Brief remarks on the HSC Extension 2 Mathematics examinations

Abstract

In this note, I comment on recent trends in the NSW Extension 2 Mathematics examination, which is often portrayed in local media as an unacceptably difficult exam. I believe that the examination problems are heading in a good direction, because they incentivise deeper problem solving strategies over memorisation of routine computations and algebraic tricks. However, students are usually exposed more frequently to the latter view of mathematics, which leads to the view that the exam questions are unfairly difficult.

NESA, the NSW Education Standards Authority, have now released the 2023 exam paper for Extension 2 Mathematics, the highest level of mathematics offered at the high school level in New South Wales. It has become a tradition that the conclusion of this particular exam is accompanied by various local media reports describing the exam as a “brutal” ordeal, with “sadistic” questions that left students feeling utterly defeated. (The quoted words are not hyperbolic, by the way. I pulled them straight out of this article). Speaking personally though, I have been quite happy with the development of the Extension 2 Maths exam in recent years (that is, since the new syllabus was first examined in 2020). Overall, it seems that the examiners are moving away from a predictable exam format, and are happy to feature questions that require ‘genuine’ problem solving — as opposed to the more routine calculations found in every high school maths textbook even at this advanced level.

From the students’ point of view, however, I can certainly see why they might be caught off-guard. As I just mentioned, exercises in the typical high school maths textbook do not leave much to the imagination. Of course, it is essential to use routine calculations to acquire a basic competence in any mathematical topic. The problem is that very often, there is not much more. Here is a simple example to fix the ideas¹ :

Problem 1A (Question 11a, HSC Ext 2, 2023) Solve the quadratic equation $z^2 - 3z + 4 = 0$ , where $z$ is a complex number. Give your answers in Cartesian form.

This is Question 11a from this year’s exam paper, and is a typical routine calculation. It requires no ingenuity to apply the quadratic formula and obtain

$z = \frac{3\pm \sqrt{7}i}{2}.$

Let’s consider an upgrade.

Problem 1B Find all values of $b$ , where $b$ is a real number, such that the quadratic $p(z) = z^2 - 3z + b$ has no real solutions.

Although this problem is still a triviality at the Extension 2 level, it does offer a bit more. Firstly, the student should recall that the number of real solutions is determined by the discriminant of the quadratic. In particular, there are no real solutions if and only if the discriminant is strictly negative. The student is then required to solve an inequality. In this case, we compute the discriminant to be $3^2 - 4b = 9-4b$ , and hence the condition for no real solutions is $b>\frac{9}{4}$ . The teacher can easily make the problem more challenging to solve with some easy adjustments (for example, we could arrange that the discriminant expression is a quadratic in the new variable $b$ , and then things would be more interesting).

Problem 1B requires combining two ideas into a coherent argument. However, we should not extrapolate from such an example to conclude that a problem with more steps in the working is necessarily more challenging. Indeed, many problems in the Extension 2 syllabus are extremely simple conceptually, but involve calculations that are quite tedious. This is especially the case in calculus questions, which form a large part of both Extension 1 and 2 Mathematics. Students who are not as confident in the basic computational skills required for Extension 2 — arithmetic, algebra, differentiation and integration — may understandably be discouraged. Naturally, this leads to the difficult questions of why and how students struggle with basic computations, but that would be a topic for another time. Instead, let’s take a look at another example from this year’s paper:

Problem 2 (Question 15a(i), HSC Ext 2, 2023) Let $J_n = \displaystyle\int_0^{\frac{\pi}{2}} \sin^n \theta\,d\theta$ where $n\ge 0$ is an integer. Show that

$J_n = \frac{n-1}{n} J_{n-2}$

for all integers $n\ge 2$ .

This is a typical HSC style reduction formula question, and has been a staple of the Extension 2 examination for decades (I believe). Now, as someone who studies differential equations, I do love my integrals, but honestly there is nothing interesting going on in this problem. To be fair, this kind of problem is most often used as a first step to prove something more interesting — for example, see Question 16b in the 2020 paper — but it is nonetheless a routine calculus computation. For this particular problem (reduction formulae involving powers of the sine or cosine functions), there is a very much standard approach. First, we split the integrand into $\sin^{n-1}\theta \sin\theta$ , and then, recognising that $\sin\theta = -\frac{d}{d\theta}(\cos\theta)$ , we can use integration by parts²:

$J_n = \displaystyle\int_{0}^{\frac{\pi}{2}} \sin^{n-1}\theta \frac{d}{d\theta}(-\cos\theta)\,d\theta = (n-1)\int_0^{\frac{\pi}{2}} \sin^{n-2}\theta \cos^2\theta\,d\theta.$

(For brevity, I have condensed several steps together: one should differentiate $\sin^{n-1}\theta$ carefully with the chain rule, and then observe that the boundary terms in the integration by parts formula are equal to 0). Finally, we use the basic identity $\cos^2\theta = 1-\sin^2\theta$ to deduce that the resulting integral is equal to $(n-1)(J_{n-2}-J_n)$ , and then the result follows³.

I have found that many students are quite comfortable with such questions. Once you remember the trick (split the integrand, integrate by parts), then there is nothing surprising, and provided your basic calculus skills are up to scratch, then it is a pleasant calculation. For the weaker students, there is good(?) news too. Since most high school textbooks are littered with many such routine integration questions, there is no shortage of practice opportunities. However, the emphasis on such problems leaves students with the mistaken impression that somehow most of advanced mathematics proceeds this way, and it is simply a matter of putting in the effort to memorise a random assortment of tricks in algebra and calculus, with no coherent bigger picture in mind. If their mathematical diet (so to speak) consists mainly of such exercises, then it is no wonder that many students are flummoxed by problems which require deeper mathematical insight, such as this interesting puzzle from last year’s (2022) paper:

Problem 3 (Question 16d, HSC Ext 2, 2022) Find all complex numbers $z_1, z_2, z_3$ that satisfy the following three conditions simultaneously:

$|z_1| = |z_2| = |z_3|, \quad z_1+z_2+z_3 = 1, \quad z_1 z_2 z_3 = 1.$

Traditionally, the most difficult problems in the exam present themselves in the infamous Question 16. Where would you begin with Problem 3? One particularly nice detail is the wording “find all” — if you happened to stumble on a solution by accident, say three quantities $A,B,C$ , you couldn’t get away with writing “by inspection we obtain $z_1 = A, z_2 = B, z_3 = C$ “, unless you could also justify why this is the only solution set. Without spoiling too much, it turns out that the numbers $z_1, z_2, z_3$ must be the roots of a specific cubic polynomial (and hence, there is indeed just one set of solutions), but how you arrive at this result is entirely up to you. In general, it is a good idea to start playing around, exploring the consequences of the given conditions in order to get more information about the numbers $z_1, z_2, z_3$ . It is also important for students to be prepared to use all the techniques at their disposal — all those routine exercises are not for nothing! For example, the second and third equations look suspiciously like the sum and product of roots of a polynomial respectively, so this is a strong hint that the numbers are related via some polynomial, and the student should recall how the coefficients of a polynomial yield information about the sum and product of roots.

Students in general (and maybe some teachers too) are much less comfortable with the apparent open-endedness of exam questions such as Problem 3 above. In my view, such a problem tests two skills which are indispensable for further study in the sciences: firstly, the ability to carry out independent inquiry. Fundamentally, it is about learning to ask good questions and also not being afraid to ask basic or even ‘stupid’ questions (if this advice is good enough for Terry Tao, it’s good enough for you and me). These are two sides of the same coin, something which I did not appreciate until I was already near the end of my bachelor studies in mathematics. The second point is the ability to combine knowledge from different areas of the syllabus. A routine exercise will usually test only one or two facts from the same mathematical topic. In contrast, a more substantial problem will call upon a variety of tools from different topics, and the student will need to combine facts in a creative way. I would say that this is a common feature of the HSC problems which the media loves to call ‘unfair’ or ‘brutally difficult’. However, if the prevailing approach to higher mathematics is one geared towards memorisation of tricks and algorithms, it is easy to see why students are frustrated and disappointed when faced with exam questions like Problem 3. Despite all their practice, they feel inadequately equipped, and naturally complain that such problems are unfair.

On the other hand, there is a kind of ‘Newton’s Third Law’ in education: as teachers, if we expect our students to develop these higher-order problem solving skills, then our teaching methods should also adapt as a consequence. (This basic principle is certainly not just confined to mathematics education!). Moreover, if the teacher has not invested time and effort into developing their own mathematical insight, it is unlikely that they will inspire their students to pursue the same goal. By the very nature of the task, I do not expect any easy tricks to help students develop higher-order reasoning skills. As a preliminary step though, I suggest some criteria for creating mathematical problems with ‘substance’, in contrast to the ‘shallow’ exercises present in so many high school learning resources.

Broad scope: a substantial problem should require knowledge from multiple areas of the syllabus. That way, students will see the utility of mastering the routine exercises. They will also begin to appreciate the connections between different mathematical topics, an aspect which is too often neglected due to the segregation of mathematical concepts into specific dot points in the syllabus.
Multi-faceted: whenever possible, a substantial problem should support different approaches to the same solution. This will promote collaboration and discussion in the classroom, improve the students’ skills in independent inquiry, as well as reinforce the first point about connections between different topics.
Creative: a substantial problem need not require complicated machinery either. Often it can be a challenge to combine simple tools in novel ways. This point cooperates naturally with the second one.
Manual labour: design, whenever possible, problems which cannot be effectively solved by brute force computation (e.g. Wolfram Alpha, Integral Calculator, ChatGPT). Of course, this is not always easy, given the accessibility and power of current computational tools. While such technology should definitely be embraced as part of the standard toolkit of mathematicians, they are rarely useful for learning more abstract concepts.

I do not believe any of my suggestions are novel or innovative in any way. In fact, for experts in mathematics education, this entire blogpost might even be trivial (oops). And yet, year after year, the same complaints about the difficulty of Extension 2 Mathematics arise, and are echoed and amplified by journalism and social media. As I hope I have made clear in this piece, my opinion is that the level of mathematics in the exam is just right. The core of the problem is that students enter the exam expecting to see textbook style questions they have practiced, and instead are shocked that they are required to use their skills to tackle unfamiliar problems. Unfortunately this entirely misses the mark, because the application of creative and insightful reasoning, built on solid logical foundations, to solve a diverse range of abstract and real-life problems is the very essence of mathematics. As Stephen Hawking allegedly said, equations are just the boring part of mathematics.

Postscript: (Note to self) If I have time later this week, I am tempted to expand on some of the ideas in this post and discuss solution strategies to selected problems in the 2023 exam paper.

That is, to make things more concrete. It sounds better in French: pour fixer les idées. This phrase is used by mathematicians a fair bit (perhaps thanks to Bourbaki?), but it seems not to be popular in conventional English ↩︎
In its full generality, integration by parts might be one of the deepest ideas in calculus… but in the context of HSC mathematics, it certainly doesn’t appear that way. ↩︎
By the way, exactly the same formula holds for $J_n = \displaystyle\int_0^{\frac{\pi}{2}} \cos^n\theta\,d\theta$ — can you prove this? There is a long way, and also a short, sneakier way 🙂 ↩︎

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