Small error → large consequences.
Real-world problems often arise when we neglect, or fail to notice, a physical feature of our environment that will cause problems. For example, if a customer has put their purse down next to their table, the waiter might not expect it, not see the purse, trip over it, and spill the platter.
A waiter can’t take the time to consider each and every step on their way between the tables. Likewise, a thinker must put some level of trust in their intuitive “system 1” to offer up the correct thoughts to their conscious, calculating “system 2.” However, every time we introduce intuition into our chain of reasoning, we create an opportunity for error.
One way to think more accurately is to limit this opportunity for error, by explicitly specifying more of the intermediate steps in our reasoning.
By analogy, a careless waiter (perhaps a robotic waiter) might use a very simplified model of the room in order to navigate: a straight line from the kitchen to the table in question. A better waiter might model the room as being filled with “static barriers,” the tables and chairs, “moving barriers,” the people who are walking around, “pathways”, the open spaces they can move through. The waiter might also model their own physical actions, from a very unsophisticated model (“moving”) to progressively more calibrated models of motion (“moving at a certain speed,” “taking individual steps,” “placing each step carefully”).
An excellent waiter might be actively selecting which spatial and motion model to employ based on the circumstances in the restaurant. A busy restaurant is a challenge to serve because it demands more sophisticated models, while also requiring faster motion. A new waiter might have to make a tradeoff between moving more slowly than the pace calls for, in order to avoid tripping, and moving fast enough, and risk spilling somebody’s food.
This analogy also applies to intellectual thought processes. When constructing an argument, we must trade off between both the level of rigorous, step-by-step, explicit detail we put into our calculations, and the speed with which we perform those calculations. Many circumstances in life place us under time pressure, while also demanding that we perform calculations accurately. A timed math exam is analogous to a waiter being forced to show off their ability to serve a busy restaurant quickly and without spilling the food or forgetting an order.
Unfortunately, students are overincentivized for speed in two ways. First, timed exams create pressure that makes students move too quickly, abandon their careful step-by-step reasoning, and make mistakes. Second, students who want to “get through the material” and get on to something more fun—even if they want to want to learn—will resist the twin requirements of greater rigor and slower speed.
This is related to the comprehension curve, the idea that there is an optimal reading speed to maximize the rate at which bits of important information are comprehended. Maximizing the comprehension rate might require slowing the reading rate substantially.
Likewise, maximizing the rate at which we come to accurate conclusions, both at the end of a problem and in the intermediate steps, might require using slower, more rigorous reasoning, and executing each reasoning step at a slower pace. When we think in terms of a “comprehension curve,” both for reading and for calculating, we no longer can assume that doing some aspect of the work more quickly will lead to us achieving our goal more quickly.
My intuition is that people tend to work too quickly, and would benefit from building a habit of doing slower, more rigorous work whenever possible. I think that this would build more confidence with their models, allowing them to later move with sufficient accuracy even at a higher speed. I also think that it would build their confidence, since they would experience fewer errors in the early stages of learning.
I think that people fail to do this partly because of hyperbolic discounting—an excessive emphasis on the short-term payoff of “getting through things” and moving on to the immediate next step. I also think that people have competing goals, such as relaxation or appearing smart in front of others, and that these also cut against slow and rigorous reasoning.
This has implications for mental training. Mental training is about learning how to monitor and control your conscious thought processes in order to improve your intellectual powers. It is not necessarily about rejecting external tools. Mental training, for example, is perfectly compatible with the use of calculators. Only if it were for some reason very useful to multiply large numbers in your head would it make sense to learn that skill as part of mental training.
However, part of mental training is about learning how to do things in your head—to visualize, calculate, think. When we do things in our head, they are necessarily less explicit, less legible, and less rigorous than the same mental activities recorded on the page. Because we are not constrained in our mental motions by the need to write things down, it becomes easier to make mistakes.
Mental training would therefore need to be centrally about how to interface with external tools. For example, if you are solving a complex math problem, mental training wouldn’t just be about how to solve that problem in your head. Instead, it would be about identifying that aspect of the work that can benefit from doing less writing, and more internal mental thinking, and about how to then record that thought on the page and make it more explicit.
Mental training is not about rejecting external tools. It’s about treating the conscious mind itself as one of the most powerful and important tools you have, and learning how to use it both independently and in conjunction with other external tools in order to achieve desired results. It is perhaps analogous to an athlete doing drills. The drills allow the athlete to isolate and practice aspects of the game or of physical activities and motions, under the presumption that during the game itself, these will be directly useful. Likewise, doing mental practice is essentially inventing “mental drills” that are presumed to be useful, in conjunction with other activities and tools, during practical real-world problem solving.
If you are a waiter carrying a platter full of food at a fancy restaurant, the small action of releasing your grip can cause a huge mess, a lot of wasted food, and some angry customers. Small error → large consequences.
Likewise, if you are thinking about a complex problem, a small error in your chain of reasoning can lead to massively mistaken conclusions. Many math students have experienced how a sign error in a lengthy calculation can lead to a clearly wrong answer. Small error → large consequences.
Real-world problems often arise when we neglect, or fail to notice, a physical feature of our environment that will cause problems. For example, if a customer has put their purse down next to their table, the waiter might not expect it, not see the purse, trip over it, and spill the platter.
A waiter can’t take the time to consider each and every step on their way between the tables. Likewise, a thinker must put some level of trust in their intuitive “system 1” to offer up the correct thoughts to their conscious, calculating “system 2.” However, every time we introduce intuition into our chain of reasoning, we create an opportunity for error.
One way to think more accurately is to limit this opportunity for error, by explicitly specifying more of the intermediate steps in our reasoning.
By analogy, a careless waiter (perhaps a robotic waiter) might use a very simplified model of the room in order to navigate: a straight line from the kitchen to the table in question. A better waiter might model the room as being filled with “static barriers,” the tables and chairs, “moving barriers,” the people who are walking around, “pathways”, the open spaces they can move through. The waiter might also model their own physical actions, from a very unsophisticated model (“moving”) to progressively more calibrated models of motion (“moving at a certain speed,” “taking individual steps,” “placing each step carefully”).
An excellent waiter might be actively selecting which spatial and motion model to employ based on the circumstances in the restaurant. A busy restaurant is a challenge to serve because it demands more sophisticated models, while also requiring faster motion. A new waiter might have to make a tradeoff between moving more slowly than the pace calls for, in order to avoid tripping, and moving fast enough, and risk spilling somebody’s food.
This analogy also applies to intellectual thought processes. When constructing an argument, we must trade off between both the level of rigorous, step-by-step, explicit detail we put into our calculations, and the speed with which we perform those calculations. Many circumstances in life place us under time pressure, while also demanding that we perform calculations accurately. A timed math exam is analogous to a waiter being forced to show off their ability to serve a busy restaurant quickly and without spilling the food or forgetting an order.
Unfortunately, students are overincentivized for speed in two ways. First, timed exams create pressure that makes students move too quickly, abandon their careful step-by-step reasoning, and make mistakes. Second, students who want to “get through the material” and get on to something more fun—even if they want to want to learn—will resist the twin requirements of greater rigor and slower speed.
This is related to the comprehension curve, the idea that there is an optimal reading speed to maximize the rate at which bits of important information are comprehended. Maximizing the comprehension rate might require slowing the reading rate substantially.
Likewise, maximizing the rate at which we come to accurate conclusions, both at the end of a problem and in the intermediate steps, might require using slower, more rigorous reasoning, and executing each reasoning step at a slower pace. When we think in terms of a “comprehension curve,” both for reading and for calculating, we no longer can assume that doing some aspect of the work more quickly will lead to us achieving our goal more quickly.
My intuition is that people tend to work too quickly, and would benefit from building a habit of doing slower, more rigorous work whenever possible. I think that this would build more confidence with their models, allowing them to later move with sufficient accuracy even at a higher speed. I also think that it would build their confidence, since they would experience fewer errors in the early stages of learning.
I think that people fail to do this partly because of hyperbolic discounting—an excessive emphasis on the short-term payoff of “getting through things” and moving on to the immediate next step. I also think that people have competing goals, such as relaxation or appearing smart in front of others, and that these also cut against slow and rigorous reasoning.
This has implications for mental training. Mental training is about learning how to monitor and control your conscious thought processes in order to improve your intellectual powers. It is not necessarily about rejecting external tools. Mental training, for example, is perfectly compatible with the use of calculators. Only if it were for some reason very useful to multiply large numbers in your head would it make sense to learn that skill as part of mental training.
However, part of mental training is about learning how to do things in your head—to visualize, calculate, think. When we do things in our head, they are necessarily less explicit, less legible, and less rigorous than the same mental activities recorded on the page. Because we are not constrained in our mental motions by the need to write things down, it becomes easier to make mistakes.
Mental training would therefore need to be centrally about how to interface with external tools. For example, if you are solving a complex math problem, mental training wouldn’t just be about how to solve that problem in your head. Instead, it would be about identifying that aspect of the work that can benefit from doing less writing, and more internal mental thinking, and about how to then record that thought on the page and make it more explicit.
Mental training is not about rejecting external tools. It’s about treating the conscious mind itself as one of the most powerful and important tools you have, and learning how to use it both independently and in conjunction with other external tools in order to achieve desired results. It is perhaps analogous to an athlete doing drills. The drills allow the athlete to isolate and practice aspects of the game or of physical activities and motions, under the presumption that during the game itself, these will be directly useful. Likewise, doing mental practice is essentially inventing “mental drills” that are presumed to be useful, in conjunction with other activities and tools, during practical real-world problem solving.