On Monday I was required to monitor at the Tokyo University undergraduate entrance exams. I shepherded 60 terrified 17 year olds through a 2.5 hour Japanese language test and then a 100 minute maths test. These tests were part of a two day examination process for those want to enter the humanities faculty of Tokyo University. About the Japanese test I can say nothing, but the maths test interested me, and can be found online (in Japanese) at the Mainichi Shinbun newspaper. In order, based on my feeble attempts at translating the exam, the four questions were:
- A straightforward but nasty calculation of the properties of a line intersecting with a cubic function, including elucidation of all minima and maxima of the products of the lengths of two line segments
- A geometry question with two proofs
- A constrained linear programming problem
- A simple Markov model with a slight twist
The students had 100 minutes, and to their credit quite a few of the students managed all four, though a lot also stumbled and didn’t get past two. I would say that for a well-trained student with good maths skills, these four questions can all be done inside their allotted 25 minutes, but it’s a pretty risky process – even a small error at the start, or misconception of how to do the problem, and you have basically lost the whole question because you only have time to attack the problem once. And these problems are probably about the same level of difficulty as the questions on a standard year 12 maths exam in Australia – where usually we would have three hours.
But these questions were for the Humanities Faculty of this university. If you want to study Japanese literature at Tokyo University, you first have to get through that 100 minutes of high level mathematics. It says something, I think, about the attitude of Japanese people towards mathematics, and towards education in general, that they would even set a mathematics test for access to a Humanities Faculty; and it says even more about the national aptitude for maths that the students could tackle this exam.
At about the same time as these exams were being held, the Guardian and the Sydney Morning Herald released articles slamming the mathematical and science abilities of the average student in the UK and Australia, respectively. The Guardian reported on a new study that found English star students were two years behind their Asian counterparts in mathematics, with 16 year old English students at the same level as 14 year old Chinese. The study also found that
The research also found England’s most able youngsters make less progress generally than those of similar abilities across the 12 other countries studied. The other countries studied were Singapore, Hong Kong, Taiwan, Japan, Australia, Slovenia, Norway, Scotland, the US, Italy, Lithuania and Russia.
Meanwhile, the Sydney Morning Herald reported on a new study showing that the proportion of students doing mathematics is falling fast, with apparently only 19% of students studying maths, science or technology in their final year of school, and a rapid fall in mathematics enrollments amongst girls especially. The corresponding figure in Japan in 2002 was 64%.
So is this a problem, why is it common to the English speaking world and viewed so differently in Asia, and what can be done about it? Obviously as a statistician I think this is woeful, and it certainly is my personal opinion that understanding mathematics is a good thing, but is it bad for a society as a whole to neglect mathematics education? I don’t know if that’s objectively verifiable. So let’s skip that question, assume for now that improving the number of people taking mathematics is good, and just jump onto the question of why it is unpopular in Australia, and why the British are so bad at it.
First, I would like to dispute the possible explanation provided in the Guardian article by “the researchers”:
In east Asian cultures education has historically been highly valued. This can be seen not only in teachers’ high salaries, but also in the heavy investment of families in private tutoring services
While it may be true that “social and cultural factors” affect maths achievement, the idea that Asians are better at maths because they value education more highly is a very weak one. If this were the case, would it not also be the case that Japanese would universally be better at foreign languages than the British or Australians? Japanese get a long exposure to English teaching but are generally woeful at it, despite all the money they sink into private tutoring services. No, there’s something else going on here, something about the Asian approach to maths and the way it is taught that is important.
It is certainly the case that private tutoring services need to be considered in the mix. When comparing a 16 year old English student to a 14 year old Japanese student, for example, you are comparing someone who does a 9 – 5 study day with very long winter and summer holidays against someone who does an 8 – 8 study day with two-week holidays, and who gets 2-on-1 or small group tutoring in key subjects for up to 3 hours a day, and on weekends. This process starts at age 10 and really ramps up at about age 15-16, just when the linked article finds the biggest gap between English and Asian students. It’s also the kind of process that benefits the “brightest” students most, and would explain the gap very nicely.
It may be that if the UK wants to compete with the sleeping giants of Asia on basic educational outcomes, it’s just going to have to face up to a simple fact: British students need to study harder. A lot harder.
There are some more nebulous cultural factors that come into play, however, and I am going to go out on a limb here and name a few factors in Japanese society (the part of Asia I am familiar with) that I believe make Japanese so much better at maths than their western counterparts.
- It isn’t about native talent: A pet hate of mine about western approaches to mathematics is the idea that some people are talented at it, and most people aren’t. I don’t think this is true at all, and I think it’s not something that Japanese believe very strongly. The reality is that getting good at maths is a long, hard slog that involves a huge amount of repetition of basic skills (things like completing the square, substitution, differentiation, interpreting graphs, sign diagrams, etc.) – just like learning a language. Sure, solving maths problems requires creativity and intuition, but these are only of any value if you know the tools you can apply them to, and are familiar enough with those tools to recognize when and how to use them. Mathematics – and especially high school mathematics – is a process of drilling, drilling, drilling, and I think that Japanese recognize this. In Japan the default assumption is that if you pay attention at school and do your homework, you will be good at maths. Sure, they recognize that advanced maths requires extra commitment and talent, but there is a fundamental assumption here that the broad body of maths (up to and including differentiation, integration, limits, and basic probability theory) are things that anyone can learn.
- The teacher is important: the flip side of the idea that education is important is an increased stress on the value of the teacher, and their role as a guide. The role of the guide is also viewed very differently if they are teaching something that they believe anyone can do, compared to if they are teaching a subject that everyone believes is impossible for most mortals to comprehend. Find me a westerner under the age of 30 who is “terrible at maths” and I will show you someone who was humiliated by an arrogant maths teacher at a crucial time in high school, usually around when they were 14. I was in the bottom class in mathematics when I was 14, expecting to drop out as soon as possible, until a good teacher put some time into teaching me, and I found that I really loved it. In Japan, teachers can be bullies and they can be cold and hard, but I would also argue that they have a much greater burden of personal care and responsibility placed on them compared to western teachers, and the failure of their students is treated more like a professional failure (rather than due to the student’s personal talents) than it is in the west. I think this is especially important with mathematics, because when you don’t get it it really hurts – like a kind of itching in the back of your brain – and the failures pile up rapidly. Just a single year between 12 and 14 in which you give up on maths is enough to make all the subsequent years ever more challenging, meaning the damage and the attendant confidence failures compound.
- Being nerdy is cool: In Japan, it’s okay to be a nerd, and being good at mathematics is admired and respected. It’s virtually unheard of to find someone here who looks down on a man who can do maths, or thinks that it is beyond the female brain, or thinks that being interested in mathematics is weird. Furthermore, the nerd world in Japan is much more gender neutral than in the west, so there’s nothing unusual about girls doing maths. Good mathematics skill – up to and including being able to rearrange equations or solve systems of equations, for example – is not seen as a weird foible, but as an admirable sign that you are a rounded human being.
- There is a social expectation of mathematical skill: In addition to nerdiness being much more acceptable, the range of mathematical abilities that qualify you as a nerd in Japan is much more esoteric and advanced than in the west. There is a general expectation that ordinary people can solve maths problems, that they understand the basic language of mathematics so that even if they can’t solve a problem they know roughly what it is and where it sits in the pantheon. Parents assume that their kids will learn mathematics, and don’t dismiss it as the too hard subject that only the special or the weird get ahead in. Whereas in Australia having a kid who is good at maths is unusual, in Japan it is unusual (and embarrassing!) to have a kid who is not good at maths.
I think these properties add up to a society in which mathematical achievement is encouraged and widespread. I think that Australia and the UK need to change some cultural factors so that the intellectual and educational landscape is closer to that in Asia if they want to keep up on mathematics and technology achievement – especially since China’s education system is maturing, and other Asian nations like Vietnam, Singapore and India are getting wealthier, with all the educational gains that implies. So what should Australia do?
- Ditch the nerd-baiting: there’s something really wrong with the way the English-speaking world treats people who do nerdy things. I’m sure it’s mellowed a lot since I was a kid but it’s still there, the kind of ugly-four-eyes assumption about anyone who is interested in anything that isn’t sport or fashion. Until this weird attitude dissipates – and until the nerd world becomes more gender-balanced, to boot – it’s going to be hard to encourage the kind of cultural changes needed to make maths achievement standard across the board
- Less intuition and initiative, more drills: I think it’s very sweet that maths teachers want to encourage their charges to think about the broader world of maths, about creative problem-solving, about applying maths to the real world, etc. But I think those are natural talents all humans possess, that cannot be unlocked without a robust background in the basic skills that make mathematics work. So leave the creativity for people who need it, and stuff kids’ heads full of “useless” rote learning of techniques and drills. It’s boring, but it’s essential to the bigger stuff. If you aren’t able to immediately see when and how to complete a square, then any problem which requires this basic technique is going to be beyond you, no matter how intuitive you are. Maths, possibly more than any other discipline, is built from the ground up, tiny block by tiny block, and all those blocks are essential. So ram them down every kid’s throat, and make every kid think that knowing the quadratic formula is not a test of some kind of obscure talent, but a basic expectation of every 12 year old
- Force mathematics at higher school levels: When I finished school our balance of subjects had to include at least one science/technology subject, but it didn’t have to include maths. This is wrong, and part of the reason that so many students in Japan do mathematics is that you can’t get into a good university if you take this approach: every one of the better universities includes mathematics in its entrance exam. My personal belief is that completion of higher school certificates should require one foreign language, mathematics, and English. That leaves two other subjects to choose from, and guarantees that you have to do some kind of mathematics to the end of school. Not only will this very quickly lead to a society where entire generations of people are generally familiar with mathematics, it will also put a real focus on the quality of teaching at the earlier years, since any student who is doing badly in years 8 – 10 is going to fail their higher school certificate. [Probably this suggestion for a national curriculum is completely unreasonable, but at the very least students could be forced to do mathematics up until year 11, for example].
- Make school more robust: The Japanese school system is about to shift to a “tougher” system that will include Saturday morning classes, because the previous system was considered “relaxed” compared to earlier years. This is, frankly, ridiculous, but so is the attitude towards education of most of the English-speaking world. Summer holidays are way too long and relaxed, there is a real lack of extension classes and tutoring, and expectations are altogether too low. Education isn’t valued enough, and until this changes anyone who wants their child to do better is going to be swimming against a strong current. Educational achievement is partly supported through the shared goals of a whole society, not just through the targets of individual families, and the expectations we hold for education are primarily set through the school system. So toughen it up – not in the sense of making teachers scarier or bringing back outdated “three Rs” educational styles, but by increasing the amount of time students spend at school, setting tougher standards for graduation and university entrance, making schools compete with each other (as Japanese schools partly do) and forcing parents to take greater responsibility for and involvement in their children’s education. This change isn’t specific to mathematics, but it would certainly help.
I don’t think there’s anything special about Asian students, or about Asian culture, that we can’t adopt. Asians’ mathematics achievements aren’t some kind of native or racial talent. It’s just a collection of attitudes towards education, mathematics and nerdiness that we can adopt if we want. Obviously there will be (potentially challenging) institutional changes required as well, and many people may judge it not worth the effort, but I personally think a world where everyone is good at mathematics is a better world, and we should be aiming for it. With these cultural changes maybe one day everyone will know the obvious thrill of being able to complete a challenging mathematics exam … and enjoying it!
fn1: Though obviously, the less people doing maths, the longer I will remain competitive in the marketplace …