Re: Learning

1

In surfing, people who go out at consistent point breaks tend to surf better than people who go out at varied beach breaks. If things are too variable, you don't learn as well.

And what does that article have to do with the 10,000 hour rule?

Alternate title "Ditch the 10% rule: use 100% of your brain to learn better with interleaved practice"


Posted by: lemmy caution | Link to this comment | 04-23-14 8:00 AM
horizontal rule
2

I'm still waiting for my 10,000th hour of commenting.


Posted by: Moby Hick | Link to this comment | 04-23-14 8:03 AM
horizontal rule
3

In surfing, people who go out at consistent point breaks tend to surf better than people who go out at varied beach breaks. If things are too variable, you don't learn as well.

You're saying that if group A were handed a different point break, they'd outperform group B?


Posted by: heebie-geebie | Link to this comment | 04-23-14 8:11 AM
horizontal rule
4

2 is probably not true.


Posted by: Moby Hick | Link to this comment | 04-23-14 8:13 AM
horizontal rule
5

learning happens when you make mistakes!

Or not. "Jonah Lehrer is back with a new blog. It is well worth reading. Welcome back Jonah." wrote Malcolm Gladwell.


Posted by: Mr. Blandings | Link to this comment | 04-23-14 8:17 AM
horizontal rule
6

@1:

I used to surf growing up. The thing is that every beach has its own characteristics and quirks. So you learn the specifics of when to start paddling to catch the wave, when to stand up & etc. for a particular beach. You pretty much have to re-learn each new break that you surf.

Also, it should take less than 10,000 hours of reading Malcolm Gladwell to realize that he's full of shit.


Posted by: AcademicLurker | Link to this comment | 04-23-14 8:24 AM
horizontal rule
7

That article says absolutely nothing about what Malcolm Gladwell said about 10,000 hours. Where on earth did that headline come from?


Posted by: Sifu Tweety | Link to this comment | 04-23-14 8:27 AM
horizontal rule
8

Also, N.B. the second author of that piece is one of the preëminent researchers of the ability to implant false memories. So Heebie has sort of a complicated relationship with him, I'm thinking.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 8:31 AM
horizontal rule
9

The beanbag thing surprises me. I played tennis in high school, and the two things I practiced the hardest, serving and backhand, are the only two skills that aren't a total disaster if I'm foolish enough to try to play now. It was repetitive drilling, serve 100 deuce side, then serve 100 ad side. Repeat. I don't know what would have happened if I'd mixed it up more (maybe more shots would still be there?), but maybe their results make more sense for learning something over a relatively short time period and relatively short (as opposed to ten years) retrieval period. It seems like drilling the basics by rote isn't as bad as this excerpt implies. I mean, we can mostly all still recall products of multiplying single digit numbers, right?


Posted by: ydnew | Link to this comment | 04-23-14 8:33 AM
horizontal rule
10

The transfer of skills from one domain to another is a highly controversial area. You tend to get transfer if the things you practice span a meaningful domain, which "tossing beanbags various distances" is but "hitting a tennis ball in different ways" is maybe not. I imagine that if you repeated the experiment but instead of different distances the varied training involved tossing a beanbag overhand vs. underhand, say, you would see no transfer.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 8:36 AM
horizontal rule
11

And with the single beanbag practice, the issue could easily be that you're over-learning the specific conditions in the training phase, such that subtle differences in the testing phase (you are standing up straighter because you're slightly more tense, say) would be enough to knock you away from the specific mechanics you learned.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 8:38 AM
horizontal rule
12

There's a bunch of results in math ed that students should struggle with the material and work out connections on their own, rather than being showed one skill at a time.

Also I'm not against implanting false memories. I believe that you can convince me I was lost at the mall at age 7, even though I wasn't. What I'm disputing is that there's ever been an implanted memory that spanned years, and was supposedly a daily occurence.


Posted by: heebie-geebie | Link to this comment | 04-23-14 9:03 AM
horizontal rule
13

9: I think the recommendation from the more forward-thinking coaches would be to work on those basic skills in less-structured environments, with the corollary that by learning them by rote you're not actually building the higher-order skills you need to really do well. Here's a blog post by one of the coaches for USA Volleyball about precisely this issue.


Posted by: Josh | Link to this comment | 04-23-14 9:14 AM
horizontal rule
14

Here's another learning idea I come up against, which I strongly disagree with, for the same reasons: you must master one level before you move on to the next. Ie, you can't do calculus before you've mastered algebra, you can't move on to Cal II before you've mastered Cal 1, etc.

The part I dispute is the word "mastered". Certainly you can't move on if you haven't understood any of the previous material, but there's a large middle ground. IME, you master algebra while taking Calculus, because all of a sudden you have to retrieve it in various novel circumstances. You master Cal 1 during Cal II, because you're having to fluently use those concepts in the midst of learning something more complicated and new.


Posted by: heebie-geebie | Link to this comment | 04-23-14 10:15 AM
horizontal rule
15

In college I felt like I really needed/mastered high school trigonometry for a bunch of electrical engineering courses, even though the math prerequisites were a couple years of calculus. Now it's all a blur. Similar to mechanics - all I remember is F=ma and you can't push a rope.


Posted by: bill | Link to this comment | 04-23-14 10:28 AM
horizontal rule
16

Ie, you can't do calculus before you've mastered algebra, you can't move on to Cal II before you've mastered Cal 1, etc.

This drives me crazy. I damn-near flunked Pre-Calculus, and was asked to repeat it. But I switched schools instead, and got A's in Calculus.


Posted by: Spike | Link to this comment | 04-23-14 10:37 AM
horizontal rule
17

This is only marginally on topic, but one of the most useful things I have learned about myself in my 30s is this: I have generally excellent patience with people who are learning things for the first time. I have almost zero patience with people who are "learning" things for the third, fourth, or fifth time.*

*Assuming that they had ample, accessible, effective opportunities to learn the first X times.


Posted by: Witt | Link to this comment | 04-23-14 10:48 AM
horizontal rule
18

I have almost zero patience with people who are "learning" things for the third, fourth, or fifth time.*

And yet, you still comment at Crooked Timber

</CheapSnark>


Posted by: NickS | Link to this comment | 04-23-14 11:00 AM
horizontal rule
19

17: In theory, I kind of agree with this. Taking it literally, though, I think of myself as fairly bright, generally, and I'm lucky if I get something right the dozenth time something walks me through it. Some things are faster than others, but actually learning something solidly on the first exposure is pretty rare.


Posted by: LizardBreath | Link to this comment | 04-23-14 11:14 AM
horizontal rule
20

people who are "learning" things for the third, fourth, or fifth time

I'm not entirely clear on what this means.


Posted by: Blume | Link to this comment | 04-23-14 11:17 AM
horizontal rule
21

95% of Calc II students have mastered Calc I, and 80% of them haven't mastered Algebra.


Posted by: Unfoggetarian: "Pause endlessly, then go in." (9) | Link to this comment | 04-23-14 11:19 AM
horizontal rule
22

1. Learn how to do something

2. Forget it.

3. Learn it again.

4. Forget it.

5. I don't realize I've forgotten it, so I do it wrong.

6. Trouble ensues.

7. Learn it again.

8. Remember it for a while until I forget it again.

This is about half of my autobiography.


Posted by: peep | Link to this comment | 04-23-14 11:20 AM
horizontal rule
23

19, 20: OK, some mundane examples.

Let's say I'm asking you to find a list of legislators' phone numbers. I tell you that they are Pennsylvania state legislators. You come back and tell me they are not on the Congressional website.

I explain the difference between state and federal government. I show you the correct website to look them up.

The next week, I give you another list. I tell you that *this* one is Congresspeople.

You come back to me and say you can't find the names. I sit down with you and realize that all you "learned" the first time was "This website right, that website wrong." So instead of looking on the Congressional website, you were looking on the state one -- because you don't actually understand the difference, nor when you would use one website instead of the other.

Second example: Let's say you're relying on me to do marketing for you. I tell you that I need the date, time, and location of the event. You tell me that you want me to "do an invitation."

I explain that we can't literally send an e-mail that is exactly the same as a paper invitation. I walk you through how the software works, and show you examples of past event announcements. The next week, I send you a sample e-mail announcement and you say "'But I thought you were going to do an invitation."


Posted by: Witt | Link to this comment | 04-23-14 11:26 AM
horizontal rule
24

So either these people are kind of dumb, or they are not paying attention. Can't fix the former, but the latter is annoying.

But I agree with LB that it can take a bunch of times for something to sink in. I end up teaching a lot of people around the office how to do things with pivot tables, and I almost always end up showing the same person how to change the value field settings at least three or four times.


Posted by: Blume | Link to this comment | 04-23-14 11:42 AM
horizontal rule
25

I learn by fitting things into patterns, which means a lot of software with absolutely no rhyme or reason to what you need to do to achieve a given result is very frustrating for me. If there's some mnemonic or other hook to help remember I learn faster, but as often as not there isn't such a device.


Posted by: togolosh | Link to this comment | 04-23-14 11:52 AM
horizontal rule
26

I learn by single combat. Can't tie a shoe without breaking arms, that's what I say.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 12:06 PM
horizontal rule
27

I learned to tie my shoes shortly before 6th grade, where velcro shoes were specifically not allowed.


Posted by: fake accent | Link to this comment | 04-23-14 12:13 PM
horizontal rule
28

I'll have to sign on to 17.1. I have a tendency to become impatient with people to whom *I have explained this* and were you not paying attention or what?


Posted by: parsimon | Link to this comment | 04-23-14 12:13 PM
horizontal rule
29

My working assumption is that I have to explain something to students at least three separate times in three separate ways before it sticks. I can't remember the last time someone only had to learn something once.


Posted by: rob helpy-chalk | Link to this comment | 04-23-14 12:16 PM
horizontal rule
30

If I'm learning complicated material I like going through it more than once. Some subjects I can really just cover the same ground an unbounded number of times and figure out more each time.

Then there's perl syntax, which I am apparently constitutionally incapable of keeping in my head, such that I pretty much have to learn it from scratch every time I try to write any perl. Not sure what's up with that.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 12:20 PM
horizontal rule
31

29: What do you think is going on there, rob: are they not paying attention, and feel, perhaps, that performing attention -- pretending to attend -- is enough (as though this will never come up again)? I certainly feel as though I see some of that. Of course you yourself are presumably explaining complex philosophical concepts to students, and those might take a few tries.

I'm a very big fan of encouraging questions: ask a question if you don't actually understand! I will not think you're stupid. I will think that you want to understand.


Posted by: parsimon | Link to this comment | 04-23-14 12:23 PM
horizontal rule
32

I'm always a little confused by learning that happens almost completely unconsciously, like how I can win the doge 2048 game pretty easily now even though at first it seemed impossible, and I didn't consciously change my strategy after adapting pretty early to the put-the-high-doge-in-the-corner thing.


Posted by: essear | Link to this comment | 04-23-14 12:25 PM
horizontal rule
33

Ime, they do understand it the first time if they say they do. They're not faking. But then they forget, and can't retrieve. My experience is like helpy-chalks - often it takes three exposures for it to stick and them to be able to retrieve it.


Posted by: heebie-geebie | Link to this comment | 04-23-14 12:26 PM
horizontal rule
34

24: I almost always end up showing the same person how to change the value field settings at least three or four times.

On this sort of thing, I insist that people take notes.


Posted by: parsimon | Link to this comment | 04-23-14 12:27 PM
horizontal rule
35

30.2: Preach it, brother. I must have learned how you do nested data structures in Perl 7 different times.


Posted by: Walt Someguy | Link to this comment | 04-23-14 12:28 PM
horizontal rule
36

Also 21 is right, which is why I think we underestimate how legitimately abstract and difficult basic algebra is because we're so familiar with it.


Posted by: heebie-geebie | Link to this comment | 04-23-14 12:28 PM
horizontal rule
37

17: One of my favorite success stories when tutoring was a kid who had failed Algebra II four times (twice in HS, twice in community college), and needed it for the major he wanted to transfer into. His mom brought me in with a little over a week to go on his fourth attempt, and with intense work that week, I got him to within three points on the final of passing the course. Next semester I worked with him all semester, and he got an A in the course.

Part of that was increased motivation on his part, part of it was me being able to explain the connections between different concepts so he could see them as an integrated body of knowledge rather than just a bunch of unrelated formulas and facts he had to memorize, and part of it was working one-on-one where I was able to see where he was having trouble and focus on that part of things. But it was a pretty strong reminder that just because someone has failed to learn something multiple times doesn't mean he or she is forever unable to do so.


Posted by: Dave W. | Link to this comment | 04-23-14 12:29 PM
horizontal rule
38

are they not paying attention

My default assumption is that they are not paying attention, because that seem to me to be the most charitable. (It's my fault for being boring.) However some people very quickly give me evidence that they are paying attention, and just don't get it.


Posted by: rob helpy-chalk | Link to this comment | 04-23-14 12:33 PM
horizontal rule
39

I'm taking sort of on-line guitar lessons [it's more like a private guitar MOOC] and the practice routines are structured so that you get a task to do, which is generally pretty complex and lengthy. But it's structured so it can be easily broken down into 5 minute chunks. The idea is that you only ever spend 5 minutes a day working on, say, arpeggios. But over weeks and months, you learn a huge amount.


Posted by: nattarGcM ttaM | Link to this comment | 04-23-14 12:37 PM
horizontal rule
40

38 and previous: are you not allowing room for "some things are legitimately difficult to understand after a single explanation"?


Posted by: essear | Link to this comment | 04-23-14 12:39 PM
horizontal rule
41

10/11: Interesting, thanks.
13: Is it really a disagreement with drilling or rote? It seems more subtle than that. His argument is that drills aren't practicing the most relevant skills (with a throw to a particular position), and the players are wasting their practice time. He speaks favorably about pass-set-hit or dig-set-hit drills. The football coach in the original talks about how they do position drills, too in addition to strategy sessions and scrimmage games. I thought it was combining the sequence of skills that was the useful bit, since in a sport, you execute one motion, then the next, and drilling skills in isolation doesn't get you ready (feet, mind, whatever) to do what's required next in a game. So, when I got back onto a court after maybe ten years, it was funny that the stuff I used to be less good at (hence the drilling) was the only stuff that stuck, and the things that came more easily and could be practiced as part of a set had all atrophied.

For more intellectual pursuits, it makes sense to have mixed problem sets, since part of solving a problem is knowing what sort of problem it is, and doing a block of identical problems is just reiterating a computation. (This is in a nutshell why students think organic chemistry is really hard - there's very little straightforward problem-solving and lots of finding patterns and strategies with many types of problems to solve.) I just don't intuitively get it when I think about sports or playing an instrument, since some level of technical proficiency is important before you can get to strategy/expression/pitch. The temporal thing (cramming vs steady studying) makes intuitive sense to me, too.


Posted by: ydnew | Link to this comment | 04-23-14 12:44 PM
horizontal rule
42

40: I was. 38.last seems to indicate that rob does as well.


Posted by: parsimon | Link to this comment | 04-23-14 12:46 PM
horizontal rule
43

39: I'm taking sort of on-line guitar lessons [it's more like a private guitar MOOC]

Sounds interesting. Link?


Posted by: Spike | Link to this comment | 04-23-14 12:55 PM
horizontal rule
44

Mental whateverness, how the fuck does that work?

Not having read the link or the gladwell, I'm perfectly positioned to opine: for some things, rote learning is great. So, for example, to learn the syntax of objective-c, copying blocks of code, then trying to recreate them from memory, without worrying about what they're doing, works great. But to learn a slightly more complicated concept like, say, nested fast enumeration statements, rote copying is useless; then you have to try changing the code, breaking it, nesting in different ways, to actually "get" it. I'd guess a lot of learning follows this skill vs. concept distinction, although I'm also certain that the distinction can be made in more interesting ways.


Posted by: ogged | Link to this comment | 04-23-14 1:00 PM
horizontal rule
45

A few months ago one of the contributors at SameFacts linked to an old Peter Drucker Harvard Business Review article, from 1999: Managing Oneself I'd give a link but I've actually got it on my thumb drive, and don't know it.

Very interesting and well-written. One thing he emphasized was the importance of learning how you learn, and that people learn differently. There are "readers," "listeners," etc., and certain types predominate in certain occupations. It you're a different kind of learner from the predominant kind where you are, you have to know it and compensate.

I think I'm a reader, the way he describes Eisenhower as being. Most politicians are listeners, and giving them things to read is a waste of time.

So perhaps people you show things to, or explain things to don't learn very well that way. Some do, some don't. I know that I'm rewarded for going way out of my way to score a copy of the manual, if there is one, and often people you're dealing with don't even know there is.


Posted by: idp | Link to this comment | 04-23-14 1:17 PM
horizontal rule
46

The first half of 21 was a typo. Though I kind of like the typoed version too.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-23-14 1:29 PM
horizontal rule
47

One thing he emphasized was the importance of learning how you learn, and that people learn differently.

A lot of the discussion of learning styles is BS, especially when learning styles are linked to sensory modalities. But to the extent that it is true that people have different learning styles, those learning styles should actually be challenged by the teacher.

If someone says to me "I'm not a visual learner," I tell them that they should learn to be. You can do this by matching up what you learn one way with what you learn another. Eventually you will be able to learn several different ways. And that's real power.

But for the most part, people don't specialize in learning styles. They actually need things to be presented several different ways. The reading/listening thing is a great example. Most people need to both read something and hear it. If someone says they only learn by listening, they probably haven't been learning much at all.


Posted by: rob helpy-chalk | Link to this comment | 04-23-14 1:31 PM
horizontal rule
48

A lot All of the discussion of learning styles is BS, to a reasonably high level of certainty.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 1:33 PM
horizontal rule
49

35. @z= sort{$y{$x}{$b} <=> $y{$x}{$a}} keys %{$y{$x};


Posted by: lw | Link to this comment | 04-23-14 1:40 PM
horizontal rule
50

49 should come with a trigger warning.


Posted by: essear | Link to this comment | 04-23-14 1:43 PM
horizontal rule
51

48: When it comes from the mouth of a self-appointed expert, sure. But if someone tells me "I'm not used to learning this way," I take their word for it. And then I tell them that they need to try it and get used to it, because its good for them.


Posted by: rob helpy-chalk | Link to this comment | 04-23-14 1:45 PM
horizontal rule
52

Whoops, typo in 49.


Posted by: lw | Link to this comment | 04-23-14 1:46 PM
horizontal rule
53

46: What did you intend to write in the first half of 21?


Posted by: peep | Link to this comment | 04-23-14 1:54 PM
horizontal rule
54

I'm not a teacher, but as I remember the situation from the student side of things, it was usually a matter where I thought I understood it, and probably genuinely did understand it better than I had five minutes ago, but it turns out that's not enough. I understood it in the sense that I've got the mnemonic down, but not why it works the way it does; or in the sense that the logic of what's in front of me seems sound, but I wasn't taking the step beyond the immediate conclusion.

In hindsight, that's not the kind of thing that could have been fixed the first time I said "I understand." It probably took trying to use it to make me realize what the actual important parts were.


Posted by: Cyrus | Link to this comment | 04-23-14 1:56 PM
horizontal rule
55

No idea this was such a sore point. Your response though, reminds me of teachers I dropped, and of the famous line of Thoreau's: "If I knew for a certainty.."

I'm not, and I'm not reading Drucker as talking cognitive theory, but as a form of self-knowledge. Of course everyone has to integrate different "sources of information." Some things will only be written, some only spoken, etc.

But you can watch yourself, and compensate for a known deficiency, or just try to learn something you're struggling with differently, instead of assuming that if you don't get it as presented you never will.


Posted by: idp | Link to this comment | 04-23-14 2:29 PM
horizontal rule
56

Re: 43

Mike Outram's Electric Campfire.

http://www.electriccampfire.com

There's some free stuff on there, but the paid lessons are much longer and richer. It's relatively new, but I'm enjoying it so far. I have had a face to face lesson with him before, but this works better at the moment as I'm stuck for practice time.


Posted by: nattarGcM ttaM | Link to this comment | 04-23-14 2:35 PM
horizontal rule
57

I'm not cynical about the 10,000-hour rule because it describes quite accurately how long it took me to feel like I had mastered a set of skills and expert knowledge after I did a midlife career switch.
However, I hope it doesn't take me 10,000 hours, or even 10,000 games, to get to 2048.


Posted by: Virga•licious | Link to this comment | 04-23-14 3:23 PM
horizontal rule
58

Everyone (the article and all y'all commenters) is conflating lots of different kinds of learning. Right? Learning how to move your fingers to tie your shoe is different than memorizing times tables is different than learning how to select bits of code and string them together to do what you want.

I mean, I assume. I don't know how to do any of those things so I'm just guessing really.

The bean bag result surprised me though.


Posted by: E. Messily | Link to this comment | 04-23-14 4:10 PM
horizontal rule
59

The argument about learning styles is an awesome meta-example of the point of the article: I totally believe that people get very used to learning in one way (by reading, or listening, or whatever) and so they practice doing that all the time and are bad at the other ways. If only they would learn how to learn, then they'd be able to learn stuff.


Posted by: E. Messily | Link to this comment | 04-23-14 4:15 PM
horizontal rule
60

58.1: not me!

58.last: people have found transfer in other cases when the tasks are really close, such that they might be assumed to span a domain (e.g.). It seems like a lot to hang a theory of learning on, although I assume that in the context of the book or their scholarly work they are basing their ideas on a lot more than that.


Posted by: Sifu Tweety | Link to this comment | 04-23-14 6:00 PM
horizontal rule
61

It sort of is interesting that Roediger is working on this stuff, in that you could imagine how it would follow from his previous work on gist in memory (which is the context in which he talked about false memory implantation, e.g.).


Posted by: Sifu Tweety | Link to this comment | 04-23-14 6:02 PM
horizontal rule
62

I'm talking about learning math. Are there other kinds?


Posted by: heebie-geebie | Link to this comment | 04-23-14 7:06 PM
horizontal rule
63

From the link in the OP:

Cramming for exams is a form of massed practice. It feels like a productive strategy, and it may get you through the next day's midterm, but most of the material will be long forgotten by the time you sit down for the final.

This stood out for me because today happened to be the final for my Yup'ik class, and that was exactly my experience with studying for the midterm and final. (There was a lot of overlap between the two.) It worked out fine in both cases, but I wonder how much I'll retain in the long term.


Posted by: teofilo | Link to this comment | 04-23-14 7:28 PM
horizontal rule
64

62: Yeah, I think our British commenters probably have more than one maths.


Posted by: essear | Link to this comment | 04-23-14 7:32 PM
horizontal rule
65

I've been thinking a lot about learning (mostly maths) lately. Have been trying to work out why the jump from GCSE (compulsory course from 14-16, exam at 16) to A level (optional course over the next two years - usual structure is exams on 3 modules taken after 1 year, 3 more the second year) is so hard for many people. The first module is about 60% GCSE material (although having to think more about it) and yet still people find it very difficult. It seems you can do well, or even very well, at GCSE by just knowing what to do.

Kid A said to me yesterday that she thought that the reason some people in her class were struggling with this course was that it was the first time they'd actually had to think about it. Whereas she learnt to think about maths when she was young (being quizzed by my dad, using more interesting maths books because of being home educated) and hasn't had to make any conceptual leaps this year.

I think there is also a certain amount of it just coming naturally too, but it's interesting to see how she perceives it.


Posted by: asilon | Link to this comment | 04-24-14 12:28 AM
horizontal rule
66

I'm talking about learning math. Are there other kinds?

Forgetting it again? I'm really good at that.


Posted by: chris y | Link to this comment | 04-24-14 1:02 AM
horizontal rule
67

65: Given that the breakline is between compulsory math and optional math-just-for-smart-people, I'd tend to suspect that a chunk of it is people's self-conception as not good at math. When it's just about doing the work, they do fine. When it means doing well in a special class for the people who are really good at math, the kids who don't think of themselves that way give up faster.


Posted by: LizardBreath | Link to this comment | 04-24-14 4:28 AM
horizontal rule
68

I suspect there's a lot of truth in 65, seriously. When I was a kid I was really good at simple maths. There were in those far off days two "O" Levels, Elementary and Additional, and I was near the top of the Additional stream all the way through algebra and trigonometry. I thought I was good at it.

Then one day they started on calculus, and I literally didn't understand what they were talking about. As suddenly as that. I'm sure some of it was that the teacher was impatient or bad at explaining concepts (although a lot of other kids seemed to grasp it), but the fact is that I had cruised up to that point by having i. a gift for learning tricks, and ii. a highly developed skill for pattern recognition, and neither of these seemed to be applicable any more. I could think about other stuff, but nobody had ever told me how to think about maths (I still don't know.)


Posted by: chris y | Link to this comment | 04-24-14 4:39 AM
horizontal rule
69

Everyone (the article and all y'all commenters) is conflating lots of different kinds of learning.

I was aware I was doing it, but hoping that there were some generalities that hold across kinds of learning.

A lot of what I was thinking of was about teaching logic, which is essentially teaching math, but I was also thinking about teaching philosophy and teaching writing. I think there are meaningful commonalities here. But my knowledge is all what Kuhn called craft knowledge. I'm just generalizing from practical experience.


Posted by: rob helpy-chalk | Link to this comment | 04-24-14 4:47 AM
horizontal rule
70

Have been trying to work out why the jump from GCSE (compulsory course from 14-16, exam at 16) to A level (optional course over the next two years - usual structure is exams on 3 modules taken after 1 year, 3 more the second year) is so hard for many people.

I bet, since it's optional, the pace and difficulty ramps up because they're expecting the filtered student group to be better at math.


Posted by: heebie-geebie | Link to this comment | 04-24-14 4:55 AM
horizontal rule
71

Math is tough!


Posted by: Opinionated Barbie | Link to this comment | 04-24-14 5:17 AM
horizontal rule
72

I think a lot of people coast in subjects on natural ability and a combination of the sorts of things Chris describes in 68: pattern recognition; tricks; having a good memory, etc.

Once you hit the 'wall' where that's no longer enough, you need to work at it [in terms of putting in the effort] and make the conceptual steps required. I think the problem is, that a lot of the time the people teaching the subject may not know how best to make those conceptual steps clear, and also, how to teach people to do the work. I suspect a lot of people have just sort of muddled through themselves in the past, and can't articulate how they did it.

I know that for me, in maths, I hit the wall at 1st year university level [linear algebra, basically]. I could still do it, but it took disproportionately larger amounts of effort than advanced high school maths [calculus, basic algebra, etc] did, and I was never quite sure I was make the right conceptual connections. I'm sure I could have done it just fine, with better instruction, and a bit of time out to 're-jig' my mental map of what I was doing and why, but encountering that wall for the first time was a real fright.

With philosophy, I never really hit that wall, except to the extent that in the latter stages of graduate school I just struggled to find the required hours in the day to put in the work. But I always understood what doing that work would involve, and how to make the right connections.

I think being able to understand the steps you'd need to take to be able to do the thing, is key, and something that some people figure out naturally for themselves, some people get taught by a good teacher, and some people never get (and not necessarily because they lack the basic ability to do it).


Posted by: nattarGcM ttaM | Link to this comment | 04-24-14 5:25 AM
horizontal rule
73

Huh, I had a similar thing to 68.2. I was generally pretty good at math, made it through Calc I and II,, and just hit a wall in Calc III. That was the shift between 2D and 3D, and I just couldn't grasp what to do with the extra variable. I was seriously lucky to pass (thanks to my father's superhuman tutoring ability). It just killed any faith I had in myself to learn it. I took Differential Equations and Matrix Algebra over the summer at community college to be sure I'd pass, and math as an independent field and I never spoke again.


Posted by: ydnew | Link to this comment | 04-24-14 5:26 AM
horizontal rule
74

Mike Outram's Electric Campfire.

Thanks. I'll try the free stuff, to see if its something I can stick with.


Posted by: Spike | Link to this comment | 04-24-14 5:27 AM
horizontal rule
75

I had something of the opposite pattern to those discussed; I always did terribly in math until I got to calculus, where finally it seemed like things made coherent sense, and from then until I stopped (not much later; never took much beyond linear algebra/diff eq.) it was pretty much smooth sailing.


Posted by: Sifu Tweety | Link to this comment | 04-24-14 5:36 AM
horizontal rule
76

75: I'm not especially good at anything requiring computation. Algebra wasn't especially easy, but I liked geometry and calculus when it was only two dimensions. The final two at community college were fine, too, although it had turned into a "just need to pass" rather than trying to learn something.


Posted by: ydnew | Link to this comment | 04-24-14 5:45 AM
horizontal rule
77

I think extrapolating to higher dimensions is just something I have an okay intuition for, for whatever reason. It doesn't seem terribly correlated with anything else. So multivariable calculus seemed at the time confusingly like it was just repeating the information from calc 2. On the other hand, I nearly failed (community college) precalculus before I got there.


Posted by: Sifu Tweety | Link to this comment | 04-24-14 5:52 AM
horizontal rule
78

Then one day they started on calculus, and I literally didn't understand what they were talking about. As suddenly as that.

Yep, same here.

"Oh! Now all the facts I have to learn are arbitrary and nothing is intuitive."


Posted by: Cryptic ned | Link to this comment | 04-24-14 6:09 AM
horizontal rule
79

17: Oh God, yes. This drives me up the wall. especially the variant where the lesson that gets learned is "call Alex/Witt".


Posted by: Alex | Link to this comment | 04-24-14 6:10 AM
horizontal rule
80

I am Sifu. Or, not exactly, but on the multivariable calc thing. High school math was annoying because I'm sloppy with calculation, so I'd understand everything but get wrong answers a fair amount. Calc was fun and intuitive, and multivariable calc was exactly the same material, just do everything one more time. (I did like getting access to new notation. That was a math thing that always felt like leveling up in a video game -- you've reached level 10: triangles with arrows over them unlocked). Linear algebra was the same, but diff eq was really hard -- I got a B, but it was a painful, dumb feeling B.

And then I ran into math in physics classes that I never successfully worked through -- tensors and such. It probably would have cleared up if I'd stuck with it, but I never did.


Posted by: LizardBreath | Link to this comment | 04-24-14 6:28 AM
horizontal rule
81

Tensors are just another "do everything one more time" thing.


Posted by: essear | Link to this comment | 04-24-14 6:35 AM
horizontal rule
82

I never really hit a wall with math but I did encounter math that made me stop caring about the physics because my intuition simply failed. That was General Relativity. I developed decent intuitions about quantum and classical mechanics, but GR just left me with a sense that things are just too weird to really grok and you just have to put your faith in the math. I'm sure bigger brains than mine have no problem with it and trust that they do have genuine intuition and insight, but it's just not me.


Posted by: togolosh | Link to this comment | 04-24-14 6:39 AM
horizontal rule
83

I'm sure they are, it just never clicked. While I've forgotten enough math by now that I can't really talk about it intelligently, what I remember as my big conceptual problem was "If it's just a rectangular array, what makes it different from a matrix?"

But the real problem was that I ran into them for the first time in a class that was just independently brutal. If I'd split up the "being introduced to new math" and "doing really hard problems with it immediately" I probably would have coped okay.


Posted by: LizardBreath | Link to this comment | 04-24-14 6:43 AM
horizontal rule
84

Seconding ttaM's 72. I'm still not sure how I'm supposed to proceed when I hit the point where native intellect isn't enough to see me through. There's something called "work," but even with jokiness aside, I'm not sure what that looks like; I always quit stuff when it became difficult, and stuck with stuff that never quite got there for me, like philosophy. The programming will be an interesting test for me, since, beyond the basics, it doesn't come to me easily.


Posted by: ogged | Link to this comment | 04-24-14 6:45 AM
horizontal rule
85

Everyone hits math they find challenging at some point. Understanding Calculus is genuinely difficult. (Of course, most high school Calc classes don't require any understanding, but college classes typically ask for some.) I remember learning Calculus rather vividly as the first actually hard thing I'd ever done.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 6:50 AM
horizontal rule
86

Upetgi talks about being frustrated with his students because they'll do immense amounts of work but won't think hard, and that's familiar from the other end. When I was stuck, I wasn't thinking, I was sort of clawing stupidly at problems -- I couldn't figure out how to crack them open enough to start thinking. I don't really know what you're supposed to do then -- get help going through things step by step, until they're familiar enough to break them open on your own?


Posted by: LizardBreath | Link to this comment | 04-24-14 6:51 AM
horizontal rule
87

Speak of the devil.


Posted by: LizardBreath | Link to this comment | 04-24-14 6:51 AM
horizontal rule
88

What I really need is on-line guitar lessons in French, so I can practice two things at once.


Posted by: Spike | Link to this comment | 04-24-14 6:54 AM
horizontal rule
89

83 "If it's just a rectangular array, what makes it different from a matrix?"

The difference is that being a tensor means having certain transformation properties with respect to rotations-- or maybe to more general changes of inertial reference frame-- so that if I hand you a tensor in one coordinate system and then ask you to work in a different coordinate system, you would want to use a different matrix but it would describe the same underlying physics.

The pedagogical problem is that to really understand this, students should have to take a class on group theory as it is most often used in physics, but most people teaching physics classes are terrified of group theory so they half-ass all the explanations. Until you get to advanced enough classes, where you just assume the students have learned it at some point in an earlier class. At no point is it ever actually taught.


Posted by: essear | Link to this comment | 04-24-14 6:55 AM
horizontal rule
90

The answer to 83.1 is genuinely subtle. I think tensor products are pretty hard, and I didn't understand them until the 3rd time I learned them.

To make a feeble attempt to answer your question: If you change your basis matrices change in one way (conjugation, in particular there's an inverse in the formula) but tensors change in a different way (no inverse in the formula).


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 6:56 AM
horizontal rule
91

Then one day they started on calculus, and I literally didn't understand what they were talking about. As suddenly as that.

"Oh! Now all the facts I have to learn are arbitrary and nothing is intuitive."

This was my experience as well. I remember in particular that polynomial derivatives didn't make sense to me.

"So you take this equation, and you do this one weird trick, and presto! You've got a different equation!"

But wait! You can't just rearrange terms like that! How can that possibly work?

"Just trust me! It works!"

But why does it work?

"[Long technical explanation that is way over my head.]"

Okay . . . .


Posted by: MAE | Link to this comment | 04-24-14 6:58 AM
horizontal rule
92

Yeah, see, that's a better answer than I got from the TAs in Classical Electrodynamics (the actual answer I recall is "It's not different. It's just that this is a tensor, rather than a matrix."). Not that I understand it anymore, but it looks like something I might have understood at the time.


Posted by: LizardBreath | Link to this comment | 04-24-14 6:58 AM
horizontal rule
93

Perhaps a good example is inner products. To give an inner product also involves a square array of numbers (the individual products (v_i, v_j)), but all the formulas for inner products look like AMA^t instead of AMA^-1. This is because matrices and inner products are different kinds of tensors.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:01 AM
horizontal rule
94

Physics curricula, especially undergrad curricula, are really just broken in extraordinarily stupid ways. It's like the list of topics to cover and the order to cover them in were set in stone in 1947 and no one ever bothered to rethink it. And all the pedagogy-obsessed weirdos focus on turning the intro classes into big group labs where students are supposed to rediscover everything for themselves, or "flipped classrooms," or whatever and no one ever does anything about the non-intro classes.


Posted by: essear | Link to this comment | 04-24-14 7:04 AM
horizontal rule
95

Huh, I didn't ever actually do much with tensors, but they didn't seem conceptually difficult. "Like vectors, but with more dimensions" or more usefully, describing a transformation from one vector to another (like crystal stress/strain), which should clearly be as independent of the coordinate space as the vectors themselves are.

I kind of ran out of steam somewhere where Einstein summation started to be a regular thing, and I just didn't have enough practice or intuition with what it implied all over the place.


Posted by: Nathan Williams | Link to this comment | 04-24-14 7:05 AM
horizontal rule
96

Indices and Einstein summation convention are so useful. I think mathematicians made a huge mistake in not adopting them.


Posted by: essear | Link to this comment | 04-24-14 7:14 AM
horizontal rule
97

I'm probably going to regret this, but I'm going to disagree with Unfoggetarian and essear. Tensors are super-trivial concept that have somehow got a reputation for being hard.

Here are two things that you can represent by a rectangular array of numbers. Let (x1, x2) and (y1, y2) be two vectors. Consider this function, which results in a scalar: a x1 y1 + b x2 y1 + c x1 y2 + d x2 y2, where a, b, c, d are coefficients. You could arrange the coefficients in a rectangular array to remind yourself whether the x gets the 1 or the 2, or the y gets the 1 or the 2. (So you put a in the first row and the first column, because it's x1 and y1.)

Now consider a function which takes a vector (x, y), and gives you a new vector, (ax + by, cx + dy). Here you organize the a, b, c, d into a matrix, and the function is given by matrix multiplication.

In both cases, you can usefully organize the information into a rectangular array of numbers, and you can turn one into the other if you want, but they're not the same thing. The first one is a tensor that's not a matrix.


Posted by: Walt Someguy | Link to this comment | 04-24-14 7:14 AM
horizontal rule
98

86 is a tricky problem. The point here is that you can't really effectively teach a whole classroom. The moment when you realize you're lost what you "should" do is stop trying to learn any new material and go back to the last spot where you did understand everything and start over from there. In a week or two when you've sorted out your confusion then you can move on. But of course in an actual classroom different students hit major confusions at different points, and there's a specified curriculum that needs to be covered, so there's no good way to get that kind of pause.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:18 AM
horizontal rule
99

In mathematics, if you ever take a basis for a vector space, you are beaten. Severely. Because of that, indices on tensors fill mathematicians with fear and horror.


Posted by: Walt Someguy | Link to this comment | 04-24-14 7:18 AM
horizontal rule
100

98: obviously MOOCs are the solution.


Posted by: Sifu Tweety | Link to this comment | 04-24-14 7:19 AM
horizontal rule
101

96: I think this is partly the fault of physicists. If the notation only involved dropping the indices under the \Sigma it would have caught on, but there's no way you're ever going to sell mathematicians on dropping the \Sigma itself. That's insanity.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:19 AM
horizontal rule
102

99: Indices do not have anything to do with picking a basis. They tell you what representations the objects are in and how you're combining those representations to make a new object.


Posted by: essear | Link to this comment | 04-24-14 7:21 AM
horizontal rule
103

I don't really disagree with the content of 97. I just think "you can turn one into the other if you want, but they're not the same thing" is an extremely deep and confusing idea and tensors are one of the first places it comes up.


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:21 AM
horizontal rule
104

I actually use the Einstein summation convention every once in a while, which consists of me saying "Fuck it, I'm not writing any more goddamn sigmas".


Posted by: Walt Someguy | Link to this comment | 04-24-14 7:21 AM
horizontal rule
105

I just think "you can turn one into the other if you want, but they're not the same thing" is an extremely deep and confusing idea and tensors are one of the first places it comes up.

Huh. I assume my intuition is way off because that doesn't seem terribly confusing to me. Maybe I'm used to it from programming?


Posted by: Sifu Tweety | Link to this comment | 04-24-14 7:24 AM
horizontal rule
106

Well, perhaps here it's better said "you can turn one into the other because they have the same information content, but you *shouldn't* because they behave differently and if you try turning one into the other you'll get everything wrong."


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:27 AM
horizontal rule
107

This is about trans* people again, isn't it?


Posted by: ogged | Link to this comment | 04-24-14 7:29 AM
horizontal rule
108

Maybe one of the confusing things is that in physics you always have an inner product that you can use to move freely between the space and the dual space, so one can and does turn one into the other all the time.


Posted by: essear | Link to this comment | 04-24-14 7:31 AM
horizontal rule
109

102 is a good point. I have a far-fetched analogy between the math and physics notation for tensors with the SKI combinatory calculus versus the lambda calculus for function composition, but probably nobody actually cares about either of these things...

103: This is something we could do a better job teaching. If you keep track of units, then it's clear that different tensors give you different things, and this is something we could spend more time teaching already with the difference between row vectors and column vectors. To take an elementary example that you could explain to everyone, a vector of prices is a row vector, while a vector of quantities is a column vector. Everybody recognizes that they need to keep track of whether 2 means 2 eggs or 2o dollars. Prices and quantities are different from each either, and it makes sense to multiply prices by quantities, but not prices by prices, or quantities by quantities.


Posted by: Walt Someguy | Link to this comment | 04-24-14 7:32 AM
horizontal rule
110

Oh boy I wish I had a better grasp of dual spaces. Now that is math that comes up all the time for me but which I never covered in my classes.


Posted by: Sifu Tweety | Link to this comment | 04-24-14 7:33 AM
horizontal rule
111

Perhaps the CS-y way of putting it is that Matrix and Tensor are different classes, both are instantiated by giving an array of numbers, but the way the methods are overloaded means that the same method will act differently on the two classes (if you just thought of them as arrays).


Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 04-24-14 7:35 AM
horizontal rule
112

98: obviously MOOCs are the solution

All joking aside, maybe something like it sometimes.

In my sophomore year, after succeeding at calculus and having the good fortune to be at more advanced level in math instruction than in physics, so that physics was easy because I could really do derivatives and integrals, therefore cut to the chase, I became aware how weak and wobbly my algebra was.

So I acquired a book that described itself as "programmed learning," or some such. Algebra step-by-step, at your pace. I worked through it pretty quickly, and found that I almost-knew most of it already, but didn't have anywhere near the confidence I had afterwards. A revelatory experience, because in high school it had been hard and baffling.


Posted by: idp | Link to this comment | 04-24-14 7:35 AM
horizontal rule
113

108 actually freaks me the fuck out, and is probably the real reason mathematicians find the physicist handing of tensors off-putting. Why do physicists do that so often? Is it to propitiate Satan? He can't be trusted, you know.


Posted by: Walt Someguy | Link to this comment | 04-24-14 7:35 AM
horizontal rule
114

Well, it looks like we've cleared up tensors for everyone. We are the greatest.


Posted by: Walt Someguy | Link to this comment | 04-24-14 12:29 PM
horizontal rule
115

I'd tend to suspect that a chunk of it is people's self-conception as not good at math. When it's just about doing the work, they do fine. When it means doing well in a special class for the people who are really good at math, the kids who don't think of themselves that way give up faster.

and

I bet, since it's optional, the pace and difficulty ramps up because they're expecting the filtered student group to be better at math.

Well, yes, because here when you're 16 you pick 3 or 4 things that you're good at and then do them for the next 2 years. So this is a self-selected group of people who have been Good At Maths for probably most of their school life, and got the best grades in their OWLs. They should all be thinking of themselves as good at maths and they should be prepared for the work to get harder, same as it is in their other subjects, but everyone says, oh, the jump in maths is really big. I don't really understand how it can be a big jump when you spend the first two months doing stuff you've already done!

I hit my maths wall at university. Which given I was doing a joint maths degree was a bit awkward. If I'd got stoned less and worked more I might have understood more, I suppose.


Posted by: asilon | Link to this comment | 04-24-14 12:31 PM
horizontal rule
116

The only thing I remember about the move to multivariable calculus was that an extra variable made it possible to avoid the discs/washers technique for calculating something or other, which always gave me trouble because I kept using the wrong sign.

I found linear algebra and differential equations somewhat more difficult than previous math classes as it was my first college class and I had trouble with the attending class where no attendance is taken part and the doing homework when not required part.


Posted by: fake accent | Link to this comment | 04-25-14 12:41 AM
horizontal rule