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 [latex]5x^{3n+1} + 5x^3y^2 + 4xy^3 – 8[/latex] is a seventh degree quadrinomial, find the value of [latex]n^2 – n + 1[/latex].”

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

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

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

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 [latex]\pi[/latex].

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 [latex]\dfrac{1}{2^n}[/latex] they drew a square and began dividing it up. Then children had to solve [latex]\dfrac{1}{2^n}[/latex] 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.