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
![\[\boldsymbol{\frac{{\rm d}h}{{\rm d}t} = \frac{4 - 5\sqrt{h}}{25\pi\left(4h^2 + 1\right)}\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-2ad72aff898d9e4871771a25174b4fd3_l3.svg "Rendered by QuickLaTeX.com")
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.
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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.
Speaking of the “curriculum” in a larger sense, I wonder if anyone has questioned why the accreditation period (which was originally 2016-2018) has now been quietly extended to 2019 for units 1&2 and 2020 for units 3&4…
Because surely there is a need for change. Soon.
Very good question. The curriculum is fundamentally unfixable, and perhaps someone on the accreditation committee has the sense to realise that and has had the guts to say it. But that would be surprising: there has been zero evidence of sense or guts in the recent past.
Steps 1 and 2 of the fix are relatively straight forward: get rid of the bitsy parts of the course that don’t really go anywhere and aren’t really Mathematics.
Then, since you now have bugger all, but what you have is quality, you rebuild until you have enough to write meaningful assessments from.
Still not sure where or why the calculators fit into all of this.
It’s all bitsy.
It’s very likely this has happened because the VCAA master plan is for Mathematica to replace CAS-calculators. Undoubtedly VCAA plans to roll this out in the next accreditation period. But ….
There have been many snags and glitches encountered by the schools who agreed to Pilot this program (all of which were the school’s responsibility to sort out for VCAA). Furthermore, VCAA is applying pressure for the exams to also be sat electronically (so that code rather than mathematics will likely be the primary assessment focus), which poses further logistical problems (not to mention being the source of various snags). It’s not very hard to see why the next accreditation period is being delayed ….
Ugh. The final Magritte nails in the coffin of a rotting corpse of a curriculum.
If VCAA goes over to Wolfram completely, will this be an end of the “show that” questions???
Proof is already dead.
A wrong exponent is one thing; hopefully it was a typo with no lasting trauma. But if the examiners really meant to write ms^{-1}, then they should’ve just written m/s. There is a kind of snobbery in books that presumes using the solidus denotes that one comes from the wrong side of the tracks.
My real beef here is the use of the generic word “change”, not just in the quoted question but in all of maths and physics. “My weight changed”; yes, well, did it go up or down? The difference is important! “My bank balance changed”; “The amount of ice in Antarctica changed”. Each of these statements is designed to be superficial, in that it omits saying whether the quantity increased or decreased. I think that use of “change” in physics has led to confusion in textbooks, whose authors are sometimes clearly unaware of whether the relevant quantity has gone up or down. As a result, they are not above inserting ad-hoc minus signs to get things to come out right at the end.
Change is defined in everyday use as change = |increase| = |decrease|. There is no reason to abandon that meaning in maths and physics, and yet these subjects have redefined change to mean increase. Why do that? We already have a word for increase: it’s “increase”.
Consider the analogy of “distance” being defined as |displacement|, in both maths/physics and in everyday speech. That’s good, and causes no ambiguities. When we want the less informative number, we calculate distance. If we want to be really quantitative, we calculate the displacement vector. Similarly, if we want to calculate an increase, we should calculate the increase; and if we don’t care whether we are dealing with an increase or a decrease, then throw away the sign and call the positive number a “change”. But don’t redefine change to mean increase.
“The change in x” is written as Delta x, but those who write Delta x clearly don’t always understand what it means. It means “the increase in x”, being “final x minus initial x”. So if Delta x = 2, then x has increased by 2. If Delta x = -2, then x has increased by -2, meaning x has decreased by 2. The question quoted is no doubt asking for the increase in momentum, the final value minus the initial value. But how many people -really- know that?
This sort of use of language comes to the fore when a student is asked to interpret the meaning of, say, dT/dh = -5 K/m in a column of gas, where T = temperature and h = height. How is this equation read? In all likelihood, the student says what he has been taught to say: “The rate of change of temperature with height is -5 kelvins per metre”. Then he crosses his fingers and hopes that somewhere, somehow, if he has to calculate something, then its sign will come out right. Otherwise he’ll just think hard and insert the correct sign at the end, based on a hopefully correct intuition of the physical situation being analysed. That doesn’t always work: I’ve seen examples in books where the author messed it up, based on this lack of knowledge of the meaning of Delta.
What the student -should- say is either one of:
“The rate of increase of temperature with (increasing) height is -5 kelvins per metre”, or, better,
“The rate of decrease of temperature with (increasing) height is 5 kelvins per metre”.
What I’ve written as “(increasing)” might be left out; I really don’t know if its absence will cause difficulties. The choice of the second alternative here especially makes the point that for each metre we climb, the temperature drops by 5 kelvins.
Simple fact is that when I see the sentence “The negative of the change in x” on the web or in a book, I tell myself that its writer clearly means -Delta x, the loss in x. But does that writer know that? I doubt it; if they did, they wouldn’t use the clumsy phrase “the negative of the change in”; instead, they would use the correct, simple language “the loss in” or “the drop in”. If “the negative of the change in x equals 12”, can they say quickly what happened to x? The answer is that x dropped by 12; but I suspect they don’t know that. Certainly a discussion I once had with two PhD’ed physicists made it clear that they had utterly no idea of these concepts.
The use of the word “change” gets passed on from generation to generation, and yet it’s clear from textbooks that its meaning is not really understood. Let’s replace it with the correct word, “increase”, and do everyone a favour in how to read and write the maths of a physical statement.
Thanks, Don. It hadn’t occurred to me before, but I share your contempt for the ms-1 formatting.
As for your objection to the use of the word “change” in mathematics, I sympathise, and would probably vote for using “increase” instead, but I’m not sure I totally agree. To begin, I’m not convinced that “change” in everyday usage necessarily means a magnitude rather than a signed quantity. Yes, people won’t typically use negatives to answer a question about change, but they will indicate a decrease by use of the word “down”, and will often indicate an increase with “up” if there is some ambiguity. (“Has the water level of the dam changed?” “It’s down two metres.”)
Of course the conflict between common usage and mathematical usage is always a problem. And yes, the problem is lessened if the mathematical usage reflects the common usage. But there will always be limits to how well precise language can reflect semi-precise or semi-universal language.
There’s no doubt that the issue of indicating or asking about signed/unsigned quantities is a mess in mathematics and maths ed. Indeed, my next post will be on a clumsy ambiguity of sign in an exam question. But, even if we were declared dictators of mathematical usage, I’m not sure there’s an easy fix we could dictate.
Good points Marty. But here is a general question: how do we transfer a spoken phrase into maths? Suppose we both have some rice, and you give me some of your rice. Then the gain [or increase, if you prefer] in my amount of rice equals the drop [or decrease] in your amount of rice. In symbols, this becomes Delta (my rice) = -Delta (your rice). This is naturally said and naturally transferred to symbols. But if you prefer “the change in”, then there’s a problem, because “the change in” seems to have no opposite phrase that translates mathematically to “minus the change in”.
In the rice example, no one ever says “The change in my amount of rice equals minus the change in your amount of rice”. Since we don’t naturally use that word “change” here, we shouldn’t translate “Delta” to mean “the change in”. Doing so only produces awkward sentences, and these become recipes for disaster.
In physics at least, I have seen many examples where the conventional use of the word “change” clearly didn’t focus its users on interpreting a physical situation mathematically. As a result, they needed to tamper with their maths. So there seems to be a disconnect between the use of Delta and its physical interpretation. In that case, I suggest that the word “change” used to represent Delta is failing its users.
Thanks, Don. I semi-agree. “Change” is too ambiguous and colloquial-sounding to work well as a technical word. But the use of the word is not going to, ahem, change.
As to your general question, how does one transfer a spoken phrase into mathematics, the answer is: badly. Most mathematical notation and technical terms, at least in “elementary” mathematics, have evolved from vague and not particularly thoughtful usage. Eventually the usage stabilises to something that works at least well enough. But, as with biological evolution, a lot of the consequent creations are jerry-rigged.