Teaching fractals

As a a volunteer for CSIRO’s Mathematicians in Schools Program, I was asked to give a talk on fractals to girls in grade 9 at a school in Sydney.

I am not an expert on fractals, but as a doctoral candidate in mathematics, I figured I knew more than the the girls would, so I thought, why not? And I agreed.
This school was participating in the MegaMenger project. Their teacher wanted me to motivate the study of fractals by talking about some real world applications. I did talk about the usual suspects – animation, modeling natural systems etc. But I also managed to sneak in a point that mathematics educators always make – that ” what is this good for”, is not the right question to ask in a maths class. Paul Lockhart in his famous essay says it better than I could. But I like to extend his lament to the whole of education – how will this help me get a job is not quite the question that should be uppermost on young minds. The gifts of education are rather more intangible – curiosity and the ability to learn things, that’s what is going to get you a job. I know that these are not things one could put in one’s CV, but then, these are teens who have not yet outgrown their love for Beiber. And yet, theirs are the loudest voices in that dreaded chorus – “why are we doing this? When are we ever going to use it?”.

My rant aside, I think the activity went well. Here’s an appraisal of what worked and what didn’t.

What worked – I am usually sceptical of giving presentations to young students, as there’s no surer way of putting them to sleep. But this one was a talk with just a picture on each slide, with me talking around the picture. I also put in a fractal paper cutting activity. I was afraid that the students might find it boring, but everyone eagerly participated.

What didn’t – I did ask them a bunch of questions – what other examples of fractals they could think of, where did they think fractals could be applied etc. In hindsight, I think it would have been better to tie their answers into a discussion. Instead of accepting one workd answers, I should have asked them to elaborate.

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What’s with the calculators?

Reaching out for the calculator has become a reflex when students see anything that looks vaguely mathematical. It is very discouraging to see students painstakingly type in “3*1/25″ when presented with the fraction “3*1/25″. How did things come to such a pass, I wonder? We are always talking about “real life” uses of mathematics. We should probably be thinking about the mathematical uses of  “real life”. I am pretty confident that the same students can easily work out how much they will have to pay for 3 dozen eggs when a dozen cost $3. I am also sure they can work out the discounts in their head, choose the cheaper option. I don’t see hordes of people walking about in the supermarket punching numbers into their calculators. I am aware of the term  “innumeracy”, I know that lottery is a thriving industry, that people scare themseleves to death about Ebola, when they are more likely to be killed on their way home. But I am not talking about that. I am talking about the urge to pull out a calculator when encountering numbers in a classroom setting – even when multiplying by 100. It is almost as if having a calculator gives them comfort, and that is deeply worrying. I don’t care if my students do not take advanced mathematics, but I do care if numbers scare them.

How can I lead them from this fearfulness to fearlessness? This occupies a lot of my thought space these days, not that I have an answer for all that. It is an evolving story that I’ll continue to transcribe faithfully.

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Nothing to be frightened of – Julian Barnes

Our lack of originality is something we usually forget as we hunch over our – to us – ever fascinating lives. My friend M., leaving his wife for a younger woman used to complain, ‘People tell me it is a cliché. But it doesn’t feel like a cliché to me.’ Yet it was, and is. As all our lives would prove, if we could see them from a greater distance – from the viewpoint, say, of that higher creature imagined by Einstein.

… there is something infinitely touching when an artist, in old age, takes on simplicity. The artist is saying: display and bravura are tricks for the young, and yes, showing off is part of ambition: but now that we are old, let us have the confidence to speak simply. For the religious, this might mean becoming as a chid again in order to enter heaven; for the artist, it means becoming wise enough and calm enough, not to hide. Do you need all those extravagances in the score, all those marks on the canvas, all those exuberant adjectives? This is not just humility in the face of eternity; it is also that it takes a lifetime to see, and say, simple things.

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The Salmon Of Doubt

By one and only Douglas Adams.

“A couple of hours later he had the answer, or at least some kind of an answer. Nothing that went so far as to make any kind of actual sense, but enough to make Dirk feel an encouraging surge of excitement: he had managed to unlock a part of the puzzle. How big a part he didn’t know. As yet he had no idea how big a puzzle he was dealing with. No idea at all.”

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Schur’s Lemma

SchursLemma

SchursLemma2

 

 

 

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Representation of finite groups – notes in pictures

Of course, we have to find sub-structure within a structure – that’s what we do! I present before you, ladies and gentlemen, submodules.

submodule

 

 

And some new structure. How about building the vector space from the group? Group algebra – vector space + ring.

groupalgebra

…and then ask the group to act on it.

regularfgmodule

 

 

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Representation of Finite Groups – Notes in pictures

A representation is simply a homomorphism between a group and the group of invertible matrices over a field, which is usually the field of complex numbers.

representation-defn

 

There are several approaches to developing the theory of representations, one of the popular ones is via FG-modules.

FGmodule-defn

 

The connection between FG-modules and representations is simple – it is that of siamese twins.

moduleToRep

 

 

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