Some Thoughts on Chinese Algebra

Watching Family Guy last night, we hit the episode where Tricia Takanawa says: “Hi! I’m here with Stephen Hawking, the only white man I’ve ever met who knows math better than me.”

We laughed. And then we almost cried.

Because here are some (translated) examples of the algebra that my son’s seventh grade (Year 7) class — that’s age 11 and 12, for the avoidance of doubt — are doing in Chinese school.

“If 5x^{3n+1} + 5x^3y^2 + 4xy^3 - 8 is a seventh degree quadrinomial, find the value of n^2 - n + 1.”

“If |z - 2| + (y - 2z - x)^2 = 0, find the value of 3(x - y) + 4(x - y) + (-z).”

“Where |a| = 3, |b| = 2, |a - b| = b - a, find the value of 3a^2b - [2a^2b - (2ab - a^2b) - 4a^2] - ab.”

“If the polynomial -\dfrac{1}{3}x^2y^{m+1} + x^2y - 3x^4 - 5 is a fifth degree quadrinomial, and the degree of the monomial \dfrac{9}{5}a^{3n}b^{3-m}c is the same as the degree of the polynomial, find the value of n.”

IF you are mathematically literate, which I’m not, these sums are actually not that hard.

Just typing these out makes my head hurt.

But here’s the thing.

IF you are mathematically literate, which I’m not, these sums are actually not that hard.

And that’s despite the fact that they use things like absolute value and variables in exponents, things that we do not introduce until A-level in the UK, as you can probably tell from my oh-so-mathematically literate use of the term “things”.

What it appears to me that the teacher is doing — and, as I’ve mentioned in my previous post on Chinese maths, I only studied maths to the age-16 level in the UK — is familiarising children with the kinds of operations they’ll need to use in much more complex contexts later.

She is drilling the concepts in, and, in doing so, taking the fear out of these apparently massively complex sums, and enabling them to eliminate noise.

Incidentally, I dread to think what level maths you need to be a high school maths teacher in China.

In China, maths is not supposed to be easy. It is supposed to be hard.

So why are Chinese 11-12 year olds so far ahead of Anglo cultures on maths?

First up, in China maths is not supposed to be easy. It is supposed to be difficult. It is intended to prepare you for doing physics, maths, engineering or computer science at university, whether or not your talents lie in the arts.

Further, concepts such as absolute value (||) are introduced at the most basic level, with drilling, and then ramped up very quickly.

Over roughly half a UK term, the class went from introducing the notion of absolute value, with simple sums and verbal multiple choice questions ensuring they understand the rules, and introducing the notion of exponents in the same way, to introducing terms like “coefficient” and “degree” and working with them in expressions such as this, not to mention similar sums using \pi.

Chinese Year 7 middle schoolers spend 45-48 hours per week in class, 10-12 hours of that doing maths, and 5-15 hours doing homework, around 3 hours of that on maths.

Then there are the school hours.

As I understand it, Chinese Year 7 middle schoolers spend 45-48 hours per week in class, 10-12 hours of that doing maths, and 5-15 hours doing homework, around 3 hours of that on maths. British Year 7 middle schoolers spend roughly 30-32 hours a week in class, around 3 hours doing homework, and around 5-6 hours doing maths.

They also get less holiday time, though not, I think, that much less. There is one long break around Chinese New Year and Spring Festival, another in the summer, and the state holidays.

Further, very able children will do extra maths on weekends and after school to get ahead. Less able children will also do extra maths on weekends and after school to enable them to keep up.

I am not, for the record, saying this is a model one should emulate in the West. But some kind of halfway house might not be a bad idea.

If your Chinese is good, these types of sums are much easier to do in Chinese than in English.

And then there is the language. Believe it or not, if your Chinese is good, these types of sums are much easier to do in Chinese than in English, because the language is easier. The single-character building blocks that make up the expressions are largely consistent, and the syntax is more natural than in English.

A “fifth degree quadrinomial” is a “five-power-four-term-expression”. “Degree” is “power-number” (AKA: “exponent”).

BUT all the words I’ve had to render here as polysyllables are monosyllables, with simple root meanings, so that big clunky ugly stream of scary words is actually only five fairly simple syllables. In Chinese, the whole expression “a fifth degree quadrinomial” takes less time to say than the single word “quadrinomial”.

A polynomial is a “many-term-expression” (but only three syllables, of course), a monomial is a “single-term-expression”, a trinomial is a “three-term-expression”, and so on.

Further, one common meaning of the character I’m rendering as “power” here (次)is “time(s)”. You’ll see it used, for example, on signs saying that you can only have one “turn” on a machine. Though, of course, like all common Chinese characters, it has a stack of common meanings, whether used on its own or as part of a “word”.

Now, the boundary between a word and a phrase in Chinese is a topic of academic debate, on which I am in no way qualified to comment, but “five-power-four-term-expression” is a block that feels, to me and to our teacher, like a single word.

There is some evidence that Chinese people’s brains work differently from people whose brains are adjusted to alphabetic languages, particularly when they are doing maths.

And, finally, without wanting to sound like some nineteenth-century Victorian pontificating in a racist fashion on “the Oriental mind”, there is some evidence that Chinese native speakers’ brains work differently from native English speaker’s brains, particularly when they are doing maths.

The visual bit of the brain lights up more when doing arithmetic for native speakers of Chinese than it does for native speakers of English.

I would suggest that one reason is because characters are visual: the more complicated characters are made up of lots of mini-characters, so the visual element of the brain is engaged a lot more at all times. And as the ONLY way to become literate in Chinese is to write the characters out lots, and lots, and lots of times (there are apps that allow you to do this on a touchscreen, but it’s still visual-kinetic learning), the visual region of the brain gets used a LOT.

And, perhaps unsurprisingly given how much more difficult maths vocabulary, and even counting vocabulary, is in English than in Chinese, English speakers need to use the word-processing part of the brain a lot more when doing maths.

A few examples? To illustrate \dfrac{1}{2^n} they drew a square and began dividing it up. Then children had to solve \dfrac{1}{2^n} using that square, and make a triangle that solves the same problem.

The books use number lines and diagrams in contexts that seem to me positively counter-intuitive, by which I mean that for my aged native-English brain with its set neural pathways, I find they confuse rather than simplify the issue. They also use geometry in contexts where, I think, we wouldn’t.

It IS harder than the Chinese average. But not, I suspect, very much harder, on the grand scheme of things.

One final note. This is maths from an academic elite school, probably top ten in the province and top five in the city.

And, no, I didn’t do this deliberately. It was the only school that seemed enthusiastic about having a foreigner on board, so that was where Zac went.

Which means poor Zac is studying alongside some of the brightest children in a province with roughly twice the population of Australia or New York State.

So it IS harder than the Chinese average. But not, I suspect, very much harder, on the grand scheme of things.

I’d really welcome thoughts on this from people who, unlike me, know maths and/or Chinese well. Also from anyone who’s currently got a child of a similar age in a non-elite Chinese school.

27 Responses

  1. Alyson says:

    Well, I can’t remember any of the maths I did in school, certainly not that sort of maths. Probably because I’ve never used it since the day I completed my o level maths paper. I certainly didn’t need it in university. I wonder if I’ll even bother with my boys, unless they particularly want to go down that road.

  2. Theodora says:

    I’d like to keep Zac’s academic options open at this stage, though, like you, I’ve not gone down a route where I’ve needed to use this sort of thing.

    Interestingly, he wants to keep up Asian-style maths, although ideally not in Chinese, and at Malaysian or Singaporean level rather than at mainland Chinese level. Which will make life a hell of a lot easier for him if he opts to do a hard science or hard maths subject in an Anglo university…

  3. Richard says:

    Well….that was an interesting mix: half the problems I could see the answer in under 5 seconds and the others would take some annoying face-smashing of algebra (Incidentally there is at least one error in q2, that missing bracket for starters). Incidentally doing IB Higher Level maths (which is pretty hard, more so since Chinese schools have started doing the IB and pushing the grade boundaries up) I only had 4-5 hours a week of classes and spent 5-6 hours a week on homework. Seems pretty brutal for Year 7 to be honest.

    • Theodora says:

      Right. I’ve fixed Q2, which is indeed missing a bracket.

      But I *think* the clue for the one that looks face-smashingly difficult, done Chinese style, is that |z-2| has to be either zero or the “opposite number” of the horrible bit on the other side, which means that a squared expression has to come out with a negative result, which it can’t do, so the squared expression has to be zero. So I can’t quite follow the logic without a pen and paper, but both sides have to equal zero, which means z has to be 2, and y-x has to be 4.

      Which are the other ones that are face-smashingly hard? I’ll go back and check them. I thought I’d already checked, but a) algebra’s not my strong suit and b) I have to use LatEx to get expressions to display, which I’ve never used before. #embarrassed.

  4. Theodora says:

    And, yeah, I think it’s too hardcore for Y7, as well. I think what I should have written is that the sums are “less hard than they look”.

    One more point? If more Chinese schools move over to IB Higher Maths, the grade boundaries are going to rocket, just by the sheer volume of numbers. I suspect it’s only the international schools that do IB Higher (maths is a bit easier in the int’l schools), b/c most Chinese schools are focused on their insanely difficult university entrance exam.

  5. Kerwin says:

    Love it!
    I’m a Math major so this stuff is easy 🙂
    Zach will be fine.

  6. Theodora says:

    I didn’t have you pinned as a math major, Kerwin. Wow.

    But that’s sort of encapsulated what I’m trying to say above, based on having close to zero maths, though having learned alongside Zac in the early stages when he was grappling with the English vocab, as well as the Chinese.

    It *is* easy if you know what you’re doing (none of the answers are complicated), and what they appear to be getting them to do is know what they’re doing very early on. So when they get to apply a similar set of rules and logic to the undoubtedly much harder algebra they tackle later on, it will be second nature.

  7. Yvette says:

    You’ve already heard my thoughts on this so I won’t repeat myself on those points. 🙂

    One question though, as a subject do Chinese kids *like* math? Because one odd phenomenon I noticed when I go to, say, Hungary and get into a conversation with the cab driver is how even people who won’t need it later in life recall math class with a lot more fondness than those from the West (where it is paradoxically accepted that one can be culturally literate without any math/science literacy). When it comes to math and physics a lot of people will tell me straight-up how much they hated it in school in the US, which always seemed a bit of an unintended insult to me, but in Eastern Europe and Asia I had more than one person tell me how OMG math was their FAVORITE…

    Just wondering if that ties into it at all, as in my experience people tend to be willing to do harder stuff for subjects they enjoy.

    • dave says:

      As i recall math classes, the problem was the teachers methods, not the subject itself. it seemed to start out poorly, like seeing an explosion and then having someone try to explain what and how it happened, but with half the evidence destroyed in the blast itself? I got along well with maths but it was almost as if a seventh sense was needed to do so

    • Elizabeth says:

      From my (limited, single summer) experience, yeah, many more Chinese kids like Science/Math than western kids. I was struck by how many said those were their favorite classes. And, most of them admitted they disliked studying Chinese.

      Compared to learning Chinese, I think math/science are easy!

  8. Theodora says:

    That’s a very good way of putting it — it is almost universally accepted, at every level from underclass to elite, at least in the UK — that you can “be culturally literate without any maths/science literacy”. People will own up quite happily to being crap at maths or science without embarrassment. Whereas you don’t get that same gung-ho attitude to English.

    I, as someone who knows close to fuckall science, really enjoy the hugely educational science museums they have here. I read all the physics stuff they have up with great interest, things like the parabolic curve and the different types of ellipses, and how light waves work and the ions in the magic balls, and so on. Zac, on the other hand, just likes to play with the activities.

    As to whether they like it or not… Chinese teachers do do things like bringing in ice lollies for the class, playing them videos in geography or history, and generally try to make learning fun. On a lesser level than in the West, but they do do it (actually, I’m not sure many Y7 teachers would bring in ice lollies, but then we’re touching on emotional maturity and that’s a whole new ball park). But I’m not sure maths is supposed to be enjoyable here.

    As to whether they “like” it… I’ve never asked. I’ll make a point of asking everyone I meet in trains and taxis whether they liked maths, and report back.

    One problem with maths in the West, FWIW, in terms of liking it, is that many people who could have enjoyed it are bored shitless because it’s too easy, and others are horrified because it’s too hard.

    • Yvette says:

      Hah- yeah trust me, I’ve been rather frustrated by the cultural illiteracy when it comes to math and science thing for a long time. Especially because geez oh man, we use science and technology more than EVER these days!

      One thought game that comes to my mind actually is my undergrad/MSc university was divided with tech stuff north of a main road, and humanities on the south, so if I wanted to offend those from the south I’d point out how very radically different things would be between the two halves should they suddenly be responsible for teaching the other side’s curriculums. If you gave the tech/science side a few weeks to put things together you would have some sort of humanities curriculum- it would have gaps, for sure, but someone would happen to be an accomplished musician, another guy would have a passion for Shakespeare, another could teach about WW2, etc. The best I ever heard when asking the humanities guys what they could do was one or two saying they could do the non-math based astronomy for poets class, MAYBE.

      Frankly I really don’t think this is a healthy thing at all in society, and I spend a bit of time thinking about how to fix it.

      • Theodora says:

        I absolutely agree with you. I debate this with travelling families all the time. About as many people will use algebra in their professional lives as will use Shakespeare in their professional lives, AKA a fraction of a percent. Which doesn’t stop either of them being good things to know, per se.

        “Cultural literacy” is a good way to frame it. One should know Pythagoras and pi-r-squared, and rather more than that.

        I was abysmal at formal logic, sadly, otherwise that would be my contribution. No, wait! I did the PreSocratics. Give me a few weeks, and I could scrub up reasonably well on the Greek part of history of science, Heraclitus, Thales and all. But, yeah, it would be a rather feeble contribution.

        • Yvette says:

          Personally I always point out to people who take the “algebra, what is it good for?” line that I’ve never encountered a pressing problem in modern society where less education was the answer. And frankly like Shakespeare I don’t think one learns stuff like algebra and basic physics so much for the content as much as how it teaches you a framework to think in which is wonderfully analytical and can apply to a lot of other things in your life as well- like how one isn’t going to write English papers all your life but it’s really useful to know how to present a coherent argument.

          But yeah, as I’ve said, somewhere things went haywire on that front. Which is weird because if you think back on those Victorian gentry (for example) we think they’re weird as hell for their bug collections or what not but the fact of the matter is everyone was a bit of a naturalist back then.

  9. Gosh what a mind melting education Zac is having! I have to say I’ve no doubt he will benefit from the additional stimulus at such a young age.

    I wish I had been subjected to a more rigorous education in math and science as a child. I did take maths all the way to A level but struggled immensely and found little sanctity in my tedious teachers.

    Like Dave I think a practical application will always appeal to me more so than a sheet of equations. I think I’m of the creative right brain format, visual rather than theoretical.

    • Theodora says:

      Well, I think one problem with the way we do maths AND science in the UK is that we teach over-simplified versions to 16, which then have to be unlearned post-16. I think he is benefiting from the stimulus overall, but it’s also insanely hard work for the poor little man…

  10. Nonplussed says:

    {sits quietly fingerpainting}

  11. Carol A says:

    Dear Theodora,

    I have two children (14, 11) and transferred my son during Year 7 from the MYP program at an IB school to a Japanese private school, I will stop lurking and share our experience.

    The MYP maths program is a great improvement from Western public schools because it requires students to do complex real-life math assignments. The IB program itself is incredibly demanding… My son wanted a career in the sciences, my husband and I felt that an Asian approach focusing on fundamental mathematics would be more suitable for him.

    When he switched over, my son used to say, “I understand the concepts but I can’t do them fast enough.” I suspect Zac may be in a similar situation. The Asian style of learning is really all about “practice is mastery.” That the more times you do something, you will do it with more accuracy and speed. Long algebra is essentially a series of simple calculations you must do quickly and with 100% accuracy. It’s similar to play a long musical piece. (Ever wondered why so many Asian kids play piano and violin…)

    What my son has learned most is diligence. This attitude has translated into higher English scores on standardized tests. The irony is that it increased when he switched from an English school to a Japanese one…. His ability to program computer games (as a hobby) has also improved because he is now so much more careful about not making small errors. The downside is that his research essay writing has gone down because the IB really hones that skill. But we are banking on how he will be able to “remember” those skills at a later age.

    To “unschool” Zac in Asian style maths, I would recommend getting him to do timed worksheets every day. At my son’s school, they have 15-mins. quizzes in the morning. Students are supposed to get 100% on them. During summer vacations, they do 15-mins. of timed worksheets every day. You should try to look for these types of workbooks at Chinese bookstores. Good luck!

    • Theodora says:

      Thanks so much for unlurking, Carol. I’d 100% agree with you about diligence, and the value of it. And, yes, also speed learned through repetition, and repetition of all permutations — so when you introduce absolute value you do so with fractions, decimals, pi, positives, negatives, algebra, all at the same time.

      Interesting that Japan does 15 minute quizzes each morning. In China they have 2-hour long tests every month, and an hourly one once a week, but the expectation is not 100%. In Zac’s school, which is an elite school, the lowest grade on one test was 16%.

      Someone is very kindly sending me a Singapore eighth grade maths book, which we’ll pick up in the UK. I like the notion of timed calculations, though, and I’ll see if there’s a way we can incorporate that, although daily won’t work with the way we live while we’re travelling.

  12. Logan says:

    Well…I’m in eighth grade, and reading those problems gave me a migraine. The Chinese exchange student living with my family looked at my algebra textbook a few minutes ago, and told me that in China, that was elementary school level math, which is really quite mind-blowing. I live in America though, and after reading that in the UK you only do three hours of homework a week, I am seriously considering moving. At my school, it averages two a night in the middle school, and four a night in high school. Might just be a private school thing.

    • Theodora says:

      Hey Logan! Thanks for your comment!

      No, we don’t get very serious about homework until our exams at 16, and again at 18 (although some do other sets at 17). We also tend to specialise in topics earlier than you guys — we can start to pick specialisms at 14 (though we have to do English, Maths and Science), and at 16 we specialise further. Which has pros and cons: I went to Oxford and am scientifically and mathematically barely literate because I studied humanities.

      I’d suspect that your homework loading is a private school thing: in the UK, too, private schools give more homework than government-run ones.

      And, yes, I’d suspect US 8th grade algebra, even in a private school, would be about elementary 4 in China. Chinese second-graders are already doing complex operations with fractions, decimals and combinations of the lot. And, in fact, we introduce algebra (simple algebra: y + 2 = 10) in elementary in the UK, with algebra-style fill in the blanks questions ( ? + 2 = 10) occurring fairly early in elementary: I think the US approach, whereby algebra is somehow separate from other maths, is rather odd.

      If you’re into maths, do check out a Singapore 8th grade maths book — that’s what Zac’s using at the moment as we travel. It’s called Discovering Mathematics by Chow Wai Keung and others — provides a daunting insight into where we Anglo speakers sit on the world maths and science stage (though I’m sure you’d cope with it perfectly well). It’s not as brutal as China but it’s a good example of the Asian approach to maths in general.

      Funny story from Zac’s school in China… Their US exchange partners came for a visit. Everything in class was painstakingly dumbed down for the lessons that the US students were looking in on so as not to make the foreign guests feel bad.

      Say 你好 to your exchange friend! And I do hope you get to go to China, it’s fab!

  13. VenX says:

    As a Chinese who have been through this, undoubtedly this have benefited me for my later works, and I do think it will do for Zac later as well (if he is interested in science as I do=] ).

    But to be honest this definitely have its drawbacks. I still could recall spending hours on solving a single question at night in high school. Even though it was quite delightful to get the answer in the end but I was very depressed if not. As I have been studying in an English speaking country for the last 3 years and I’ve been very proud of my advanced mathematic skills. But due to those days I have put into self-study, I also find myself, sometimes, hard to be involved in a group event like group discussions and such. I’m still very confused by the way they educated you as a product on a flow-line rather than a person. Too much concerns have been put onto how good the grades you have got and how accurate your answer is. My feeling towards this is still very mixed.

    But maths is essential for life. It is not only about solving the questions, it do alter the way you think and process problems. Believe me, difficult maths is not a bad thing at all. What worse is sometimes your creativity get killed by a set answer.

    • Theodora says:

      Thanks for your thoughtful response. I too have mixed feelings about the Chinese school system. I believe if you could have a school system that taught maths and sciences like the Chinese do, and English and humanities — group topics and discussions and creativity — like the English do, it would be the perfect mix.

      What interested me in our experience of school was that the teachers did very much see the children as individuals, as teachers and educators do everywhere, but seemed also constrained in the flowline of production that you identify. Zac, as the only native English speaker in his school, and the only non-Chinese but for one Korean, really confused his English teacher by writing an essay when an essay was requested. What he was supposed to do was use the template and just fill in different words. Of course, being English, he used his imagination and creativity to write his own answers, in his own words, as he’d been trained to do. The teacher was very confused.

      What I find is very sad, though, about the Chinese childhood is how very, very much of it is devoted to school and classes. We lived in a lower middle class area, so we’d occasionally — rarely! — see children playing in the street. In the wealthier areas, they’re invisible.

  14. Kevin Song says:

    I’m a student at an American college, and all this is easy even in western math

    • Theodora says:

      Yep. But this is the math eleven-year-old children do in China (first year of middle school).