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Thursday, October 30, 2008
Symposium en garde!
The challenge is on! The gauntlet is cast! I feel like broadening my mind, and I'd like some help.
Donalbain, one of our delightful regular posters, has stated that it's possible for anyone to learn mathematics and science, while writing is something only certain people can learn. I'm of the opinion that mathematics and science are, similarly, things that certain people block on past a certain point, no matter how hard they try, and that feeling it's something anyone can learn suggests an instinctive talent for the subject: I base this on the experience of being taught overly advanced maths from the age of fifteen to sixteen and spending an entire year in bewilderment, even though I had a good teacher and sat next to a helpful friend who did understand it.
Well, that sounds like an experiment waiting to happen.
So, here's what I propose. Donalbain, if willing, or someone else if Donalbain prefers to decline, will present me with a mathematical problem it will take a certain amount of understanding to solve, and explain the theory to me. I'll try to solve it.
Similarly, anybody else who feels they have a communicable skill and wants to play, please make an effort to teach me and anybody else who's reading this, how to do whatever it is you do.
At the same time, I'll give some pointers on how to write a six word short story, and anyone who wants to teach me something will, in exchange, have a go at that.
Let's all try to learn each others' skills.
Rather than concluding that anybody could win this debate, which would be confrontational, hard to measure and kind of silly, let's instead fool around for a while, and then I'll do another post where we can report on the experience of trying to learn a new skill, what difficulties we encountered, how it compared to doing the stuff we actually have skills in, and what we feel we've learned.
So, I'll kick off. Here's the best advice I can offer on how to write a six word short story.
The original six word short story was Hemingway's 'For sale: baby shoes. Never worn.' - a really superb piece of writing. The Wired collection is here. I'll quote some examples to illustrate what makes a six word story work.
The most basic, I'd say: think about when in time the story takes place.
Many of the best ones are written post facto, in a way that tells the whole story by implication. Hemingway's is a good example. Something has happened off-stage, in this case the death of a baby, and presumably to people who either can't bear to think of having another one (otherwise they'd keep the shoes for a later child), or are poor enough to need even the small money a second-hand pair of baby shoes would raise. The baby's birth was anticipated, because the shoes were bought in advance. There is, in fact, a beginning, middle and an end: the anticipation of the birth, the death, the grieving aftermath. All of this can be intuited - but it doesn't have to be present. There are some other examples of this: Brian Herbert's 'Epitaph: He shouldn't have fed it', for instance, has the beginning - the joining of man and scary creature - the middle - he fed it - and the end - it killed and possibly ate him. One of my favourites on this blog comes from Robert: 'Professor - re: zombie incident: You're fired.' The story there is clear: the professor tried to create zombies, there was a terrible incident, and while it's now been sorted out nobody's happy with him. These are stories that comment on a 'story' that's already taken place, not summarising it but showing in the fallout what must have happened.
An alternative method is to use reported speech. In this case, you're not looking back on a story, but hearing from somebody at the turning-point: in effect, you're telling the story in the present tense. Rockne S. O'Bannon, for instance, goes with 'It's behind you! Hurry before it' - and the breaking-off indicates exactly how that sentence ends. Ursula K. LeGuin's 'Easy. Just touch the match to' is another example of the same; so is 'Computer, did we bring batteries? Computer?' by Eileen Gunn. One of mine is 'Let go, I told them nothing.', which is similarly spoken out of the middle of a crisis. In such cases, you need to consider who's being addressed and under what pressing set of circumstances. A variation is Orson Scott Card's 'I saw, darling, but do lie.'; this is a post facto speech as in the examples above, but the question of who's being addressed is key to the story.
There are certain contexts that lend themselves to six-word stories. Hemingway's is written in a format that naturally would be terse, to wit, an advert. This makes it naturalistic in style even though constrained in form. While this isn't compulsory, it's a good technique to consider: think about formats which naturally would be terse. Margaret Atwood's 'Corpse parts missing. Doctor buys yacht.' and 'Starlet sex scandal. Giant squid involved.', and David Brin's 'Dinosaurs return. Want their oil back.', for instance, have the feel of headlines; the result is that the body text - the full story - is implied. One of mine was 'Wanna break up. Ur 2 emotionl.', which is in text-speak. E-mails, epitaphs, memos, slogans, telegrams, notes to self: all are methods worth considering.
'Wanna break up. Ur 2 emotionl.' is an example of another thing to consider, which is the unreliable narrator. In that instance, the irony plays off the fact that anyone who breaks up in a text message is no judge of how emotional is '2 emotionl'. Raising questions about the reliability of the speaker is a good way to expand the story beyond its narrow margins.
There's another way of making the story seem bigger, which is playing off established storylines or common human experiences. Steven Meretzky's 'Dorothy: "Fuck it, I'll stay here."' is an example of the former, though such stories are not my personal favourites, as they seem closer to gags. Margaret Atwood's 'Longed for him. Got him. Shit.' is an example of the latter, a terse summary of a situation felt by many, as is Bruce Sterling's 'It cost too much, staying human.', and my own 'How hard can it be? Oh.' If you can think of universal stories, they can sometimes be quickly summarised.
Consider, too, the rhythm of your piece. Six words has a finite number of possible combinations: a six-word phrase, a five and a one, three twos, two threes, a two and a four, a one, a two and a three ... oh, heck, I'm not the mathematician, but you see what I'm getting at. Rhythm is an important carrier of story. End on a single-word sentence, like 'Longed for him. Got him. Shit.', and you're likely to get an ironic punch; have a complete sentence, and it's likely to feel more contemplative; two threes or three twos will feel punchier than, say, a two and a four. Think about how the rhythmic possibilities could carry the structure of your story.
Think about the emotional tone. Given the constrictions, writers often go for humour - a quick, ironic piece works well with the form. You can, on the other hand, go for outright tragedy if you choose, as with Hemingway, in which case the brevity stops being the soul of wit and becomes a sad, nothing-else-to-say terseness. Either works fine, but the story needs to have some kind of mood if it's going to be interesting.
That's probably enough instruction to be going on with: think about when in time your story is taking place, who's speaking and how, why they're speaking and what's being implied off-stage. Read as many examples as possible and see if you can get a feel for them.
So, you all do that. What shall I do? Come on, hit me.
Donalbain, one of our delightful regular posters, has stated that it's possible for anyone to learn mathematics and science, while writing is something only certain people can learn. I'm of the opinion that mathematics and science are, similarly, things that certain people block on past a certain point, no matter how hard they try, and that feeling it's something anyone can learn suggests an instinctive talent for the subject: I base this on the experience of being taught overly advanced maths from the age of fifteen to sixteen and spending an entire year in bewilderment, even though I had a good teacher and sat next to a helpful friend who did understand it.
Well, that sounds like an experiment waiting to happen.
So, here's what I propose. Donalbain, if willing, or someone else if Donalbain prefers to decline, will present me with a mathematical problem it will take a certain amount of understanding to solve, and explain the theory to me. I'll try to solve it.
Similarly, anybody else who feels they have a communicable skill and wants to play, please make an effort to teach me and anybody else who's reading this, how to do whatever it is you do.
At the same time, I'll give some pointers on how to write a six word short story, and anyone who wants to teach me something will, in exchange, have a go at that.
Let's all try to learn each others' skills.
Rather than concluding that anybody could win this debate, which would be confrontational, hard to measure and kind of silly, let's instead fool around for a while, and then I'll do another post where we can report on the experience of trying to learn a new skill, what difficulties we encountered, how it compared to doing the stuff we actually have skills in, and what we feel we've learned.
So, I'll kick off. Here's the best advice I can offer on how to write a six word short story.
The original six word short story was Hemingway's 'For sale: baby shoes. Never worn.' - a really superb piece of writing. The Wired collection is here. I'll quote some examples to illustrate what makes a six word story work.
The most basic, I'd say: think about when in time the story takes place.
Many of the best ones are written post facto, in a way that tells the whole story by implication. Hemingway's is a good example. Something has happened off-stage, in this case the death of a baby, and presumably to people who either can't bear to think of having another one (otherwise they'd keep the shoes for a later child), or are poor enough to need even the small money a second-hand pair of baby shoes would raise. The baby's birth was anticipated, because the shoes were bought in advance. There is, in fact, a beginning, middle and an end: the anticipation of the birth, the death, the grieving aftermath. All of this can be intuited - but it doesn't have to be present. There are some other examples of this: Brian Herbert's 'Epitaph: He shouldn't have fed it', for instance, has the beginning - the joining of man and scary creature - the middle - he fed it - and the end - it killed and possibly ate him. One of my favourites on this blog comes from Robert: 'Professor - re: zombie incident: You're fired.' The story there is clear: the professor tried to create zombies, there was a terrible incident, and while it's now been sorted out nobody's happy with him. These are stories that comment on a 'story' that's already taken place, not summarising it but showing in the fallout what must have happened.
An alternative method is to use reported speech. In this case, you're not looking back on a story, but hearing from somebody at the turning-point: in effect, you're telling the story in the present tense. Rockne S. O'Bannon, for instance, goes with 'It's behind you! Hurry before it' - and the breaking-off indicates exactly how that sentence ends. Ursula K. LeGuin's 'Easy. Just touch the match to' is another example of the same; so is 'Computer, did we bring batteries? Computer?' by Eileen Gunn. One of mine is 'Let go, I told them nothing.', which is similarly spoken out of the middle of a crisis. In such cases, you need to consider who's being addressed and under what pressing set of circumstances. A variation is Orson Scott Card's 'I saw, darling, but do lie.'; this is a post facto speech as in the examples above, but the question of who's being addressed is key to the story.
There are certain contexts that lend themselves to six-word stories. Hemingway's is written in a format that naturally would be terse, to wit, an advert. This makes it naturalistic in style even though constrained in form. While this isn't compulsory, it's a good technique to consider: think about formats which naturally would be terse. Margaret Atwood's 'Corpse parts missing. Doctor buys yacht.' and 'Starlet sex scandal. Giant squid involved.', and David Brin's 'Dinosaurs return. Want their oil back.', for instance, have the feel of headlines; the result is that the body text - the full story - is implied. One of mine was 'Wanna break up. Ur 2 emotionl.', which is in text-speak. E-mails, epitaphs, memos, slogans, telegrams, notes to self: all are methods worth considering.
'Wanna break up. Ur 2 emotionl.' is an example of another thing to consider, which is the unreliable narrator. In that instance, the irony plays off the fact that anyone who breaks up in a text message is no judge of how emotional is '2 emotionl'. Raising questions about the reliability of the speaker is a good way to expand the story beyond its narrow margins.
There's another way of making the story seem bigger, which is playing off established storylines or common human experiences. Steven Meretzky's 'Dorothy: "Fuck it, I'll stay here."' is an example of the former, though such stories are not my personal favourites, as they seem closer to gags. Margaret Atwood's 'Longed for him. Got him. Shit.' is an example of the latter, a terse summary of a situation felt by many, as is Bruce Sterling's 'It cost too much, staying human.', and my own 'How hard can it be? Oh.' If you can think of universal stories, they can sometimes be quickly summarised.
Consider, too, the rhythm of your piece. Six words has a finite number of possible combinations: a six-word phrase, a five and a one, three twos, two threes, a two and a four, a one, a two and a three ... oh, heck, I'm not the mathematician, but you see what I'm getting at. Rhythm is an important carrier of story. End on a single-word sentence, like 'Longed for him. Got him. Shit.', and you're likely to get an ironic punch; have a complete sentence, and it's likely to feel more contemplative; two threes or three twos will feel punchier than, say, a two and a four. Think about how the rhythmic possibilities could carry the structure of your story.
Think about the emotional tone. Given the constrictions, writers often go for humour - a quick, ironic piece works well with the form. You can, on the other hand, go for outright tragedy if you choose, as with Hemingway, in which case the brevity stops being the soul of wit and becomes a sad, nothing-else-to-say terseness. Either works fine, but the story needs to have some kind of mood if it's going to be interesting.
That's probably enough instruction to be going on with: think about when in time your story is taking place, who's speaking and how, why they're speaking and what's being implied off-stage. Read as many examples as possible and see if you can get a feel for them.
So, you all do that. What shall I do? Come on, hit me.
posted by Kit Whitfield # 8:21 am 
Comments:
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This was good -- maybe I'll attempt a six word story myself later.
As for the math, it's tough to come up with a good problem without knowing what you already know. One of the coolest little tricks is doing the sum
1 + 1/2 + 1/4 + 1/8 + 1/32 + 1/64 + ... (it's an infinite sum.)
But there's a good chance you already know what that adds up to, and why. If not, though, would you want to try that one?
There are some other good infinite sums to do if you happen to know that one, but aren't comfortable with the general idea. Unfortunately they're more work to explain...
-Mary
As for the math, it's tough to come up with a good problem without knowing what you already know. One of the coolest little tricks is doing the sum
1 + 1/2 + 1/4 + 1/8 + 1/32 + 1/64 + ... (it's an infinite sum.)
But there's a good chance you already know what that adds up to, and why. If not, though, would you want to try that one?
There are some other good infinite sums to do if you happen to know that one, but aren't comfortable with the general idea. Unfortunately they're more work to explain...
-Mary
No, I don't know that one. Let's see...
Okay, one plus a half is one and a half,
then one and three quarters,
then one and five eighths,
then one and, um, four eights are thirty-two so that makes 4 x 5 = 20/38 plus the extra which makes one and 20/38
then one and 41/68...
But if it goes on for ever, then surely there isn't a final answer?
Okay, hang on. (I'm showing my working-out here, so you can get a picture of my limits.) Maybe there's supposed to be some sort of equation, something to the power of something. You haven't included 1/16, but as everything else is double the predecessor I'm going to assume that's a typo.
So, it's always 1/2 to the power of ... let's try some algebra...
Let 1 = x.
The number is x + (x over x to the power of 2) - oh nuts, no, 1 to the power of 2 is 1. Scratch that.
Okay:
The answer is, let's call it a. So a = 1 + 1/2 + 1/2 to the power of two + 1/2 to the power of three, and so on.
But that's just identifying the basic pattern, which is pretty obvious.
Nope, sorry, unless I hear some more theory, all I've got is 'one and a bit' - slightly less than two.
Okay, one plus a half is one and a half,
then one and three quarters,
then one and five eighths,
then one and, um, four eights are thirty-two so that makes 4 x 5 = 20/38 plus the extra which makes one and 20/38
then one and 41/68...
But if it goes on for ever, then surely there isn't a final answer?
Okay, hang on. (I'm showing my working-out here, so you can get a picture of my limits.) Maybe there's supposed to be some sort of equation, something to the power of something. You haven't included 1/16, but as everything else is double the predecessor I'm going to assume that's a typo.
So, it's always 1/2 to the power of ... let's try some algebra...
Let 1 = x.
The number is x + (x over x to the power of 2) - oh nuts, no, 1 to the power of 2 is 1. Scratch that.
Okay:
The answer is, let's call it a. So a = 1 + 1/2 + 1/2 to the power of two + 1/2 to the power of three, and so on.
But that's just identifying the basic pattern, which is pretty obvious.
Nope, sorry, unless I hear some more theory, all I've got is 'one and a bit' - slightly less than two.
No, I don't know that one. Let's see...
Okay, one plus a half is one and a half,
then one and three quarters,
then one and five eighths,
then one and, um, four eights are thirty-two so that makes 4 x 5 = 20/38 plus the extra which makes one and 20/38
then one and 41/68...
But if it goes on for ever, then surely there isn't a final answer?
Okay, hang on. (I'm showing my working-out here, so you can get a picture of my limits.) Maybe there's supposed to be some sort of equation, something to the power of something. You haven't included 1/16, but as everything else is double the predecessor I'm going to assume that's a typo.
So, it's always 1/2 to the power of ... let's try some algebra...
Let 1 = x.
The number is x + (x over x to the power of 2) - oh nuts, no, 1 to the power of 2 is 1. Scratch that.
Okay:
The answer is, let's call it a. So a = 1 + 1 over 2 + 1 over 2 to the power of two + 1 over 2 to the power of three, and so on.
But that's just identifying the basic pattern, which is pretty obvious.
Nope, sorry, unless I hear some more theory, all I've got is 'one and a bit' - slightly less than two.
Okay, one plus a half is one and a half,
then one and three quarters,
then one and five eighths,
then one and, um, four eights are thirty-two so that makes 4 x 5 = 20/38 plus the extra which makes one and 20/38
then one and 41/68...
But if it goes on for ever, then surely there isn't a final answer?
Okay, hang on. (I'm showing my working-out here, so you can get a picture of my limits.) Maybe there's supposed to be some sort of equation, something to the power of something. You haven't included 1/16, but as everything else is double the predecessor I'm going to assume that's a typo.
So, it's always 1/2 to the power of ... let's try some algebra...
Let 1 = x.
The number is x + (x over x to the power of 2) - oh nuts, no, 1 to the power of 2 is 1. Scratch that.
Okay:
The answer is, let's call it a. So a = 1 + 1 over 2 + 1 over 2 to the power of two + 1 over 2 to the power of three, and so on.
But that's just identifying the basic pattern, which is pretty obvious.
Nope, sorry, unless I hear some more theory, all I've got is 'one and a bit' - slightly less than two.
Double post because my notation in the first wasn't clear, and Blogger won't let me delete it, for some reason. Please ignore the first one.
I don't know what I could teach over the internet, in a comments thread. That calls for definite writing skills, and a gift for terseness, which I am still trying to learn.
So. I'll try applying your suggestions to a six word story:
"Daddy? May I sleep now? Please?"
(Verification word: "pegrap", which is that annoying feature in Word of automatically spacing between paragraphs, whether you want to or not)
So. I'll try applying your suggestions to a six word story:
"Daddy? May I sleep now? Please?"
(Verification word: "pegrap", which is that annoying feature in Word of automatically spacing between paragraphs, whether you want to or not)
# posted by 
: 2:46 pm
: 2:46 pm
I am totally willing to win this thread! But first, as Mary says above, we need to know what you are interested in. What area of mathematics do you remember from the olden days that you would like to have a go at learning? Or, on the other hand, how are you at science? Would you like to play with some science?
And, just to get my excuses in first; you have the advantage because this is a medium of words, where as to do fun maths and science you need more than plain text! But we can play!
And, just to get my excuses in first; you have the advantage because this is a medium of words, where as to do fun maths and science you need more than plain text! But we can play!
Well, my understanding of maths is based on GCSE-level, which is to say up to the age of sixteen; I'm now thirty-one, so I'm pretty rusty. (As you can see from my struggles above: I was pretty much racking my brains for everything I could remember.) I also stopped studying science then - basic physics, chemistry and biology being my subjects - so my education in all subjects is about at the same level.
In the interests of integrity, I think I ought to volunteer for the subjects I hate and fear most, since they're the ones I'm convinced I can't learn. (Not that I hate and fear them in themselves, you understand, I just hate and fear having to learn them myself. I am always grateful for the existence of people who are good at them, meaning humanity gets the benefits and I don't have to beat my brains.)
So: the two subjects I struggled worst with are mathematics and physics. In most areas, I have difficulty learning an abstract concept when it's explained as a concept, but can work out a fundamental principle quite fast if I'm given an illustrative example; maths and physics are the areas that lend themselves worst to this, in my experience. What can you do me for that?
(I'm going out for the evening in an hour or two, and won't be back till late, so if it takes a while for me to address posts, I'm not ignoring you, I'm just elsewhere.)
In the interests of integrity, I think I ought to volunteer for the subjects I hate and fear most, since they're the ones I'm convinced I can't learn. (Not that I hate and fear them in themselves, you understand, I just hate and fear having to learn them myself. I am always grateful for the existence of people who are good at them, meaning humanity gets the benefits and I don't have to beat my brains.)
So: the two subjects I struggled worst with are mathematics and physics. In most areas, I have difficulty learning an abstract concept when it's explained as a concept, but can work out a fundamental principle quite fast if I'm given an illustrative example; maths and physics are the areas that lend themselves worst to this, in my experience. What can you do me for that?
(I'm going out for the evening in an hour or two, and won't be back till late, so if it takes a while for me to address posts, I'm not ignoring you, I'm just elsewhere.)
Ah, you see, that's making good use of the techniques! Also good use of punctuation, including the paragraph break, which is something I forgot to mention: punctuation can create a lot of space when used correctly. Congratulations.
Oooh! I had another idea...
"Nine pints... but I saw it!"
Is using type face cheating?
This reminds me of a task we sometimes do at my drama group. The idea is to start with a random script of a couple of lines for each character, and each iteration, you shorten the play, but keeping as much of the story as you can. It can't be done through the medium of mime. You just have to pack the whole story down into as short a time as possible.
Verification word: Nestl. Needs no comment on a cold, fireside day like this!
"Nine pints... but I saw it!"
Is using type face cheating?
This reminds me of a task we sometimes do at my drama group. The idea is to start with a random script of a couple of lines for each character, and each iteration, you shorten the play, but keeping as much of the story as you can. It can't be done through the medium of mime. You just have to pack the whole story down into as short a time as possible.
Verification word: Nestl. Needs no comment on a cold, fireside day like this!
See, you're actually good at this! Much better, I suspect, than I'll be at a science/maths problem.
Improvisational drama's a very good training for writing, I find. Writing is basically acting on the page where you play all the parts.
I want a nestl.
Improvisational drama's a very good training for writing, I find. Writing is basically acting on the page where you play all the parts.
I want a nestl.
In the mean time, I will try to put together a physics problem based on the A Level syllabus that I will teach you to do! It will probably involve you doing some stuff.. so be prepared to gather things and obey my instructions! I will probably be ready to do it tomorrow morning.
donalblain, I think some of the "counter-intuitive" physics might be a good example -- like the classic one involving shooting a pool ball across a spinning circular surface.
Speaking of spinning, there's always Newton's bucket.
(Verification word: "rastede", which is how I feel after a good nestl)
Speaking of spinning, there's always Newton's bucket.
(Verification word: "rastede", which is how I feel after a good nestl)
# posted by : 4:17 pm
: 4:17 pm
In my head at the moment is something more basic. I think just a simple lesson on momentum.
Verification: login and so I did.
Verification: login and so I did.
donalblain: Is using type face cheating?
I don't think it's cheating, any more than punctuation is.
I used it in mine, but now that I see it, I don't like it. I think it might be better -- more understated -- without it.
I'm thinking now of shorter stories.
There's the classic (and usually humorous) "No shit, there I was..." for five.
And the even more classic, if frequently tragic, "Hey, guys! Watch this!"
Actually I kind of like that one, as a six word story, too:
Texas toast: "Hey, guys! Watch this!"
but that's more of a gag setup than a story, I suppose.
(Word verification: "gismstsi" which refers to the hurricane-like flurry of nifty but useless electronic gadgets which flood the stores during holiday shopping time)
I don't think it's cheating, any more than punctuation is.
I used it in mine, but now that I see it, I don't like it. I think it might be better -- more understated -- without it.
I'm thinking now of shorter stories.
There's the classic (and usually humorous) "No shit, there I was..." for five.
And the even more classic, if frequently tragic, "Hey, guys! Watch this!"
Actually I kind of like that one, as a six word story, too:
Texas toast: "Hey, guys! Watch this!"
but that's more of a gag setup than a story, I suppose.
(Word verification: "gismstsi" which refers to the hurricane-like flurry of nifty but useless electronic gadgets which flood the stores during holiday shopping time)
# posted by : 4:37 pm
: 4:37 pm
Now that I have had my two goes.. I am starting to struggle. Everything is starting to feel contrived and self concious. Case in point: I started with I begged her, but she left. which was far too meh for words. So I editted it and came up with I begged her, but she stayed which just sounds ugly and contrived. Hmmm... now I am going all stream of conciousness and just came up with
She believed me, but still left.
I have also just noticed that all of mine have been two parters.. I wonder if that is significant. The next task is to avoid that and get it to a single thought.
Kit: Can you do this problem?
Due to a horrible, horrible mistake, two maglev trains are heading towards one another on the same track! Thankfully they are both runaways and nobody is aboard.
Train A is headed north at a speed of 10km/s and has a mass of 2000kg.
Train B is headed south and has a mass of 4000kg.
After the collision, nothing is broken off the trains, but they are both come to a dead stop!
At what speed was train B travelling?
Verification: Feroor. The sound made by the engineer just before he jumps off the runaway train.
She believed me, but still left.
I have also just noticed that all of mine have been two parters.. I wonder if that is significant. The next task is to avoid that and get it to a single thought.
Kit: Can you do this problem?
Due to a horrible, horrible mistake, two maglev trains are heading towards one another on the same track! Thankfully they are both runaways and nobody is aboard.
Train A is headed north at a speed of 10km/s and has a mass of 2000kg.
Train B is headed south and has a mass of 4000kg.
After the collision, nothing is broken off the trains, but they are both come to a dead stop!
At what speed was train B travelling?
Verification: Feroor. The sound made by the engineer just before he jumps off the runaway train.
You're right, I did leave out 1/16th by accident.
The cute explanation (which I did not come up with) of how to do the sum will work best if you play along by drawing a rectangle on a piece of paper. Make it twice as wide as it is tall.
Now you have a rectangle which is two units wide and one unit tall. I'm inventing a new measurement called a "unit" which is exactly as long as your rectangle is tall, by the way. The rectangle's area is two square units, which is its width times its height.
Draw a vertical line down the middle of your rectangle, dividing it into two squares, each one unit tall by one unit wide. The area of each square is one square unit, or 1 (units)^2.
Okay, now draw a horizonal line through one of the squares, dividing it in half. If the area of the square was one square unit, then naturally the area of each of the little rectangles you just made is 1/2 (units)^2.
Now please draw a vertical line dividing one of those little rectangles in half. Now you have two new little squares, each with an area of 1/4 (units)^2. (You can easily see that four of them together will fill your original 1x1 square.)
Now please draw a horizontal line dividing one of those little squares into two very small rectangles. Since each rectangle has half the area of that little square, each must have an area of 1/8th (units)^2. (And again, you can see that 8 of them will fill your original 1x1 square.)
Draw a vertical line dividing one of the very small rectangles into two very small squares. Each of these squares has an area, naturally, of 1/16th (units)^2.
Just once more, I think, will be enough: Draw another horizontal line through one of the very small squares, dividing it into two tiny rectangles. Each tiny rectangle will have an area of 1/32 (units)^2.
Okay, obviously, we could go on like this forever, but let's just count up what we've got. We've got a big square of area 1, plus a little rectangle of area 1/2, plus a little square of area 1/4, plus a very small rectangle of area 1/8, plus a very small square of area 1/16, and two tiny rectangles of area 1/32.
Clearly 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/32 = 2, because all of those areas have to add together to give you the area of your original rectangle.
But I could divide one of my tiny rectangles again, and show that
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64 = 2
Then I could divide one of my little 1/64 size squares into rectangles of size 1/128 again, and I'd have
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/128 = 2
The idea is that I can continue this process forever, dividing the last little piece in half, but never, ever, am I going to get anything bigger than "2" by adding up pieces of my original rectangle. In fact, no matter how many times I do this, I'm always going to get extactly two. Mathematically, we say that this series "converges."
Since the sum of 1/2^n is 2 no matter how big "n" gets, we say tha the limit of the sum of 1/2^n is 2, as n approaches infinity. It's not clear what it means to say that n "equals" infinity, but if it did, somehow, the sum would still be two.
Okay, end of explanation. How did I do? Now I just have a couple of comments I want to make about this problem...
I want to point out that I could do a really similar experiment where I start by dividing a 1x1 square into ten pieces, and then take one of them and divide it into ten smaller pieces. That would show that .9 (the area of nine of the ten original pieces) + .09 (nine of the ten smaller pieces) + .01 (the last of the smaller pieces) = 1. But I could divide that last little piece into ten more, and show that .9 + .09 + .009 + .001 = 1. And then I could divide that last piece, and so on. This is a proof that shows, very surprisingly, that .999999999... = 1! (As it should, if you happen to believe that 1/3 = .33333... and 2/3 is .666666... and 1/3 + 2/3 = 1) It's just a sort of funny thing about the way we choose to write numbers that those seemingly different expressions actually represent the same number.
You can do lots of different infinite sums this way, just by drawing squares and rectangles, but although it's a good trick, it doesn't work for *all* infinite sums. For instance 1 + 2 + 3 + 4 + 5 + 6 + ... does not equal a finite number, obviously!
How about the sum of 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... That one is *not* obvious. Maybe it's a finite number, maybe it's not! You can try drawing squares and rectangles and fooling around, but since there's no technique that's guarenteed to work for all cases, that won't necessarily get you the answer. (As it turns out, the infinite sum of 1/n does not add up to a finite number. It "diverges." But 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... the infinite sum of 1/n^2, *does* add up to a finite number.) Finding out what a series converges to is an example of Real Math, because sometimes they converge, and sometimes they don't, and no one method is going to tell you how do it for every kind of series. But if you have followed the explanation in this comment, then you have learned how to do many different infinite sums, if not all of them.
If you haven't followed it, please ask questions, and don't think that my failings as a teacher necessarily mean math is hard to learn!
Incidentally I do this kind of math all the time for my research (although I usually just use the formula, and don't bother with the squares.) I have to do problems that have to do with light bounching between two mirrors, you see. If ninety percent of the light is reflected each time (it's not a perfect mirror), then the amount of light I have between the mirrors is my original light times (.9 + .9*.9 + .9*.9*.9 + .9*.9*.9*.9 + ...) That is, it's the sum of .9^n, as n goes to infinity. That converges, thankfully, and the answer that I get is one that can be and has been experimentally tested many times. The world really does work this way -- infinite reflections add up to a finite amount of light.
-Mary (my word verification is "comento". Makes me feel like I should've written this in Spanish)
The cute explanation (which I did not come up with) of how to do the sum will work best if you play along by drawing a rectangle on a piece of paper. Make it twice as wide as it is tall.
Now you have a rectangle which is two units wide and one unit tall. I'm inventing a new measurement called a "unit" which is exactly as long as your rectangle is tall, by the way. The rectangle's area is two square units, which is its width times its height.
Draw a vertical line down the middle of your rectangle, dividing it into two squares, each one unit tall by one unit wide. The area of each square is one square unit, or 1 (units)^2.
Okay, now draw a horizonal line through one of the squares, dividing it in half. If the area of the square was one square unit, then naturally the area of each of the little rectangles you just made is 1/2 (units)^2.
Now please draw a vertical line dividing one of those little rectangles in half. Now you have two new little squares, each with an area of 1/4 (units)^2. (You can easily see that four of them together will fill your original 1x1 square.)
Now please draw a horizontal line dividing one of those little squares into two very small rectangles. Since each rectangle has half the area of that little square, each must have an area of 1/8th (units)^2. (And again, you can see that 8 of them will fill your original 1x1 square.)
Draw a vertical line dividing one of the very small rectangles into two very small squares. Each of these squares has an area, naturally, of 1/16th (units)^2.
Just once more, I think, will be enough: Draw another horizontal line through one of the very small squares, dividing it into two tiny rectangles. Each tiny rectangle will have an area of 1/32 (units)^2.
Okay, obviously, we could go on like this forever, but let's just count up what we've got. We've got a big square of area 1, plus a little rectangle of area 1/2, plus a little square of area 1/4, plus a very small rectangle of area 1/8, plus a very small square of area 1/16, and two tiny rectangles of area 1/32.
Clearly 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/32 = 2, because all of those areas have to add together to give you the area of your original rectangle.
But I could divide one of my tiny rectangles again, and show that
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64 = 2
Then I could divide one of my little 1/64 size squares into rectangles of size 1/128 again, and I'd have
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/128 = 2
The idea is that I can continue this process forever, dividing the last little piece in half, but never, ever, am I going to get anything bigger than "2" by adding up pieces of my original rectangle. In fact, no matter how many times I do this, I'm always going to get extactly two. Mathematically, we say that this series "converges."
Since the sum of 1/2^n is 2 no matter how big "n" gets, we say tha the limit of the sum of 1/2^n is 2, as n approaches infinity. It's not clear what it means to say that n "equals" infinity, but if it did, somehow, the sum would still be two.
Okay, end of explanation. How did I do? Now I just have a couple of comments I want to make about this problem...
I want to point out that I could do a really similar experiment where I start by dividing a 1x1 square into ten pieces, and then take one of them and divide it into ten smaller pieces. That would show that .9 (the area of nine of the ten original pieces) + .09 (nine of the ten smaller pieces) + .01 (the last of the smaller pieces) = 1. But I could divide that last little piece into ten more, and show that .9 + .09 + .009 + .001 = 1. And then I could divide that last piece, and so on. This is a proof that shows, very surprisingly, that .999999999... = 1! (As it should, if you happen to believe that 1/3 = .33333... and 2/3 is .666666... and 1/3 + 2/3 = 1) It's just a sort of funny thing about the way we choose to write numbers that those seemingly different expressions actually represent the same number.
You can do lots of different infinite sums this way, just by drawing squares and rectangles, but although it's a good trick, it doesn't work for *all* infinite sums. For instance 1 + 2 + 3 + 4 + 5 + 6 + ... does not equal a finite number, obviously!
How about the sum of 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... That one is *not* obvious. Maybe it's a finite number, maybe it's not! You can try drawing squares and rectangles and fooling around, but since there's no technique that's guarenteed to work for all cases, that won't necessarily get you the answer. (As it turns out, the infinite sum of 1/n does not add up to a finite number. It "diverges." But 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... the infinite sum of 1/n^2, *does* add up to a finite number.) Finding out what a series converges to is an example of Real Math, because sometimes they converge, and sometimes they don't, and no one method is going to tell you how do it for every kind of series. But if you have followed the explanation in this comment, then you have learned how to do many different infinite sums, if not all of them.
If you haven't followed it, please ask questions, and don't think that my failings as a teacher necessarily mean math is hard to learn!
Incidentally I do this kind of math all the time for my research (although I usually just use the formula, and don't bother with the squares.) I have to do problems that have to do with light bounching between two mirrors, you see. If ninety percent of the light is reflected each time (it's not a perfect mirror), then the amount of light I have between the mirrors is my original light times (.9 + .9*.9 + .9*.9*.9 + .9*.9*.9*.9 + ...) That is, it's the sum of .9^n, as n goes to infinity. That converges, thankfully, and the answer that I get is one that can be and has been experimentally tested many times. The world really does work this way -- infinite reflections add up to a finite amount of light.
-Mary (my word verification is "comento". Makes me feel like I should've written this in Spanish)
# posted by : 4:53 pm
: 4:53 pm
Um - 5 m/s? On the grounds that they must meet with equal force, but one is twice as heavy as the other, so presumably is travelling at half the speed?
I'm supposed to be going out now, so I'll cudgel my brain against the other problem later...
I'm supposed to be going out now, so I'll cudgel my brain against the other problem later...
You have the answer right! Well done! Although, for force, you should be using momentum.. but that can be worked on!
It seems I need to step up a level to confuse the Kit!
It seems I need to step up a level to confuse the Kit!
A six word story attempt, inspired by the "headlines" suggestion:
Man Bites Dog: Dog Files Suit
(A bit obvious, I know, but I'll try again later...)
-Mary
Man Bites Dog: Dog Files Suit
(A bit obvious, I know, but I'll try again later...)
-Mary
# posted by : 5:36 pm
: 5:36 pm
"So, I'm imagining the vampires, eh?"
That's the sum of my lunch break creativity today.
I could give you a fairly good walkthrough on painting miniature gaming figures, but it'll have to wait until I've left work.
That's the sum of my lunch break creativity today.
I could give you a fairly good walkthrough on painting miniature gaming figures, but it'll have to wait until I've left work.
I had a brilliant method for painting my Warhammer models.
1) Hand model to friend.
2) Wait
3) Take model from friend.
4) Give friend beer.
1) Hand model to friend.
2) Wait
3) Take model from friend.
4) Give friend beer.
I am doing this post in real time. it is now 1:30am.
The saddest thing? I meant it.
Damn.. I am stuck in the same rythmn as for all my other stories. And the same two sentence structure. Plus, that doesnt tell a story. Bah! Curse your creative brain Kit! Curse it!
"Who's there?"
"Its me."
"Fuck off."
A change of structure and rythmn. The structure is different, but mainly for the sake of being different.
Verification: jambly. How you feel after a couple of beers in the half term holiday at 1:37pm
The saddest thing? I meant it.
Damn.. I am stuck in the same rythmn as for all my other stories. And the same two sentence structure. Plus, that doesnt tell a story. Bah! Curse your creative brain Kit! Curse it!
"Who's there?"
"Its me."
"Fuck off."
A change of structure and rythmn. The structure is different, but mainly for the sake of being different.
Verification: jambly. How you feel after a couple of beers in the half term holiday at 1:37pm
# posted by : 1:37 am
: 1:37 am
Also!
Kit.. is there any part of science that you wanted to be able to do/understand? And also, therein lies the difference. With your challenge we can all read each other's little attempts and wince or laugh or cry and so get immediate feedback. With science, the most important thing is not being able to give an answer to a problem that can be ticked or crossed. It is actually having the understanding and that is far harder to evaluate..
verification: nedri.. an obvious anagram of "I nerd" (which reminds me of the story of the Best Fancy Dress Costume Ever!)
Kit.. is there any part of science that you wanted to be able to do/understand? And also, therein lies the difference. With your challenge we can all read each other's little attempts and wince or laugh or cry and so get immediate feedback. With science, the most important thing is not being able to give an answer to a problem that can be ticked or crossed. It is actually having the understanding and that is far harder to evaluate..
verification: nedri.. an obvious anagram of "I nerd" (which reminds me of the story of the Best Fancy Dress Costume Ever!)
# posted by : 1:43 am
: 1:43 am
Short short stories, hmmm? Like that old favorite, the death notice posted in the local paper by the grieving widow:
"Fred died. Boat for sale."
Everything I can come up with seems to be rather drearily domestic:
"She knew he didn't mean it."
"Mom? Don't worry, no one's hurt..."
Maybe I can expand my horizons after a night's sleep.
Not sure what I could try to teach anyone, especially in a comment. If you want to develop an algorithm for a computer program, or bake a loaf of yeast-raised bread, there are cookbooks for both.
You know what I wish someone could teach me?
"I always wished I could sing."
My word is byefq, the frequently-asked-questions section on an etiquette site discussing the proper ways to bid farewell.
I remain
your obedient servant,
Amaryllis
"Fred died. Boat for sale."
Everything I can come up with seems to be rather drearily domestic:
"She knew he didn't mean it."
"Mom? Don't worry, no one's hurt..."
Maybe I can expand my horizons after a night's sleep.
Not sure what I could try to teach anyone, especially in a comment. If you want to develop an algorithm for a computer program, or bake a loaf of yeast-raised bread, there are cookbooks for both.
You know what I wish someone could teach me?
"I always wished I could sing."
My word is byefq, the frequently-asked-questions section on an etiquette site discussing the proper ways to bid farewell.
I remain
your obedient servant,
Amaryllis
# posted by : 3:14 am
: 3:14 am
oh yes, I'd meant to add one more "mom" story:
"One bite won't kill you...Oops."
New word: bulld.
"Nice doggie, a good bulld--- owww!"
"One bite won't kill you...Oops."
New word: bulld.
"Nice doggie, a good bulld--- owww!"
# posted by : 3:31 am
: 3:31 am
A horror story? A romance? A religious allegory?
"Before I knocked, the door opened."
This evening I discovered I could NOT teach functions to a twelve-year-old.
(word verification: "unfari", the petulant cry of the the dyslexic poster who couldn't think of anything clever to do with it)
"Before I knocked, the door opened."
This evening I discovered I could NOT teach functions to a twelve-year-old.
(word verification: "unfari", the petulant cry of the the dyslexic poster who couldn't think of anything clever to do with it)
# posted by : 4:07 am
: 4:07 am
Re Mary's problem: see, here's the thing. I just about understood the rectangles issue. I tried, and thought about it, and almost saw how it worked; I seemed to have learned something.
Then I went out for the evening, and halfway through, somebody mentioned maths - and I returned to the explanation, thought about it, and realised, 'Nope, don't understand it any more.'
Students used to say that a lecture from Richard Feynman was like a Chinese meal: half an hour later, you wondered what you'd learned. He could make it clear, but they couldn't reconstruct it. That's the experience I had with that problem: I could just about jimmy the understanding into my head, but it didn't stay there. Possibly I could memorise an answer by rote, but I don't think that's real understanding: if I'd properly understood the principles behind it, I would have retained it better. Some people have more of a feel for numbers than others, and I have very little.
With science, the most important thing is not being able to give an answer to a problem that can be ticked or crossed. It is actually having the understanding and that is far harder to evaluate.
I'd very much disagree that 'ticked or crossed' is something you can apply to writing. Some pieces may be more liked than others, but that's not necssarily a measure of quality, as I'm sure you'll agree if you've ever gotten into one of those arguments where someone cites box office figures to prove you're wrong when you say a movie was bad. Subjectivity is a big issue in art.
As to wanting to understand, I think that's the nub of the matter. People who understand science or maths tend to assume that anyone who's interested can learn them - but nobody's interested in something they can't learn. Which comes first, the inability to understand or the lack of interest? I'd vote for the former. Obviously practice makes you better at something, but without a minimum level of feel for something, the point is going to elude you, and if you can't see the point, it's harder to learn. From the outside, it looks like a refusal to see the point, but from the inside, it's more like an inability: I try to grasp the point, but at best it's like holding up something heavy, and if my attention wanders I tend to let it drop.
(Word: lattelu. It sounds warm and comforting, and on this cold morning, I want one.)
Then I went out for the evening, and halfway through, somebody mentioned maths - and I returned to the explanation, thought about it, and realised, 'Nope, don't understand it any more.'
Students used to say that a lecture from Richard Feynman was like a Chinese meal: half an hour later, you wondered what you'd learned. He could make it clear, but they couldn't reconstruct it. That's the experience I had with that problem: I could just about jimmy the understanding into my head, but it didn't stay there. Possibly I could memorise an answer by rote, but I don't think that's real understanding: if I'd properly understood the principles behind it, I would have retained it better. Some people have more of a feel for numbers than others, and I have very little.
With science, the most important thing is not being able to give an answer to a problem that can be ticked or crossed. It is actually having the understanding and that is far harder to evaluate.
I'd very much disagree that 'ticked or crossed' is something you can apply to writing. Some pieces may be more liked than others, but that's not necssarily a measure of quality, as I'm sure you'll agree if you've ever gotten into one of those arguments where someone cites box office figures to prove you're wrong when you say a movie was bad. Subjectivity is a big issue in art.
As to wanting to understand, I think that's the nub of the matter. People who understand science or maths tend to assume that anyone who's interested can learn them - but nobody's interested in something they can't learn. Which comes first, the inability to understand or the lack of interest? I'd vote for the former. Obviously practice makes you better at something, but without a minimum level of feel for something, the point is going to elude you, and if you can't see the point, it's harder to learn. From the outside, it looks like a refusal to see the point, but from the inside, it's more like an inability: I try to grasp the point, but at best it's like holding up something heavy, and if my attention wanders I tend to let it drop.
(Word: lattelu. It sounds warm and comforting, and on this cold morning, I want one.)
Okay, if you really believe you're fundamentally unable to grasp math and science..? Why? Is it because you're female? Because you're "right brained" and read a lot? Is it genetic? Is it the way you were raised? Why?
The problem for me is, everyone who says "some people just can't learn math" usually ends up coming up with reasons that offend me, because nearly all of them imply that I shouldn't be able to learn it either. I'm a bookish female with non-mathy parents who did badly at it in childhood. But it's my career now, for better or worse.
I'd also like to say that the experience of feeling like I've grasped something only to have it slip out of my head is a very common on in my own education, and I think in everyone's math education. A lot of learning is re-learning...
Finally -- if you just compared me to Richard Feynman, I'm flattered. :-)
(My word is "serillec" which is just another way or writing Russian.)
The problem for me is, everyone who says "some people just can't learn math" usually ends up coming up with reasons that offend me, because nearly all of them imply that I shouldn't be able to learn it either. I'm a bookish female with non-mathy parents who did badly at it in childhood. But it's my career now, for better or worse.
I'd also like to say that the experience of feeling like I've grasped something only to have it slip out of my head is a very common on in my own education, and I think in everyone's math education. A lot of learning is re-learning...
Finally -- if you just compared me to Richard Feynman, I'm flattered. :-)
(My word is "serillec" which is just another way or writing Russian.)
I think anyone who is interested ENOUGH can learn mathematics and physics to an undergraduate level. At that level there simply ARE certain techniques that will work. There are certain procedures you can use that will get you to where you want to go. You need to apply a little creativity, yes, but I think that the creativity is a product of knowing the techniques.
The question then becomes "is Kit interested ENOUGH?" Now.. that probably sounds a little mean and aggressive, but is not meant to. Honest! The only person who can decide if she is interested enough is Kit. Yes, to get the techniques down pat can be hard. It can also be boring (Linear Algebra 1 at uni was the single most boring time of my life except for woodwork at school). This may well make it not worth someone's time to plough through and gather the techniques needed. There is nothing wrong with that at all.
Now, for no reason, I will tell two tales of my attempts at being creative. Both happened at school.
Tale the first:
My first ever woodwork project at High School was to draw then cut out the shape of an animal in wood and mount it on a plinth. I did a child's drawing of a fish.
A few weeks later after we had filed, sanded and varnished the thing, the teacher asked us all for a pound to pay for the wood so that we could take it home. I declined the opportunity to do so, and so was asked to stay behind. Mr MadeUpName took me into his office and said "Look Donalbain, if you can't afford the pound.. don't worry about it. You can take it home for free."
I shook my head and replied. "No sir. I have the pound. I just don't want the fish, because it is rubbish."
His reply will stay with me forever; "Yeah, it is really isn't it?"
The fish went in the bin, I kept the pound and for the next three years I was allowed to read a book in woodwork instead of doing any work!
Tale the Second
In art class, we have been asked to draw a picture of the plant on my desk. I am determined to do this well. I simply ,ust create a piece of art that looks like the object I am drawing and so I study the plant. I look at the shapes it makes. I appreciate the light that falls on it. I become one with plant. Finally I know this plant and I look down to draw it. Carefully I draw the outline of a leaf. Then another. Then one more. Then I look up to get a new appreciation of the plant and it looks nothing like the piece of plant on my paper. Grumbling at my severe lack of discipline, I erase my drawing and study the plant again. Then I look down and draw once more. I take another glance at the plant and again, I have completely failed to capture its likeness. Forced to erase once more, I begin the process again, only to have the same result. And again my misery repeats itself. "One more try" I promise myself. I will get it this time. But as I place my pencil on the paper, I am gripped by paranoia and I need to take one more glimpse of the plant. And as I do, I catch my best friend Paul, who drew a photorealistic image of the plant in the first thirty seconds of the lesson, swapping my plant for another. Just like he had done every time I dropped my head to look at my paper.
"You fucking bastard!" I shout at the top of my voice.
For the next three years, I was not even allowed inside the art room.
The question then becomes "is Kit interested ENOUGH?" Now.. that probably sounds a little mean and aggressive, but is not meant to. Honest! The only person who can decide if she is interested enough is Kit. Yes, to get the techniques down pat can be hard. It can also be boring (Linear Algebra 1 at uni was the single most boring time of my life except for woodwork at school). This may well make it not worth someone's time to plough through and gather the techniques needed. There is nothing wrong with that at all.
Now, for no reason, I will tell two tales of my attempts at being creative. Both happened at school.
Tale the first:
My first ever woodwork project at High School was to draw then cut out the shape of an animal in wood and mount it on a plinth. I did a child's drawing of a fish.
A few weeks later after we had filed, sanded and varnished the thing, the teacher asked us all for a pound to pay for the wood so that we could take it home. I declined the opportunity to do so, and so was asked to stay behind. Mr MadeUpName took me into his office and said "Look Donalbain, if you can't afford the pound.. don't worry about it. You can take it home for free."
I shook my head and replied. "No sir. I have the pound. I just don't want the fish, because it is rubbish."
His reply will stay with me forever; "Yeah, it is really isn't it?"
The fish went in the bin, I kept the pound and for the next three years I was allowed to read a book in woodwork instead of doing any work!
Tale the Second
In art class, we have been asked to draw a picture of the plant on my desk. I am determined to do this well. I simply ,ust create a piece of art that looks like the object I am drawing and so I study the plant. I look at the shapes it makes. I appreciate the light that falls on it. I become one with plant. Finally I know this plant and I look down to draw it. Carefully I draw the outline of a leaf. Then another. Then one more. Then I look up to get a new appreciation of the plant and it looks nothing like the piece of plant on my paper. Grumbling at my severe lack of discipline, I erase my drawing and study the plant again. Then I look down and draw once more. I take another glance at the plant and again, I have completely failed to capture its likeness. Forced to erase once more, I begin the process again, only to have the same result. And again my misery repeats itself. "One more try" I promise myself. I will get it this time. But as I place my pencil on the paper, I am gripped by paranoia and I need to take one more glimpse of the plant. And as I do, I catch my best friend Paul, who drew a photorealistic image of the plant in the first thirty seconds of the lesson, swapping my plant for another. Just like he had done every time I dropped my head to look at my paper.
"You fucking bastard!" I shout at the top of my voice.
For the next three years, I was not even allowed inside the art room.
# posted by : 1:53 pm
: 1:53 pm
Okay, if you really believe you're fundamentally unable to grasp math and science..? Why? Is it because you're female? Because you're "right brained" and read a lot? Is it genetic? Is it the way you were raised? Why?
Well, I don't believe I'm fundamentally unable to grasp the basics, I just believe that there's a limit to my facilities that education couldn't penetrate. It's not a crippling limit. I can do some basics, I just get lost when it becomes too advanced - and my definition of 'advanced' is probably less advanced than many. If you're asking why I believe this to be the case, it's based on experience: I've just run into enough stuff that went over my head.
Or if you're asking why I think I have limits ... well, everyone's good at some stuff and bad at others. Any more detailed generalisation would be a bit silly, as I doubt the most advanced neurologists and psychologists could explain it. I'm just sceptical at suggestions that anyone can learn anything, if that means learning them up to a level of proper competence; they're usually made by people who have enough talent that they take it for granted.
Well, I don't believe I'm fundamentally unable to grasp the basics, I just believe that there's a limit to my facilities that education couldn't penetrate. It's not a crippling limit. I can do some basics, I just get lost when it becomes too advanced - and my definition of 'advanced' is probably less advanced than many. If you're asking why I believe this to be the case, it's based on experience: I've just run into enough stuff that went over my head.
Or if you're asking why I think I have limits ... well, everyone's good at some stuff and bad at others. Any more detailed generalisation would be a bit silly, as I doubt the most advanced neurologists and psychologists could explain it. I'm just sceptical at suggestions that anyone can learn anything, if that means learning them up to a level of proper competence; they're usually made by people who have enough talent that they take it for granted.
I think anyone who is interested ENOUGH can learn mathematics and physics to an undergraduate level.
Yes, but that's the point I'm suggesting: below a certain level of aptitude, interest is unlikely. A tone deaf person is not going to be very interested in music. People are generally interested in stuff they can do. Of course, if you're not interested, you're less likely to study, so it's self-perpetuating.
Yes, but that's the point I'm suggesting: below a certain level of aptitude, interest is unlikely. A tone deaf person is not going to be very interested in music. People are generally interested in stuff they can do. Of course, if you're not interested, you're less likely to study, so it's self-perpetuating.
But it seems to me you are saying two mutually exclusive things. You are saying that you COULD learn it but arent interested, but then you are saying that you CANT learn it because it is beyond your limits.
# posted by : 2:18 pm
: 2:18 pm
Not really. I'm saying that there's a limit to what I can learn - as I said, I'm not saying I'm completely incapable of learning, just that my competence stops at a certain level, and I don't think I could really learn past that - and that, in consequence, I find it hard to interest myself in it, because it's hard to be interested in something confusing.
No -- it's hard to be interested in something that's not confusing!
1+1=2? Boring.
(infinite terms) = 2? Interesting!
1+1=2? Boring.
(infinite terms) = 2? Interesting!
Also, I said something wrong up there. I said "since the same of 1/2^n is 2 no matter how big "n" gets..." But really the sum is just slightly less than 2 for any finite number of terms... Just as you originally said.
It's only because it gets closer and closer to 2 as you include more and more that we say it equals to when you include all infinitely many. I hope I didn't confuse you with that...
It's only because it gets closer and closer to 2 as you include more and more that we say it equals to when you include all infinitely many. I hope I didn't confuse you with that...
Yes, that did confuse me; I was assuming that there was some final answer that could be expressed by an equation or power-of or some other mathematical notation. So 'slightly less than two' was actually right?
For a finite number of terms, exactly right. The more terms you include, the closer it gets to two.
And there is a formula for it, in fact. If
s = 1 + q + q^2 + q^3 + ...
and we multiply both sides by q..
qs = q + q^2 + q^3 + ...
And then subtract the second equation from the first, we get
s - qs = 1
so
s = 1 / (1 - q)
That's the formula. (Derivation cribbed from my favorite math site
You'll notice that when q = 1/2, you get 1 / (1/2) which is 2.
But for me, this is a much less satisfying answer to the question than drawing the squares is. Drawing the squares shows you why you can add infinite terms and come up with a finite answer. The formula is mere symbol manipulation, and I don't trust symbol manipulation for complex problems.
If you do find yourself getting interested, Wikipedia actually has a nice page on this type of sum (which is called a "geometric series" by the way.)
And there is a formula for it, in fact. If
s = 1 + q + q^2 + q^3 + ...
and we multiply both sides by q..
qs = q + q^2 + q^3 + ...
And then subtract the second equation from the first, we get
s - qs = 1
so
s = 1 / (1 - q)
That's the formula. (Derivation cribbed from my favorite math site
You'll notice that when q = 1/2, you get 1 / (1/2) which is 2.
But for me, this is a much less satisfying answer to the question than drawing the squares is. Drawing the squares shows you why you can add infinite terms and come up with a finite answer. The formula is mere symbol manipulation, and I don't trust symbol manipulation for complex problems.
If you do find yourself getting interested, Wikipedia actually has a nice page on this type of sum (which is called a "geometric series" by the way.)
Donalblain: Finally I know this plant and I look down to draw it. Carefully I draw the outline of a leaf. Then another. Then one more.
Hmm. Let me say, then, you had a terrible art teacher.
When I teach drawing, I emphasize that "You are not trying to reproduce the reality of the object. You are trying to express what it LOOKS LIKE."
Nobody looks at a plant and sees every leaf individually. I would have tried to teach you to look at that plant and see the cylinder of the container, the sort of sphere-shape of this bunch of leaves, how overlaps the egg-shape of that bunch of leaves, how the leaves between them are a small dark mass like an ice cream cone...
Then I would have shown you how the light along a few leaf edges shines white, not green, and to indicate those curves among those masses...
And so forth. (it's hard to describe in words)
But drawing each leaf is exactly the WRONG way to go. Your friend did you a favor.
Hmm. Let me say, then, you had a terrible art teacher.
When I teach drawing, I emphasize that "You are not trying to reproduce the reality of the object. You are trying to express what it LOOKS LIKE."
Nobody looks at a plant and sees every leaf individually. I would have tried to teach you to look at that plant and see the cylinder of the container, the sort of sphere-shape of this bunch of leaves, how overlaps the egg-shape of that bunch of leaves, how the leaves between them are a small dark mass like an ice cream cone...
Then I would have shown you how the light along a few leaf edges shines white, not green, and to indicate those curves among those masses...
And so forth. (it's hard to describe in words)
But drawing each leaf is exactly the WRONG way to go. Your friend did you a favor.
# posted by : 4:40 pm
: 4:40 pm
Kit of the White Field: A tone deaf person is not going to be very interested in music.
Beg to differ. Anyone who knows me would describe me as hopelessly musically inept, and yet I am very interested indeed. Interested enough to spend years painfully learning the techniques to pick out a tune on a piano, and to read the score of an opera.
And there are many people who are literally deaf from birth who take great interest in music, both in written form and feeling it through the vibrations.
I'll never be able to play or sing WELL. But that doesn't mean I cannot appreciate the aesthetics thereof.
(Oh, and I forgot my last word verification. This one is "vjessi", a small furred Roumanian house demon, that resembles a silver weasel)
Beg to differ. Anyone who knows me would describe me as hopelessly musically inept, and yet I am very interested indeed. Interested enough to spend years painfully learning the techniques to pick out a tune on a piano, and to read the score of an opera.
And there are many people who are literally deaf from birth who take great interest in music, both in written form and feeling it through the vibrations.
I'll never be able to play or sing WELL. But that doesn't mean I cannot appreciate the aesthetics thereof.
(Oh, and I forgot my last word verification. This one is "vjessi", a small furred Roumanian house demon, that resembles a silver weasel)
# posted by : 4:45 pm
: 4:45 pm
Anyone who knows me would describe me as hopelessly musically inept
Well, examples tend to fall down. Are you literally tone deaf, unable to tell one note from another, or do you just struggle to play it right?
Well, examples tend to fall down. Are you literally tone deaf, unable to tell one note from another, or do you just struggle to play it right?
Well, I can tell one note from another when I hear it. I mean, I take enormous delight in the relationship and interplay of notes and rhythms -- to me, it's extremely visual, a matter of interlocking mathematical proportions.
But the ability to reproduce it -- to produce a recognizable (or even pleasant) sound from any instrument or voice -- the only way I can do it is to strictly apply the painstakingly memorized techniques of "hit THIS button for a count SO long" and hope that I did it right.
Today's verification word -- "resse" -- is a long forgotten Latin verb, meaning "the state of being That Thing."
But the ability to reproduce it -- to produce a recognizable (or even pleasant) sound from any instrument or voice -- the only way I can do it is to strictly apply the painstakingly memorized techniques of "hit THIS button for a count SO long" and hope that I did it right.
Today's verification word -- "resse" -- is a long forgotten Latin verb, meaning "the state of being That Thing."
# posted by : 7:33 pm
: 7:33 pm
the trick to singing is figuring out if you match the pitch of the person standing next to you. This involves befriending someone who does sing, standing next to them and listening very carefully all the while watching for grimaces and smiles. Train the ear, then the sound.
Legally blind; perfect pitch? You're hired!
Hapax? Have you tried percussion instruments? Or, you could start with the Kodaly method, or borrow Orff instruments from a friendly elementary school music teacher. It's a great way to start with building blocks and work your way up.
Also, find a copy of the score for Terry Riley's In C. Very fun.
Legally blind; perfect pitch? You're hired!
Hapax? Have you tried percussion instruments? Or, you could start with the Kodaly method, or borrow Orff instruments from a friendly elementary school music teacher. It's a great way to start with building blocks and work your way up.
Also, find a copy of the score for Terry Riley's In C. Very fun.
# posted by : 9:14 pm
: 9:14 pm
OK, as a math person, I want to point out that there are not "levels" of math. People seem to keep referring to this concept, but, well, it doesn't really exist. If Praline got stuck on calculus, why not do, well, something else? Especially as there's so much basic stuff the schools neglect, like basic number theory... well, I guess it could be different in Britain, I have no idea.
I mean, so many people go to college and learn calculus, but, well, they don't really learn math. OTOH you consider something like the Ross program or the PROMYS program where they make you work out basic number theory from scratch - you make it through that and you've learned a lot about how to think like a mathematician.
(Not that there's anything special about number theory - but I think it's probably especially accessible. I mean, everyone knows about the integers, right? We all know how to count, right?)
I don't think you can really teach much in this medium - certainly not a whole "work this out yourself" thing - but I think I can probably fit some small thing here, and I think I know what; I'll return later as I need to figure out exactly how I'm going to present it...
Also:
"Obviously that's no drill. It's a-"
I think that one might be a bit unclear, though...
I mean, so many people go to college and learn calculus, but, well, they don't really learn math. OTOH you consider something like the Ross program or the PROMYS program where they make you work out basic number theory from scratch - you make it through that and you've learned a lot about how to think like a mathematician.
(Not that there's anything special about number theory - but I think it's probably especially accessible. I mean, everyone knows about the integers, right? We all know how to count, right?)
I don't think you can really teach much in this medium - certainly not a whole "work this out yourself" thing - but I think I can probably fit some small thing here, and I think I know what; I'll return later as I need to figure out exactly how I'm going to present it...
Also:
"Obviously that's no drill. It's a-"
I think that one might be a bit unclear, though...
# posted by : 11:32 pm
: 11:32 pm
Have you tried percussion instruments?
Yep. No sense of rhythm, either.
Besides, most percussion instruments have tones and notes, too.
Honestly, I come from a very musical family, and my in-laws include professionals. LOTS of people have tried to teach me this stuff.
Like I said, I can be taught to press the right key, and count to three. That's about it.
Yep. No sense of rhythm, either.
Besides, most percussion instruments have tones and notes, too.
Honestly, I come from a very musical family, and my in-laws include professionals. LOTS of people have tried to teach me this stuff.
Like I said, I can be taught to press the right key, and count to three. That's about it.
# posted by : 11:54 pm
: 11:54 pm
OK, let's try this...
Consider a clock. A clock is labeled with 12 numbers, from 1 to 12. We can define a way to add the numbers on the clock. We'll simply go around the clock in the obvious way. So 10+3=1, 8+6=2, 11+11=10, etc. Note that for any of these numbers x, x+12=12+x=x, so it makes sense to replace 12 with 0.
(Though, it's not really right to think of it as "there is no 12"; rather, there is a 12, it's just that 12 and 0 are the same; 24 is also 0; 1, 13, 25, and -11 are all the same; etc. But if this confuses you, you don't have to think about it this way.)
Now we've been talking about a clock with 12 hours on it, from 0 to 11, but there's no reason it has to have 12 hours; we could just as well consider a clock with 7 hours, from 0 to 6. Then 5+5=3, 3+4=0, etc. (It may help you to draw a picture of an actual 7-houred clock.)
We denote the clock of m hours by Z_m. (Z is the symbol for the integers, from the German "Zahlen". Also, that m should be in subscript, but it won't let me do subscripts, so I'll use underscores instead.)
Now I've talked about addition, but in fact we can multiply as well. Let's consider Z_5. We multiply in the obvious way, so just as 3+3=6=1, 2*3=6=1. 3*3=9=4. And 4*3=12=7=2, you see? In Z_m, every number is just reduced to its remainder when you divide by m. 4*4=16=1. There are always precisely m elements in Z_m, represented by the numbers 0, 1, 2, ..., m-1. Every other number is equivalent to one of these. (To repeat what I said earlier, it's not that those numbers don't exist in the system; it's just that they coincide with other numbers in the system.)
So we've got this system, Z_m - or rather this collection of systems, as it's different for every m. And it (rather, they; I'm going to stop doing this now) has this operation we call addition, and this operation we call multiplication.
So we want to consider "subtraction" and "division". What do these mean? Well, to say a-b=x is really to say that x+b=a. To say that -b=x is to say that x+b=0. [Terminology note: When x+b=0, we say that x and be are additive inverses.] Now in Z, regardless of a and b, it is always true that there is an x such that x+b=a, and what's more, there is only one such x.
Of course (still talking about Z), this is not true of multiplication. We want x=a/b to mean that bx=a; but this equation does not always have a solution. We can find a meaning for, say, 4/2, and 6/2, but if we try to talk about 5/2, we have a problem. There is no x such that 2x=5. Neither is there an x such that 2x=1. [Terminology note: when xb=1, we say x and b are multiplicative inveres.] And if we try to consider 0/0, well, there is an x such that 0x=0; but unfortunately there's more than one, so it's not clear what 0/0 should mean - after all a symbol can only stand for one thing.
Now we are going to consider similar questions in Z_m.
So, here are some questions to get you started - first, the easy part:
Which of the following exist/make sense? (And for that matter, what are they?)
-1, -2, -3, 4-1, 4-5, 12-6, 3-3
in
Z_2, Z_3, Z_4, Z_5, Z_6
[Don't actually waste your time doing all these if you don't see a reason to... the point of these problems is just to be food for thought, to get you thinking about a-b in Z_m.]
In general: When does a-b make sense in Z_m, and how can you compute it?
OK, now for the not-so-easy part.
Which of the following make sense? (And for that matter, what are they?)
1/2, 1/3, 1/4, 2/3, 2/4, 4/2, 3/3, 3/2, 5/3, 0/2, 0/4, 3/6, 6/3, 2/6, 6/2, 1/5, 1/6, 2/1, 3/1.
in
Z_4, Z_5, Z_6, Z_7, Z_8, Z_9
[Again, just more food for thought.]
Remember: Z_m only has finitely many elements, so if you don't see an easy way to do the calculation, you can just brute-force the problem by trying all the elements to see which ones work.
Unless you're crazy-smart, you probably will not be able to quickly figure out in general when a/b makes sense in Z_m. However, you should notice some patterns, even if you have a pretty incomplete idea what they are, or can't really describe them.
Also, note that multiplicative inverses - the 1/b problems - are especially important; why?
Are there any m for which things work out particularly nicely?
Consider a clock. A clock is labeled with 12 numbers, from 1 to 12. We can define a way to add the numbers on the clock. We'll simply go around the clock in the obvious way. So 10+3=1, 8+6=2, 11+11=10, etc. Note that for any of these numbers x, x+12=12+x=x, so it makes sense to replace 12 with 0.
(Though, it's not really right to think of it as "there is no 12"; rather, there is a 12, it's just that 12 and 0 are the same; 24 is also 0; 1, 13, 25, and -11 are all the same; etc. But if this confuses you, you don't have to think about it this way.)
Now we've been talking about a clock with 12 hours on it, from 0 to 11, but there's no reason it has to have 12 hours; we could just as well consider a clock with 7 hours, from 0 to 6. Then 5+5=3, 3+4=0, etc. (It may help you to draw a picture of an actual 7-houred clock.)
We denote the clock of m hours by Z_m. (Z is the symbol for the integers, from the German "Zahlen". Also, that m should be in subscript, but it won't let me do subscripts, so I'll use underscores instead.)
Now I've talked about addition, but in fact we can multiply as well. Let's consider Z_5. We multiply in the obvious way, so just as 3+3=6=1, 2*3=6=1. 3*3=9=4. And 4*3=12=7=2, you see? In Z_m, every number is just reduced to its remainder when you divide by m. 4*4=16=1. There are always precisely m elements in Z_m, represented by the numbers 0, 1, 2, ..., m-1. Every other number is equivalent to one of these. (To repeat what I said earlier, it's not that those numbers don't exist in the system; it's just that they coincide with other numbers in the system.)
So we've got this system, Z_m - or rather this collection of systems, as it's different for every m. And it (rather, they; I'm going to stop doing this now) has this operation we call addition, and this operation we call multiplication.
So we want to consider "subtraction" and "division". What do these mean? Well, to say a-b=x is really to say that x+b=a. To say that -b=x is to say that x+b=0. [Terminology note: When x+b=0, we say that x and be are additive inverses.] Now in Z, regardless of a and b, it is always true that there is an x such that x+b=a, and what's more, there is only one such x.
Of course (still talking about Z), this is not true of multiplication. We want x=a/b to mean that bx=a; but this equation does not always have a solution. We can find a meaning for, say, 4/2, and 6/2, but if we try to talk about 5/2, we have a problem. There is no x such that 2x=5. Neither is there an x such that 2x=1. [Terminology note: when xb=1, we say x and b are multiplicative inveres.] And if we try to consider 0/0, well, there is an x such that 0x=0; but unfortunately there's more than one, so it's not clear what 0/0 should mean - after all a symbol can only stand for one thing.
Now we are going to consider similar questions in Z_m.
So, here are some questions to get you started - first, the easy part:
Which of the following exist/make sense? (And for that matter, what are they?)
-1, -2, -3, 4-1, 4-5, 12-6, 3-3
in
Z_2, Z_3, Z_4, Z_5, Z_6
[Don't actually waste your time doing all these if you don't see a reason to... the point of these problems is just to be food for thought, to get you thinking about a-b in Z_m.]
In general: When does a-b make sense in Z_m, and how can you compute it?
OK, now for the not-so-easy part.
Which of the following make sense? (And for that matter, what are they?)
1/2, 1/3, 1/4, 2/3, 2/4, 4/2, 3/3, 3/2, 5/3, 0/2, 0/4, 3/6, 6/3, 2/6, 6/2, 1/5, 1/6, 2/1, 3/1.
in
Z_4, Z_5, Z_6, Z_7, Z_8, Z_9
[Again, just more food for thought.]
Remember: Z_m only has finitely many elements, so if you don't see an easy way to do the calculation, you can just brute-force the problem by trying all the elements to see which ones work.
Unless you're crazy-smart, you probably will not be able to quickly figure out in general when a/b makes sense in Z_m. However, you should notice some patterns, even if you have a pretty incomplete idea what they are, or can't really describe them.
Also, note that multiplicative inverses - the 1/b problems - are especially important; why?
Are there any m for which things work out particularly nicely?
# posted by : 2:43 am
: 2:43 am
"My helmet? Crap, I left it-"
This "disaster befalls the character" technique is pretty useful, I have to say. :D
This "disaster befalls the character" technique is pretty useful, I have to say. :D
# posted by : 9:51 am
: 9:51 am
Thinking about it, a good six-word story was part of "2001: A Space Odyssey"
"My God, it's full of stars!"
"My God, it's full of stars!"
No comment about math or otherwise, but just had to share today's verification word: "nuckslyc" -- I've met that guy. You know, That Guy. With too much hair gel and too-tight pants, who tries to pick you up with the sleaziest, most insulting come-ons straight out of '70s sitcoms...
I just didn't know what he was called before. A Nuckslyc.
Six word story about him:
"Hey, sexy... holy shit, what's THAT?"
I just didn't know what he was called before. A Nuckslyc.
Six word story about him:
"Hey, sexy... holy shit, what's THAT?"
# posted by : 9:18 pm
: 9:18 pm
The webcomic Sketchies (www.sketchies-comic.com) runs what they call, "Six-Word Superhero Summaries Shown Saturdays." Here are the last six:
V for Vendetta: Ah, alliteration… An amazingly accessible activity.
Dr. Doom: World Conqueror… but ugly. RICHARDS' FAULT!
Captain Marvel: A wizard did it. No, seriously.
Invisible Girl: It's how Reed sees me anyway…
Spawn: Capes and chains are SO DOPE.
Cyclops: Nailing Emma Frost… Who's lame now?
V for Vendetta: Ah, alliteration… An amazingly accessible activity.
Dr. Doom: World Conqueror… but ugly. RICHARDS' FAULT!
Captain Marvel: A wizard did it. No, seriously.
Invisible Girl: It's how Reed sees me anyway…
Spawn: Capes and chains are SO DOPE.
Cyclops: Nailing Emma Frost… Who's lame now?
# posted by : 4:29 am
: 4:29 am
I say one can be interested in things for which one has very little innate aptitude, but that an important middle step is learning the language. Some folks with talent will pick up the language, i.e. the techniques that allow one to do math, music, drawing, dancing, combat, etc., quickly, but most of us need to be taught by someone who is both skilled and a good teacher.
Sniffoy makes good points about math.
Sometimes people confuse innate talent with early-life exposure and encouragement. They are not always easy to tell apart.
The "writing is acting all the parts thing" is why I enjoy writing and not acting or role-playing. I want to be in charge.
Let's see if Blogger eats this post, like it did my lovely and thoughtful post on the Writing Diet thread.
Sniffoy makes good points about math.
Sometimes people confuse innate talent with early-life exposure and encouragement. They are not always easy to tell apart.
The "writing is acting all the parts thing" is why I enjoy writing and not acting or role-playing. I want to be in charge.
Let's see if Blogger eats this post, like it did my lovely and thoughtful post on the Writing Diet thread.
# posted by : 11:31 pm
: 11:31 pm
Soldier: Glory good trade for arm.
I'm not good at these, but they're fun. Practice makes skill. Next thing you know I'll be writing sestinas to practice.
I'm not good at these, but they're fun. Practice makes skill. Next thing you know I'll be writing sestinas to practice.
# posted by : 11:36 pm
: 11:36 pm
A six-word story inspired by Bareback:
"Hi honey! Torturers treating you okay?"
Word verification: rearizer. For those real buttheads.
"Hi honey! Torturers treating you okay?"
Word verification: rearizer. For those real buttheads.
Kit: I'm a math tutor at our local community college. Are you still interested in tackling a math question or two?
Six word story: "Shoes stopped fitting in month 8." (Inspired by current events! Well, current events in *my* life, anyway.)
Word Ver: lemme see if I get how this game works. "pretru" is a slang term referring to telepathic criminal investigators in the 29th Century.
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Six word story: "Shoes stopped fitting in month 8." (Inspired by current events! Well, current events in *my* life, anyway.)
Word Ver: lemme see if I get how this game works. "pretru" is a slang term referring to telepathic criminal investigators in the 29th Century.
# posted by : 6:38 pm
: 6:38 pm<< Home
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