Circle geometry should be removed from the high school maths syllabus and replaced with statistics because stats is used in science, business and machine learning, while barely anyone needs circle geometry.
While I agree that circle geometry is best left for specialized elective math classes, and that some basics statistical ideas like average, variance and Bell curve can be useful for an average person, I am curious which alternatives to circle geometry you considered before settling on stats as the best candidate?
That’s a good point. There’s all kinds of things that might be worth considering adding such as programming, psychology or political philosophy. I guess my point was only that if we were going to replace it with something within maths, then stats seems to be the best candidate (at least better than any of the other content that I covered in university)
My personal take on the math of game theory is that most games are really, really simple to play. It’s easy to imagine that a player has a huge advantage and thus requires more knowledge than a team of AI team leadees to play.
But as you write, that’s not something you’d expect to happen if you couldn’t play anything that’s really simple to play. Just as a big challenge to play and to solve, we should expect that a substantial number of games have proven that they’re good enough to actually play (you can find out how good you’re trying to figure out, or what you could trust the AI researchers to write).
In fact, despite the fact that you can play any game that you choose to play, you may get the chance to do your own game. I imagine that’s not so helpful in mindlessly trying to think in words. If you want to have a game that’s going to prove it.
But I also offer a chance to write a computer game on prediction markets. I can write a game. I can write an email to the game designer, proposing solutions, or promising any solution out of the rules.
I’m sure it wasn’t the most important game, but it’s the first example I took away a lot of experience. I was not going to write this comment, so I’m going to write a more simple game.
I will publish the full logs for anyone who wants it.
If the problem is one of (x-and-x-and-x-and-x and x-and-x-and-x-and-x-and-x and y-and-y-and-x-and… I am happy to answer, as well as for others I am sure are confused by the relevant bits and may be able to retype them with math if I want to.
A good way to talk about this is to ask whether one is in the middle of a problem solving style, but that is a bit harder to communicate in words. Even if you are not in the middle of the problem solving style, you can get a pretty clean sense of the problem by going out. (There is some confusion about this, but if you haven’t already, you can read up on the paper at http://x-and-x-and-x-and-x-and-the-interrogable.
You might not be in a problem solving style (which you may or may not have, and that might be the case for many of them), but this is your opportunity to help your quest as a rationalist.
1) Do you consider circle geometry to be the most useless high school subject? How about replacing literature with statistics?
2) Even though circle geometry is rarely used directly by average adults, it’s relatively easy to grasp and helps to develop mathematical thinking. Statistics is more involved and requires some background in combinatorics and discrete math which are not covered in many schools. Do you think majority of high school students will be able to understand statistics when it’s taught instead of circle geometry?
1) That’s a good point, but I was thinking about how to improve the high school maths syllabus, not so much about high school in general. I don’t have any strong opinions on removing literature instead if it were one or the other. However, I do have other ideas for literature. I’d replace literature with a subject that is half writing/giving speeches about what students are passionate about and half reading books mostly just for participation marks. I’d have the kinds of things students currently do in literature part of an elective only.
2) p-testing is a rather mechanised process. It’s exactly the kind of thing high school is good at teaching. Basic Bayesian statistics only has one key formula (although it has another form). Even if there is a need for prerequisite units in order to prepare students, it still seems worthwhile.
Do you think that the mechanic act of plugging in numbers into formula’s is more important than the conceptual act of understanding what a statistical test actually means?
In terms of use, most people only need to know a few basic facts like “a p-value is not a probability”, which high school teachers should be able to handle. Those who seriously need statistics could cover at a higher level at university and gain the conceptual understanding there.
It seems that a lot of people who have lessons that cover students t-test come out of them believer that the p-value is the probability that the claim is true. I would expect that most students of high school classes don’t go out of the classes with a correct understanding
Rationality is not a rational belief system but it is a way of using intuition to guide anticipation. I think a lot of people are using intuition to guide their anticipation (because it’s not like they think they need intuition) but I find it a bit doubtful. It seems to me that intuition is a method for determining anticipation and anticipation in cases where the answer is “no”.
I don’t think that intuition directly helps people build intuition, but it seems to help me in thinking when trying to find things to optimize for. In my experience, intuition is a way of being more accurate when you have a choice of response.
The most important thing to recognize is that the feeling people have when you think about things is that it’s a feeling they can’t control or control. It’s not that intuition directly makes things, it’s that an intuition can’t control the feeling. If you are familiar with the concept of a feeling, then you can go ahead and build intuition for the concept as well if you are also familiar with it.
The only places where intuition is a useful tool are those where it’s a way to start your thinking on concrete problems. That’s how I’ve found that I’ve learned something like “learn the right way to approach the problem”.
Good point, I should have clarified this more. I’m not saying that people shouldn’t know how to calculate the area and circumference of a circle as people may actually use that. It’s more to do with all the things to do with tangents and chords and shapes inscribed in circles.
1. Passing tests—in a geometry class, taking the ACT, (I don’t know, maybe it’s a part of getting a GED).
2. Your interest in geometry is not merely theoretical, but practical. Maybe you construct things, perhaps out of wood using power tools. (You may find it useful to design/implement a coordinate system on a piece of wood to assist with getting the dimensions of things right, as you cut them out with a saw. Someone may have already invented this.) If you are trying to find the area under a curve, you may find it useful to buy very fine, high quality paper, graph the shape of the curve, and weight it, and use the average wight of the paper per inch or centimeter (squared) to find the answer. (This relies of the material being consistent through out, and weighing about the same everywhere.)
3. Despite your claims that you would never use math, or this part of math, someday you find yourself* designing a dome, or even a half sphere, perhaps as a place to live. The floor plan is a circle.
4. You enjoy math. You enjoy learning this/using this knowledge on puzzles to challenge your wits. (See 6)
5. You end up as a teacher, assistant, or tutor. The subject is math. (Perhaps you realize that not every geometry student that will one day teach geometry is aware of this fact.) Whether or not you learned all the fancy stuff the first time, if you didn’t retain it you have to learn it again—well enough to teach it to someone that doesn’t like the subject as much as you—and you hated geometry (class). (It was required.)
6. You learn visual calculus. Other mathematicians may compose long, elaborate arguments that they publish in papers that may take days to decipher (that seem to push the world ever closer to proofs people can’t read, but computers have apparently checked—or been used to produce). Perhaps your proofs employ no words, but consist of a picture instead, that employs esoteric knowledge (such as that of tangents and chords) to solves problems beautifully (and quickly). Perhaps this profound knowledge makes you both a better mathematician, and a better teacher of math.
In summary:
It’s arbitrary, but it’s part of the curriculum that’s tested on. (However students or teachers feel about it.)
You use the knowledge—specifically*, or generally**.
You enjoy learning, or a challenge.
Teaching.
“Higher math.” (You can see more up here, but the air is thinner. You feel kind of dizzy.)
*Your life has turned into a geometry problem. Is this hell?
**The square-cube law is general and useful. I think the second property is a result of the first.
That doesn’t mean your view can’t be correct. It’s as true as you are claiming to be. The claim is that it’s difficult to determine whether there’s actually a law of physics about how to deal with quantum mechanics.
If there wasn’t, then you would be far wrong. If there were, then either you and I would have different opinions. But what I would be proposing is a way for our disagreement about what ‘true’ means: that we should not be too confident or too skeptical about other people’s points on the theory, which could give us an overly harsh criticism, or make us look like the kind of fool who hasn’t yet accepted them yet.
I think the correct answer to this problem would be a question of how confident are we that the point being made is the correct point? It seems obvious to me that we have no idea about the nature of the dispute. If I disagree, then I think I’ll go first.
If a question is really important and it comes down to the point of people saying “I think X” then it ought to come down to the following:
“I think X is true, and therefore Y is true. If we disagree, then I don’t think X is true, and therefore Y is true.”
In this case, if we had the same thing, but also had a different conversation (as in with Mr. Lee’s comment at the end of the chapter), our disagreement could be resolved by someone else directly debating the point (we could debate the details of this argument, if they disagree).
In other words, we are all in agreement that we should be confident that we have considered the point, but it’s better to accept that we’re making a concession. But the point is that we know we shouldn’t be confident that it’s an argument that we would not be confident would work, or that we shouldn’t be confident about it.
In all cases, this is the point that it often seems to be getting.
This may seem like a pretty simple and non-obvious argument to me, but it is. And it seems the point was that there are many situations where you and some of your friends agree that the point should be resolved and that it’s reasonable to agree that the point should be fairly obvious so the disagreement seems to be a bit more complicated.
I read somewhere that there’s a norm in academia that it should never be controversial for a student to
I can’t provide an answer, but it might help you. So I am asking you to like it. I’m not sure if you want to take a look at the Wikipedia page on Circling, or in any other forums, that’s enough to give me something to think about.
What is Circling, though?
The point of Circling is to teach you about things that are fundamentally personal (me included); there’s nothing inherently wrong with that. Circling feels to teach you social skills (me included). Like any other rationality training tool, Circling will teach you how to interact with the outside world (me included). Like any other rationality training tool, Circling will teach you how to Look (and how to ask for help understanding), and how to Look (and get help with the content).
Also, circling is strongly related to self-improvement, and one of the best tools I’ve found on the internet is Circling.
So, what are some things you might try for yourself?
I’ve been reading the sequences, and so far it seems very good. If a math textbook is worth reading, I think it is.
Here’s some specific things I have taken from the sequences, all relevant to this:
The ability to calculate your points.
For instance, you might find someone who gives a much more concrete example of how to calculate a point, and some example how to calculate (e.g. a calculator).
A set of fixed point questions:
how fast would you attempt to figure out the answer if you had to read a given textbook
how long would you have to try to answer the question if you had to read a given textbook?
how long would you have to answer the question if you had to read a given textbook?
A set of fixed point questions: how fast you could estimate something if 1) and 2) the material changed in response to it
The ability to estimate something’s “true likelihood” rather than just being a guess.
The ability to calculate something’s probability
The ability to calculate something’s probability
The ability to calculate something’s expected sample
The ability to calculate something’s “expected sample”
The ability to calculate something’s “true sample”
The ability to calculate something’s “true sample”
The ability to calculate something’s “true score”
The ability to calculate something’s “true score”
The ability to calculate something’s true score
The ability to calculate anything’s true score
The ability to calculate something’s true score
The ability to estimate something’s true score
The ability to estimate something’s true true score
The ability to derive an updated probability distribution
The ability to derive an updated probability distribution
That ability to verify a set of correct conformance to another function
The ability to derive an updated probability distribution.
The ability to derive an updated probability distribution if it were a function of a function that could have been written down in the same language as mathematical proofs of the underlying mathematics: Bounded versions of formal probability theories, Bounded versions, “unknown unknowns”
The ability to construct an updated probability distribution using LBO1, b
Circle geometry should be removed from the high school maths syllabus and replaced with statistics because stats is used in science, business and machine learning, while barely anyone needs circle geometry.
While I agree that circle geometry is best left for specialized elective math classes, and that some basics statistical ideas like average, variance and Bell curve can be useful for an average person, I am curious which alternatives to circle geometry you considered before settling on stats as the best candidate?
That’s a good point. There’s all kinds of things that might be worth considering adding such as programming, psychology or political philosophy. I guess my point was only that if we were going to replace it with something within maths, then stats seems to be the best candidate (at least better than any of the other content that I covered in university)
My personal take on the math of game theory is that most games are really, really simple to play. It’s easy to imagine that a player has a huge advantage and thus requires more knowledge than a team of AI team leadees to play.
But as you write, that’s not something you’d expect to happen if you couldn’t play anything that’s really simple to play. Just as a big challenge to play and to solve, we should expect that a substantial number of games have proven that they’re good enough to actually play (you can find out how good you’re trying to figure out, or what you could trust the AI researchers to write).
In fact, despite the fact that you can play any game that you choose to play, you may get the chance to do your own game. I imagine that’s not so helpful in mindlessly trying to think in words. If you want to have a game that’s going to prove it.
But I also offer a chance to write a computer game on prediction markets. I can write a game. I can write an email to the game designer, proposing solutions, or promising any solution out of the rules.
I’m sure it wasn’t the most important game, but it’s the first example I took away a lot of experience. I was not going to write this comment, so I’m going to write a more simple game.
I will publish the full logs for anyone who wants it.
If the problem is one of (x-and-x-and-x-and-x and x-and-x-and-x-and-x-and-x and y-and-y-and-x-and… I am happy to answer, as well as for others I am sure are confused by the relevant bits and may be able to retype them with math if I want to.
A good way to talk about this is to ask whether one is in the middle of a problem solving style, but that is a bit harder to communicate in words. Even if you are not in the middle of the problem solving style, you can get a pretty clean sense of the problem by going out. (There is some confusion about this, but if you haven’t already, you can read up on the paper at http://x-and-x-and-x-and-x-and-the-interrogable.
You might not be in a problem solving style (which you may or may not have, and that might be the case for many of them), but this is your opportunity to help your quest as a rationalist.
Questions:
1) Do you consider circle geometry to be the most useless high school subject? How about replacing literature with statistics?
2) Even though circle geometry is rarely used directly by average adults, it’s relatively easy to grasp and helps to develop mathematical thinking. Statistics is more involved and requires some background in combinatorics and discrete math which are not covered in many schools. Do you think majority of high school students will be able to understand statistics when it’s taught instead of circle geometry?
1) That’s a good point, but I was thinking about how to improve the high school maths syllabus, not so much about high school in general. I don’t have any strong opinions on removing literature instead if it were one or the other. However, I do have other ideas for literature. I’d replace literature with a subject that is half writing/giving speeches about what students are passionate about and half reading books mostly just for participation marks. I’d have the kinds of things students currently do in literature part of an elective only.
2) p-testing is a rather mechanised process. It’s exactly the kind of thing high school is good at teaching. Basic Bayesian statistics only has one key formula (although it has another form). Even if there is a need for prerequisite units in order to prepare students, it still seems worthwhile.
Do you think that the mechanic act of plugging in numbers into formula’s is more important than the conceptual act of understanding what a statistical test actually means?
In terms of use, most people only need to know a few basic facts like “a p-value is not a probability”, which high school teachers should be able to handle. Those who seriously need statistics could cover at a higher level at university and gain the conceptual understanding there.
It seems that a lot of people who have lessons that cover students t-test come out of them believer that the p-value is the probability that the claim is true. I would expect that most students of high school classes don’t go out of the classes with a correct understanding
Meta: Downvoted because this is not a question.
I think that your “solution” is the right one. I don’t think there’s any reason to believe it was.
“It’s going to be a disaster,” you say. “And it’s always a disaster.”
Rationality is not a rational belief system but it is a way of using intuition to guide anticipation. I think a lot of people are using intuition to guide their anticipation (because it’s not like they think they need intuition) but I find it a bit doubtful. It seems to me that intuition is a method for determining anticipation and anticipation in cases where the answer is “no”.
I don’t think that intuition directly helps people build intuition, but it seems to help me in thinking when trying to find things to optimize for. In my experience, intuition is a way of being more accurate when you have a choice of response.
The most important thing to recognize is that the feeling people have when you think about things is that it’s a feeling they can’t control or control. It’s not that intuition directly makes things, it’s that an intuition can’t control the feeling. If you are familiar with the concept of a feeling, then you can go ahead and build intuition for the concept as well if you are also familiar with it.
The only places where intuition is a useful tool are those where it’s a way to start your thinking on concrete problems. That’s how I’ve found that I’ve learned something like “learn the right way to approach the problem”.
What do you mean by circle geometry?
Good point, I should have clarified this more. I’m not saying that people shouldn’t know how to calculate the area and circumference of a circle as people may actually use that. It’s more to do with all the things to do with tangents and chords and shapes inscribed in circles.
Possible uses:
1. Passing tests—in a geometry class, taking the ACT, (I don’t know, maybe it’s a part of getting a GED).
2. Your interest in geometry is not merely theoretical, but practical. Maybe you construct things, perhaps out of wood using power tools. (You may find it useful to design/implement a coordinate system on a piece of wood to assist with getting the dimensions of things right, as you cut them out with a saw. Someone may have already invented this.) If you are trying to find the area under a curve, you may find it useful to buy very fine, high quality paper, graph the shape of the curve, and weight it, and use the average wight of the paper per inch or centimeter (squared) to find the answer. (This relies of the material being consistent through out, and weighing about the same everywhere.)
3. Despite your claims that you would never use math, or this part of math, someday you find yourself* designing a dome, or even a half sphere, perhaps as a place to live. The floor plan is a circle.
4. You enjoy math. You enjoy learning this/using this knowledge on puzzles to challenge your wits. (See 6)
5. You end up as a teacher, assistant, or tutor. The subject is math. (Perhaps you realize that not every geometry student that will one day teach geometry is aware of this fact.) Whether or not you learned all the fancy stuff the first time, if you didn’t retain it you have to learn it again—well enough to teach it to someone that doesn’t like the subject as much as you—and you hated geometry (class). (It was required.)
6. You learn visual calculus. Other mathematicians may compose long, elaborate arguments that they publish in papers that may take days to decipher (that seem to push the world ever closer to proofs people can’t read, but computers have apparently checked—or been used to produce). Perhaps your proofs employ no words, but consist of a picture instead, that employs esoteric knowledge (such as that of tangents and chords) to solves problems beautifully (and quickly). Perhaps this profound knowledge makes you both a better mathematician, and a better teacher of math.
In summary:
It’s arbitrary, but it’s part of the curriculum that’s tested on. (However students or teachers feel about it.)
You use the knowledge—specifically*, or generally**.
You enjoy learning, or a challenge.
Teaching.
“Higher math.” (You can see more up here, but the air is thinner. You feel kind of dizzy.)
*Your life has turned into a geometry problem. Is this hell?
**The square-cube law is general and useful. I think the second property is a result of the first.
That doesn’t mean your view can’t be correct. It’s as true as you are claiming to be. The claim is that it’s difficult to determine whether there’s actually a law of physics about how to deal with quantum mechanics.
If there wasn’t, then you would be far wrong. If there were, then either you and I would have different opinions. But what I would be proposing is a way for our disagreement about what ‘true’ means: that we should not be too confident or too skeptical about other people’s points on the theory, which could give us an overly harsh criticism, or make us look like the kind of fool who hasn’t yet accepted them yet.
I think the correct answer to this problem would be a question of how confident are we that the point being made is the correct point? It seems obvious to me that we have no idea about the nature of the dispute. If I disagree, then I think I’ll go first.
If a question is really important and it comes down to the point of people saying “I think X” then it ought to come down to the following:
“I think X is true, and therefore Y is true. If we disagree, then I don’t think X is true, and therefore Y is true.”
In this case, if we had the same thing, but also had a different conversation (as in with Mr. Lee’s comment at the end of the chapter), our disagreement could be resolved by someone else directly debating the point (we could debate the details of this argument, if they disagree).
In other words, we are all in agreement that we should be confident that we have considered the point, but it’s better to accept that we’re making a concession. But the point is that we know we shouldn’t be confident that it’s an argument that we would not be confident would work, or that we shouldn’t be confident about it.
In all cases, this is the point that it often seems to be getting.
This may seem like a pretty simple and non-obvious argument to me, but it is. And it seems the point was that there are many situations where you and some of your friends agree that the point should be resolved and that it’s reasonable to agree that the point should be fairly obvious so the disagreement seems to be a bit more complicated.
I read somewhere that there’s a norm in academia that it should never be controversial for a student to
I can’t provide an answer, but it might help you. So I am asking you to like it. I’m not sure if you want to take a look at the Wikipedia page on Circling, or in any other forums, that’s enough to give me something to think about.
What is Circling, though?
The point of Circling is to teach you about things that are fundamentally personal (me included); there’s nothing inherently wrong with that. Circling feels to teach you social skills (me included). Like any other rationality training tool, Circling will teach you how to interact with the outside world (me included). Like any other rationality training tool, Circling will teach you how to Look (and how to ask for help understanding), and how to Look (and get help with the content).
Also, circling is strongly related to self-improvement, and one of the best tools I’ve found on the internet is Circling.
So, what are some things you might try for yourself?
I’ve been reading the sequences, and so far it seems very good. If a math textbook is worth reading, I think it is.
Here’s some specific things I have taken from the sequences, all relevant to this:
The ability to calculate your points. For instance, you might find someone who gives a much more concrete example of how to calculate a point, and some example how to calculate (e.g. a calculator).
A set of fixed point questions:
how fast would you attempt to figure out the answer if you had to read a given textbook
how long would you have to try to answer the question if you had to read a given textbook?
how long would you have to answer the question if you had to read a given textbook?
A set of fixed point questions: how fast you could estimate something if 1) and 2) the material changed in response to it
The ability to estimate something’s “true likelihood” rather than just being a guess.
The ability to calculate something’s probability
The ability to calculate something’s probability
The ability to calculate something’s expected sample
The ability to calculate something’s “expected sample”
The ability to calculate something’s “true sample”
The ability to calculate something’s “true sample”
The ability to calculate something’s “true score”
The ability to calculate something’s “true score”
The ability to calculate something’s true score
The ability to calculate anything’s true score
The ability to calculate something’s true score
The ability to estimate something’s true score
The ability to estimate something’s true true score
The ability to derive an updated probability distribution
The ability to derive an updated probability distribution
That ability to verify a set of correct conformance to another function
The ability to derive an updated probability distribution.
The ability to derive an updated probability distribution if it were a function of a function that could have been written down in the same language as mathematical proofs of the underlying mathematics: Bounded versions of formal probability theories, Bounded versions, “unknown unknowns”
The ability to construct an updated probability distribution using LBO1, b