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.



  1. Daniel Kim · September 24, 2016

    Understanding quadratics allows one to grasp convexity, which can be mapped to many areas of life.


    • Chuan-Zheng Lee · September 24, 2016

      Hmm, maybe. Quadratic functions are a simple example of a convex function, so are useful for helping build intuition. But I’m not sure that I see the link between properties of quadratics (e.g. the factor theorem, the determinant) and convexity. I totally agree that convexity is a very insightful and applicable concept though (albeit one not really covered in the high school syllabus, except for an in-passing mention about twice-differentiable functions).


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