The funning this, I decided to do this for fun. I spent hours cutting out all of these things (I still have them). Again, it was just for fun as I was probably just listening to an audiobook or something (perhaps a little scotch thrown in there). But... after I did all that cutting did I actually read the post. There was no point! There was no game to play. Doh! I wasn't aware it was only a concept and was unfinished.
I wonder if we can compare what happens over the long therm with people who intuitively understand math vs. people who learn to understand it with the help of metaphors such as above?
Are such un-abstractions helpful for teaching basic math knowledge? Advanced math skills? Both?
I certainly learn abstractions better when I have a solid concept to work from. I understand it is a crutch/bridge/scaffold for a true understanding, but such a thing really helps me learn anything. This has been true for me of maths, various programming concepts, human language grammar rules, and on.
I really really dig the power abstractions and abstract thought provide, but for whatever reason, I can't grasp a lot of abstracts without first learning how to apply them to a specific case and working out from there. As I have gotten older, I have been able to see quicker how to apply the abstractions from the initial case study, and regularly can apply abstraction in interesting ways - I am particularly good at seeing where various design patterns can be used on some software we are architecting, and doing the mental refactorings on the fly. Yet when I learn about a new design pattern, I still need to see it in a working piece of code, and preferably a step-by-step transformation to grasp it.
I want think these things are great for a start, to reach that first epiphany which clicks a concept in place, but I'm thinking they should be applied selectively and discarded as soon as possible.
I base this pretty much entirely on my own frustrating experience being "taught" with methods that focused entirely on and metaphors and procedures for "how" of things, not the "why" which I believe is essential to any real understanding.
In my experience teaching science and math, I've found this sort of analogy to be of limitted utility. It is often possible to get a whole class on board with the story and understanding the "rules" and able to come to the right conclusions. They're thrilled. Then you try to bring up the more abstract version and connect it and you get blank stares. Some kids make the connection quickly, but they tend to be the ones that wouldn't have needed the story version in the first place.
I have found that people who understand a topic well often mistake frameworks of understanding that they come up with after they understand for good ways to teach the concept to people who don't yet understand. It's like people think, "Oh I get it! Here's the big picture!" And they explain it that way. But the big picture they came up with was the result, not the cause, of their understanding.
It's pretty clear to me that this is an attempt to explain the mechanisms of lambda calculus using familiar metaphors. I think it could be very helpful for introducing the concepts.
Really? When I first read a text book about lambda calculus, it was fairly straightforward.
When I read _this_, I was puzzled. What the heck are we doing? Why do I care about alligators? What's the point of this?
And it only made sense when he said "it's like lambda calculus". It's a sad explanation if you need to understand the concept to understand the explanation.
I've never read a textbook about lambda calculus. When I red this, it was fairly straightforward, and I didn't need a pre-existing knowledge of lambda calculus to get it. There are plenty of people who don't understand the point of what they read in math textbooks. Not everything is one-size-fits-all.
The point of a metaphor is that it helps you make guesses about the behavior of something you don't yet understand by extrapolating from something you do understand. If there's no extrapolation possible, then it's lost its power. At that point it's just a theme.
If you are familiar with combinatory logic, perhaps an explanation featuring enchanted forests full of songbirds would seem unnecessarily complicated. Nevertheless, "To Mock a Mockingbird" is one of the most delightful books I've ever read.
Perhaps, in the end, it's just a form of recreation. One thing that interests me about such fantastic metaphors is to what extent they trigger or inhibit the "math reflex" of recognizing a problem one has already solved.
If there isn't a spoiler title, how many people familiar with Lambda Calculus have an "Aha! This is the Lambda Calculus" moment before it is revealed?
The author of "To Mock a Mockingbird", Raymond Smullyan, has written extensively
on logic, and he has books all across the spectrum of difficulty. He even wrote
what some consider to be "The Hardest Logic Puzzle Ever".
I'm speechless. I realize how far I've been reading his articles without knowing it. Only since his reactive variable binding in js I've known his name though.
The funning this, I decided to do this for fun. I spent hours cutting out all of these things (I still have them). Again, it was just for fun as I was probably just listening to an audiobook or something (perhaps a little scotch thrown in there). But... after I did all that cutting did I actually read the post. There was no point! There was no game to play. Doh! I wasn't aware it was only a concept and was unfinished.
I read it wondering about the game dynamics. It felt like a trick to waste my time when it wasn't a game, just a lesson about something else.
This has been posted before, but HN has grown since then and it deserves a wider audience.
Previously: http://news.ycombinator.com/item?id=600736
Why would anyone go to so much trouble to make lambda calculus difficult?
I wonder if we can compare what happens over the long therm with people who intuitively understand math vs. people who learn to understand it with the help of metaphors such as above?
Are such un-abstractions helpful for teaching basic math knowledge? Advanced math skills? Both?
As a single datum:
I certainly learn abstractions better when I have a solid concept to work from. I understand it is a crutch/bridge/scaffold for a true understanding, but such a thing really helps me learn anything. This has been true for me of maths, various programming concepts, human language grammar rules, and on.
I really really dig the power abstractions and abstract thought provide, but for whatever reason, I can't grasp a lot of abstracts without first learning how to apply them to a specific case and working out from there. As I have gotten older, I have been able to see quicker how to apply the abstractions from the initial case study, and regularly can apply abstraction in interesting ways - I am particularly good at seeing where various design patterns can be used on some software we are architecting, and doing the mental refactorings on the fly. Yet when I learn about a new design pattern, I still need to see it in a working piece of code, and preferably a step-by-step transformation to grasp it.
I'm not sure exactly how I feel about this...
I want think these things are great for a start, to reach that first epiphany which clicks a concept in place, but I'm thinking they should be applied selectively and discarded as soon as possible.
I base this pretty much entirely on my own frustrating experience being "taught" with methods that focused entirely on and metaphors and procedures for "how" of things, not the "why" which I believe is essential to any real understanding.
In my experience teaching science and math, I've found this sort of analogy to be of limitted utility. It is often possible to get a whole class on board with the story and understanding the "rules" and able to come to the right conclusions. They're thrilled. Then you try to bring up the more abstract version and connect it and you get blank stares. Some kids make the connection quickly, but they tend to be the ones that wouldn't have needed the story version in the first place.
I have found that people who understand a topic well often mistake frameworks of understanding that they come up with after they understand for good ways to teach the concept to people who don't yet understand. It's like people think, "Oh I get it! Here's the big picture!" And they explain it that way. But the big picture they came up with was the result, not the cause, of their understanding.
Abstract vs. Global learning.
As scary of an idea it is, there are people in the world who have difficulty learning just by looking at the inputs and outputs.
It's pretty clear to me that this is an attempt to explain the mechanisms of lambda calculus using familiar metaphors. I think it could be very helpful for introducing the concepts.
Really? When I first read a text book about lambda calculus, it was fairly straightforward.
When I read _this_, I was puzzled. What the heck are we doing? Why do I care about alligators? What's the point of this?
And it only made sense when he said "it's like lambda calculus". It's a sad explanation if you need to understand the concept to understand the explanation.
I've never read a textbook about lambda calculus. When I red this, it was fairly straightforward, and I didn't need a pre-existing knowledge of lambda calculus to get it. There are plenty of people who don't understand the point of what they read in math textbooks. Not everything is one-size-fits-all.
Familiar metaphors like an egg hatching into the prey of its parents?
The point of a metaphor is that it helps you make guesses about the behavior of something you don't yet understand by extrapolating from something you do understand. If there's no extrapolation possible, then it's lost its power. At that point it's just a theme.
If you are familiar with combinatory logic, perhaps an explanation featuring enchanted forests full of songbirds would seem unnecessarily complicated. Nevertheless, "To Mock a Mockingbird" is one of the most delightful books I've ever read.
http://www.amazon.com/gp/product/0192801422/ref=as_li_ss_tl?...
Perhaps, in the end, it's just a form of recreation. One thing that interests me about such fantastic metaphors is to what extent they trigger or inhibit the "math reflex" of recognizing a problem one has already solved.
If there isn't a spoiler title, how many people familiar with Lambda Calculus have an "Aha! This is the Lambda Calculus" moment before it is revealed?
The author of "To Mock a Mockingbird", Raymond Smullyan, has written extensively on logic, and he has books all across the spectrum of difficulty. He even wrote what some consider to be "The Hardest Logic Puzzle Ever".
http://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever
Add monkeys that throw eggs and you have a brainfuck interpreter.
This reminds me of an afternoon I spent with my then-officemate Carl de Marcken at the MIT AI Lab, recasting Ackermann's function (http://en.wikipedia.org/wiki/Ackermann_function) as a pyramid scheme for the weekly "Girl Scout Benefit / Graduate Student Beer" announcement (http://projects.csail.mit.edu/gsb/old-archive/gsb-archive/gs...)
Re-reading it now, I'm struck by the fact that Larry Ellison actually owns a Hawaiian island (though not Kauai). The system works!
I thought it felt familiar when I was reading the rules of the game! :p
Be right back, off to yoink this into a Tablet app ..
Whoa, that post made the rounds of the functionalerati long before Bret became famous for his kill math and kill programming stuff.
I'm speechless. I realize how far I've been reading his articles without knowing it. Only since his reactive variable binding in js I've known his name though.