How should mathematics be taught to non-mathematicians? (2012) • Hacker News Board

Re: How should mathematics be taught to non-mathematicians? (2012)

#71

Post by M_Grey » Sat, Oct 08, 2016, 6:23 PM UTC

wolfgke wrote: post #28

M_Grey wrote: post #13

Earlier quoted context omitted.

Plenty of advertising, car sales, consumer financial products, lotteries, etc... rely on people not having an innate understanding of math, and especially probabilities. Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college!

> Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college! It's not obvious whether this is good or bad (too many cooks spoil the broth).

The cooks are in the kitchen with their votes already, how could it be worse to have them actually know what the hell they're doing there?

Re: How should mathematics be taught to non-mathematicians? (2012)

#72

Post by kkylin » Sat, Oct 08, 2016, 6:34 PM UTC

tikhonj wrote: post #69

imgabe wrote: post #46

Do other disciplines ask similar questions? How should physics be taught to non-physicists? How should writing be taught to non-writers? How should car maintenance be taught to non-mechanics? I guess my point is, why should we teach mathematics any differently to "non-mathematicians" than we do to "mathematicians"? I mean, at the point when you're first teaching someone, how do you even know if they're a "non-mathema…

Yes, all the time. I've heard it for physics, I've heard it for CS and I've heard it for programming, all with good reason: the way you teach somebody deeply invested in your field is different than the way you teach somebody deeply invested in a different field, learning yours as a supplement. Or maybe a better way of putting it is that the way you teach somebody interested in a field in and of itself is different t…

Absolutely agree with this comment. If you get beyond the title, what Gowers is talking about specifically is "If everyone were compelled to take 2 more years of mathematics, what should it be?" It is very different from what we would teach undergraduates getting engineering degrees; it would also be very different from what we would teach math majors. If we were to replace "mathematics" by, say, "programming," it would also not be surprising that one might come up with a list of topics differing from what we would want an aspiring computer scientist to know.

I did not find anything really surprising in the first part of Gowers's essay, but I thought the list of questions was great, independent of whether or not it is really appropriate for a required course.

Re: How should mathematics be taught to non-mathematicians? (2012)

#73

Post by baby » Sat, Oct 08, 2016, 6:37 PM UTC

I think music is the best example. There are two ways of learning music:

* learning solfege

* learning how to play directly

Most great school of musics will teach you solfege first, and you will have to go through hardcore solfege classes while you start learning how to play an instrument. Some teacher will do the same, or some family will make their kid do the same.

Now I can tell you this is not fun, but this is the way to become a great musician. You need the theory, you need to know how to read that stuff like you're reading English. But this is not fun, and I've known many in my youth who gave up learning an instrument because of this.

Now if you would teach every kid to play an instrument first, and have fun with it, and actually producing music with their fingers/mouth then... they would maybe enjoy it enough to get interested into taking solfege classes and music theory later on.

Re: How should mathematics be taught to non-mathematicians? (2012)

#74

Post by kkylin » Sat, Oct 08, 2016, 7:01 PM UTC

I really like the collection of problems, but I'm not sure it is easy to teach this kind of problem solving. I, for one, would love to try someday. (I'm a mathematician teaching at a large research university in the US, and most of the courses I get to teach are not anything like this.)

I'm also not sure this can completely replace the more "traditional" way we teach math, which is not to say I don't think it has problems (there are lots). If I may make an imperfect team-sport analogy, traditional classroom teaching of mathematics is all drilling and very little scrimmage / play. These problems are sort of on the other extreme. If we are to (1) equip students with intellectual tools that they can use, and (2) convey, to at least a fraction of the students, the sense of beauty and joy that attracted many of us to mathematics in the first place, we would need a balance between the two. I get the impression that this is something like what Gowers is actually advocating, but I haven't had a chance to read all his blog posts on this topic to find out (will have to do that later)...

Re: How should mathematics be taught to non-mathematicians? (2012)

#75

Post by jimmydddd » Sat, Oct 08, 2016, 7:12 PM UTC

baby wrote: post #73

I think music is the best example. There are two ways of learning music: * learning solfege * learning how to play directly Most great school of musics will teach you solfege first, and you will have to go through hardcore solfege classes while you start learning how to play an instrument. Some teacher will do the same, or some family will make their kid do the same. Now I can tell you this is not fun, but this is th…

I agree that music is a good analogy. For example, in the US there is a guy who has "shortcut" piano courses. His point is that yes, if you aspire to be a concert pianist, then you should spend years learning scales, reading music, and playing music from classical composers. But if you just want to play a few Billy Joel or Beatles songs, you can skip all that and cut to the chase. You won't have that great "foundation" needed to be a master player, but most people don't want that in the first place.

Re: How should mathematics be taught to non-mathematicians? (2012)

#76

Post by danarmak » Sat, Oct 08, 2016, 7:12 PM UTC

yodsanklai wrote: post #68

These problems are examples of questions that can be tackled with the help of mathematical tools. They can show students some applications of mathematics, but IMHO they don't really teach mathematics. Maths have to do with statements and proofs about abstract objects. I'd rather learn about euclidean geometry than trying to figure out the number of piano tuner in manhattan (I'm sure most students will be bored either…

Do applications motivate students? Personally I would expect examples like these to de-motivate, since they make clear that the subject does not, in fact, apply to the students' idea of real life.

If you teach abstract mathematics and make vague promises they will be really useful, some students may believe you. But if you try to demonstrate how they're useful, and the best examples you can come up with are estimating the number of piano tuners in Chicago or the number of air molecules in the room, things students know they won't ever need to do in their lives, they should become less interested.

Re: How should mathematics be taught to non-mathematicians? (2012)

#77

Post by danarmak » Sat, Oct 08, 2016, 7:33 PM UTC

Fermi estimates require at least some knowledge of the quantities you're using as inputs. Are the students expected to know these, and tested on that, or are they allowed to state any values they like for the inputs? Do the teachers and the test-writers think only the process is important, or the results as well?

For example:

> How many molecules from Socrates’s last breath are in the room?

We can estimate the number of air molecules in the room by using Avogadro's number (6.022e23), the rough chemical composition of the air (roughly 80% nitrogen, element 7, and 21% oxygen, element 8), and the room's volume (which can be hard to estimate by eye unless it's a very small room).

We also need the total mass of the atmosphere; we can estimate it from the Earth's surface are and atmospheric pressure, but we need to convert 1 atm to kilos per square meter. Would an average student remember that conversion rate? I certainly don't. (Turns out that 1 atm ~ 10,330 kg/m^2).

That's quite a few constants to remember from physics and chemistry.

(I'm aware that Fermi estimates are only one of many kinds of example questions in the post.)

Re: How should mathematics be taught to non-mathematicians? (2012)

#78

Post by danarmak » Sat, Oct 08, 2016, 7:35 PM UTC

imgabe wrote: post #46

Do other disciplines ask similar questions? How should physics be taught to non-physicists? How should writing be taught to non-writers? How should car maintenance be taught to non-mechanics? I guess my point is, why should we teach mathematics any differently to "non-mathematicians" than we do to "mathematicians"? I mean, at the point when you're first teaching someone, how do you even know if they're a "non-mathema…

People keep asking this about programming all the time. There are many movements that aim to teach everyone (kids or adults) to program.

Re: How should mathematics be taught to non-mathematicians? (2012)

#79

Post by c22 » Sat, Oct 08, 2016, 7:48 PM UTC

striking wrote: post #48

az0xff wrote: post #37

Earlier quoted context omitted.

A course designed around using a particular "modern" web technology stack will have to change too often (every time it doesn't become "modern" anymore) for it to be sustainable. Imagine that in 2016 you have a course centered around using what was modern in 2006. That would inevitably happen with a course like that. I'd rather teach programming from 0 to making a really basic 2D game (be it in C++ or Python or whatev…

I love games as an introduction to programming. But then you have to teach kids how to do collisions (or physics), you have to teach them how to keep track of multiple sprites that behave the same but are in different places at the same time (I'm talking about classes, yes), and it's harder to point to a "real world" usage of game programming... which makes it harder to get your course approved. I know, teaching toda…

When I taught myself basic as a child I did so by building my own single-user dungeon. In my first c class in high school the project was to create a playable terminal-based hangman game. Neither of these engaging projects required me to learn anything about physics or sprites. Though I did build an ascii-animated hanging sequence for those who lost at hangman.

Re: How should mathematics be taught to non-mathematicians? (2012)

#80

Post by wolfgke » Sat, Oct 08, 2016, 8:16 PM UTC

M_Grey wrote: post #71

wolfgke wrote: post #28

Earlier quoted context omitted.

> Hell, if people understood those issues, they might even have something to say about Gerrymandering and the electoral college! It's not obvious whether this is good or bad (too many cooks spoil the broth).

The cooks are in the kitchen with their votes already, how could it be worse to have them actually know what the hell they're doing there?

Perhaps I am a little bit biased by the situation in Germany. In the past years there was an increasing number of non-voters in elections (typically people who are dissatisfied with the political system). Now in the last years a rather right-wing populist party (AfD) appeared that convinced many of these non-voters to vote for them.

Compare this a little bit to the fictional situation as if Donald Trump would bring many people who did not vote in the last presidential elections back to the ballot boxes to vote for him.