Probability and basic statistics, he argues, are not only more generally useful than calculus, they are also more fun.
False dilemma. Probability and statistics involve calculus. Areas under curves, anyone?
And I’ve always found calculus more fun. Probability and statistics were about lists of data pertaining to experiments on rats, or tricky combinatorial problems that I can’t do; calculus was about cool stuff like limits and infinity. (It’s no coincidence that calculus was something I taught myself from books at age 13, and statistics was a class I flunked in school at age 17.)
Statistics was never explained to me in a way I could understand. I had a similar experience with physics. Later, I realized this was because the explanations weren’t abstract enough.
False dilemma. Probability and statistics involve calculus. Areas under curves, anyone?
Really? I don’t need to know how an engine works to drive a car. I also don’t need to know how to integrate exp(-x^2) in order to be able to check whether a variable follows a Normal distribution.
And I’ve always found calculus more fun. Probability and statistics were about lists of data pertaining to experiments on rats, or tricky combinatorial problems that I can’t do; calculus was about cool stuff like limits and infinity.
This almost certainly makes you massively abnormal (I’m abnormal in approximately the same direction). We should not be optmising a general school curriculum for weird people who think stuff like limits is cool and prefer abstract explanations to concrete ones.
Normal distributions and exp(-x^2) are sort of the exceptional case. Any reasonable study of probability and statistics will include probability density functions, which you can’t talk about at all unless you explain integrals.
Of course, exp(-x^2) is harder to integrate than most pdfs (naturally occurring or artificial) that you’d run into. I wouldn’t expect someone who learned enough calculus to understand integrals to know enough to integrate it. But before teaching someone to look values up in a table, I would want them to understand that the probabilities they’re finding are an integral of the pdf, for intuition purposes.
I agree that the year-long calculus sequence that is the norm at most colleges is probably overkill for non-mathematicians, even ones that need to know some calculus. But the basic facts of calculus enhance understanding of a whole bunch of related ideas.
I wonder if it would work well to teach a calculus class which only focused on concepts and excluded any calculation whatsoever of derivatives and integrals—given that Internet access is sufficient to integrate or differentiate any function you come across, those skills seem less relevant now.
I...don’t need to know how to integrate exp(-x^2) in order to be able to check whether a variable follows a Normal distribution.
(1) Yes you do. Seriously. You (or your computer) needs to compute some kind of (approximation to an) integral or derivative in order to do this. Or someone has to have done it for you, in which case…
This almost certainly makes you massively abnormal… We should not be optmising a general school curriculum for weird people...
That’s exactly what I was told for my whole childhood, as I was being flunked.
Yes, sanity is massively abnormal, isn’t it? So what conclusion do we draw from this? Don’t bother trying to spread sanity, and instead punish the sane ones?
Just what exactly is the optimization target here?
You (or your computer) needs to compute some kind of (approximation to an) integral or derivative in order to do this. Or someone has to have done it for you.
Well, yes, and you (or your computer) needs to be able to compute the reciprocal eigenvector of a large matrix in order to be able to use the Pagerank algorithm to search the internet. Should everyone be learning advanced scientific computing techniques and basic linear algebra before they use Google?
You are allowed to do some things without fully understanding how they work. You say elsewhere in this thread that you have no idea how programming works—does this mean you shouldn’t be allowed to alter your search engine preferences?
It is both unnecessary and undesirable for everyone to understand everything about everything—specialism works. Knowing how to compute the integrals involved in deriving a Normal distribution table is unnecessary for being able to make good use of the table, just like knowing how to compute eigenvectors is unnecessary in order to make good use of Google.
Well, yes, and you (or your computer) needs to be able to compute the reciprocal eigenvector of a large matrix in order to be able to use the Pagerank algorithm to search the internet. Should everyone be learning advanced scientific computing techniques and basic linear algebra before they use Google?
This is the car analogy again, and my point was that the car analogy fails. Unless, that is, you also think that the ability to parrot back the sentence “light is a wave” is the legitimate goal of education in physics.
School is not for learning lessons, it’s for learning meta-lessons, if it has any purpose at all other than babysitting. If for some reason someone needs to acquire the actual procedural knowledge of looking something up in a specific kind of table, they can learn it on the job. What they need to learn in school are the meta-lessons that magic doesn’t exist, curiosity is a virtue, and that they need to be wondering what parts things are made of. But if all you do is repeatedly teach them to follow sets of instructions without the appropriate intellectual context, then they will learn the exact opposite meta-lessons: that it’s okay to have magical nodes in one’s model of the world, and that they shouldn’t ask questions.
You say elsewhere in this thread that you have no idea how programming works
Not exactly. What I actually meant by “I don’t know anything about programming” was “I don’t know any programming languages, and don’t understand how instructions written in programming languages affect computer hardware.”
It is both unnecessary and undesirable for everyone to understand everything about everything
My position is not “it is desirable for everyone to understand everything about everything”. It is “if you don’t know what an integral is, you cannot understand the subject of statistics”.
But this whole discussion was clearly premised on the assumption that some other purpose might be found. (Otherwise, it doesn’t matter what the curriculum is.)
This is the car analogy again, and my point was that the car analogy fails. Unless, that is, you also think that the ability to parrot back the sentence “light is a wave” is the legitimate goal of education in physics.
I’m sorry, but this is just a total non-sequitir. Parroting back “light is a wave” without having some idea of what this predicts is not useful. Being able to make use of a computer to do basic statistical analysis which makes predictions about the real world is useful, whether or not you can compute the underlying integrals. There are skills which are useful to have in and of themselves, without fully understanding how the underlying mechanisms work, and I think it quite likely that basic statistical analysis is one of them.
On the other hand, I think we basically agree that Paul Graham’s view of compulsory education as essentially a giant creche to keep kids busy while their parents go to work is roughly accurate, so this really is a discussion abot what colour we should paint the bikeshed.
Being able to make use of a computer to do basic statistical analysis which makes predictions about the real world is useful, whether or not you can compute the underlying integrals.
Maybe it is “useful”, but it’s quite literally Artificial Arithmetic. As I’ve been arguing, I don’t consider “usefulness” in this sense to be a worthwhile purpose of education. As I said above, if a person really needs to learn this kind of ad-hoc skill, they can learn it when they actually need it.
On the other hand, I think we basically agree that Paul Graham’s view of compulsory education as essentially a giant creche to keep kids busy while their parents go to work is roughly accurate, so this really is a discussion abot what colour we should paint the bikeshed.
Just what exactly is the optimization target here?
I’m tempted to say “fun”.
(Could be an availability heuristic at work: I’m working through Smullyan’s “To Mock a Mockingbird” and having lots of fun. Still, if you make math fun, people will want more of it than if you make it dry, boring and utilitarian.)
False dilemma. Probability and statistics involve calculus. Areas under curves, anyone?
And I’ve always found calculus more fun. Probability and statistics were about lists of data pertaining to experiments on rats, or tricky combinatorial problems that I can’t do; calculus was about cool stuff like limits and infinity. (It’s no coincidence that calculus was something I taught myself from books at age 13, and statistics was a class I flunked in school at age 17.)
Statistics was never explained to me in a way I could understand. I had a similar experience with physics. Later, I realized this was because the explanations weren’t abstract enough.
Really? I don’t need to know how an engine works to drive a car. I also don’t need to know how to integrate exp(-x^2) in order to be able to check whether a variable follows a Normal distribution.
This almost certainly makes you massively abnormal (I’m abnormal in approximately the same direction). We should not be optmising a general school curriculum for weird people who think stuff like limits is cool and prefer abstract explanations to concrete ones.
Normal distributions and exp(-x^2) are sort of the exceptional case. Any reasonable study of probability and statistics will include probability density functions, which you can’t talk about at all unless you explain integrals.
Of course, exp(-x^2) is harder to integrate than most pdfs (naturally occurring or artificial) that you’d run into. I wouldn’t expect someone who learned enough calculus to understand integrals to know enough to integrate it. But before teaching someone to look values up in a table, I would want them to understand that the probabilities they’re finding are an integral of the pdf, for intuition purposes.
I agree that the year-long calculus sequence that is the norm at most colleges is probably overkill for non-mathematicians, even ones that need to know some calculus. But the basic facts of calculus enhance understanding of a whole bunch of related ideas.
I wonder if it would work well to teach a calculus class which only focused on concepts and excluded any calculation whatsoever of derivatives and integrals—given that Internet access is sufficient to integrate or differentiate any function you come across, those skills seem less relevant now.
(1) Yes you do. Seriously. You (or your computer) needs to compute some kind of (approximation to an) integral or derivative in order to do this. Or someone has to have done it for you, in which case…
(2) Review Two More Things to Unlearn from School, Fake Explanations, Guessing the Teacher’s Password, Truly Part of You, Understanding your Understanding, and numerous other LW posts in order to simmer in the idea that this way of thinking is Bad.
That’s exactly what I was told for my whole childhood, as I was being flunked.
Yes, sanity is massively abnormal, isn’t it? So what conclusion do we draw from this? Don’t bother trying to spread sanity, and instead punish the sane ones?
Just what exactly is the optimization target here?
Well, yes, and you (or your computer) needs to be able to compute the reciprocal eigenvector of a large matrix in order to be able to use the Pagerank algorithm to search the internet. Should everyone be learning advanced scientific computing techniques and basic linear algebra before they use Google?
You are allowed to do some things without fully understanding how they work. You say elsewhere in this thread that you have no idea how programming works—does this mean you shouldn’t be allowed to alter your search engine preferences?
It is both unnecessary and undesirable for everyone to understand everything about everything—specialism works. Knowing how to compute the integrals involved in deriving a Normal distribution table is unnecessary for being able to make good use of the table, just like knowing how to compute eigenvectors is unnecessary in order to make good use of Google.
This is the car analogy again, and my point was that the car analogy fails. Unless, that is, you also think that the ability to parrot back the sentence “light is a wave” is the legitimate goal of education in physics.
School is not for learning lessons, it’s for learning meta-lessons, if it has any purpose at all other than babysitting. If for some reason someone needs to acquire the actual procedural knowledge of looking something up in a specific kind of table, they can learn it on the job. What they need to learn in school are the meta-lessons that magic doesn’t exist, curiosity is a virtue, and that they need to be wondering what parts things are made of. But if all you do is repeatedly teach them to follow sets of instructions without the appropriate intellectual context, then they will learn the exact opposite meta-lessons: that it’s okay to have magical nodes in one’s model of the world, and that they shouldn’t ask questions.
Not exactly. What I actually meant by “I don’t know anything about programming” was “I don’t know any programming languages, and don’t understand how instructions written in programming languages affect computer hardware.”
My position is not “it is desirable for everyone to understand everything about everything”. It is “if you don’t know what an integral is, you cannot understand the subject of statistics”.
The purpose of school, many suspect, is the creation of a compliant populace.
I am in fact one of those many.
But this whole discussion was clearly premised on the assumption that some other purpose might be found. (Otherwise, it doesn’t matter what the curriculum is.)
I’m sorry, but this is just a total non-sequitir. Parroting back “light is a wave” without having some idea of what this predicts is not useful. Being able to make use of a computer to do basic statistical analysis which makes predictions about the real world is useful, whether or not you can compute the underlying integrals. There are skills which are useful to have in and of themselves, without fully understanding how the underlying mechanisms work, and I think it quite likely that basic statistical analysis is one of them.
On the other hand, I think we basically agree that Paul Graham’s view of compulsory education as essentially a giant creche to keep kids busy while their parents go to work is roughly accurate, so this really is a discussion abot what colour we should paint the bikeshed.
Maybe it is “useful”, but it’s quite literally Artificial Arithmetic. As I’ve been arguing, I don’t consider “usefulness” in this sense to be a worthwhile purpose of education. As I said above, if a person really needs to learn this kind of ad-hoc skill, they can learn it when they actually need it.
Yes, that’s probably right.
I’m tempted to say “fun”.
(Could be an availability heuristic at work: I’m working through Smullyan’s “To Mock a Mockingbird” and having lots of fun. Still, if you make math fun, people will want more of it than if you make it dry, boring and utilitarian.)