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.
The Four Colour Theorem is a famous problem in mathematics. It is very simple to express, but proved very difficult to solve. The method of solution was both new and controversial, to the point where some mathematicians still consider the question unanswered.
In 1852, a graduate student by the name of Francis Guthrie noticed something interesting when he was colouring in a map of the English counties. He discovered that he could colour it all with only four colours, so that no two regions which were adjacent (that is, shared a common border) had the same colour:
As any good mathematician would do in this situation, he generalised the question – is this true of all maps? If some catastrophic occurrence befell the English counties and all the lines were redrawn, would he still only need four colours? What about a completely abstract partitioning of the plane, such as this:
Yep, still only four colours!
He simplified things down a little bit – countries that had more than one connected piece were disallowed, to avoid situations like these:
There is no way to colour this map with four colours, if both the regions marked A are to be the same colour.
It’s easy to show that at least four colours are needed (try it yourself!), but he couldn’t answer why only four were needed. No doubt he tried many different combinations, scribbling in notebooks and trying to find an easy counterexample. Instead, he passed the problem on to someone else.
Augustus De Morgan was a rather famous mathematician – anyone who has studied complex numbers will no doubt remember De Morgan’s Laws, not to mention his work on solidifying the method of proving by induction. He was able to prove a simpler result, which he hoped to generalise: any map with five (or fewer) regions could be coloured with only four colours. However, he couldn’t prove it for an arbitrary number of regions.
After the problem was published in the journal Proceedings of the London Mathematical Society, a man by the name of Arthur Kempe struck upon a brilliant idea.
First he proved that any map has a region with five or fewer neighbours (which is quite remarkable, considering the generalities in which we’re talking). He described a process of shrinking which he claimed was reversible: if a region has three or fewer neighbours, we can shrink it away so that the map now has one less region to colour. We shrink everything down until our map only has four regions, colour those arbitrarily, and reverse the process:
The grey regions are shrunk one at a time until we have only four regions – they are coloured, and the shrinking process is reversed.
Everyone was happy for 11 years, until Percy (!) Heawood found that the process wasn’t as reversible as everyone had assumed: a region with five neighbours couldn’t be shrunk like this (so that you could colour the rest and put it back nicely), and that was an unavoidable for some maps. The proof was sunk.
And yet, no-one could find any map that required more than four colours. They tried everything they could think of, but mathematicians are always (or should be) mindful that they are rather meagre minds in the light of the massive order and reason they are wrangling with – perhaps people just weren’t clever enough to come up with the required counterexample!
The language of graph theory was being developed, and was nicely adapted to this problem. Instead of considering countries and regions, the graph theorists considered coloured points, with lines between them representing adjacency:
With the results of graph theory now at their disposal, mathematicians quickly dispatched the Five Colour Theorem – I read the proof, and actually reproduced it rather faithfully in my exam, when I took Graph Theory in second year. However, the Five Colour Theorem wasn’t much help – moving from a graph that is five-coloured to one that is four-coloured often requires a lot of recolouring, as can be seen in this example:
A five coloured map.
The same map, now four coloured – note that the central blue region is now green, and its lower left neighbour is now blue.
In 1922 a man by the name of Philip Franklin proved that any map with 26 or fewer regions could be four-coloured. His proof used an idea called a reducible configuration, first introduced in Kempe’s proof (though not named as such). The idea was similar – take a number of adjacent regions, remove them, and four-colour the remainder. If you can put the regions back, regardless of how you four-coloured the remainder, then the regions are said to form a reducible configuration. Note that this is a generalisation – we are removing groups of regions, rather than one-at-a-time like Kempe proposed.
The idea was therefore formulated that would later be the starting point of the proof: prove that every possible map has a reducible configuration. Then, when you remove it from a particular map, the new (and simpler) map will also have a reducible configuration, and you can repeat the process and we have our result.
But what do these reducible configurations look like? It was demonstrated that a region that has four or fewer neighbours was reducible, but was that it? Could this turn into a proof?
A new age of proof
In 1970, Wolfgang Haken began his search for the full list of reducible configurations. Most estimates put the number at about 10,000, but the number might have been anything up to infinite (therefore frustrating his search for an exhaustive list). Working with another mathematician, Kenneth Appel, they developed computer software to help them find this list. They wrote their program, fired it up, and hoped that it would one day terminate and say “Here it is, the full list!”.
In just under 1,000 hours (about 6 weeks), the process stopped at 1,936 reducible configurations. They had found the list. Every map had at least one of these configurations – if you removed it to make a simpler map, the new map would contain one of these 1,936 configurations which you could take out. Eventually you could get down to just four regions, colour those and reverse the process. Thus the theorem was proved.
It was one of the first computer-assisted proofs to be presented, and as such gained a lot of attention due to the problem’s age – computers had achieved something humans were unable tackle with over a hundred years of pondering.
Scepticism and acceptance
Modern computers can do Haken and Appel’s computation in just under an hour, but the computation is so hard that no human could ever hope to do it, even if they devoted their entire life to this theorem. Mathematicians were very wary of this result – it comes down to trusting that the computer has done things the way you wanted it to. Computers are rather predictable machines, so most mathematicians do accept the proof today, but there are still some that remain sceptical.
Perhaps a bigger problem is that the proof doesn’t give us a very good understanding of what’s going on – what is it about a plane that it can be coloured in this way? So far, the only answer we have is that we’ve tested all the possible cases. Mathematicians want to be able to understand things like this, to assist in our generalisations.
If we are colouring a sphere or a cylinder, the result is the same. But what if we were to colour a torus, or donut shape? In that case, we actually need seven colours:
Seven regions, each with six neighbours, if we were to roll it together like so:
…so we know we need at least seven colours. Is seven all we need? Again, we’d need to feed all of this into a computer, which would probably give us an answer. The method of proof is therefore “give the computer a surface, and it will test it for you”, rather than a nice formula to describe how many colours you need, no matter how your object is sitting in space. What about a donut with two holes? What about a Mobius strip? The answer is the same – ask the computer.
What about higher dimensions? Again, ask a computer. We don’t have a good knowledge of what the connection is between the shape of an object and how we colour it – we just know that if we do a whole lot of calculations, a computer will give us an answer.
The fact that four colours is the maximum number of colours we need is now clear. However, the way that the fact was proven makes it almost useless to help us in our understanding of what’s really going on. The problem is therefore still being worked on to this day, so that one day humanity may be able to understand without the aid of a computer. We’re working towards a better answer, even though we already know what it is.
(I predict that computer assisted proofs play an increasingly prominent role in modern mathematics. As computer power begins to outstrip the capacity of the human brain, we will want to answer questions beyond our own power to solve. We won’t be able to help ourselves!)
Mathematics is often portrayed as a black-and-white, right-or-wrong subject. Whilst this may be true for much of high school mathematics, it is clearly not when it comes to problems like these. What it means to be right, how you’re right, how you go about finding answers and whether you can use your methods to answer bigger questions is far more important than simply knowing whether your attempts agree with the proverbial answers in the back of your textbook!