What is wrong with mathematics education?

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.

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.

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.

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.

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.

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.

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

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.

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

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.

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

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.

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.

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?

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?

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.

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.

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.

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.

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.

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.

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

Though, statistics is probably more useful.

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?

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.

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.

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