## How the Other Maths Lives

A few days ago, the Sydney Morning Herald had an article on “the hardest question” from this year’s NSW Extension 2 Exam. The question is worth 5 marks, which equates to 9 minutes of a 3 hour exam (accessible here). The question, and another question (6-ish minutes), which apparently came along for the ride, are posted below. Readers’ homework exercise is to Compare and Contrast. Continue reading “How the Other Maths Lives”

## WitCH 75: Car Crash

It had to be done. Feel free to transcribe your comments from the Exam 2 discussion post.

## WitCH 10: Malfunction

It’s a long, long time since we’ve had a WitCH. They have been not-so-slowly accumulating, however. And now, since we’re temporarily free of the Evil Mathologer, it is the WitCHing hour. Due mostly to the hard work of Damo, all of the outstanding WitCHes have been resolved, with the exception of WitCH 8. That one will take time: it’s a jungle of half-maths. Our new WitCHes are not so tricky, although there is perhaps more to be said than indicated at first glance. The first of our new batch of WitCHes is from the VCE 2018 Specialist Exam 1:

The Examiners’ Report gives the answer as . The Report also indicates that the average score on this question was 1.3/5, with 98% of students scoring 3 or lower, and over a third of students scoring 0. Happy WitCHing.

### Update (16/02/20)

This problem is ridiculous and, more importantly, it is wrong. First, the wrongness.

As indicated by the examination report, the examiners imagined that they were, in essence, asking for students to determine the speed function of the particle. The distance is given by , and a non-trivial calculation gives . Then, the coefficients can be read off.

That is not, however, the question the examiners asked. What did the examiners really ask? They asked for integers for which . But

and

So, multiplying out the fractions and cancelling out a 3, what the examiners really asked for were integer solutions to the equation

This equation has infinitely many integer solutions, meaning the examination report is missing infinity minus one valid solutions.

This is a flat out, undeniable error (which the Trumpian VCAA will never concede), but is it a problem? As commenters here have noted, there is little chance of a VCE student being actively misled to chase the infinitely many solutions. In, particular, the method to find all solutions requires first finding the particular solution the examiners had in mind. We are not convinced such direct concerns should be so quickly dismissed, and we discuss this further below. Still, the extra solutions require thought to even contemplate, and significant work to compute, which is an important point.

Whatever the immediate practical concerns, however, mathematicians are aghast at this error. They are aghast because the exam question is simply not testing mathematics. Yes, the students went through the ritual and attempted to compute what was intended and were graded accordingly. And, yes, teachers can now coach current and future students on the required ritual. But none of that is mathematics and, indeed, it is worse: it is antimathematics. It is teaching students to ignore mathematical meaning, to see no value in mathematical precision, to respect only ritual.

OK, that is the awful wrongness of the exam question. Now, the sundry ridiculousnesses:

• The question is badly and needlessly opaque. There is no a priori reason to imagine the distance as being given by the integral of a quadratic. Asking for (more accurately, attempting to ask for) the speed function in this overly cute manner adds no value, only confusion. The confusion is enhanced by the arbitrariness of the 3/4 limit and, especially, by the pointless specification that the coefficients of the quadratic be integers.
• Independent of the opacity, the wording of the question is lazy and clumsy. The distanced travelled “in three-quarters of a second” is not the same as the distance travelled in the first three-quarters of a second and, indeed, is not anything. The phrases “moving along a curve” and “travels along a curve” are just verbiage. The units are pointless.
• The question would be much more natural as an arc-length question, rather than a distance question.
• The answer in the examination report is incorrect, even in the intended terms. The question asked for the values of the coefficients, not the integral. Yes, this is a nitpick, but it is exactly the kind of nitpick that the examiners routinely employ in their sanctimonious whacking of VCE students. So screw ’em. Sauce for the gander.
• Last, and far from least, there is something very strange about the score distribution for the question. The average score was 1.3/5, which is depressing, although not surprising: computing the speed (without CAS) requires a level of care and facility beyond most CAS-drunk students, and the question contains a hidden absolute value to negotiate. What is strange is that, whereas 2% of students received the full 5/5 for the question, apparently 0% of students received 4/5. It is difficult to see how that could occur with any sensible grading scheme.

## WitCH 7: North by Southwest

Our new WitCH, below, comes courtesy of Charlie the Enforcer. Once again, this WitCH is from the 2018 SCSA Mathematical Methods Exam (here and here): it’s the gift that keeps on giving. (And a reminder, WitCH 2 and WitCH 3 still require attention are still unresolved.)

Question 11 and the solution in SCSA’s marking key are below. Happy hunting.

## Update

John has pretty much caught it all. The killer issue is the use of the term “deceleration” in part (c) which, the solution implies, refers to the drone speeding up in the southerly direction. This is arguably permissible, since deceleration can be (though is far from universally) defined as a negative acceleration, and since way back in part (a) it was implied that North coincides with the positive x direction.

Permissible acts, however, can nonetheless be idiotic: voting Liberal or Republican, for example. And, to use “deceleration” on a high stakes exam to refer implicitly to increasing speed is idiotic. Moreover, to use “deceleration” in this manner immediately after explicitly indicating the “due south” direction of motion is truly ruly idiotic. Still not as idiotic as voting Liberal or Republican, but genuinely special-effort idiotic.

That’s enough to condemn the question, even by SCSA standards. But, the question is also awful in many other ways:

• The question is boring and butt ugly.
• No indication is given whether exact or numerical solutions are permitted or required.
• Having a drone an arbitrary 5m up in the sky for a 1D problem is asking for trouble. For example:
• The “displacement” of x(0) = 0 for a drone 5m up is pretty stupid.
• “Where is the drone in relation to the [mysterious] pilot?” Um, kind of uppish?
• “How far has the drone travelled …” is needlessly wordy and ambiguous. If you want a distance, for God’s sake say “distance”.
• Given the position function x(t) is at hand, part (c) can easily and naturally be solved by hand. But of course why think about things when you can do mindless calculator crap?

## A Loss of Momentum

The VCE maths exams are over for another year. They were mostly uneventful, the familiar concoction of triviality, nonsense and weirdness, with the notable exception of the surprisingly good Methods Exam 1. At least two Specialist questions, however, deserve a specific slap and some discussion. (There may be other questions worth whacking: we never have the stomach to give VCE exams a close read.)

Question 6 on Specialist Exam 1 concerns a particle acted on by a force, and students are asked to

Find the change in momentum in kg ms-2 …

Doh!

The problem of course is that the suggested units are for force rather than momentum. This is a straight-out error and there’s not much to be said (though see below).

Then there’s Question 3 on part 2 of Specialist Exam 2. This question is concerned with a fountain, with water flowing in from a jet and flowing out at the bottom. The fountaining is distractingly irrelevant, reminiscent of a non-flying bee, but we have larger concerns.

In part (c)(i) of the question students are required to show that the height h of the water in the fountain is governed by the differential equation

The problem is with the final part (f) of the question, where students are asked

How far from the top of the fountain does the water level ultimately stabilise?

The question is typical in its clumsy and opaque wording. One could have asked more simply for the depth h of the water, which would at least have cleared the way for students to consider the true weirdness of the question: what is meant by “ultimately stabilise”?

The examiners are presumably expecting students to set dh/dt = 0, to obtain the constant, equilibrium solution (and then to subtract the equilibrium value from the height of the fountain because why not give students the opportunity to blow half their marks by misreading a convoluted question?) The first problem with that is, as we have pointed out before, equilibria of differential equations appear nowhere in the Specialist curriculum. The second problem is, as we have pointed out before, not all equilibria are stable.

It would be smart and good if the VCAA decided to include equilibrium solutions in the Specialist curriculum, along with some reasonable analysis and application. Until they do, however, questions such as the above are unfair and absurd, made all the more unfair and absurd by the inevitably awful wording.

********************

Now, what to make of these two questions? How much should VCAA be hammered?

We’re not so concerned about the momentum error. It is unfortunate, it would have confused many students and it shouldn’t have happened, but a typo is a typo, without deeper meaning.

It appears that Specialist teachers have been less forgiving, and fair enough: the VCAA examiners are notoriously nitpicky, sanctimonious and unapologetic, so they can hardly complain if the same, with greater justification, is done to them. (We also heard of some second-guessing, some suggestions that the units of “change in momentum” could be or are the same as the units of force. This has to be Stockholm syndrome.)

The fountain question is of much greater concern because it exemplifies systemic issues with the curriculum and the manner in which it is examined. Above all, assessment must be fair and reasonable, which means students and teachers must be clearly told what is examinable and how it may be examined. As it stands, that is simply not the case, for either Specialist or Methods.

Notably, however, we have heard of essentially no complaints from Specialist teachers regarding the fountain question; just one teacher pointed out the issue to us. Undoubtedly there were other teachers bothered by the question, but the relative silence in comparison to the vocal complaints on the momentum typo is stark. And unfortunate.

There is undoubted satisfaction in nitpicking the VCAA in a sauce for the goose manner. But a typo is a typo, and teachers shouldn’t engage in small-time point-scoring any more than VCAA examiners.

The real issue is that the current curriculum is shallow, aimless, clunky, calculator-poisoned, effectively undefined and effectively unexaminable. All of that matters infinitely more than one careless mistake.

## Update (24/02/19)

The exam Reports are now out, here and here. There’s no stupidity so large or so small that the VCAA won’t remain silent.