About that maths exam…

Virtually all of the media discussion about last week’s NCEA level 1 algebra MCAT paper has been about how “difficult” it was. Teachers lined up to criticise it for disillusioning hard-working students; Hutt-based list MP Chris Bishop took the opportunity to reminisce about the time he too was angry about an “unbelievably difficult” maths exam. Yet the most striking aspect of NZQA’s new style of questioning has remained strangely untouched.

Flick through the paper year 11 students sat last week and you’ll stumble across questions like this:

The area of a rectangle is x^2-x-2.

[…] (b) What do you know about the value of x for this rectangle? Explain your answer.

Or this:

Jason writes down 4 numbers: 1, 3, 5 and 7. He adds the pairs of numbers to form a triangle, as shown below. He stops when he gets to a single number at the bottom of the triangle.


[…] (ii) Find, using algebra, the relationship of the numbers in the first line to the numbers in the fourth line when he changes the order of the numbers in line 1. Explain your answer.

Even just five years ago, the exercise comprised questions more reminiscent of previous decades:

A milk drink costs $1.50 more than a fruit drink. 5 fruit drinks and 4 milk drinks cost a total of $24. What is the cost of 1 milk drink?

Solve \frac{x^2-2x-3}{x^2-7x+12}=2.

Of course, you shouldn’t trust that I’m not cherry-picking questions to prove a point, because I probably am. You should flick through a couple of others and judge for yourself. It seems to me that, in the last few years, NZQA has been nudging high school maths assessments from the mechanistic, rote-learnt thinking that dominated maths education a generation ago, to questions examining a more conceptual understanding.

For this reason, anyone who uses this debacle as an opportunity to gloat about how bad at maths they were, or about how incapable modern teenagers are (this Civilian article has more than an ounce of truth in it), is missing a much more important trend. The level of raw algebraic manipulation required to complete this year’s paper is not that different to previous years. But in times gone past, if you spent enough time rehearsing memorised procedures for a short list of stock questions, you could get through the paper almost brainlessly. Now, you can’t. Some of those template questions are still there. But increasingly, New Zealand high school maths examiners want you to prove that you actually understand what you’re doing—and why you’re doing it.

Someone’s bound to write a comment about “standards” or “fairness” or whatever, so let’s get that out of the way. Given the shift, whether the paper was “fair” is a function of what guidance and training teachers were given. No-one’s really commented on this, but I’m guessing it wasn’t much. That being the case, I think students can feel justifiably upset, and I think it’s reasonable to demand that NZQA make allowance this year.

I just hope that doesn’t discourage examiners from setting a similar test again next year.

What is maths?

Disdain for mathematics runs strongly in popular culture. Journalists proudly joke about being able to do none of it, knowing many of their readers do the same. People associate arithmetic and algebra with nerds, and novels and films with leisure, despite the fact that these topics are studied by teenagers at the same time. It’s not cool to be fascinated by mathematics, not even among the educated.

This culture, I suspect, derives from the perception that it’s some sort of obscure and irrelevant art, which in turn probably comes from its being taught as a mechanistic topic with only right and wrong answers and no underlying ideas. Yet nothing could be a bigger misrepresentation. It’s true that, without the complexities of nature in your way, things can be proven to be right or wrong more reliably than in the sciences. But the core ideas of mathematics are deep, informative, and not as rigid as you might think.

For example: The point of calculus isn’t that you can differentiate or integrate some expression I put in front of you. In fact, in most real-world cases you don’t even have an expression to differentiate. The biggest insights of calculus—in my opinion; others can reasonably differ—are that we can analyse phenomena in terms of rates of change (differentiation), that we can analyse the accumulation of processes by “adding” infinitesimally small pieces (integration), and, in an astonishing coincidence, that you can do the latter by doing the former in reverse (the fundamental theorem of calculus). These ideas in some way underpin most modern technology and a good grasp can help inform how you see the world.

So when Wallace Chapman ranted on Breakfast (starting at 37:15) about how quadratic equations are “outdated”, well, he’s wrong—they’re as relevant as they’ve ever been—but he has something of a point. I don’t solve quadratic equations anymore. My computer does that much faster than I ever could. But I know that if something can be described by one, it might have two, one or no real solutions depending on its coefficients, and that some phenomena can vary drastically depending on which of these is the case. And if it does have two solutions and I know both of them, I can write the equation as (x-a)(x-b) = 0, to help me figure out other things. These ideas are what makes mathematics relevant. But they’ve long been underemphasised in high school curricula, everywhere in the world.

If such goals sound like too much of a dream, I’ll concede that it is, to a degree. After all, other subjects struggle with it too: I didn’t really understand what the study of English literature was about until after I left school (at least, I think I do now), and I suspect I wasn’t alone. Indeed, we couldn’t get rid of solving equations entirely: just as essay-writing is an important skill in many subjects, raw procedural manipulations are an essential skill in maths, as the NZ Initiative argued last year for the primary level. But the humanities and sciences seem to do better at persuading their students that there’s some meaning to them.

Much to my chagrin, I don’t have concrete, practical steps in mind for classrooms, beyond some obvious ones that are too vague to be useful. But the introduction of questions that require students to think about the ideas behind what they’re doing is a promising sign.

Why this matters

People have been lamenting the state of recruitment into science, technology, engineering and mathematics (STEM) for years now; I trust I don’t have to explain that this is a problem. The salient point here is that, as long as mathematics is seen as that “other” subject that people either get or are clueless about, persuading teenagers to enter STEM disciplines is always going to be an uphill battle.

In this light, concerns about this exam demotivating students away from STEM subjects are understandable. But by obsessing with the exam’s difficulty, we risk losing sight of the forest. The teaching of subjects doesn’t just exist on a one-dimensional spectrum from “easy” to “hard”. The rote-learnt procedures that used to dominate secondary maths education have their place, but they need to be complemented by the motivations behind mathematical ideas and the insights they bring. This would go some way to helping the next generation understand the relevance of maths to, well, everything—something becoming even more true as the knowledge economy replaces the industrial one.

I’m glad that New Zealand’s maths examiners are taking steps in that direction. Whatever happens with this year’s algebra MCAT, I hope the rest of the maths education community will back them in continuing the shift in years to come.

About Asians. Again.

On occasion, my American friends wonder out loud to me: “Is there racism in New Zealand?” It’s a great story. Sure, I tell them. In recent years, our centre-left party’s been running on anti-Asian rhetoric, allowing our governing centre-right party to take the moral high ground. The shock on their faces is priceless. The right, the moral high ground against racism? Isn’t that the wrong way round?

That’s only half the story, of course. Labour vehemently denies that it’s racist. But then, so do Donald Trump and Ted Cruz. No-one thinks they’re racist, because everyone thinks racism is bad, and no-one thinks they’re bad. And anyway, personally, I try to avoid using the word “racist” as a shortcut for criticism. I just think it’s lazy, like the phrase “politically correct”. I mean, sure, Andrew Little’s words seem to target certain groups of people. Sure, that probably makes people of those races feel a little bit upset. But the real question is whether those opinions are justified, or policies sound. That was the stance I took towards their housing “analysis” last July: that I would look past the race relations, and towards the statistics. (I concluded their conclusions were bogus.)

What, then, do I take of this most recent episode about chefs?

Here’s the problem. Labour can say as much as it likes that its anti-foreigner policies “aren’t about race”. It can be proudly nationalist without being racist. It can publish policies that don’t mention the words “Chinese” or “Asian” once, and just refer to “foreign” or “non-residents”. It would apply just as much to Canadians as to Croatians as to Chinese. I get the concept, really, I do.

But when their rhetoric consistently centres around the Chinese, it’s very hard to take their pleas of race-neutrality seriously. In the 2014 election campaign, they constantly made reference to investors from China. When they talked about housing last July, they focussed, again, on Chinese buyers. This isn’t some sort of one-off out-of-context quoting. This is a theme in Labour’s rhetoric, and it’s been going on for years, long preceding Mr Little’s leadership. Their policy might be race-neutral, sure. But their rhetoric most certainly isn’t.

And they might have noticed that they get the same reactionevery time. Mr Little can plead about how “baffling” it is that his comments “may have offended anyone”. But I mean, really? After what happened last time, and the time before? I realise that people are often misconstrued, and I empathise. But there’s a limit to how many times you can plead this before you should probably start wondering if it’s actually you.

Personally, I’d rather that we threw this whole anti-foreigner sentiment out altogether. But if non-racist anti-foreign sentiment is going to be a cornerstone of Labour Party policy, I have a challenge for Labour. Next time a Canadian investor strikes a $1 billion deal, or an American company increases its stake in our third-largest forestry estate, or just any time Canada, America, Australia or Europe do any of their 59% of all foreign direct investment, or America does any of its 46% of all land acquisitions by foreigners (both statistics from 2013–2014)—would they mind making some noise about how Canadians or Americans are taking opportunities away from hard-working Kiwis?

It’d still take some work to convince me that a New Zealand economy less open to the world would be better off. But I’d at least be able to take them seriously when they try to tell me their desire to reduce immigration and foreign investment holds for all foreigners equally.

Where did Red Peak come from?

By Chuan-Zheng Lee

Two months ago, I wrote in this blog professing indifference to the flag change question. I did, though, have just one plea. Follow basic principles of vexillology, I implored. Among other things, keep it simple, “so simple that a child can draw it from memory”. I considered it a fairly routine suggestion. Surely, if it’s worth doing, it’s worth doing well?

Alas, it was too hard. Three of the flags on the shortlist have renditions of the silver fern that I can’t draw as an adult, let alone as a child. Just one meets basic vexillological criteria. Cutely labelled “Hypnoflag“, it’s the least popular of the four. I don’t mind it, but I don’t care much for it, either.

Then I stumbled across the Red Peak. I had glanced at it a few weeks ago and, I admit, was largely apathetic to it. But as I looked at it again, reflected on it and read more about it, its brilliance progressively dawned on me. It has everything in that vexillological handbook: simplicity, meaningful symbolism, distinctiveness. It nods to our Māori origins and our geography. It’s clear even when made small: redpeak-small. I generally think symbols are arbitrary, but I can imagine this one in more than just a hypothetical sense. It fits.

I still don’t mind our current flag. As I wrote in July, I don’t understand the aversion many have towards the Union Jack. Our flag isn’t an epitome of good design, but it ticks the boxes. Red Peak is the first design that would motivate me to vote “yes” in March, not because I want change, but because it would be such a great flag to have.


It won’t happen, of course. We can sign whatever petitions we want, and it’s theoretically possible to get Red Peak on the ballot, but practically it’s too little too late. One might rightly ask, where was all this support before the shortlist was decided, when it mattered?

There are a few things that could have happened. One is that, like me, people may have reflected and changed their opinions. The post that went viral was penned by software investor Rowan Simpson, previously an advocate of the silver fern. Others too, may have learnt about flag design and grown to like Red Peak as they came to understand it. It is just as well that the process is long enough for public discussion.

But why would the tide come only after the shortlist release? It’s possible that people were pushed into action by the shortlist. Motivation to understand what’s good rises when what’s bad becomes a concrete possibility. The Flag Consideration Panel didn’t actively solicit feedback on the long list (which included Red Peak) like they did for “what we stand for“, nor should they have felt obliged to. The whole point of delegating to a committee is that they can devote more effort to their deliberations than the rest of us have time to, and picking a shortlist was their one job. And, to be fair to those who hadn’t thought much about the options beforehand, there were no hints that the panel was going to centre on silver fern designs. Of the forty on the long list, just eleven had detailed silver ferns. Voters might have quite reasonably assumed that the shortlist would reflect this, and not thought about it much more deeply. And when the final options started staring them in the face, they started wondering: does it have to be these?


That would be the innocent explanation. More cynically, a fair amount of Red Peak support seems to be coming from people who previously criticised the flag referendum as a “distraction” from “more important issues”. The tension between this stance and their newfound passion seems not to have occurred to them. Often vehemently opposed to John Key, they take up nearly any cause that contradicts him.

One can’t help but wonder: if Red Peak had been on the shortlist, would that endorsement have triggered a backlash, just like the four on the shortlist now? Some of Red Peak’s proponents are genuine, no doubt, like Mr Simpson. But it’s hard to imagine such strong support if Red Peak had been always been a front-runner, or worse, Mr Key’s favourite.

It’s easy to dismiss the whole movement on this basis. That would be a mistake. The first reason is that its designer puts forward a very respectable case for it. There are surely arguments against it, but the fact that some of its supporters are government opponents is not one of them.

The second, more pragmatic, reason is that Red Peak has pulled people out of indifference, apathy and cynicism towards a flag change in a way that no other design has. For passionate change advocates, this is surely a good thing. Perhaps now they have a chance of convincing sceptics that it’s not a “waste of money” or a “distraction”. It offers an opportunity to move a flag change away from negative reasons, like “this doesn’t reflect us today”, to positive reasons, like “this is a cool flag”.

The fact that Red Peak has slowly won people over should give hope that it can continue to change minds. It’ll take time—too much time, perhaps, to get enough support before the November vote, even if it were on the ballot. If so, the true test is whether Red Peak can keep its groundswell growing after (here is my prediction) New Zealand votes to retain its current flag in March. With an alternative not just for the sake of being an alternative, a renewed flag change effort might get much more support than current polls indicate.

The half Labour got right and the half they got wrong

By Chuan-Zheng Lee

I get that racism stirs emotions, but I try to give attempts at statistical analyses a fair hearing. About half of the work released by Labour on Saturday is actually sound. This half is also the half that has received the most criticism. Bayesian inference is a perfectly good means of developing probabilistic models about things like “based on their name, of what ethnicity is this person?” Reading Rob Salmond’s explanation of it yesterday, there’s nothing obviously untoward about this part of their methodology. I know a lot of people have felt offended about an apparent conflation of last name with role in the housing market, but strictly speaking, Labour’s analysis doesn’t imply it.

In statistical jargon, what Mr Salmond’s Bayesian analysis computes is an expectation (over Bayesian probabilities) of the number of buyers of each ethnicity, among those who bought a house between February and April with an unidentified agency representing 45% of the market in that period. You can think of this as, “if you picked 3,922 people with these surnames but otherwise at random, and repeated this experiment lots of times, on average, any such 3,922 people would have this many of each ethnicity: 40.7% of them would be European, 39.5% of them would be Chinese,” and so on.

If both of these two assumptions hold:

  1. The 3,922 in their data set are representative of the 8,790 who bought houses in that period.*
  2. Knowing whether someone bought a house tells you nothing statistically more about their ethnicity than their surname does. For example, a Wang who bought a house is no more or less likely to be Chinese than a Wang who did not. (In probability speak, ethnicity and buying a house are conditionally independent given one’s surname.)

then the 3,922 people in their data set are “otherwise (than surnames) at random” for the purposes of determining the ethnicity of house buyers in Auckland between February and April.

Even intuitively, this should look fine. Whenever we see names before faces, we’ll often make guesses about ethnicity based on last name. Sometimes we’ll be wrong—for example, someone in an interracial marriage who changed her last name—but the odd error doesn’t stop us from guessing. We keep doing this because it’s a fairly good heuristic. If you think you don’t guess ethnicities based on names, I politely suggest that you ask yourself again.


Where, then, did Labour go so horribly wrong? In pretty much all of the other half, the part where they tried to make the leap from ethnicity to residency.

Here are Phil Twyford’s and Mr Salmond’s claims, emphasis added:

  1. “39.5% of the Auckland houses sold went to people who appear to be ethnically Chinese. This is a large discrepancy from the 9% of the Auckland population who are ethnically Chinese.” (ref)
  2. “It’s staggering evidence that strongly suggests there’s a significant offshore Chinese presence in the Auckland real estate market. It could not possibly be all Chinese New Zealanders buying; that’s implausible.” (ref)

Claim 1 follows from the Bayesian analysis and it’s fine. Claim 2, however, is useless. Depending on how you read it, the italicised part is either trivially true or flatly wrong.

If he meant it literally, as in, you only need to find one Chinese foreigner to prove him right, then that’s obviously true but meaningless (i.e., trivially true). Presumably what he actually meant is that it’s implausible that claim 1 can be explained by anything other than a “significant offshore Chinese presence”. Not just unlikely, but implausible.

Yet, despite the rigour involved in arriving at claim 1, neither Mr Twyford nor Mr Salmond give any precision to what they mean by “implausible”. To me, this means some very high posterior probability (if you’re a Bayesian), like 0.999 or something, or some extraordinarily low p-value (if you’re a frequentist), like 0.001 (well below the typical 0.05). But maybe they had something else in mind, and that’s okay. They also neglected to specify what they meant by “significant”—say, “enough to affect prices”, or “comprises 10% of the market”.

Now, I know what they’ll say. It’s not possible to quantify this sort of hypothesis. Well, firstly, it is: I just did, in two different ways. They might not have the data to reach that criterion, but that’s another matter. Secondly, that’s no excuse for the sort of magic trick they performed, especially if the claim is that something is “implausible”. If we want to give useful political sound-bites, at least keep to “it merits further investigation” or something like that. That’s defensible. The sweeping statement Mr Twyford paraded on Saturday is not.


The leap from claim 1 to claim 2 that Messrs Twyford and Salmond made requires an implicit assumption. You have to believe that the propensity of a resident to buy a house is roughly independent of ethnicity. That is, ethnically-Chinese residents are just as likely to buy a house as Indian, European, Māori, Pasifika and other residents.

Keith Ng and Thomas Lumley, among others, have offered plenty of reasons to believe this assumption might be false. Maybe recent migrants tend to buy houses and have cash, for example. Maybe Chinese just prefer real estate to stocks, tend to save more, move more often, get more help from their parents. There are all sorts of hypotheses that Labour failed to rule out.

In response, Mr Salmond tried to address two of them with a comparison between the resident Chinese and resident Indian populations. Even this, though, only “contra-indicates” at most two such explanations (relating to recent migrants), and at best in a way that suggests they can’t be responsible for the whole difference between 39.5% and 9%. Here’s the thing though: with so many variables and alternative hypotheses, it could easily be that all of them are responsible, each in a small way, that add up to the difference observed in claim 1. And, to be sure, there might also be an “offshore Chinese presence” somewhere in there too. The problem is that we don’t know, based on this data set, what it is. For this reason, it’s perfectly “plausible” that the “offshore Chinese presence” Mr Twyford asserts is not “significant” (whatever that means).

Is it a tall ask for Labour to rule out all alternative explanations in combination? Yes, of course it is. And that is precisely the point: there are too many competing explanations for the metric Labour have used—ethnicity—to tell us anything useful about residency. The root problem is that Labour’s analysis doesn’t measure residency. It measures ethnicity. As Professor Lumley put it:

If you have a measure of ‘foreign real estate ownership’ that includes my next-door neighbours and excludes James Cameron, you’re doing it wrong, and in a way that has a long and reprehensible political history.

It may be frustrating for Labour to find the data it wants, but that doesn’t entitle them to present a half-baked “analysis” to try to back claims that Chinese foreigners are responsible for soaring house prices. And this is putting aside the artificial restriction of housing supply through zoning and height restrictions, and a continued concerning tendency for Labour to talk a lot about Chinese people and not very much about Canadian, British and American foreign investors.

* Even if they’re not, it’s worth noting that 45% is such a large share of the whole market that even if every ethnically-Chinese buyer used this agency, there would still be an overrepresentation of ethnic Chinese.^

On the meaning and design of a flag

2015-06-30 01.15.44 (800x450)

Someone will need to send me a new one if the referendum passes

By Chuan-Zheng Lee

There’s something about being away from home that makes you cling on to anything about it. My keyring has a kiwi on it, photos of One Tree Hill fill my desktop background, and a flag hangs boldly on my bedroom wall. Clearly, the flag means something to me.

Yet, I’ve struggled to be moved by either side of the flag debate. Don’t get me wrong—I’ve tried to care, really, I really have. But the more I think about it, the harder it gets to escape from this conviction: Symbols have exactly and only the meaning that we give to the symbol. We can adopt whatever symbol we want, and it’ll stand for us if we want it to.* The New Zealand flag means something to me because I grew up believing that it was “the New Zealand flag”. If we want to adopt a new one, I can just make that mean something to me too.


That’s pretty much my position on the flag. At this point, I suppose people will start throwing arguments at me to try to persuade me I should care. I’ll address a few.

“It looks like Australia’s flag.” Our flag first came into use in 1869. Australia’s had no such form until a flag competition after they federated in 1901. They didn’t copy us, but we still have first claim, and I refuse to cave just because Australia came up with a similar flag after us. They don’t get the flag as a consolation prize for losing pavlova.

More to the point, I don’t care if our flags have similarities. We’re quite similar countries, and I don’t mind enlightening people about the difference. The Commonwealth Star and different-coloured stars are obvious enough if you know what the flags are. Frankly, I spend more time educating people I meet in America that my accent isn’t Australian.

“It has the Union Jack on it.” I don’t have a problem with this association either. It’s empirically true that our culture largely derives from Britain’s. I don’t see why this needs to signify anything more than some aspect of our history that’s still visible today.

Now, this doesn’t mean we have to have it on our flag—its removal wouldn’t bother me at all, either. We might choose to emphasise other aspects of our history or identity—our Māori heritage, for example—and that’s fine too.

“Our soldiers fought under it.” Our soldiers fought for New Zealand; the flag is just an identifier. You don’t fight for a flag, you fight for a country. It’s not clear to me why we can’t change this identifier, and treat it as a continuation of the previous flag.

“It’s a waste of money.” As far as the government budget goes, $26 million isn’t exactly life-threatening, so this is a waste of money if, and only if, the exercise isn’t worthwhile in the first place. If we think it’s worth getting right, it’s a cost worth bearing.

Okay, okay, I know what you’re thinking. How can I say I don’t care about the question, and then defend the cost of asking it? My point is that “it’s a waste of money” doesn’t show we shouldn’t hold the referendum, because it relies on the premise that we shouldn’t hold the referendum. If my compatriots really want to settle this, that’s fine by me. I’ll go along for the ride. But if people have reasons for believing it’s a waste, they should make those arguments instead.


Flag of Milwaukee, Wisconsin (Wikimedia Commons)

Now, if we are going to redesign the flag, then there are a few things probably worth keeping straight. It’s not as if there’s no such thing as a bad flag: for a vexillological disaster, see the flag of Milwaukee, a city in Wisconsin.

Firstly, although I said that the meaning of symbols is arbitrary, we should still give some thought to the meanings themselves. Specifically, meanings of elements should be factually, not just aspirationally, connected to New Zealand. Virtually every design in the flag gallery does this, so I’m picking it’s uncontroversial.

Secondly, it’s worth keeping some basic principles in mind, like the ones that Ted Kaye of the North American Vexillological Association compiled in Good Flag, Bad Flag. The flag should be kept simple, “so simple that a child can draw it from memory.” With respect to their creators, this should rule out elaborate patterns like the one on the left below, or any of these. For the same reason, if there must be a silver fern on the flag, it should be stylised like the one in the middle, not the one on the right.

Too complicated

Good stylised fern

Not-so-good stylised fern

Please don’t

It should also contain no lettering. Specifically, it should avoid the letters “NZ”—the flag shouldn’t embed a reminder about who it belongs to! This means the 1974 Commonwealth Games logo should be left out. And this attempt (right) to combine “NZ” into a complicated rendition of the fern is, in the nicest way possible to its creator, a definite no-no.

Here’s a TED talk about these and other basic principles, interleaved with some hilarious commentary on some American cities’ flags. You really should watch it. Before I saw it, I really didn’t care about flags. Now, I have at least some minimal appreciation for good flag design.

* Obviously there are some limits to this. There are some symbols already “taken” by others whose meaning you just can’t escape, for example, the flag of the Confederate States of America. ^
† We formally adopted our flag in 1902, but that formality doesn’t change the fact that it was widely used before then. ^
‡ The fifth principle in Good Flag, Bad Flag is “be distinctive or be related”, and while Kaye recognises that the Union Jack symbolises connection to the Commonwealth, he criticises the flag of Manitoba, a province in Canada, for giving too much emphasis to it. I think relegating it to one quarter makes it distinctive enough—at least as distinctive as, say, Malaysia’s and Liberia’s are from the United States’. Then again, I’m not a vexillologist. ^