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steven0461 comments on Open Thread June 2010, Part 3 - Less Wrong

6 Post author: Kevin 14 June 2010 06:14AM

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Comment author: steven0461 13 July 2010 11:51:00PM *  6 points [-]

In mathy fields, how much of it is caused by insufficiently deep understanding and how much of it is caused by taboos against explicitly discussing intuitive ways of thinking that can't be defended as hard results? The common view seems to be that textbooks/lectures are for showing the formal structure of whatever it is you're learning, and to build intuitions you have to spend a lot of time doing exercises. But I've always thought such effort could be partly avoided if instead of playing dignified Zen master, textbooks were full of low-status sentences like "a prisoner's dilemma means two parties both have the opportunity to help the other at a cost that's smaller than the benefit, so it's basically the same thing as trade, where both parties give each other stuff that they value less than the other, so you should imagine trade as people lobbing balls of stuff at each other that grow in the lobbing, and if you zoom out it's like little fountains of stuff coming from nowhere". (ETA: I mean in addition to the math itself, of course.) It's possible that I'm overrating how much such intuitions can be shared between people, maybe because of learning-style issues.

Comment author: sketerpot 14 July 2010 01:15:59AM *  13 points [-]

I think you've got something really important here. If you want to get someone to an intuitive understanding of something, then why not go with explanations that are closer to that intuitive understanding? I usually understand such explanations a lot better than more dignified explanations, and I've seen that a lot of other people are the same way.

I remember when a classmate of mine was having trouble understanding mutexes, semaphores, monitors, and a few other low-level concurrency primitives. He had been to the lectures, read the textbook, looked it up online, and was still baffled. I described to him a restroom where people use a pot full of magic rocks to decide who can use the toilets, so they don't accidentally pee on each other. The various concurrency primitives were all explained as funny rituals for getting the magic toilet permission rocks. E.g. in one scheme people waiting for a rock stand in line; in another scheme they stand in a throng with their eyes closed, periodically flinging themselves at the pot of rocks to see if any are free. Upon hearing this, my friend's confusion was dispelled. (For my part, I didn't understand this stuff until I had translated it into vague images not too far removed from the stupid bathroom story I told my friend. The textbook explanations are just bad sometimes.)

Or for another example, I had terrible trouble with basic probability theory until I learned to imagine sets of things that could happen, and visualize them as these hazy blob things. Once that happened, it was as if my eyes had finally opened, and everything became clear. I was kind of pissed off that all the classes I'd been in that tried to teach probability focused exclusively on the equations, so I'd had to figure out the intuitive stuff without any help.

As a side-note, this is one reason why I'm optimistic about online education like Salman Khan's videos. It's not that they're inherently better, obviously, but they have the potential for much more competition. I can imagine students in The Future comparing lecturers, with the underlying assumption that you can trivially switch at any time. "Oh, you're trying to learn about the ancient Roman sumptuary laws from Danrich Parrol's lectures? Those are pretty mind-numbing; try Nile Etland's explanations instead. She presents the different points of view by arguing vehemently with herself in several funny accents. It's surprisingly clear, even if she does sound like a total nutcase."

[Side-note to the side-note: I think more things should be explained as arguments. And the natural way to do this is for one person to hold a crazy multiple-personality argument-monologue. This also works for explaining digital hardware design as a bunch of components having a conversation. "You there! I have sent you a 32-bit integer! Tell me when you're done with it!" Works like a charm.]

Man, the future of education will be silly. And more educational!

Comment author: NancyLebovitz 14 July 2010 06:12:37AM 6 points [-]

Man, the future of education will be silly. And more educational!

It wouldn't surprise me if a big part of the problem now is the assumption that there's virtue to enduring boredom, and a proof of status if you impose it.

Comment author: Matt_Simpson 14 July 2010 06:33:39AM *  2 points [-]

It wouldn't surprise me if a big part of the problem now is the assumption that there's virtue to enduring boredom

If by boredom you mean dominance and inequality, then Robin Hanson has been riffing on this theme lately. The main idea is that employers need employees who will just accept what they're told do instead of rebelling and trying to form a new tribe in a nearby section of savannah. School trains some of the rebelliousness out of students. See e.g., this, this, and this.

Comment author: NancyLebovitz 14 July 2010 11:50:07AM *  1 point [-]

No, by boredom I mean lack of appropriate levels of stimulus, and possibly lack of significant work.

Dominance and inequality can play out in a number of ways, including chaos (imagine a badly run business with employees who would like things to be more coherent), physical abuse, and deprivation. Imposed boredom is only one possibility.

Causing people to have, or feel they have, no alternatives is how abusive authorities get away with it.

Comment author: Alicorn 14 July 2010 01:19:26AM 2 points [-]

She presents the different points of view by arguing vehemently with herself in several funny accents.

That sounds like such fun!

Comment author: sketerpot 14 July 2010 01:31:03AM *  2 points [-]

It's every bit as fun as you imagine. And it works great.

Comment author: Emile 14 July 2010 07:29:26PM 0 points [-]

Heh, this reminds me of this discussion of Plain Talk on a wiki I participated in years ago. I must have drawn those little characters, what, ten years ago? Not quite (more like six or seven), but it feels like ages ago.

Comment author: Nisan 16 July 2010 04:29:55PM 1 point [-]

I agree with this. It is also true that people's intuitions differ, and people respond differently to different kinds of informal explanation. steven0461's explanation of Prisoner's Dilemma would be good for someone accustomed to thinking visually, for example. For this reason, your vision of individual explanations competing (or cooperating) is important.

Comment author: [deleted] 14 July 2010 12:27:44AM 5 points [-]

One of the things I've always disliked about mathematical culture is this taboo against making allowances for human weakness on the part of students (of any age.) For example, the reluctance to use "plain English" aids to intuition, or pictures, or motivations. Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches. But it seems to get in the way of effective communication.

You can go too far in the other direction -- I've found that it can also be hard to learn when there's too little rigorous formalism. (Most recently I've had that experience with electrical engineering and philosophy.) There ought to be a happy medium somewhere.

Comment author: JoshuaZ 14 July 2010 12:45:35AM *  4 points [-]

Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches.

This isn't really a signaling issue so much as a response to the fact that mathematicians have had centuries of experience where apparent theorems turned out to be not proven or not even true and the failings were due to too much reliance on intuition. Classical examples of this include how in the 19th century there was about a decade long period where people thought that the Four Color Theorem was proven. Also, a lot of these sorts of issues happened in calculus before it was put on a rigorous setting in the 1850s.

There may be a signaling aspect but it is likely a small one. I'd expect more likely that mathematicians err on the side of rigor.

ETA: Another data point that suggests this isn't about signaling; I've been too a fair number of talks in which people in the audience get annoyed because they think there's too much formalism hiding some basic idea in which case they'll ask questions sometimes of the form "what's the idea behind the proof" or "what's the moral of this result?"

Comment author: [deleted] 14 July 2010 02:28:28AM 4 points [-]

Just to be clear: I'm not against rigor. Rigor is there for a good reason.

But I do think that there's a bias in math against making it easy to learn. It's weird.

  1. Math departments, anecdotally in nearly all the colleges I've heard of, are terrible at administrative conveniences. Math will be the only department that doesn't put lecture notes online, won't announce the correct textbook for the course, won't produce a syllabus, won't announce the date of the final exam. Physics, computer science, etc., don't do this to their students. This has nothing to do with rigor; I think it springs from the assumption that such details are trivial.

  2. I've noticed a sort of aesthetic bias (at least in pure math) against pictures and "selling points." I recall one talk where the speaker emphasized how transformative his result could be for physics -- it was a very dramatic lecture. The gossip afterwards was all about how arrogant and salesman-like the speaker was. That cultural instinct -- to disdain flash and drama -- probably helps with rigorous habits of thought, but it ruins our chances to draw in young people and laymen. And I think it can even interfere with comprehension (people can easily miss the understated.)

Comment author: SilasBarta 14 July 2010 12:53:42AM 2 points [-]

Over 99% of students learning math aren't going to be expected to contribute to cutting-edge proofs, so I don't regard this as a good reason not to use "plain English" methods.

In any case, a plain English understanding can allow you to bootstrap to a rigorous understanding, so more hardcore mathematicians should be able to overcome any problem introduced this way.

Comment author: JoshuaZ 14 July 2010 12:56:55AM 1 point [-]

I agree that this is likely often suboptimal when teaching math. The argument I was presenting was that this approach was not due to signaling. I'm not arguing that this is at all optimal.

Comment author: SilasBarta 14 July 2010 03:37:43PM *  1 point [-]

I don't think this problem is limited to math: it's present in all cutting-edge or graduate school levels of technical subjects. Basically, if you make your work easily accessible to a lay audience[1], it's regarded as lower status or less significant. ("Hey, if it sounds so simple, it must not have been very hard to get!")

And ironically enough, this thread sprung from me complaining about exactly that (see esp. the third bullet point).

[1] And contrary to what turf-defenders like to claim, this isn't that hard. Worst case, you can just add a brief pointer to an introduction to the topic and terminology. To borrow from some open source guy, "Given enough artificial barriers to understanding, all bugs are deep."

Comment author: Douglas_Knight 14 July 2010 08:02:56AM 0 points [-]

The common view seems to be that textbooks/lectures are for showing the formal structure of whatever it is you're learning

I thought that writing was for that and lectures were supposed to be informal, the kind of thing you were asking for. And, I thought everyone agreed that lectures work much better.

Comment author: steven0461 14 July 2010 08:46:42PM *  0 points [-]

I thought that writing was for that and lectures were supposed to be informal, the kind of thing you were asking for.

I think you're right, but only to a limited (varying) degree. I also think it's not just a matter of being informal, but a matter of just stating explicitly a lot of insights that you're "supposed" to get only through hard mental labor.

Comment author: SilasBarta 14 July 2010 12:56:29AM 0 points [-]

I don't have an answer, but I can attest to not mimicking a textbook when I try to explain high school math to someone. Rather, I first find out where gap is between their understanding and where I want them to be.

Of course, textbooks don't have the luxury of probing each student's mind.