Today is very hot - I couldn't resist posting this, the ultimate water and maths experience! (MC Escher again of course).
Monday, February 28, 2011
Water
Today is very hot - I couldn't resist posting this, the ultimate water and maths experience! (MC Escher again of course).
Sunday, February 27, 2011
Postmodern(ish) Literature and Cricket
I've just finished reading an extremely unusual novel by Graham Greene called Travels With My Aunt. Greene usually writes thrillers. This novel is very different.
Published in 1969, it's a surreal travel novel attempting to come to grips with the realities of a postmodern, post colonial era. The hero is Henry Pulling, a boring British retired bank manager who grows dahlias as a hobby. A less exciting Leopold Bloom I suppose. Less exciting at first, anyway...
Well I won't ruin the plot, but this novel goes off in almost as many unexpected directions as its almost aggressively normal hero.
Ultimately, Henry Pulling loses his confidence in the ideal of British Best and British Empire, despite mistrusting the supposed alternatives. Nonetheless, he clings to what he sees as typically British conventions of honour and appropriate moral behaviour.
This irony is echoed by today's ICC World Cup Cricket match. India is playing England. Given these two nations' colonial history, the fact that they now meet to compete at "the gentleman's game" in (now independent) India seems to represent the ultimate conflict between Imperial ideals and traditional values.
Which is the true image of the long gone British empire? Probably both, or neither. Our only sources of information are multiple, kaleidoscopic images which leave us no less confused than when we started.
Thursday, February 24, 2011
Fractals
Fractals are one of the few visual tastes of infinity available to us...
And amazingly, they are present in astonishing quantities throughout nature...
Though maybe not this one!
George Herbert through the centuries
I STRUCK the board, and cry’d, No more ;
I will abroad.
-George Herbert, "The Collar"
Have you ever felt like that? I certainly have...
What's fascinating about this very famous poem is that it was written by a humble clergyman (i.e. minister/priest) who also happened to be a very brilliant poet. George Herbert gave up a very lucrative political/academic career to follow his religious convictions... and the poetry he wrote about this and other opinions and decisions is still with us today.
Personally, I find it enormously comforting to know that even such a morally upstanding gentleman occasionally experienced such (unfortunately familiar) frustration.
If you're interested, the rest of the poem (which is well worth reading) can be found here:
http://www.luminarium.org/sevenlit/herbert/collar.htm
Wednesday, February 23, 2011
Definitions and Theorems
So no-one was brave enough to take a guess?
Well, the problem lies in the fact that I failed to differentiate between definitions and theorems. And the difference is crucial. Absolutely crucial.
A definition is basically the naming of a certain property or group of properties. So for example, say we observe that pretty often the order of multiplication in the natural numbers does not change the value of the product (i.e. 3x4=4x3). So instead of saying that whole shpiel each time, we decide to define the term "commutative" (which in "real maths" loosely means a x b = b x a or a + b = b + a). You can't prove what commutative means. That's just what you've decided to call that particular property.
If you are alert, you will observe that this is pretty similar to how I explained what an axiom was. I promise that there is an important distinction. An axiom is something you assume to always be true. A definition is simply the name you give to something which is sometimes true in some situations.
So, back to theorems and definitions.
So we've defined commutativity. Now the more we look at the natural numbers, the more we get the feeling that all natural numbers will be commutative under multiplication. Now we're getting into theorem territory. The minute we make the claim "natural numbers are commutative under multiplication (i.e. it doesn't matter which order you multiply natural numbers in)", we are saying that our definition of commutativity applies to a very large group of numbers.
This has to be proved. And I'm not going to tackle that enormous topic now. Suffice it to say that it is important to distinguish clearly (in Maths notes or elsewhere) between theorems and definitions.
Briefly (and slightly poetically):
Well, the problem lies in the fact that I failed to differentiate between definitions and theorems. And the difference is crucial. Absolutely crucial.
A definition is basically the naming of a certain property or group of properties. So for example, say we observe that pretty often the order of multiplication in the natural numbers does not change the value of the product (i.e. 3x4=4x3). So instead of saying that whole shpiel each time, we decide to define the term "commutative" (which in "real maths" loosely means a x b = b x a or a + b = b + a). You can't prove what commutative means. That's just what you've decided to call that particular property.
If you are alert, you will observe that this is pretty similar to how I explained what an axiom was. I promise that there is an important distinction. An axiom is something you assume to always be true. A definition is simply the name you give to something which is sometimes true in some situations.
So, back to theorems and definitions.
So we've defined commutativity. Now the more we look at the natural numbers, the more we get the feeling that all natural numbers will be commutative under multiplication. Now we're getting into theorem territory. The minute we make the claim "natural numbers are commutative under multiplication (i.e. it doesn't matter which order you multiply natural numbers in)", we are saying that our definition of commutativity applies to a very large group of numbers.
This has to be proved. And I'm not going to tackle that enormous topic now. Suffice it to say that it is important to distinguish clearly (in Maths notes or elsewhere) between theorems and definitions.
Briefly (and slightly poetically):
Theorem = Claim
Definition = Name
Sunday, February 20, 2011
By Definition
When I was young and foolish I used to make the most beautiful Maths notes you've EVER seen. I had different coloured pens, and each chapter had a colour with matching headings. Theorems and definitions were done in a certain colour so that I knew they were important and there was yet another colour for examples and helpful hints...
What was I doing wrong?
Give it a guess, oh reader! I'll post the answer next time...
What was I doing wrong?
Give it a guess, oh reader! I'll post the answer next time...
Tuesday, February 15, 2011
Constructive Daydreaming?
I was going to edit my old post with this new thought, but then I decided that it really was a different (though related) thought...
So I was wondering exactly why constructive activity (like learning about Maths or anything else) IS useful. You could say that it runs society, but in my opinion its debatable whether our society actually benefits the human race much at all.
My best answer is the constructive activity develops your capacity for daydreaming in a more interesting, and (dare I say it?) constructive way. When I was a baby I doubt my daydreaming went further than wondering when mum would come and give me my next meal. Yet Albert Einstein, who presumably had fairly similar baby-daydreams to me, went on to dream up E=mc2! At some point along the line, development must have taken place.
Was that development the result of constructive activity in some sense? Who knows. But that's my best suggestion...for now.
So I was wondering exactly why constructive activity (like learning about Maths or anything else) IS useful. You could say that it runs society, but in my opinion its debatable whether our society actually benefits the human race much at all.
My best answer is the constructive activity develops your capacity for daydreaming in a more interesting, and (dare I say it?) constructive way. When I was a baby I doubt my daydreaming went further than wondering when mum would come and give me my next meal. Yet Albert Einstein, who presumably had fairly similar baby-daydreams to me, went on to dream up E=mc2! At some point along the line, development must have taken place.
Was that development the result of constructive activity in some sense? Who knows. But that's my best suggestion...for now.
Hurry
The terribly sad thing is that so often I just don't have a chance to daydream. I guess a lot of us have that problem. I could even go so far as to say if we all daydreamed more regularly then maybe the world would be a much better place.
The problem is we're all in such a hurry to do everything. I'm guilty of this myself. I'm even guilty of pushing others (mainly my students) to daydream less and hurry more. Isn't that just sad?
Of course without a bit of focused activity the world wouldn't really go anywhere. The trick I suppose is to achieve a balance between focused activity and the daydream which creates new worlds.
The problem is we're all in such a hurry to do everything. I'm guilty of this myself. I'm even guilty of pushing others (mainly my students) to daydream less and hurry more. Isn't that just sad?
Of course without a bit of focused activity the world wouldn't really go anywhere. The trick I suppose is to achieve a balance between focused activity and the daydream which creates new worlds.
Saturday, February 12, 2011
The Art of Losing Gracefully
On Friday the U14 waterpolo team I manage played against the top waterpolo school in South Africa. The score was [this information has been withheld for the protection of the parties concerned]. Carnage!
Being my normal competitive self, this was hardly the start to the weekend that I would have chosen. But talking to the girls afterwards I found myself in the middle of an action packed tutorial (in which I was both the teacher and a student) on losing gracefully.Some of the modules follow:
Rule Number One: believe in your team till the bitter end
Rule Number Two: encourage each other, even when things are going badly
Rule Number Three: still believe in your team after the bitter end
Rule Number Four: give respect and honour to the opponents
Rule Number Five: don't blame the ref!
I could go on. But the tutorial isn't over yet (though I hope that next week's match provides a chance to learn the art of winning gracefully).
You may want to know what this post has to do with daydreaming in maths. The truth is that a similar state of mind is attached to losing in all spheres of life. And from my experience in the classroom, a lot of people feel like they're losing badly to Mathematics, one of the toughest opponents of their school career.
If you are one of those people: it isn't over until its over - you can still conquer Maths!
And once it's over, even if you lost - believe in yourself!
You aren't a failure because you lost. Take it from someone who knows.
Being my normal competitive self, this was hardly the start to the weekend that I would have chosen. But talking to the girls afterwards I found myself in the middle of an action packed tutorial (in which I was both the teacher and a student) on losing gracefully.Some of the modules follow:
Rule Number One: believe in your team till the bitter end
Rule Number Two: encourage each other, even when things are going badly
Rule Number Three: still believe in your team after the bitter end
Rule Number Four: give respect and honour to the opponents
Rule Number Five: don't blame the ref!
I could go on. But the tutorial isn't over yet (though I hope that next week's match provides a chance to learn the art of winning gracefully).
You may want to know what this post has to do with daydreaming in maths. The truth is that a similar state of mind is attached to losing in all spheres of life. And from my experience in the classroom, a lot of people feel like they're losing badly to Mathematics, one of the toughest opponents of their school career.
If you are one of those people: it isn't over until its over - you can still conquer Maths!
And once it's over, even if you lost - believe in yourself!
You aren't a failure because you lost. Take it from someone who knows.
Nick's Mathematical Puzzles
I found something!
http://www.qbyte.org/puzzles
This is an interesting site... All the puzzles are supposed to be accessible to high school level maths. I can't solve all of them - but at least hints, answers and solutions are provided!
http://www.qbyte.org/puzzles
This is an interesting site... All the puzzles are supposed to be accessible to high school level maths. I can't solve all of them - but at least hints, answers and solutions are provided!
Friday, February 11, 2011
Language, Axioms and Bananas
Talking about Algebra with my grade 8 class today, I was struck again by how confusing and apparently random Maths can be. I remember being totally mystified by Maths at various points in my relationship with the subject. Occasionally I still am mystified.
But something that has helped me to deal with my mystification is to remember that Maths as we know it is not the be all and end all and only conceivable possible way. It could (theoretically anyway) have all happened completely differently.
How? I hear you cry.
Well, Maths is a language, just like other languages (despite having a few extra features and lower comprehensibility ratings than most). The fact that "banana" represents banana does not say something innate about the relationship between word and object (this is Wittgenstein and those guys, but watered down). It just happens to be the word that history and society and chance and whatever other factors have attached to our familiar yellow fruit.
Same with Maths. All of our Mathematical system is based on certain axioms (like the fact that a point has no dimensions). These axioms (stated most famously by Euclid) are assumed to be true in the proofs and ways of thinking about things which are so entrenched in us. It is very difficult to imagine them not being true. Yet in actual fact this is only the case because its convenient that it should be, or because we're used to it being so.
Plus, of course, in order to do anything interesting or useful we have to have a common frame of reference. So by all means lets argue about something more interesting than whether we should write 2x2 as 2^2 or 2_2; or whether 3ab really does mean 3 x a x b!
The other alternative is very much like saying that banana is a very bad word and difficult to pronounce, so from now on the yellow fruit shall be known only as Ba.
But something that has helped me to deal with my mystification is to remember that Maths as we know it is not the be all and end all and only conceivable possible way. It could (theoretically anyway) have all happened completely differently.
How? I hear you cry.
Well, Maths is a language, just like other languages (despite having a few extra features and lower comprehensibility ratings than most). The fact that "banana" represents banana does not say something innate about the relationship between word and object (this is Wittgenstein and those guys, but watered down). It just happens to be the word that history and society and chance and whatever other factors have attached to our familiar yellow fruit.
Same with Maths. All of our Mathematical system is based on certain axioms (like the fact that a point has no dimensions). These axioms (stated most famously by Euclid) are assumed to be true in the proofs and ways of thinking about things which are so entrenched in us. It is very difficult to imagine them not being true. Yet in actual fact this is only the case because its convenient that it should be, or because we're used to it being so.
Plus, of course, in order to do anything interesting or useful we have to have a common frame of reference. So by all means lets argue about something more interesting than whether we should write 2x2 as 2^2 or 2_2; or whether 3ab really does mean 3 x a x b!
The other alternative is very much like saying that banana is a very bad word and difficult to pronounce, so from now on the yellow fruit shall be known only as Ba.
Thursday, February 10, 2011
Prime
Prime numbers are far more interesting than I ever realised. When I was first introduced to them I really didn't see what all the fuss was about; and its taken many years to gradually get a very rudimentary idea of some of their awesomeness...
So you probably know that a prime number is one which only has two factors: 1 and itself. In other words, it can't be broken down into smaller numbers. An example of a prime number is 23.
When you compare 23 to 24, a composite (non-prime) number, you get two totally different pictures. There are lots of ways of breaking 24 down into smaller pieces: 8x3, 6x4, 12x2 are some examples. If you break it down into the very smallest possible pieces (in fact, into prime pieces or factors) you end up with 2x2x2x3.
By contrast, 23 can't be broken down into smaller factors at all!
Of course 23 is quite a boring example. But when you find an extremely large prime number, like 243,112,609 − 1 (which has 12,978,189 digits) well then the fact that it has absolutely no factors other than 1 and itself is really quite impressive.
According to Charlie Epps, this has amazing applications in computer coding. According to Scarlett Thomas (in PopCo, which you HAVE to read immediately if you haven't already) it has exciting applications in codes as in secret codes.
From my extremely lay-person's point of view, it seems the key to this is that when you multiply two very large prime numbers together to make an even larger composite number, it is very difficult to work backwards and work out what those original two prime numbers were. Hence some serious encryption...
However, if like me you don't really care too much about the details of applications, it's still incredible to think of such an enormous number, and then to try get your mind around the fact that it is not made up of any smaller pieces.
12,978,189 digits?
Wow.
So you probably know that a prime number is one which only has two factors: 1 and itself. In other words, it can't be broken down into smaller numbers. An example of a prime number is 23.
When you compare 23 to 24, a composite (non-prime) number, you get two totally different pictures. There are lots of ways of breaking 24 down into smaller pieces: 8x3, 6x4, 12x2 are some examples. If you break it down into the very smallest possible pieces (in fact, into prime pieces or factors) you end up with 2x2x2x3.
By contrast, 23 can't be broken down into smaller factors at all!
Of course 23 is quite a boring example. But when you find an extremely large prime number, like 243,112,609 − 1 (which has 12,978,189 digits) well then the fact that it has absolutely no factors other than 1 and itself is really quite impressive.
According to Charlie Epps, this has amazing applications in computer coding. According to Scarlett Thomas (in PopCo, which you HAVE to read immediately if you haven't already) it has exciting applications in codes as in secret codes.
From my extremely lay-person's point of view, it seems the key to this is that when you multiply two very large prime numbers together to make an even larger composite number, it is very difficult to work backwards and work out what those original two prime numbers were. Hence some serious encryption...
However, if like me you don't really care too much about the details of applications, it's still incredible to think of such an enormous number, and then to try get your mind around the fact that it is not made up of any smaller pieces.
12,978,189 digits?
Wow.
Cunning Passages
History has many cunning passages, contrived corridors and issues.
T.S. ELIOT, Gerontion
TS Eliot's poetry is difficult (to say the least). Difficult to follow that is - I have no information as to whether or not it was also difficult to write. If you're unfamiliar with his work, then you should be able to have a look at some of his poems via this link:Black Cat Poems - TS Eliot.
A wise lecturer once explained to me that the chief difficulty with discerning what is going on in Eliot's poetry is that he has removed all the scaffolding with which we usually find our way around a poem. The window frames have been removed, leaving the glass floating mysteriously in mid-air.
Difficult old reprobate. It's also been said that he did this quite deliberately, to prevent the "plebs" from understanding his "oh so sophisticated" work...
Yet the poems themselves are still enormously powerful.
I guess jealousy makes me nasty.
***
I have heard the key
Turn in the door once and turn once only
We think of the key, each in his prison
Thinking of the key, each confirms a prison
Wednesday, February 9, 2011
Field Trips
After school today I marked 87 scripts. Some scripts were almost flawless, other struggled to make their meaning even vaguely accessible. Some covered fairly standard grade 9 work, others dealt with fairly advanced grade 12 material.
But they all had one thing in common. Each of them took me down a unique (and often alarming) journey through Mathematics.
The geography of Mathematics is far from dull.
There are the mountains when a wonderful insight comes from an unexpected source. The light of clarity shines on beautiful, competent (and legible) work.
There are also valleys, where it is dark and murky. Here the intrepid traveller peers through the mist, trying to find the path, or at least enough of the path to credit with some marks.
Sometimes the traveller happens upon a sparkling stream of new ideas. Sometimes she trips over a rock of a silly mistake. Occasionally she sinks into a bog of general misunderstanding and confusion...
Amazingly, each of these features (and doubtless many others that my metaphor has not yet uncovered) can be found in almost every single script at one point or another.
Not an afternoon wasted then?
But they all had one thing in common. Each of them took me down a unique (and often alarming) journey through Mathematics.
The geography of Mathematics is far from dull.
There are the mountains when a wonderful insight comes from an unexpected source. The light of clarity shines on beautiful, competent (and legible) work.
There are also valleys, where it is dark and murky. Here the intrepid traveller peers through the mist, trying to find the path, or at least enough of the path to credit with some marks.
Sometimes the traveller happens upon a sparkling stream of new ideas. Sometimes she trips over a rock of a silly mistake. Occasionally she sinks into a bog of general misunderstanding and confusion...
Amazingly, each of these features (and doubtless many others that my metaphor has not yet uncovered) can be found in almost every single script at one point or another.
Not an afternoon wasted then?
Maths Is Everywhere
This is my most favourite TV show at the moment, and thanks to You-Tube I can steal a little tiny bit of it to show to my classes and you, oh reader...
Proof and Imagination
A few years ago I saw a movie called Proof. It's about an elderly Maths professor and his daughter. He spends his whole life trying to prove a single theorem, and eventually gets Alzheimers or something like that. His daughter sacrifices her academic career to care for him... Anyway, I won't give the plot away, but the idea of the film centres around the great value of proof.
Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.
Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.
It's imagination that has a shape.
Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.
Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.
It's imagination that has a shape.
Tuesday, February 8, 2011
Mathecaticians
This is not a misspelling (though my spelling has always been a little bit suspect). It's a profound thought.
Gauss and Pascal are our kittens. They behave much as I imagine the original famous old mathematicians would have done. In other words:
They poke their noses into everything!
My favourite story about Herr Gauss (original) is probably entirely urban legend. I have not verified it, and if you know it to be false please keep that to yourself. It's a wonderful story.
So when he's about 8 or 9 (maybe 10, let's stretch a point) Gauss is sitting in his Maths class. As usual, he's completed all his work. And his homework. And the next day's homework. And his poor teacher is going crazy, probably partially because Gauss is pulling Heidi's hair and throwing thumbtacks at Kurt.
Finally, the teacher tells Gauss that his next assignment (I expect he was told to complete it before he went out to break) was calculate the sum of the first 100 natural numbers.
So, being eager to get to his bratwurst sandwich before Kurt stole it, Gauss does just that. Except, as he's a lazy little brat who doesn't feel like taking the long way round, he invents and proves a formula for calculating the sum of any arithmetic series.
No, I don't think you were paying sufficient attention: he proved the formula for calculating the sum of any arithmetic series.
The same elegant, easy-peasy-once-you-see-how-to-do-it proof that we use today...
Long live curiosity!
Daydreaming in Maths
I feel like a fraud.
Actually, I just feel like the same grade 9 girl who swore never EVER to study Maths for a single second longer than she absolutely had to.
Or maybe like the grade 12 girl who wrote sonnets, sestinas and villanelles in Maths lessons. In between absentmindedly creating calculus doodles.
Or like the university student who liked to sit next to the window during Maths lectures, watching the human traffic go by on Jammie plaza and mulling over one of John Donne's more obscure puns.
So this isn't a blog about Maths (despite the fact that I'm a Maths teacher, *gasp*). Nor is it a blog about English.
It's a blog about daydreaming.
Actually, I just feel like the same grade 9 girl who swore never EVER to study Maths for a single second longer than she absolutely had to.
Or maybe like the grade 12 girl who wrote sonnets, sestinas and villanelles in Maths lessons. In between absentmindedly creating calculus doodles.
Or like the university student who liked to sit next to the window during Maths lectures, watching the human traffic go by on Jammie plaza and mulling over one of John Donne's more obscure puns.
So this isn't a blog about Maths (despite the fact that I'm a Maths teacher, *gasp*). Nor is it a blog about English.
It's a blog about daydreaming.
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