My DET miseries are almost at an end! I have only to do the following: mail Casual Direct registration forms, give resumes to schools, and then sit back and wait for work to come rolling in.
This is going to be awesome, because I will be GETTING OUT OF THE HOUSE, and making fat stacks of cashdollars. I know this isn’t exciting for those of you with real, grown-up jobs (Tom and Heather I am looking at YOU), but apparently my time is worth $300 a day. This is fairly huge for me. I have earmarked some of this potential money for certain things, and since we LOVE consumerism and lists here at AP5, I thought I’d write some of my plans down.
1. Lady-based cocktail evenings.
I am partially convinced that being a young woman of independant means can only be demonstrated if you occasionally go out for a few cocktails after work with other young women of independant means. I’m not talking $5 cocktails and all the bar fights you can dodge at the Ashfield Hotel, either – I mean fancy ones, at Marble Bar.
I have some fairly serious wanderlust at the moment, and making some fat teaching dollars would allow Tom and I to skip the country occasionally and have intercontinental adventures (different to incontinent adventures). Vietnam, Argentina, Italy… these are all places on my list.
I want to get me a fancy digital SLR. I bought my Canon Powershot about a year ago, and I’ve been spending the time redeveloping my eye. You’re all very patient about me getting out my camera at every available moment, so, thank you. But there is only so much this little clickything can do, and I crave something with a lens I can fiddle with. I am some kind of lenssexual (not lensexual, which is being attracted to dudes named Len). Then I will go on adventures to places
4. Moar Ikea
Oh man, the plans I have. MOAR Ikea. I want, like, the weird towels which are velour on one side, and that are the same colours as my bedspread. I want more kitchen stuff, like the Mjod glasses Dan picked up. I want my house to be awesome and beautiful and have good shit that looks amazing and does what it’s supposed to do. Maybe one day I will even buy a couch. A new couch, of my very own which isn’t some kind of futon.
Oh, Tom will play ALL the GTA4. All of it.
A Pleasing Fiveness, let’s have at it!
Okay, so, obviously some people are uninspired to post. I was fairly uninspired this week, and hence, a list. Sorry, guys. But I have some ideas!
– Maybe sometimes we could have a week where everyone blogs on the same topic, but from personal experience. Like, blog about a favourite childhood memory, or something like that. I think that could be really interesting.
– Requests! Is there anything that you guys would like me to write about? Anything where you go “That thing Julia does, I would like to hear her reasons for it”.
– Commenting! Guys, I think we need to try to comment more. Even if you just write something brief, I think it would help all of us to know that what we’re writing is being read and being appreciated.
I also have a second post coming for you, but it will have to wait for tomorrow, for Secret Reasons.
For the past few weeks, my life has been devoted to the teaching of mathematics. I’ve been lecturing Calculus at the Summer School, and teaching school leavers 2 unit mathematics in the form of an intensive bridging course. I’ve had a chance to interact directly with over 100 students, and their attitudes towards mathematics were similar to those I’ve encountered while tutoring privately. I thought I’d blog about the difficulties many students face, and how I’ve attempted to tackle them.
It’s counterintuitive and (therefore) hard
Mathematics attempts to deal with some rather abstract concepts. Most people will know a bit of mental arithmetic, but even something as simple as the “times tables” are drilled into us at a young age as part of their mathematical training. People tend to remember some very well, but not others – the 6’s, 7’s and 8’s are particularly confusing for most, since their ‘pattern’ is not as easy to identify (even though the pattern they make is rarely emphasised – rote learning is the key, for some reason).
Asking students to add fractions is an order of magnitude more difficult – it makes sense (according to how they think about addition) that 1/2 + 1/3 = 2/5, rather than the correct answer of 5/6. The necessity of generating a ‘common denominator’ is not something students understand. The fact that 2/5 is less than 1/2, and that adding a third to a half shouldn’t make it less than what you started with, or what 2/5 represents, doesn’t occur to them. What’s worst is when they don’t even know whether they’re wrong or right. They just write stuff down and hope it turns out to be correct, using rules that are often completely inappropriate for the given situation. This stage, which I encounter frequently, is a classic case of a mathematical collapse. The student is left feeling helpless and stupid, and the more you feel like that when you’re engaged in any activity, the more likely you will be to avoid it in the future.
And that’s just fractions, something you should have mastered (according to the syllabus) by the end of year 7. Once we start talking about the trigonometric functions like sine and cosine, students can’t put all the pieces together. It’s a function, rather than a number – something students cannot have automatic intuition for. Functions are mathematical machines, invented in our brains. They don’t exist anywhere else but on our calculators (who have a computer-designed approximation anyway). Thus you cannot add sine or divide by sine without knowing what you’re doing – applying the rules designed for numbers aren’t going to work here.
What is sine? It’s a function of angle that gives the height of a triangle inscribed on a unit (radius 1) circle with the given angle. Here’s a nice .gif:
What sine does is not so hard to explain, and cosine is nicely symmetric – it gives the horizontal distance rather than the height of the same triangle. It’s apparently hard enough, though, that most teachers will resort to “It’s a button on your calculator that you push. It’s equal to Opposite over Hypotenuse”. People who get this explanation are receiving explanations on the order of “World War 2 happened because Nazis were evil”. Easy, but oh so wrong. Perhaps you are dumb enough to need that kind of oversimplified explanation, but I sincerely doubt it.
The tools that mathematicians have developed to understand things such as massive quantities, probability, algebra and calculus are deeply unintuitive. Every method is developed with particular objects in mind, and generalising to other objects is hard – if it works for one thing, why doesn’t it work all the time? These questions are hard to answer, and even harder to answer in a 40 minute period with 30 other children in the class. This leads me nicely into:
Mathematics relies on generalisations of previous knowledge
Whenever I start tutoring students, whether they’re in year 7 or year 12, I always start with fractions. I’ve never had a student, not once, who could perform all the (rather elementary) operations without any mistakes. When you ask students to deal with something like fractional indices, you’re assuming that they can work with fractions. How can a teacher even begin to talk about fractional indices while their students can’t add a half and a third?
Many students get a blank look on their face when they find out they’re wrong after attempting to apply their reason to a problem. They have no idea why they’re wrong, but they don’t know what they don’t know. They gaze off in a helpless stupor as you try to explain to them what they should have done. They’re sure it all makes sense, but they no longer have any idea of why one approach will give the answer, whilst another will not.
Mathematics relies on the foundations that were (or should have been) laid in previous years. If you miss out on those lessons in fractions, or spend an entire period thinking “I hate my teacher” rather than concentrating on the concept of negative numbers, you start to fall behind. Most students never recover to a point where they can enjoy maths again, because it will forever be an uphill battle.
It’s my challenge as a teacher to locate these misunderstandings, correct them and press on with the more difficult material before the stupor sets in.
Mathematics is poorly motivated
“When are we ever going to use this?” is a question I get asked all the time. Mathematics is the kind of subject where the motives aren’t always clear, but the question always seems like an excuse for not working or succeeding whenever my students ask it. Clearly, if it’s all just fanciful nothingness, it’s not so bad if I can’t do it, right?
Sure, students might not be able to see exactly how to use mathematics to improve their life, but I don’t see how it’s any different to how other subjects work. Any benefit you will get out of learning most subjects will be tangential. You never hear:
“This perfectly executed sentence got me my job. Thank you English!”
“Thank god I knew about the way castles were fortified against sieges – the zombies would have eaten me otherwise! Thank you History!”
…and so forth. In reality, it’s more:
“I’m a more critical thinker now, and have good analytical techniques when it comes to messages being communicated to me. Thank you, English/History!”
…which is not something you can quantify.
Mathematics is sometimes represented as learning to be like a calculator or a computer. Whilst mental computation is a useful skill, it’s certainly not what the subject is from about year 7 when we start talking pronumerals and make claims about all numbers. I think this is an attempt to quantify the skills you are learning, rather than appreciating the abstract and tangential benefits you will learn to master.
The response in recent years seems to have been to make mathematics very connected to the real world. Taking a look at the General Mathematics syllabus, you can see their attempts – there is a large portion devoted to financial mathematics, the most boring kind of maths. Seriously, depreciation models. Any interesting subjects (such as spherical geometry, probability and the normal distribution) are dumbed down to the point where virtually no mathematical understanding is required – simply some real-world flavour coating on easy substitution (put x into the equation in the formula sheet) problems. I can’t tell you how frustrated it makes me when I flick through the textbooks of my students and feel so uninspired. If I can’t feel excited about this stuff, how can they?
I think there’s a problem at the heart of mathematics that makes it all seem a little pointless, and it goes deeper than just the “when will I use it in the supermarket” question. Mathematics is all about deductive logic – making conclusions from a given set of premises:
Premise 1: Whimples are awesome.
Premise 2: I am a whimple.
Conclusion: I am awesome.
In a sense, the conclusion was already hidden among the premises, that is, in order to communicate the three statements above, I only need to communicate the first two. A mathematical problem is therefore about discovering true things (that are very often profound) by looking at what premises and conclusions you have already reached or discovered, and about the economy of expression.
This is very different to inductive logic, which is what science is. It attempts to take what you know and generalise outwards from there, for example:
Premise: All the whimples I’ve ever met were awesome.
Conclusion: All whimples are awesome.
I find deductive reasoning all the more fascinating and wonderful, which is why I chose to become a mathematician (even though science is WAY AWESOME). However, I can see why many students might feel a sense of “why bother?” when it comes to mathematics. There is only one answer, it’s there, and someone else can figure it out, really. Or you can just copy it from Bob – the Back of the book – and it’s not really that big a deal. The answers have no real meaning in and of themselves, so what’s the harm in copying?
The great profundity that I find in mathematics is its ability to say such powerful things about such a general class of objects. By starting with a few premises, so much can be artfully deduced. Mathematics truly is an art of elegance, much like poetry – it’s all in the selection and expression of thought. How we learn mathematics is also similar to the way you learn to play a musical instrument, or even learn to dance – drills and scales and boring repetition are all designed to get you used to the tool (the instrument or your body) that you’ll be using for profound expression later on.
… but it’s hard to tell that to the girl who’s got to pass MATH1011 so she can go be a vet and never think about x’s and y’s ever, ever again. Whilst I think that mathematics is one of the most essential things in life, and that it teaches you about proof, how to be sure, how to listen and how to deal with so many problems you might face in life (be they logical conundrums or even moral quandaries), deals with the secrets of the universe and holds the most exquisite beauty possibly conceivable by the human mind… I can honestly (and grudgingly) say that you could do without formal training. In the same way that some people just don’t like music or reading or television or the internet and instead find something else to fill their time, I can see why you might just not enjoy anything about mathematics. I am confused by this response, but I can accept it. Give me 10 minutes, though, and I think I could tell you something mathematical that you’ll find quite awesome (I hope to do this for my next blog post!).
Mathematics is Inhuman
Mathematics is all about discipline. It teaches you to ignore your intuition on many levels, instead forcing you to use your reason in very specific ways in very specific contexts. It teaches you to keep all the facts and assumptions in mind when you’re talking about anything, as the most basic properties may hold the key to your success. It really does stretch the brain out to places humans really haven’t needed to go (in terms of evolution), so the fact that we can do it is truly amazing.
However, it is inhuman in the sense that there is never any grey areas. Your answer is either right or wrong – at least in all mathematics before third year pure. You cannot argue your way out of mathematics or ‘fudge’ it with natural talent the same way you can with many of the humanities – remember those times you didn’t read the book but still got a good mark on your book review? Or wrote an essay that was kinda crap but you made it sound impressive with your words? Not so in mathematics. If you don’t know SOHCAHTOA, and what it means, then there is no way you can get any marks in a trigenometry test. It’s not so much about the inherent difficulty of the subject, but more about what questions are asked. I try and encourage in my students an open, inquiring mind that forms its own mental models of what’s going on, their own mental methods for solving problems. I try and ask questions that have multiple answers, using open questions rather than “what is s when t=1”. By encouraging them to actively engage with it in a more organic way, they need less help in staying motivated.
Those are some preliminary thoughts. I’ll work more on this stuff for more postings in the future, perhaps when I’m not feeling so sick and rundown. I sure do want to talk about general numeracy as well as the wonders of mathematics a whole bunch. Please comment and let me know of your own personal feelings about and encounters with mathematics, because I’m always, always interested!
In the meantime, I’d like to keep you abreast of the awesome internet phenomena, such as kittens inspired by kittens:
….that video has exploded across the internet in just a few days. Funny how a video like that can lie dormant for months, and then the right connections are made and the right sites hear about it and before you know it your video has over 2 million views.
Also, sorry about the late post. Next Sunday, we’ll have Finn delivering a post, and Aidan the Sunday after that.
I really, really wanted to talk about maths today. I’ve been teaching it for (I kid you not) about 14 hours every weekday for the last few weeks, and it’s been pretty intense.
I feel like I’m starting to get a feeling for what most people find difficult, frustrating and boring about mathematics. I think I’ve also got some pretty awesome insight into the world of maths, how people learn it and how it should be taught.
However, I have 78 (!) new non-spam emails in my inbox and a class to teach very soon. When I’m done, I am going to go home and catch up on my sleep and try not to have a brain haemorrhage. I’ll save my insighty goodness for another day, and instead post some awesome mathematical images that I’ve found for your amusement and delight.
The first is a mathematical diagram representing what is called the projective space PG(3,2), and I think it has a pleasing fiveness to it:
I got this from a paper called “Pretty pictures of Geometries” by B. Polster. He says:
“Why are good pictures important? Two of the main reasons that come to mind are the following:
-To convey some of the abstract beauty of the objects we study to people outside our field. This seems to be especially important today as it becomes more and more important to “justify” and “sell” the kind of research we are fascinated by.
-Many of us think in terms of pictures of various degrees of abstraction. The kind of pictures we want to concentrate on in this note are immediately accessible and can serve to lure students into studying incidence geometry and as a first step in teaching students pictorial thinking in geometry.”
The kinds of mathematics I’m engaged in deals with abstraction as well. I study Algebra, not Geometry, so the beauty I find there is almost always unable to be drawn, though there are a few interesting diagrams. Here is a picture representing the “root system” of the algebra G_2, the object I studied for my Honours thesis:
What it ‘really means’ is rather complex, but for the moment, it’s enough to say “oooh, pleasing sixness! Hey, isn’t that the Star of David?” or something like that.
Here are some more pictures, starting with fractals:
(A fractal called “God’s Own Pentagram”)
(The famous “Mandelbrot Set”
(I don’t know what this one is called, but I like how organic it looks)
Each of these fractals have a property called “Self-similarity”, which means that you can find copies of the whole or parts of the image, in miniature, throughout the fractal. If you look closely, you’ll see the self-similarity in each of those fractals above. You can even download and try this absolutely fantastic fractal zoomer called XaoS, which lets you both zoom in on parts of a fractal to see this self-similarity in action, and form Julia sets like this one:
Here’s how to represent a 4-dimensional object using a 3D image, utilising time as a replacement for a fourth spacial dimension:
I find this rather hypnotic! I’ve heard that those who study geometry get a ‘feeling’ for how things look in 4D space by doing things like this in their brain for all kinds of 4D shapes. Wat.
I have to go to class now, so I’d better cut this short. I’d like to leave off with a question, though. Consider this image:
Do you see her rotating clockwise or anti-clockwise? It’s important to note that nothing is rotating – it’s just a moving 2D silhouette. Our brains interpret the image as rotating, but you can get her to reverse direction if you concentrate hard enough. You have to force your brain to interpret the data it’s getting in a completely different way, and it helps to figure out what parts of the image are making your brain say “this is a rotating woman!”.
Tell me what you see, and if you can reverse her rotation in your brain! Next week, I’ll talk more about maths.
On Tuesday I had a pretty epic attack of the sads, and I started a post here about it, which made me feel better, and then yesterday I saw a number of very good people, and that made me feel fantastic, so the post became pretty irrelevant (I can tell you all about it IRL anyway), so I’ve scrapped that post on the grounds that it is no longer necessary. This is good! Just as an aside, before I get on with my Real Post, and as a reply to Tab’s post yesterday: January was probably the best month of my life, mental-health wise, and I have you guys to thank for that in a pretty huge way. Obviously, some of that was me going “I will do this. I will see these people. I will accomplish this small goal” but having you guys around to be with, to talk to, has been so important to me. So thank you!
Now here is a post about propaganda.
I’m a pretty visual learner. One day I will write stuff about learning styles and multiple intelligences, because I think that shit is pretty fascinating, but suffice it to say that I like having a lot of pictures. I don’t have a problem with words, but having a picture to drive home the point works wonders on me. I also grew up in a house that had a newspaper published out of the front room, a newspaper that made its money from advertising. I have inherited, so my family tell me, the Kelly Advertising Gene. (Grandfather was in advertising/PR, or what they called “exploitation” before WWII, my dad runs a newspaper with advertising, and always organises/promotes local events, my sister is some kind of PR goddess), but I seem to have used it primarily for promoting Sutekh (which was fun, and easy for me) and for my unholy love of propaganda, the best kind of advertising in the world.
Propaganda can tell you a lot about the events of the time, and the attitudes of the society that produces it. The art, the language, has to be highly emotive for it to work, so it has to play upon the values and sentiments of its intended audience.
This poster is from WWI, and it’s distinctly Australian. Even though it would’ve been seen by men living in the cities, it evoked the knowledge of one of the biggest and most Australian dangers, the bushfire. Something that Britain didn’t have, that threatened life and property and livelihood. Something that required courage, strength and physical prowess to defeat. This poster causes the viewer to feel shame that he is not helping the men risking their lives, especially because the bushfire metaphor implies that all of us are in danger, here. The rhetoric is pretty fantastic – it speaks to both the national sense of self (which had been created pretty heavily 14 years before with Federation), and the ideals of the time surrounding masculinity. The poster discusses both what it means to be Australian, and what it means to be a man, and tells you that someone who won’t enlist is neither, which is basically the early-century version of calling someone a scrub n00b.
This next picture is one of the most famous Australian war images, painted, rather hilariously, by Norman Lindsay, who wrote the Magic Pudding. It depicts Germany as a hulking beast (see the little spiked German helmet? Yeah.) with blood on its hands, blood which is spilling out of Europe – but dripping down, about to touch Australia. This poster is fascinating because it’s actually quite unlike most Australian and British propaganda posters, which tended to have quite a lot of text, and usually relied on the character and ego of the person reading the poster. This, on the other hand, was created right at the start of the war, in 1914, and I think it’s probably a very good summation of how a lot of people were feeling at the time – this violent beast, threatening the whole world. Man, stick a big beard and Islamic garb on it, and you’d have something relevant to the fear and uncertainty people felt right after the WTC attacks. This poster represents people who have recieived a big shock, and haven’t quite recovered from it, yet. Images like this were pretty prevalent during the first couple of years of the war, but as it dragged on with no end in sight, the images focused on things like the first poster, a sense of duty to the men who were fighting. The Germans aren’t really mentioned in Australian propaganda after about 1916, because, quite frankly, people didn’t care anymore. It wasn’t worth it to fight Germans, but it was worth it to stand up next to the men already risking their lives. This poster is actually much more like the German and Russian posters from the same period, which I think is pretty interesting.
Now! Onto WWII! The sense of Australianness goes even further during this conflict. But it’s really clear that a lot of attitudes have changed. There are very few posters about helping Britain, because people don’t care, and a lot of people are still cranky about the British fucking us over at Gallipoli. It’s pretty obvious to see that some values have changed as well.
Check it out: if you join the AIF, attractive ladies will pay attention to you. I kind of love this style of message, because they’ve sort of worked out what really motivates young men to do dangerous things. Pride in one’s country, helping out mates, making your parents proud, sure, all these things matter, but not quite as much as impressing the ladies.
The interesting thing about this poster is that even though it’s been sexed up a bit, those ladies still look like very nice girls, the kind you DO take home to Mother. So there’s still a bit of morality shining through. It might also send messages to ladies – look, at these nice, well-brought-up girls. They get to hang out with nice, clean boys in uniform! Yeah. Or maybe it’s his girlfriend and his mum. Either way, your womenfolk will be impressed, if you join up. It’s important to note that the guy in this isn’t the tallest, most perfect specimen of Australian manhood, either – he’s got a touch of the Chesty Bonds chin, but he’s fairly normal looking. But yet, ladies, all over his junk.
Here we’ve got another poster featuring ladies, but instead, these are immoral, sexy ladies who are ACTUALLY spies. Hot spies. This is warning against discussing secret information, even in front of ladies – of course, ladies LOOK like they’re stupid and can’t understand anything, but some of them (especially the foxy ones who sleep around) are insidious tools of the Fuhrer. The art in this is really gorgeous. So much of the artwork for posters was done by people requisitioned from advertising agencies, magazines, etc, and so that’s where you get that very normalised look (like the AIF recruitment poster above), but sometimes there were actual, real artists doing these posters, and there are some absolute gems, like this one. I love the light on her dress and shoulders.
Japan entering the war in late 1941 was basically the best thing that happened to propaganda writers, because for the first few years of the war, most of Australia didn’t care. Messages about helping Britain, or defeating evil Germany, invader of small, helpless nations, had worn out their welcome during WWI, and it was still a sore point for a lot of Australians. It’s important to realise that for most of WWII, the atrocities of Nazi Germany were widely unknown by the Allies. They knew they were persecuting Jews, but they didn’t know the complete extent of it. For the early part of the war in Australia, trying to get men to enlist based on The Evil Hun was pretty pointless. But then! Japan enters the war, enters it right in the Pacific, and takes the closest British naval base to Australia (Singapore). Then they bomb Darwin, and send submarines into Sydney Harbour, and sit, for a while, outside the heads, shelling the Eastern Suburbs. Finally, the government can say “You need to join up because Australia is directly threatened”. Finally, we get a big, bad villain for the posters.
Even better, for an Anglo-Australian 1940s audience, the villain was a different colour, with strange, unfamiliar facial features. Sometimes, Japanese soldiers (and Tojo in particular) were caricatured into laughable imbiciles (similar to a lot of the anti-Chinese cartoons from the turn of the century), but particularly in the early years of the War in the Pacific, they’re scary and serious. Another boon for propaganda artists was the sun from the Japanese Flag – as you can see in this poster, it works very well as something rising imposingly over the world, or over Australia. In this case, it’s rising over the Pacific.
This also has the famous slogan “Fight, work, or perish”, which was a favourite of the government. It was an emotive slogan that could be used on many, many posters – posters showing slogans, posters showing factory workers, posters showing the enemy. After the war, an altered version – Populate Or Perish – became the catchcry for increasing immigration to Australia and promoting an increase in the birthrate.
One of the best things about showing propaganda to students in a history class is that you can show them things they know are offensive. You can encourage discussion about the ideas, and you get the students interested (like when there’s something sexy going on). It’s one of the most engaging historical sources, and it’s also one of the most useful ones in terms of helping students empathise with different opinions and perspectives. It’s also a type of source which demonstrates both social/cultural history (which is my bag) and political/military history. And it’s funny. Sometimes it’s rude, sometimes it appeals to low aspects of your character, but that’s good too. It’s real, it’s unpolished, and I think that’s important to show.