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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...

(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.

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?

bentarm14y60

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 t... (read more)

0Morendil14y

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.)

See in context

23 What is wrong with mathematics education?

by Perplexed

20th Mar 2011

1 min read

23

This guy says that the problem is that high-school math education is structured to prepare people to learn calculus in their freshman year of college. But only a small minority of students ever takes calculus, and an even smaller minority ever uses it. And not many people ever make much use of pre-calc subjects like algebra, trig, or analytic geometry.

Instead, high-school math should be structured to prepare people to learn statistics. Probability and basic statistics, he argues, are not only more generally useful than calculus, they are also more fun.

I have to agree with him. What do the people here think?

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[-]Costanza14y110

A couple of weeks ago, somebody I know took a basic civil-service type math test as part of a local government job application. Maybe fifty candidates were seated at desks in an auditorium. Before the test began, the instruction page explained how to answer a sample question, which happened to involve multiplying two negative numbers. One jobseeker looked at the sample question with surprise, and whispered to the guy sitting next to him: "Multiply a negative and a negative? Can you even do that?"

I'm all for optimizing the educational experience of the most talented students. But "high-school math education" implies math education for the bottom half of the bell curve as well. I'd agree with the original post, as long as its understood that, for most students, we're talking about very, very basic probability and statistics. The difference between the Bayesian and frequentist approach isn't even on the table.

[-]Oscar_Cunningham14y100

People seem to be asking the question "We need to teach children mathematics, what mathematics should we teach them?"

What we should be asking is just "What should we teach them?" If we later find that we can lump some of the subjects together and call it Mathematics, so be it.

[-]LewisHall14y10

I like that you are looking at this basic question. I know when I was learning math, I would get hung up on things such as why is a negative number times a negative number equal to a positive number. And I couldn't move forward until I grasped this concept. Seeing math as concepts to be understood is a better approach than rote teaching.

Also kids need to understand that having a question is more important than knowing the answer. Einstein once said that it is "better to be amazed that to understand." Kids can go farther in life AND in their education if they feel comfortable in not knowing something and are excited about the quest of searching for the answer, even if they don't find it. Too much stress is placed in school on right and wrong answers, in my opinion.

[-]bentarm14y90

If you want good ideas for revolutionising mathematics education, you picked the wrong TED talk. Summary of Wolfram's talk: we still teach people maths as if we were living 50 years ago, when the ability to actually do this stuff on paper was useful. It would be much more sensible to be teaching them how to use computers optimally, as that's what people who use maths in real life actually do.

My summary of Wolfram's talk (which he might not endorse): if it's a choice between teaching people maths and teaching them programming, teach them programming.

[-]mutterc14y40

Speaking of Wolfram, I learned calculus (in my freshman year of college) in a then-new, Mathematica-based course called Calculus & Mathematica. (My backwoods high school had no pre-calc or calculus).

I really liked it; we were freed of the boring mechanical stuff, leaving time to better learn the underlying concepts. There were plenty of examples that non-programmer-types could copy, paste and tweak. (As a side effect, the then-shiny, new NeXTStations got me interested in then-new Linux).

It seems to still be going today, though I've not kept up with the people.

[-]Curiouskid13y00

I thank you for posting this. I took AP CS instead of calculus this year after I watched Wolfram's TED talk. I was worried that eventually I would have to learn calculus the normal way.

Do they offer online classes? How am I going to get around taking calculus the normal way?

[-]noitanigami14y70

There is a more fundamental flaw in maths eduction than calculus vs, statistics. Very few people come out of school knowing what a maths problem is. If you ask most people to identify a maths problem, they will point to the exercises in a book or a work sheet. Very, very few will point to the world or their experiences. There is no place in the current curriculum that shows people how to frame a question about the world in a way that maths can be applied. Neither calculus or statistics is useful if you don't know how to frame the problem in the first place.

[-]komponisto14y70

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.

[-]bentarm14y60

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.

[-][anonymous]14y40

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.

[-]komponisto14y10

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.

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?

[-]bentarm14y60

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.

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.

[-]komponisto14y00

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".

[-]Morendil14y20

School is not for learning lessons, it's for learning meta-lessons, if it has any purpose at all other than babysitting.

The purpose of school, many suspect, is the creation of a compliant populace.

[-]komponisto14y00

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.)

[-]bentarm14y10

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.

[-]komponisto14y20

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.

Yes, that's probably right.

[-]Morendil14y00

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.)

[-]Johnicholas14y60

The reason you should study calculus isn't to learn to integrate and differentiate; it's to overlearn basic algebra. If you are designing a proposed statistics-based calculus-replacer, it needs to give extensive experience with basic algebra.

My memory of statistics classes is fuzzy, but my impression was that it didn't give as many opportunities to practice algebra - algebra did occurr in proofs, but the proofs were beyond the freshman/sophomore level, and not routinely assigned as homework.

[-]sketerpot14y30

That's an intriguing idea, and now that you mention it, there are other classes that seem to have similar purposes. Physics gives you practice with basic calculus, and electromagnetics involves more multivariate calculus than you can shake a stick at.

[-]magfrump14y50

Can we have high school students learn number theory instead? There are things like this which are absolutely awesome.

[-]falenas10814y50

I think the problem with this is that if I wanted to do anything involving calculus I would need the background that the current type of math education gives me, whereas statistics is fairly easy to learn.

I'm for adding more statistics to regular math courses, as long as we keep most of the old math as well.

[-]bentarm14y20

I'm for adding more statistics to regular math courses, as long as we keep most of the old math as well.

Which requires teaching more maths. Holding constant the amount of maths teaching, would you be for replacing some of what currently goes on in pre-calc for more statistics?

[-]falenas10814y20

I'm not intimately familiar with the standard math curriculum, so I don't want to guess at what could be replaced without losing too much.

[-]sketerpot14y40

Euclid-style geometry could probably be compressed or skipped entirely without much loss. It's got some interesting stuff, and it's the only introduction most people will ever have to mathematical proofs, but I think the emphasis on geometry in the curriculum is largely an accident of history. I would be in favor of replacing that with basic statistics.

Although, considering how breathtakingly messed up the bulk of math instruction is, this discussion feels like redecorating the staterooms in the Titanic. Find a typical high school senior and ask them a question about basic geometry, or to solve a system of two simple linear equations, or to figure out the height of a tall building using a measuring tape and some angle measuring device and a calculator. These are all skills they will have learned at one point, and they will probably have completely forgotten. If indeed they ever truly understood any of it.

If there's a way to teach math better, we could probably get big gains there. I don't know how, though.

[-]nazgulnarsil14y40

neither. the time would be better spent on logic, both formal and informal.

[-]Richard_Kennaway14y30

So far, all of the following have been suggested as really important, either more important than the others or at least as important. In alphabetical order:

algebra
calculus
logic
probability and statistics
programming
trigonometry

And number theory got an honorable mention for being awesome.

How are people judging what is more important than what? I hope not by looking at how often they use these things themselves.

Not a month goes without my having occasion to use all of these, and 3D geometry, and other mathematical stuff as well. But as an applied mathematician and programmer, I would, and they all seem to me like just a step beyond basic arithmetic. It would be easy, too easy, for me to say that everyone should know this stuff, so I won't.

Instead, FYI and FWIW, here's the current high school curriculum in the UK for A-level maths.

[-]AlexMennen14y30

Most people never use their math education higher than arithmetic. I don't see the point of teaching people math skills that they almost certainly will never use.

I'm not quite sure I agree that high school math is focused towards preparing people for calculus as opposed to statistics. All either one of them requires is a basic understanding of algebra. However, I would agree that mathematically competent people should be encouraged more to take statistics (particularly Baysian statistics).

Also: http://www.maa.org/devlin/LockhartsLament.pdf

[-]buybuydandavis14y20

I wouldn't say statistics as much as data analysis. Statistics too often means conventional hypothesis testing, which I doubt the Bayesian Conspiracy is too hot on.

I remember one very clever fellow who designed speaker systems shared an anecdote about the company he worked in. Apparently, they had a computer program that would generate the frequency and spatial response of speaker systems. With enough use, people would develop an intuition about system design that went far beyond the usual state of design. They had a feel for the right answers that went beyond knowing how to set up the calculations.

What people largely lack is any feel for data distributions. So much of statistics are analytic methods for dealing with data that can be completed more straightforwardly with non parametric methods. Let the computers do the data crunching, and teach people to make decent estimates based on data. That's so much more useful for life, and would provide the added benefit of teaching students that the real world has better and worse answers, not right answers.

[-]anonym14y20

I think discrete math is another tragically under-represented math topic that would be of more benefit to most people than calculus. Within that broad topic, I'd single out the essentials of logic, set theory, combinatorics, and basic discrete probability: those are the sorts of things that most people, regardless of what they do for a living, could profitably use to do their jobs better from time to time and to solve the problems that day-to-day life throws at them.

[-]Chala14y20

The biggest barrier for me personally in my mathematics education was that it was supposed to be boring, it was supposed to be hard and thirdly you were forced to do an inordinate number of repititions of the same problem untill it nearly killed your brain. Nevermind that spaced repitition would be far more effective.

Before you even debate what to teach you'd better decide how to teach or you're just wasting your time.

Though I suppose my experience may not have been representative, so lesswrong was your mathematics education effective at teaching you what it was trying to teach you?

[-]Plasmon14y20

Why not teach the basics of calculus and statistics in high school? I was taught both subjects (and more) in high school, which was not at all unusual in this country (though education reforms have now reduced the amount of mathematics taught in high school).

But only a small minority of students ever takes calculus, and an even smaller minority ever uses it. And not many people ever make much use of pre-calc subjects like algebra, trig, or analytic geometry.

If you had lived in an era before literacy became common, and someone made this argument to argue against teaching literacy to everyone, what would you have answered?

[-]Alicorn14y10

Before literacy became common, were there institutions in place to educate everyone in a population on a set of topics? (I do not know the historical answer to this question, but suspect not.)

[-]DanArmak14y00

I think the answer is 'yes' simply because the best way to educate everyone is to make them literate first. The institutions that taught children to read and write were always the same ones that taught them anything else.

Historically, there were educational environments that didn't teach literacy - like vocational training for various professions - but none of those were, or could be, universal.

[-][anonymous]14y00

Religion, right?

[-]Plasmon14y00

True, the lack of infrastructure may have been a valid argument against attempting to make everyone literate.

In general, people who want any subject "X" taught in schools will argue against arguments of the form "X will only be usefull for a small fraction of students; therefore, we should teach less X" by saying that the benefits of a society in which a large fraction of people understand "X" outweigh the apparent lack of utility of "X" to most individuals. The less practical use "X" has, the more strongly "X"-proponents tend to argue this. Let's call this the anti-utilitarian heuristic.

Let me carefully distinguish between calculus as (1) "laborious integration or differentiation by hands using various techniques that are usually memorized instead of understood" as opposed to (2) "actually understanding the concept of integration and differentiation". Almost no one would deny that (2) has a large amount of practical use, and that, alas, a large amount of people lack even that. I think that a basic understanding of calculus, i.e. (2), is sufficiently useful (to individuals and to society), that we should take the anti-utilitarian heuristic seriousely.

[-]jsalvatier14y20

I am surprised that people think that currently taught statistics is fun. Bayesian statistics is really fun and satisfying, but I found classical statistics rather uninspired. I do agree that on average statistics is far more useful.

[-]lukeprog14y20

Definitely. Statistics and probability theory are more important, more widely useful, and more fun to learn than calculus.

But that's not the only problem with math education. Salman Khan has some good suggestions.

[-][anonymous]14y10

I have had the same thought, except that statistics is also taught in a lousy way to prepare people who will do publishable research, which is almost nobody. I also have the separate concern that since we teach kids new math skills every year, we never have high expectations about their ability to apply them, leading to weak problem-solving skills.

Just have everybody take a class called "Math for Being Smart" in their senior year of high school. First half is solid word problems; second half is probability and statistics, only you learn how to think about it instead of how to do a t-test by rote. Require it for graduation, and pass people on the basis of a skills checkoff instead of test average.

[-]Armok_GoB14y10

This discussion raises mind killer flags for me. Just a warning to those caught in it.

Also, there are people who can do without trig? To me that seems among the most concretely usable math there is in all kinds of situations. if you have to measure any distances or angles you can't do without it and I can't think of many things where you can do without that outside of compsci.

[-]Nic_Smith14y40

Economics has calculus all over the place, but trig much more rarely.

[-]Raemon14y00

I've used trig in my adult life so far exactly twice. I felt awesome for having used it, then a little sad that years of schooling was justified by two brief moments.

[-]timtyler14y10

He claims that "the laws of nature are written in the language of calculus". If he means the actual laws, IMO, that is likely to be wrong - they are more likely to be discrete maths.

[-]calef14y00

The laws of nature that physicists actually use, however, are written in the language of Calculus. And understanding the laws in Calculus form is ridiculously useful.

Hell, everyone shouldn't just have to take Calculus, they should have to take Differential Equations as well. You cannot understand reality without calculus knowledge, or without an understanding of differential equations.

[-]timtyler14y10

The laws of nature that physicists actually use, however, are written in the language of Calculus. And understanding the laws in Calculus form is ridiculously useful.

Not compared with stats and other discrete maths. I studied calculus at college and university - and it was definitely one of the more useless things I learned. It's main use has been teaching it to other people so they can get through their own calculus maths exams.

[-]paulfchristiano14y10

Hell, everyone shouldn't just have to take Calculus, they should have to take Differential Equations as well. You cannot understand reality without calculus knowledge, or without an understanding of differential equations.

Really? Understanding reality arguably requires understanding the notion of differential equations, but understanding under what precise conditions solutions to differential equations exist, or techniques which can solve particular differential equations, doesn't really seem critical to someone who doesn't actually need to solve them.

[-]Morendil14y10

I don't know as much math as I should, but I often have occasion to wish that more programmers and software engineers knew about things like probability densities and calibration, because that would reduce some crucial inferential distances. I rarely wish more programmers knew more calculus.

[-]komponisto14y20

I often have occasion to wish that more programmers and software engineers knew about things like probability densities...I rarely wish more programmers knew more calculus.

Does anyone else notice the contradiction here? This is a perfect illustration of my point: a "probability density" is a function whose integral gives you the probability. In fact, not only is the very definition of the object logically dependent on calculus, but understanding why the object exists requires knowledge of measure theory (specifically the Radon-Nikodym theorem).

...which, come to think of it, is not surprising given that measure theory is required to define "probability" in the first place!

Yes, mathematical education is extremely screwed up. But the usual complaints and controversies don't even begin to get to the real issue, which is that people can go through education in mathematics without appreciating the power of abstraction or understanding the need for the ideas in their head to form a coherent logical structure.

Ironically (in view of the parent comment), the solution is probably to teach computer programming! I don't know anything about programming myself, but my impression is that this is exactly the kind of thing one needs to "get" in order to be a good programmer.

[-]Morendil14y60

I meant that at, perhaps, a more basic level. Taboo "calculus".

If you ask a programmer for an "estimate" (most often, of the time a given task will take) everyone thinks it natural to give you a single number.

That's what I did along with everybody else. It came as a shock to me the first time I stumbled across the idea of expressing estimates with degrees of confidence: that is, my 50% confidence estimate should be such that half of the time I'd be early and half of the time I'd be late. (To many a programmer, the notion of finishing early compared to an estimate is counter-intuitive in its own right.)

Still later I hit upon (more accurately, got it hammered into my head) the idea that you could represent an estimate most accurately as a distribution of frequencies of finishing a similar task in time t. Taking t as a continuous parameter you could represent the estimate as a curve.

And still later I figured out that you could interpret the same curve as a probability density for a single task.

This series of insights unlocked for me a great deal of clarity about why project planning so often went wrong, and practical ideas for doing something about it.

In all this time I don't think I learned anything about calculus that I didn't know before. I had and still have no idea what a Radon-Nikodym theorem is (but I'll check it out), and only the vaguest notion of what measure theory is.

I've rarely had occasion to wish that more programmers could recall offhand, say, what the integral of e^x is, or how to apply the product or chain rules, or how to get a Taylor series expansion.

the need for the ideas in their head to form a coherent logical structure

That's actually what turned me off math years ago. As a computer programmer I'm most comfortable with the style where you first define something before you use it, and build gradually toward higher levels of abstraction. In school this was often seriously compromised in favor of "memorize this and never mind how to construct it in the first place".

[-]atucker14y10

Calculus has shown me more mind-blowing examples of mathematical elegance than my (frequentist) Statistics class.

Though, statistics is probably more useful.

[-]Curiouskid13y00

Here's a practical application of all this debating:

I haven't taken calculus yet. I don't know what the best way to learn it is. BUT I do know that it isn't through my current school. How do you think MY math education should be?

[-]timtyler14y00

I have to agree with him. What do the people here think?

He's right at the end: calculus is outdated, maths should go digital.

[-]komponisto14y00

calculus is outdated, maths should go digital.

Calculus is digital. The conceptual reduction of the continuous to the discrete so that it could be subjected to computation was the whole frickin' point.

[-]timtyler14y00

Right - but not really. Discrete maths is what you get with computer science. Calculus is very low on the curriculum there - and rightly so. So: pushing calculus back in schools makes perfect sense.

[-]Normal_Anomaly14y00

I have taken both subjects, and statistics is far more fun. I expect to use both in real life, but I much prefer taking stats and would prefer that the general public know stats. Your mileage may easily vary, though: I prefer statistics because it involves fewer long paper-and-pencil computations, and thus fewer opportunities to make trivial mistakes.

[-]jwhendy14y00

Absolutely -- I saw that quite a bit ago and agree. He summarizes why. Plus, why wouldn't you believe a real live mathemagician?

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