[call-to-arms] Computer-based Math Education

25th Feb 2012

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21 comments, sorted by Click to highlight new comments since: Today at 6:10 PM

If you want this project to succeed, it's probably best to JFDI. Take some small area, show how you think it should be taught. If people like it,I'm sure you could find people to help you. Even if you find enough volunteers now, it will be hard to agree on a common strategy / tools / programming languages etc. -- esp. for open-source projects, it's good to lead by example.

Well, I do get the impression that your project requires tools that do not exist yet -- so I suppose those have to be written. Anyway, if you just consider the parts that do not require programming, it would still be a good thing to create something people can look at, contribute to, etc.

It's not at all clear what the best way to do computer-based math education is (at least not to me!). I think a good starting point would be for you to create some material to show your vision. If you can convince people with that, it will be much easier to find contributors.

Some comments on the "use free stuff" argument:

- Wolfram is very good about providing copies of Mathematica to students. I think I've been offered at least five free copies of the software for various reasons.
- Many schools also have access to Mathematica or Maple or whatnot (although they presumably pay for it; but the students don't).
- It might be more useful to teach a computer-based math course that
*doesn't*rely on specific software, not even Sage. Yes, this is harder. But being able to look things up in a manual is a necessary skill anyway if you're using any of these tools. - Furthermore, basing the course on specific software threatens to turn it into an explanation of the useful commands to use, rather than of the math involved.

Wolfram is very good about providing copies of Mathematica to students. I think I've been offered at least five free copies of the software for various reasons.

Hm? The (non-time-limited) student copy I see on the website is $139.

Many schools also have access to Mathematica or Maple or whatnot (although they presumably pay for it; but the students don't).

That will be relevant when it's true in India, China, and Africa, not just in America or Europe.

Furthermore, basing the course on specific software threatens to turn it into an explanation of the useful commands to use, rather than of the math involved.

This can also be used as an argument against teaching a programming language in a CS101 course.

Of course, CS101 courses *do* occasionally fall into the "commands not concepts" pitfall, (like courses that use C or Scheme, for instance) but IMO that's still better (= more understandable for students) than teaching programming *without* teaching a language along with it. (... I think. Has that ever been tried?)

Mathematica is often given out at summer camps and math competitions and such. Not available to everyone, but then not everyone would be interested in such a textbook either. But actually I don't think this was a very good point, sorry.

However, I think the places you mention that don't have Mathematica available at schools also don't have computers available at schools, so free software would hardly be a benefit there.

As far as teaching math using software without teaching a language: I'm basing this on my experience with a few classes that *did* use such a model. Notably, there was a number theory course which asked us to do things like send each other RSA messages manually, or test numbers for primality, using an unspecified CAS.

Of course, this is not quite the same as the suggestion in the post, because nobody currently makes students compute GCDs by hand for practice (or do they?) But I think that this worked quite well at the time and so it would work quite well for a calculus course. Instead of using a CAS to do modular arithmetic with large integers, you use a CAS for derivatives and algebra. In either case, you need only learn a couple of commands, and the syntax for entering in algebraic expressions (which is probably nearly the same across all such software).

Teaching a programming language in an introductory CS course is different, I think, because you need to run (and debug) actual code. I think an alternative *does* exist: occasionally teachers will explain everything in pseudocode, and expect pseudocode back. I don't know if anyone does this for beginning CS students, though.

However, I think the places you mention that don't have Mathematica available at schools also don't have computers available at schools, so free software would hardly be a benefit there.

They do. Or rather, small tracts of them do (and that's still *millions* of students) and governments are trying to get computers to the rest of them. And *colleges* everywhere definitely have computers.

As far as teaching math using software without teaching a language: I'm basing this on my experience with a few classes that did use such a model. ... (the rest of the post)

You're right, my analogy was off. If what you care about is the final output, (like in math) you can afford to be language-agnostic. On the other hand, if your concern is writing code that *gets* some specified output... well, of course you need to teach a language.

You probably meant to link to computerbasedmath.org.

Let's build an open-source open-content one!

Most successful OSS products are a result of dogfooding. I am having difficulty figuring out how a person who still needs to learn the relevant math would be able to create anything like what you are suggesting.

Are there enough (a) mathematically literate LWers with (b) tons of free time who (c) think computer-based math education is a good idea and (d) are willing to work for free?

...Seriously? Why don't you estimate the odds of a AND b AND c AND d, as your first math exercise.

a: around 650 of 1050, or about 60%

In order of frequency, we include 366 computer scientists (32.6%), 174 people in the hard sciences (16%) 80 people in finance (7.3%), 63 people in the social sciences (5.8%), 43 people involved in AI (3.9%), 39 philosophers (3.6%), 15 mathematicians (1.5%), 14 statisticians (1.3%), 15 people involved in law (1.5%) and 5 people in medicine (.5%).

b: One dozen to several hundred

Of those people willing to admit the time they spent on Less Wrong... the mean was 21 minutes and the median was 15 minutes. There were at least a dozen people in the two to three hour range, and the winner (well, except the 40579 guy) was someone who says he spends five hours a day.

c: Probably 10%-90% (rough guess)

d: Probably 1-20% (again, a rough guess)

Lower bound:

10 with free time*60% mathematically adept*10% think it's a good idea*1% willing to work for free = .006 LWers

Upper bound:

300 with free time*60% mathematically adept*90% think it's a good idea*20% willing to work for free = 32 LWers

Anyone want to do a more detailed analysis, this didn't yield a definite result.

Most successful OSS products are a result of dogfooding. I am having difficulty figuring out how a person who still needs to learn the relevant math would be able to create anything like what you are suggesting.

Perhaps someone who wants something to teach their kids with? (At least for the lower-level stuff).

Hmm? If I squint, I can maybe sorta see what this has to do with what I was talking about, but it seems likelier to be a misunderstanding.

Did you click through to Wolfram's rant/read the part where I said *"stop teaching kids how to take derivatives; that's what MathematicaTM is for. Just teach them what a derivative is, so we can move on to more interesting problems."*?

This is definitely not about "use computers to teach maths as it's usually taught!" (what Khan Academy does quite well.) This is about "let the computers do the calculating while we do the math! ... I mean, the drudgery-free parts of math."

... I guess I need to emphasize that more.

You probably meant to link to computerbasedmath.org.

Fixed, thanks.

...Seriously? Why don't you estimate the odds of a AND b AND c AND d, as your first math exercise.

Why would I want to estimate it if I can *just ask*? If it turns out there aren't any, I start getting creative. If it turns out there are... well, so far so good, and thank sanity I didn't jump to the harder solutions even though a simpler one was available.

I am having difficulty figuring out how a person who still needs to learn the relevant math would be able to create anything like what you are suggesting.

By "open-source", I meant "using pre-existing open-source software", not "building our own open-source software". I thought the post made that clear. The aim of the possible project is to write textbooks (or maybe wiki articles?), not software.

**ETA**: Took "open-source" off the TL;DR and replaced "implementation" with "curriculum" and similar terms. People were beginning to think I was talking about software, which I wasn't.

Which means that we won't get any "math, not computation!" courses/textbooks until they can find a taker.

My statistics courses are getting close to this. We use R for almost all the computations (even in quizzes, although not the final exam unfortunately), about a third of each lecture is spent looking at R code/output. This is not at all perfect though, since most people have had no experience programming a computer or even interacting with one via a line-based interpreter.

Obviously, we'd have to use free stuff... Sage instead of Mathematica, for instance. I don't know if there's a workable open-source alternative to CDF, but that's really a secondary concern at this point. The main concern is... courses/textbooks.

This free lin-alg textbook references Sage and seems to encourage its use. Also, Sage workbooks files are similar-ish to CDF, in that they can contain interactive elements.

Are there enough (a) mathematically literate LWers with (b) tons of free time who (c) think computer-based math education is a good idea and (d) are willing to work for free?

In other words, can LW form a modern-day Nicolas Bourbaki group?

There are many other groups who would be more than qualified and possibly interested: e.g. Hacker News, some sections of Reddit, the Khan Academy.

Part of the problem with this is that software like Mathematica or even SAGE is quite opaque. So you can teach folks what a derivative is, tell them that a black box called Mathematica/SAGE will calculate derivatives for them (as long as they don't hit a bug in the implementation) but the connection between the two is lost, unlike with teaching the calculation algorithm as part of the course.

AFAICT, the best way of preserving this connection while mechanizing the calculational parts of math is to employ proof assistants such as Coq and Matita, which can extract a mathematical algorithm from a formally-verified construction, and run it on input data. However, both Coq and Matita are currently limited to rudimentary tasks, such as manipulating expressions in simple algebraic structures such as rings and fields. They are a long way from having even the equivalent of a simple CAS, partly because CAS themselves have little in the way of formal foundations--this is quite evident as soon as one tries to deal with such issues as branch cuts, real vs. complex quantities, provisos, dummy variables and the like.

In short, this idea has plenty of long-run potential but is not quite ready for prime time.

Sort of. Ideally every algorithm would yield output which is formally proven to be correct wrt. the underlying mathematics. So you'd be able to select either a portion of the algorithm or a step in the printout, and see how that relates to the mathematical theory of e.g. derivatives. This is quite feasible for CAS tasks such as simplifying expressions, solving equations, computing derivatives and integrals. It is less so for numerics and graphics, but these are not a part of traditional 'math', so little is lost: these would be studied under numerical analysis and computer graphics.

TL;DR= There doesn't exist a course/curriculum/general textbook based on Conrad Wolfram's "Computer-Based Math Education" idea. Let's create an open-content one! .... if we canBy computer-based math, I

mean "math as usual, now taught through a computer!" (a la Khan Academy) I mean "math where we let computers do the calculation drudge-work, while we do the interesting parts."don'tOr, paraphrasing Conrad Wolfram: "stop teaching kids how to take derivatives; that's what Mathematica

^{TM}is for. Just teach them what a derivative is, so we can move on to more interesting problems. Like, you know, the ones in the real world." (Here's Wolfram's original polemic about the issue.)Obviously, this is controversial, and Wolfram spends most of his talk rebutting arguments against it. If, after reading them, you're still not convinced that this is a good idea, then

start another threadto discuss it. I don't intend this thread to become a blues-vs-greens battleground. Seriously, just start another thread.On the other hand, if you

areconvinced that Wolfram is on to something...My problem with this whole venture is that it's too important (IMO) to be left to the Wolframs.

I mean, come on. Wolfram's basic thesis might be true, but it's no coincidence that this particular truth is being spouted by the brother of the guy who created Mathematica.

And, unfortunately, the Wolframs seem to be the only ones pushing for it. Which means that we won't get any "math, not computation!" courses/textbooks until they can find a taker.

Now I'm guessing that most LWers would want to reap the benefits of Wolfram's basic idea without having to pay his family a fortune for it, and before however long it takes them to convince an education board about it. (How many "How do I go about learning useful math?" threads have we had so far?)

So why don't we give the world a leg-up on the path to the widespread mathematical literacy that Wolfram promises? Why don't

weput out a computer-based math course for the world?Obviously, we'd have to use free stuff... Sage instead of Mathematica, for instance. And whatever we put out would have to be free, because... well, if you could write textbooks that people are likely to pay for, you wouldn't need to be part of an LW community venture to do it.

My major questions, therefore, are:

Are there enough (a) mathematically literate LWers with (b) tons of free time who (c) think computer-based math education is a good cause and (d) are willing to work for free toward a good cause?