I have a somewhat embarrassingly big love for mathematics. I find myself talking about it all the time in a variety of social contexts – birthday parties, drinks with my brother’s friends, I seem to take any excuse to talk about the wonders of my field by the horns. I get the feeling that most people don’t understand mathematics as I do (a strangely creative and incredibly powerful area that bridges the gulf between philosophy and science, the study of reason itself for its own sake) and I feel compelled to correct them. Does the subject have an unavoidably narrow appeal? How much is the state of mathematics education to blame? Why do people flinch when I say things like “equation”?
I sat down and read the draft Mathematics syllabus a while ago, but read it again in response to an article I read a few days ago. Though it only addressed the K-10 syllabus, my first stop is the new 11-12 course called Essential Mathematics. Here’s the rationale:
Mathematics is the study of function and pattern in number, geometry and data. It provides both a framework for thinking and a means of communication that is powerful, logical, concise and precise. Essential mathematics focuses on using mathematics to make sense of the world. The emphasis is on providing students with the mathematical skills and understanding to solve problems and undertake investigations in a range of workplace, personal, training and community settings. There is an emphasis on the use and application of information and communication technologies in the course. The course includes investigation of the application of mathematical understanding and skills in workplaces or community settings.
Reading through the meat of the syllabus, you find exactly what you’d expect – measurement, money, pie charts, basic statistics, simple probability. I think this “practical” mathematics is a great course to have available to students, as it fosters the necessary numeracy that is required to navigate today’s complicated world. Innumeracy is just as unacceptable as illiteracy, and is far more prevalent. But I am shocked at the simplicity of the syllabus. How many students are unable to convert between 12-hour and 24-hour time before starting year 11? How many are unfamiliar with the concept of an “average”, or be able to reason through that the chances of getting “two heads” on two fairly-flipped coins is one in four? Are we really making such knowledge optional?
There is almost no content within this syllabus that is not directly addressed in earlier years. It’s as though they’ve stripped things down to “the basics” and want to give failing students another go. The essential problem, however, is that this is not a course that most people would like to teach, and not a subject that most students would want to learn. I’m sure that almost no-one wants to sit in class and learn how to adjust for the GST or draw a stem-and-leaf plot, and I’m just as sure that there are few who would want to be responsible to make these kids do so.
The essential problem, I think, is a misconception of what these basics should be. Students who end up in this course are those who haven’t been engaged with the subject in previous years, and this course isn’t going to change that. It’s like teaching failing kids to read by getting them to read job ads, without telling them about the stories they can read to themselves and others, the world that is opened to them when they can finally understand what those letters actually mean. There is little attempt to make it interesting, as though the needs for such mathematics in society should provide all the motivation students require.
So let’s go back a bit and look at a relatively simple example of something that wouldn’t be taught to a kid doing Essential Mathematics. It’s the Quadratic Formula:
Chances are you had at least one test during school where your ability to use this equation earnt you marks. Now learning to use this equation is hard work; there’s a lot of bits to remember, there’s a “plus or minus” and you have to do that before you divide, and that’s after you’ve done this square root business (surds are tricky). Even then, your calculator might tell you “MA ERROR” when you try to make the equation work, and you don’t know whether you’ve done something wrong or there’s something wrong with the question.
Why on earth do we go to the bother of teaching you how to do this? Almost no-one will ever use the quadratic theorem on even an occasional basis, and those that do already have calculators and/or software that will do it for them. When I teach mathematics to university students enrolled in the bottom two of the five possible levels of mathematics at my university, at least half the students don’t know how to use this equation anyway.
Maybe this like learning to brew your own beer – you don’t need to know how it all works, you can just buy it from the store, but it’s good to know “just in case” and those that get good at it can make lots of money. Hell, it can be a fun hobby and maybe you can earn some money, but if you don’t get it, then it’s not really that big a deal. This is certainly the line most mathematics teachers I’ve come into contact with use when defending their subject. The students are told that you’ll get a better job if you know it, somehow, probably. Also, it scales really well in the HSC, so you should do maths.
But I think that misses the mark. The obvious parry heard from many students is that while it may be useful for some people, it’s not going to be useful to them. They’re going to be an artist, a novelist, a landscape gardener, work in a call centre, when are they going to use this formula? And unfortunately for the state of mathematical education today, I don’t blame them for thinking this way. The mathematics syllabus is becoming more and more skills-based, meaning more and more disconnected pieces of highly specialised mathematical knowledge most useful for the job market. This means lots of calculator use at lower levels, and only statistics and hardcore calculus (both very good for making money) for those who are more interested in the subject.
Above all, the emphasis is on usage. This objection can be applied equally to many other disciplines – when does knowledge of Shakespeare, or the Napoleonic Wars, or the fact that we’re made of carbon affect your daily life? When are you going to use that knowledge? But as far as I can tell from my own personal experience, this objection is not as prevalent elsewhere. It’s as though the importance of learning the subject for its own sake is so embedded in the way the subject is taught that you don’t even need to ask, that the subject is engaging and interesting enough that the students don’t want to ask.
An additional and related complication is quite obvious to me in the way these problems are often phrased:
Not only is this boring, but it’s also unrealistic. Who decides what area their path should have? Whilst being able to take words and turn them into mathematical formulae and then solving is an incredibly important skill, I can see why this kind of question really turns students off. They’re technically presented with exactly the amount of information they need to solve the problem, and no more, though situations are rarely like this. Instead of engaging their problem-solving skills in an interesting way, showing how mathematics can make sense of complex situations, the single skill of “turn words into formula” is tested in a simplistic and frankly dull way.
Mathematicians try and demonstrate how their knowledge can be used in the real world, but all it serves to do is to show how out of touch they are. It fosters a belief that the mathematical world is a separate, Platonic world, where there is RIGHT and WRONG and nothing in between, ugly and inhuman. Who thinks like this? Who asks these stupid questions?
We need to start thinking differently about what mathematics can and should inspire in people. Imagine you’ve invited your friends over and cracked open a brand new board game.
So good, this game.
You grab the rulebook, and start reading. After you’re done, you’ll hand it to someone else, until everyone’s up to speed. You pick your colours and away you go.
Even though you all start from the same ground state (you all know the rules and haven’t read any strategy guides), one of you will win. Even if there’s an element of luck, good players will win more often on subsequent playthroughs. There is, somehow, not enough information – and yet you’re expected to do the best with what you’ve got. Players who play enough may start to create “house rules”, alterations of the foundation to maximise play experience. What is going on here? What makes one foundation better than another? What makes one person better at the game than another?
There’s an analogous skill, one that is equally important – what happens if you’re presented with lots of information (in the form of conflicting strategy guides and advice from experience players), and you have to decide what’s relevant? How do you decide what makes an effective model for victory?
Whatever the answers to these questions are, they are what mathematicians like myself want to find, and a desire for such answers is what we want to inspire in others. The applications are instantly infinite – we almost always have too much or too little information, and we have very little sense of what’s going on “under the hood” and what that means for the way things play out. The aim of this approach is to help people correctly identify the foundational assumptions (“the rules”) of a situation, asses the possibilities, and make the most of it. Everybody does this every day, to varying degrees of effectiveness. The skills of argument, of comparison and contrast, of quantifying and extrapolating, pattern recognition and modelling; in short, the ability to make distinctions using reason are so important and universal, and mathematics is poised and ready to help people learn how to do this better.
Now other subjects teach this love of logic too. I loved history for the same reasons I loved mathematics – you tried to get a sense of the past, of people’s motivations and actions from the texts and facts available, where weighing up evidence (establishing the foundation) was intimately linked with constructing the best argument (the finished product). What drew me away from history and humanities in general was that I couldn’t stand the cheating, so to speak – the conflation between medium and message, where gifted speakers or writers could have undue influence on others through careful window-dressing of bad arguments. As a shamefully gifted cheater, I found my basic essay skills plus a light drizzling of facts could pass me through a book review with a minimum of time spent actually reading the book.
Some people have an “ear” for languages, and some people can barely speak their own, and the analogy holds just as well for mathematics. Mathematics requires a lot of discipline, and that doesn’t help its attractiveness, but it really is like learning a language. Add to that the fact that the “rules” to a lot of things we grapple with nowadays are very complicated and not very intuitive – most of the time, they’re counter-intuitive. Here’s two nice examples where we go nowhere near the world of physics.
Suppose you’re on a game show and you’re given the choice of three doors – behind one is a car, and behind the other two there is nothing. The car was placed randomly behind the winning door before the show. You pick your door, but the host doesn’t open it yet – instead, he opens one of the doors you didn’t choose with nothing behind it. Now the car is either behind the door you originally chose, or the other unopened door. You’re given the chance to switch your preference. Should you do so?
This is the famous Monty Hall problem. Identifying what is important in this problem is hard, and the answer is even more surprising – switching doors doubles your chances of success!
Suppose you’re in a room with a group of people, and you each call out your birthday in turn. How many people would you need to get into a room to make it 50% likely that at least two people share a birthday?
This is the equally famous Birthday problem, and the answer (23) is quite surprising – especially considering you need 367 people to make it a sure thing!
Even though you don’t need words for numbers to have an innate sense of number, the level of abstraction required to do this kind of thinking is quite intimidating, and humans are not custom-built to do this kind of manipulation. You have to do it every day to make your brain think in these new, unnatural ways, and nothing is going to make it easier. Mental discipline is a good thing to encourage, it must be said, but sometimes students only see the stick and don’t get to see the carrots that keep people like me going.
For example, I was having a discussion with a (non-maths) friend of mine about how cool maths is while we were out for dinner (any excuse will do, remember?). He recalled the story of Isaac Newton, who invented Calculus. He just made it up. At the same time as Leibniz, sure, but he invented it. How cool is that? It’s one of the most powerful analytical tools today, and it wasn’t around before he was. Why did he do it? What possessed him to make it the way it is – could he have made it a different way? Why is the way it is today “the best” way? What do we even mean by that?
How about the statement “This statement is a lie”? What is going on here – linguistic trickery, or something more fundamental and interesting?
How about fractals – Benoit Mandelbrot asked the seemingly simple question “How long is the coastline of Britain?” and came up with the answer “Infinitely long“, and he could say it with a straight face. Paradoxes and puzzles are compelling, and fractals have the bonus features of being really pretty and engage students with technology!
Watch this in HD with the lights off!
Maybe you knew something of these wonderful ideas, and maybe you learnt about them in high school, but I doubt it. I was blown away by this stuff when I got to university, and I can’t understand why students don’t learn about this. This kind of backwards thinking is so stunning, and appears to be unique to mathematics. They don’t teach you the cool stuff, or even give you a hint that it even exists. Teachers battle to tell students how their learning will earn them money or get them better marks, and that’s the only way the conversation is developing – and the syllabus along with it. I’m going to quote from the article I linked before. I agree with it in its entirety.
“Just as children best learn to read by experiencing the joy of great stories, they best learn mathematics by experiencing its beauty and the joy of mathematical play. But in this curriculum there is little sense of the fun and the beauty of mathematics. Not a hint of infinity, of the fourth dimension, of Moebius bands, of puzzles or paradoxes.
Why? If mathematics can be taught as ideas, as something beautiful and fun, then why is it not being proposed? Because it is difficult to do. To teach real mathematics makes demands on the teacher, and it is risky.
What is proposed is little more than a cowardly version of current curriculums, a codification of the boring, pointless approach – which is “safe” but which has already failed a generation of students.
The draft curriculum begins by declaiming the beauty and intrinsic value of mathematics, and the elegance and power of mathematical reasoning. But as a means of unfolding all this before our students, the proposed curriculum is a feeble tool indeed.”
If you look at what engaged the founders of mathematics (Euclid and his geometry, Newton and physics, Gauss and number theory), you’ll find the means to encourage students to participate. You could talk bright students through the Millenium prizes, to show them what captivates mathematicians to this day – even get them to look at Hilbert’s problems, which guided the course of mathematics throughout the twentieth century. For other students, teaching them about puzzles and games and paradoxes would be easy, fun and valuable. Instead, we’re teaching them how to use a calculator and calling it a day. Trying to engage students by focussing on how society sees and uses mathematics is going to force this negative feedback of disengagement to continue.
There is such a shortage of mathematics teachers in Australia that you can become a high school mathematics teacher with six months of training, having only completed 2 unit mathematics yourself (source). You will be called on to teach classes that you yourself only need to have passed, and there is no requirement for any higher training whatsoever.
I say again, you can become a high school teacher, having only passed 2-unit. Who thinks this will solve anything? If you’re running out of mathematics teachers, the solution cannot be to scrape the bottom of the barrel, so to speak. Something more drastic needs to happen.
Listening to trainee teachers talk about mathematics in the classroom was one of the main reasons I dropped out of my Masters of Teaching and went into tertiary study instead. There was such a dearth of passion for the subject. Most students were surprised that I had done advanced level mathematics, saying they found that stuff “too boring”. I cannot understand how making it easier for poor quality maths teachers to enter the system is going to solve anything, and with the syllabus in its present state, I can only see this trend continuing.
Mathematics is beautiful, fun, compelling, dramatic, creative and human. Mathematics education is crude, oriented towards the job market and painfully dull. And things are only going to get worse.
Well, not if I can help it. I’m going to spend my life trying to fix the perception of mathematics at every level of society. It is my mission, my calling, and if I can make it work, my career.
Okay so guys! Long time etc etc.
I have been working full-time recently, and I have a post coming about what that’s been like, but first I would like to tell y’all about another project I’ve got going on at the moment.
I have embarked upon a diet. There are two main reasons for this:
1. I am desperately unfit, and far above a healthy weight. I would like to NOT develop diabeetus, and also I would like to be a healthy weight before Tom and I decide to make a tiny little person.
2. I am lazy as shit (see causes of #1) and working fulltime has completely sapped my ability to care about food preparation.
I have a complex relationship with food. On one hand, it’s a beautiful thing that I closely associate with celebration and good times. On the other, I was bulimic for about 6 years in my teens and early 20s, so sometimes I can get a bit crazy and out-of-control with it. This has hindered weight loss efforts in the past – calorie counting, other diets, etc, all involve CONSTANTLY THINKING about food, and it is usually only a couple of weeks before the urge to purge shows up in full-force and sends me mad.
But I found, as I was teaching, that I had ceased to care about food. I wasn’t thinking about lovely things I wanted to cook, or even eat. This is generally a sign that my stress levels are at their limit – in good times, I love the whole process of preparing and eating food. However, I would come home from work, take a nap, and then when Tom got home I was too exhausted to either cook food, or even to actually care what we ate at all. This led to a LOT of takeaway. Unhealthy, expensive food.
So currently, I am eating food that comes in nutritionally balanced bar form. The discount chemist near my work sells them cheaply, I buy them (berry, chocolate or cappucino flavour), and I have one at recess, and one when I get home from school. Then, at night, I have a microwaveable meal, which provides me with enough savoury so I don’t get grumpy.
Tom and I are calling this diet “fatchelor chum” because it is the weight-loss equivilent of Stagg Chilli, or Bachelor Chum. Pre-prepared food for lazy people.
The laziness aspect makes me really love this diet (as does the fact that I feel a lot healthier since I started it, except for right now because I’ve been eating pizza all weekend). On weekdays, it is so lovely to just chuck a couple of bars in my bag, knowing that I’ll have time to eat them even if I have playground duty, that I won’t have to join the line at the staffroom microwave, and that I can eat what is essentially chocolate for lunch. I love the fact that when I get home from a day of watching intellectually disabled students jump up and down on tuna sandwiches in their socks (“I don’t like tuna, miss”), I do not have ANOTHER job to do. My epic laziness is also why exercise plans don’t work for me – I kind of hate gyms, and sweating. But recently, I am more okay with walking longer distances. Last week, for community access, we went on a half-hour-each-way walk that we’d done at the beginning of term. Last time, I was red in the face and breathless (there are a few hills). This time, I was fine. That’s fantastic, for me.
I feel, a little bit, like I’m giving the finger to societal perceptions about dieting and weightloss. I have had, for a long time, a theory (possibly I read it during uni, but I can’t remember now) that since it is generally okay for women in western society to have and enjoy sex, we must now prove our virtue by removing a different sensual pleasure from our lives; food. Food is frequently marketed to women as “sinful”, “guilty (or guilt-free)”, “naughty” – rather than just using the paradigm of “healthy all the time” “healthy some of the time” “only healthy in small doses”, which is basically how food rolls.
It is okay to eat food. It is okay to enjoy food. It is even okay – on a moral level – to eat too much food. For me, the amount of food I was eating was not okay in terms of my health. In terms of whether or not I was a good person, well, it had no effect at all. I still did nice things for my parents. I still went off to my social-justice job. I still voted against Tony Abbott. Having a Big Bondi Burger with bacon doesn’t make me an immoral person, it makes me an unhealthy one. And sure, there’s an ethical argument to be made that I owe it to my loved ones to not develop heart failure, but that’s not the argument that the media makes. The argument they make is that to be “good”, if you are a woman, you must be shown to be denying yourself pleasure. It is about self-sacrifice and hard work and control over one’s baser urges (like the urge to nom on some bacon). It is weirdly puritanical.
So, even on this diet, I am not really being the good, hardworking, virtuous person that whoever decides these things wants me to be. I am not working hard to do this. This is actually less effort than eating badly. I am not denying myself good times – I eat regular food around my friends, and try to not be crazy about it. I cut my coffees down first to skim lattes, then to long blacks (with faux sugar) – because it means I can have a beer in the evenings without pushing my calorie level up too high. That’s probably the only real sacrifice I’ve made thus far, and it was only swapping full-fat milk out for beer.
And worst of all – this diet means that I am no longer in charge of what Tom eats. Fantastic wifefail there, on my part. He cooks bachelor chum for himself – and most nights, even microwaves my strange freezermeals for me. And then sometimes I SIT IN BED and eat it. I am the very pinacle of laziness and unfemininity, and it is helping me become healthier. Suck it, dominant paradigm!
Interestingly enough, I think I’ll stick with the fatchelor chum when this term ends. It’s still the easiest thing in the world (although I might just keep the bars and do a lot of greek and caesar salads in the evenings, now that summer’s on its way). This weekend I have eaten pizza and an enormously greasy schnitzel, and the end result is that I feel a bit shit. It feels uncomfortably heavy in my stomach, and I’m regretting it a bit – not because of the extra calories, but because it wasn’t awesome. I didn’t feel this way last weekend, when I ate a million tapas. Obviously my body is becoming acclimatised to healthier food, which can only be a good thing. I find myself craving apples, rather than mint slice. And I don’t seem to want to throw up. I also don’t hate my body – I can still look supercute sometimes (my red polkadot dress, let me show you it), and this is because I surround myself with rad people of all shapes and sizes who love me for reasons that are not at all about what shape I am. This kind of emotional support – knowing that even if I don’t lose weight, I am still seen as a worthwhile person – is what is making it possible for me to try this project out. One of the things I absolutely hate is when people who subscribe to the belief that thin = better person notice that I’m losing weight, because the way they talk about it makes me hate myself. I do not like my body being seen as public property. Essentially: if I would not be comfortable talking about my former eating disorder with someone, chances are I am not comfortable with them commenting on my body or eating habits AT ALL.
I want y’all to know that I am in no way pushing my lifestyle choice on you. This is currently working very well for me, but it may not for all people. Everybody gotta do what they gotta do. I’m not going to tell you about the calorie levels in food that you eat, or how many kilos I’ve lost, or anything like that, because that shit is BORING.
This has been the first thing I’ve written in a long, long time, and so I apologise for the lack of style. I’m going to try, like Percy, to update on a more regular basis. And it’s my birthday in a few weeks, so you can expect my yearly musings about where my life is going, and what I want to accomplish over the next year.
Recently, I went to the 10th Representation Theory of Algebraic Groups and Quantum Groups conference, held in Nagoya, Japan. Afterwards, I travelled to Tokyo to meet a friend of mine, who gave me a massive bonus to my Adventuring (Tourist) roll. Here follows my tale of of culture shock (and awe), detailing my delightful discoveries in the strangest land I have ever travelled.
The critical level representation – efforts in the arena of the Geometric Langlands Program
It was a great way to find out what was important to other people. As a student, I want to know what others find interesting; that way, my own research efforts might stay relevant to others. When I find someone I want to work with in the future (read: apply for a job), I want to have something substantial to offer my collaborators!
The conference was entirely in English, and the atmosphere was warm and welcoming. Also, the air conditioning and free wireless were pretty great! At the conclusion of the conference, the organising professors sang Karaoke for us, which was a pleasant surprise – though slightly alarming. Not to be outdone, the younger attendees all went out to an 8-story karaoke complex, paid ~$23AUD for two hours of singing and all-you-can-drink fun. I sang some Silverchair, the smartest guy in the room (not me, btw) mangled Amy Winehouse so badly that I cried with laughter, and the Italians gave a great rendition of Killing Me Softly. We got very drunk, and learnt about how Japanese can make a pretty decent beer. Not wine, though, omg.
A few of us kicked on to a lovely sake bar, where you could get awesome drinks and a little bit of food, all delivered to your table. I really liked what little of the “drinking house” culture in Japan I saw in Nagoya.
Before that, I was mostly on my own, in a strange city where nobody could understand a single word I was saying. I was determined to not eat fast food, so I tried as many restaurants as I could. The Japanese accent is everywhere in their cuisine, and emphasis on non-Japanese authenticity is low – the “Authentic Southern Fired Chicken” I ordered from a fully-decked-out-even-with-cowboy-costumes-and-Cheers-playing-on-the-TV American-themed bar was regular Chicken Karaage with chilli powder. Most of my meals were achieved through pointing and hand gestures, but delicious, cheap food is abundant. I never spent more than ~$9 on any meal, and it was all extremely filling!
But the Internet was very difficult to find. There was a coin-operated internet machine in the hotel lounge:
…but it was terribly old and slow. I decided to go in search of better internet, and experienced racism directed towards me for the first time in my life. I followed signs (in English!) to Internet cafés, and was told on two separate occasions that their establishment was “No gaijin, Japanese only”. The third cafe I went to was reached by lift, and the lift doors opened to reveal… another locked door with an intercom, right up against the lift. I pressed the button and tried to gain access, but lost my nerve when the doors started closing and I still hadn’t heard any English. Eventually I found a place that would take me, but it was just so surreal.
Like this sign:
People were somewhat reluctant to sit next to me on trains Nagoya, and not speaking the language was so, so hard. It was very isolating.
Oh, back to the technology – I think I was expecting Japan to be like Australia, only at a higher technology level; what I found was a higher prevalence of somewhat gimmicky technology:
Happy poopy time
…but overall, technology didn’t seem substantively more advanced. The trains were great, though, once you puzzled out where you were and where you wanted to be and what platform you had to be on to get the right coloured train – they came about once every five minutes, and moved a lot of people.
I also saw a castle that was being rebuilt, having been destroyed in WWI. It was pretty impressive:
….and the moat was cool:
And so my time in Nagoya drew to a close, as does this post. I’ll continue my Japan adventures in the coming weeks, still to come:
- A dance club catering to the geeks of Japan – they use .gifs to enhance their music, for real.
- Engrish, oh lol
- Bullet trains!
- More of my clashes with the oddities of Japanese culture -“They don’t have street signs? What do you mean, they don’t have street signs?”
Next part will be up soon. Stay tuned!
We here at A Pleasing Fiveness have been gone a very long time.
I enjoyed writing and reading this blog – I have very fond memories, and looking back over our stuff, I think we’re worth trying again (take a look at our stats!).
My plan: I’m going to start blogging here again. I’m going to try for once a fortnight. I’m going to keep up the pretence regarding this blog’s audience, though I’m not going expect anything from the old authors in return.
It would be awesome if you, dear reader, also joined me in this re-blogging adventure! But if not, that’s cool, I still think you’re pretty rad.
Over and out.