In this WitCH we will again pick on the Cambridge text Specialist Mathematics VCE Units 3 & 4 (2019): see the extract below. (We’d welcome any email or comment with suggestions of other generally WitCHful texts and/or specific WitCHes.) And, a reminder that there is still plenty left to discover in WitCH 2 , WitCH 3 and Tweel’s Mathematical Puzzle still have room for comment.
Have fun.
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-6c82e31b4bc6b201a95e00be21e812b1_l3.png "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.
********************
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.
We’re sure we’ll live to regret this post, but yesterday’s VCE Methods Exam 1 looked like a good exam.
No, that’s not a set up for a joke. It actually looked like a nice exam. (It’s not online yet). Sure, there were some meh questions, the inevitable consequence of an incompetent study design. And yes, there was a minor Magritte aspect to the final question. And yes, it’s much easier to get an exam right if it’s uncorrupted by the idiocy of CAS, with the acid test being Exam 2. And yes, we could be plain wrong; we only gave the exam a cursory read, and if there’s a dodo it’s usually in the detail.
But for all that the exam genuinely looked good. The questions in general seemed mathematically natural. A couple of the questions also appeared to be difficult in a good, mathematical way, rather than in the familiar “What the Hell do they want?” manner.
What happened?
This one is long, a real Gish gallop. Question 4, Part 2 from the 2017 VCE Specialist Mathematics Exam 2 is a mess. The Examiners’ Report is, predictably, worse.
Part (a) of Question 4 is routine, requiring students to express 
in polar form. One wonders how a quarter of the students could muck up this easy 1-mark question, but the question is fine.
The issues begin with 4(b), for which students are required to
Show that the roots of 
are 
and 
.
The question can be answered with an easy application of completing the square or the quadratic formula. So, why did almost half of the students get it wrong? Were so many students really so clueless? Perhaps, but there is good reason to suspect a different source of the cluelessness.
The Examiners’ Report indicates three general issues with students’ answers. First,
students confused factors with solutions or did not proceed beyond factorising the quadratic.
Maybe the students were confused, but maybe not. Maybe some students simply thought that, once having factorised the quadratic, the microstep to then write “Therefore z = …”, to note the roots written on the exam in front of them, was too trivial in response to a 1 mark question.
Second, some students reportedly erred by
not showing key steps in their solution.
Really? The Report includes the following calculation as a sample solution:
Was this whole tedious, snail-paced computation required for one measly mark? It’s impossible to tell, but the Report remarks generally on ‘show that’ questions that
all steps that led to the given result needed to be clearly and logically set out.
As we have noted previously, demanding “all steps” is both meaningless and utterly mad. For a year 12 advanced mathematics student the identification of the roots is pretty much immediate and a single written step should suffice. True, in 4(b) students are instructed to “show” stuff, but it’s hardly the students’ fault that what they were instructed to show is pretty trivial.
Third, and by far the most ridiculous,
some students did not correctly follow the ‘show that’ instruction … by [instead] solely verifying the solutions given by substitution.
Bullshit.
VCAA examiners love to worry that word “show”. In true Princess Bride fashion, however, the word does not mean what they think it means.
There is nothing in standard English usage nor in standard mathematical usage, nor in at least occasional VCE usage (see Q2(a)), that would distinguish “show” from “prove” in this context. And, for 4(b) above, substitution of the given values into the quadratic is a perfectly valid method of proving that the roots are as indicated.
It appears that VCE has a special non-English code, in which “show” has a narrower meaning, akin to “derive“. This cannot alter the fact that the VCE examiners’ use of the word is linguistic and mathematical crap. It also cannot alter the fact that students being penalised for not following this linguistic and mathematical crap is pedagogical and mathematical crap.
Of course all the nonsense of 4(b) could have been avoided simply by asking the students to find the roots. The examiners declined to do so, however, probably because this would have violated VCAA’s policy of avoiding asking any mathematical question with some depth or difficulty or further consequence. The result is a question amounting to no more than an infantile and infantilising ritual, penalising any student with the mathematical common sense to answer with the appropriate “well, duh”.
***************************
Onwards we trek to 4(c):
Express the roots of 
in terms of 
.
Less than a third of students scored the mark for this question, and the Report notes that
Misunderstanding of the question was apparent in student responses. Many attempts at solutions were not expressed in terms of 
as required.
Funny that. The examiners pose a question that borders on the meaningless and somehow this creates a sea of misunderstanding. Who would’ve guessed?
4(c) makes little more sense than to ask someone to write 3 in terms of 7. Given any two numbers there’s a zillion ways to “express” one number “in terms of” the other, as in 3 = 7 – 4 or whatever. Without further qualification or some accepted convention, without some agreed upon definition of “expressed in terms of”, any expression is just as valid as any other.
What was expected in 4(c)? To approach the question cleanly we can first set 
, as the examiners could have and should have and did not. Then, the intended answers were 
and 
.
These expressions for the roots are simple and natural, but even if one accepts a waffly interpretation of 4(c) that somehow requires “simple” solutions, there are plenty of other possible answers. The expressions 
and 
and 
and 
are all reasonable and natural, but nothing in the Examiners’ Report suggests that these or similar answers were accepted. If not, that is a very nasty cherry on top of an incredibly silly question.
***************************
The pain now temporarily lessens (though the worst is yet to come). 4(d) asks for students to show that the relation 
has the cartesian form 
, and in 4(e) students are asked to draw this line on an Argand diagram, together with the roots of the above quadratic.
These questions are routine and ok, though 4(d) is weirdly aimless, the line obtained playing no role in the final parts of Q4. The Examiners’ Report also notes condescendingly that “the ‘show that’ instruction was generally followed”. Yes, people do tend to follow the intended road if there’s only one road.
The final part, 4(g), is also standard, requiring students to find the area of the major segment of the circle |z| = 4 cut off by the line through the roots of the quadratic. The question is straight-forward, the only real trick being to ignore the weird line from 4(d) and 4(e).
***************************
Finally, the debacle of 4(f):
The equation of the line passing through the two roots of 
can be expressed as 
, where 
. Find 
in terms of 
.
The Report notes that
This question caused significant difficulty for students.
That’s hilarious understatement given that 99% of students scored 0/1 on the question. The further statements acknowledging and explaining and apologising for the stuff-up are unfortunately non-existent.
So, what went wrong? The answer is both obvious and depressingly familiar: the exam question is essentially meaningless. Students failed to comprehend the question because it is close to incomprehensible.
The students are asked to write b in terms of a. However, similar to 4(c) above, there are many ways to do that and how one is able to do it depends upon the initial number a chosen. The line through the two roots has equation 
. So then, for example, with a = -4 we have b = 0 and we can write b = a + 4 or b = 0 x a or whatever. If a = -5 then b = 1 and we can write b = -a – 4, and so on.
Anything of this nature is a reasonable response to the exam question as written and none of it resembles the answer in the Report. Instead, what was expected was for students to consider all complex numbers a – except those on the line itself – and to consider all associated complex b. That is, in appropriate but non-Specialist terminology, we want to determine b as a function f(a) of a, with the domain of f being most but not all of the complex plane.
With the question suitably clarified we can get down to work (none of which is indicated in the Report). Easiest is to write 
. Since 
must be symmetrically placed about the line 
, it follows that 
. Then 
. This gives 
, and finally
which is the answer indicated in the Examiners’ Report.
In principle 4(f) is a nice question, though 1 mark is pretty chintzy for the thought required. More importantly, the exam question as written bears only the slightest resemblance to the intended question, or to anything coherent, with only the slightest, inaccurate hint of the intended generality of a and b.
99% of 2017 Specialist students have a right to be pissed off.
***************************
That’s it, we’re done. One more ridiculous VCE exam question, and one more ridiculously arrogant Report, unsullied by an ounce of self-reflection or remorse.
Our previous post was on good guys making a silly, funny and inconsequential mistake. This post is not.
Question B1 of Exam 2 for 2018 Northern Hemisphere Specialist Mathematics begins innocently enough. In part (a), students are required to graph the function 
over its maximal domain. Then, things begin to get stupid.
In part (b), the graph of f is rotated around the y-axis, to model a vase. Students are required to find the volume of this stupid vase, by setting up the integral and then pushing the stupid buttons on their stupid calculators. So, a reasonable integration question lost in ridiculous pseudomodelling and brainless button-pushing. Whatever. Just standard VCE crap. Then, things stay stupid.
Part (c) is a related rates question. In principle a good problem, though it’s hard to imagine anyone ever requiring dh/dt when the water depth is exactly 
cm. Whatever. Standard VCE crap. Then, things get really, really stupid.
Part (d) of the problem has a bee climbing from the bottom of the vase to the top. Students are required to find the minimum distance the bee needs to travel.
Where to begin with this idiotic, 1-mark question. Let’s begin with the bee.
Why is it a bee? Why frame a shortest walk question in terms of a bug with wings? Sure, the question states that the bug is climbing, and the slight chance of confusion is overshadowed by other, much greater issues with the question. But still, why would one choose a flying bug to crawl up a vase? It’s not importantly stupid, but it is gratuitously, hilariously stupid.
Anyway, we’re stuck with our stupid bee climbing up our stupid vase. What distance does our stupid bee travel? Well, obviously our stupid, non-flying bee should climb as “up” as possible, without veering left or right, correct?
No and yes.
It is true that a bottom-to-top shortest path (geodesic) on a surface of revolution is a meridian. The proof of this, however, is very far from obvious; good luck explaining it to your students. But of course this is only Specialist Mathematics, so it’s not like we should expect the students to be inquisitive or critical or questioning assumptions or anything like that.
Anyway, our stupid non-flying bee climbs “up” our stupid vase. The distance our stupid bee travels is then the arc length of the graph of the original function f, and the required distance is given by the integral
![\[\boldsymbol{{\Huge \int\limits_{\frac12}^{\frac32}}\sqrt{1+\left[\tfrac{20}{1 - (2-2x)^2}\right]^2}}\ {\bf d}\boldsymbol{x}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-64eadc75f69049237f16d859e2cc7d56_l3.png "Rendered by QuickLaTeX.com")
The integral is ugly. More importantly, the integral is (doubly) improper and thus has no required meaning for Specialist students. Pretty damn stupid, and a stupidity we’ve seen not too long ago. It gets stupider.
Recall that this is a 1-mark question, and it is clearly expected to have the stupid calculator do the work. Great, sort of. The calculator computes integrals that the students are not required to understand but, apart from being utterly meaningless crap, everything is fine. Except, the calculators are really stupid.
Two brands of CAS calculators appear to be standard in VCE. Brand A will readily compute the integral above. Unfortunately, Brand A calculators will also compute improper integrals that don’t exist. Which is stupid. Brand B calculators, on the other hand, will not directly compute improper integrals such as the one above; instead, one first has to de-improper the integral by changing the limits to something like 0.50001 and 1.49999. Which is ugly and stupid. It also requires students to recognise the improperness in the integral, which they are supposedly not required to understand. Which is really stupid. (The lesser known Brand C appears to be less stupid with improper integrals.)
There is a stupid way around this stupidity. The arc length can also be calculated in terms of the inverse function of f, which avoid the improperness and then all is good. All is good, that is, except for the thousands of students who happen to have a Brand B calculator and who naively failed to consider that a crappy, 1-mark button-pushing question might require them to hunt for a Specialist-valid and B-compatible approach.
The idiocy of VCE exams is truly unlimited.
The 2018 Northern Hemisphere Mathematical Methods exams (1 and 2) are out. We didn’t spot any Magritte-esque lunacy, which was a pleasant surprise. In general, the exam questions were merely trivial, clumsy, contrived, calculator-infested and loathsomely ugly. So, all in all not bad by VCAA standards.
There, was, however, one notable question. The final multiple choice question on Exam 2 reads as follows:
Let f be a one-to-one differentiable function such that f (3) = 7, f (7) = 8, f′(3) = 2 and f′(7) = 3. The function g is differentiable and g(x) = f –1(x) for all x. g′(7) is equal to …
The wording is hilarious, at least it is if you’re not a frazzled Methods student in the midst of an exam, trying to make sense of such nonsense. Indeed, as we’ll see below, the question turned out to be too convoluted even for the examiners.
Of course f –1 is a perfectly fine and familiar name for the inverse of f. It takes a special cluelessness to imagine that renaming f –1 as g is somehow required or remotely helpful. The obfuscating wording, however, is the least of our concerns.
The exam question is intended to be a straight-forward application of the inverse function theorem. So, in Leibniz form dx/dy = 1/(dy/dx), though the exam question effectively requires the more explicit but less intuitive function form,
![\[\boldsymbol {\left(f^{-1}\right)'(b) = \frac1{f'\left(f^{-1}(b)\right)}}.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-7b74170a56b51abf6a274b4cbc305433_l3.png "Rendered by QuickLaTeX.com")
IVT is typically stated, and in particular the differentiability of f –1 can be concluded, with suitable hypotheses. In this regard, the exam question needlessly hypothesising that the function g is differentiable is somewhat artificial. However it is not so simple in the school context to discuss natural hypotheses for IVT. So, underlying the ridiculous phrasing is a reasonable enough question.
What, then, is the problem? The problem is that IVT is not explicitly in the VCE curriculum. Really? Really.
Even ignoring the obvious issue this raises for the above exam question, the subliminal treatment of IVT in VCE is absurd. One requires plenty of inverse derivatives, even in a first calculus course. Yet, there is never any explicit mention of IVT in either Specialist or Methods, not even a hint that there is a common question with a universal answer.
All that appears to be explicit in VCE, and more in Specialist than Methods, is application of the chain rule, case by isolated case. So, one assumes the differentiability of f –1 and and then differentiates f –1(f(x)) in Leibniz form. For example, in the most respected Methods text the derivative of y = log(x) is somewhat dodgily obtained using the chain rule from the (very dodgily obtained) derivative of x = ey.
It is all very implicit, very case-by-case, and very Leibniz. Which makes the above exam question effectively impossible.
How many students actually obtained the correct answer? We don’t know since the Examiners’ Report doesn’t actually report anything. Being a multiple choice question, though, students had a 1 in 5 chance of obtaining the correct answer by dumb luck. Or, sticking to the more plausible answers, maybe even a 1 in 3 or 1 in 2 chance. That seems to be how the examiners stumbled upon the correct answer.
The Report’s solution to the exam question reads as follows (as of September 20, 2018):
f(3) = 7, f'(3) = 8, g(x) = f –1(x) , g‘(x) = 1/2 since
f'(x) x f'(y) = 1, g(x) = f'(x) = 1/f'(y).
The awfulness displayed above is a wonder to behold. Even if it were correct, the suggested solution would still bear no resemblance to the Methods curriculum, and it would still be unreadable. And the answer is not close to correct.
To be fair, The Report warns that its sample answers are “not intended to be exemplary or complete”. So perhaps they just forgot the further warning, that their answers are also not intended to be correct or comprehensible.
It is abundantly clear that the VCAA is incapable of putting together a coherent curriculum, let alone one that is even minimally engaging. Apparently it is even too much to expect the examiners to be familiar with their own crappy curriculum, and to be able to examine it, and to report on it, fairly and accurately.
It’s been a long, long time. Alas, we’ve been kept way too busy by the Evil Mathologer, as well as some edu-idiots, who shall remain nameless but not unknown. Anyway, with luck normal transmission has now resumed. There’s a big, big backlog of mathematical crap to get through.
To begin, there’s a shocking news story that has just appeared, about schools posting “wrong Year 12 test scores” and being ordered to remove them by the Victorian Tertiary Admissions Centre. Naughty, naughty schools!
Perhaps.
The report tells of how two Victorian private schools had conflated Victoria’s VCE subject scores and International Baccalaureate subject scores. The schools had equated the locally lesser known IB scores of 6 or 7 to the more familiar VCE ATAR of 40+, to then arrive at a combined percentage of such scores. Reportedly, this raised the percentage of “40+” student scores at the one school from around 10% for VCE alone to around 25% for combined VCE-IB, with a comparable raise for the other school. More generally, it was reported that about a third of IB students score a 6 or 7, whereas only about one in eleven VCE scores are 40+.
On the face of it, it seems likely that the local IB organisation that had suggested Victorian schools use the 6+ = 40+ equation got it wrong. That organisation is supposedly reviewing the comparison and the two schools have removed the combined percentages from their websites.
There are, however, a few pertinent observations to be made:
None of the sense or substance of the above is hinted at in the schools-bad/VTAC-good news report.
Of course the underlying issue is tricky. Though the IBO tries very hard to compare IB scores, it is obviously very difficult to compare IB apples to VCE oranges. We have no idea whether or how one could create a fair and useful comparison. We do know, however, that accepting VTAC’s cocky sanctimony as the last word on this subject, or any subject, would be foolish.
Late last year we posted on Madness in the 2017 VCE mathematics exams, on blatant errors above and beyond the exams’ predictably general clunkiness. For one (Northern Hemisphere) exam, the subsequent VCAA Report had already appeared; this Report was pretty useless in general, and specifically it was silent on the error and the surrounding mathematical crap. None of the other reports had yet appeared.
Now, finally, all the exam reports are out. God only knows why it took half a year, but at least they’re out. We have already posted on one particularly nasty piece of nitpicking nonsense, and now we can review the VCAA‘s own assessment of their five errors:
So, the VCAA responds to five blatant errors with five Trumpian silences. How should one describe such conduct? Unprofessional? Arrogant? Cowardly? VCAA-ish? All of the above?
Here’s a quick but telling nugget of awfulness from Victoria’s 2017 VCE maths exams. Q9 of the first (non-calculator) Methods Exam is concerned with the function
![\[\boldsymbol {f(x) = \sqrt{x}(1-x)\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-8265e5fa83aad4fcb434dc902291a3fb_l3.png "Rendered by QuickLaTeX.com")
In Part (b) of the question students are asked to show that “the gradient of the tangent to the graph of f” equals 
.
A normal human being would simply have asked for the derivative of f, but not much can go wrong, right? Expanding and differentiating, we have
![\[\boldsymbol {f'(x) = \frac{1}{2\sqrt{x}} - \frac32\sqrt{x}=\frac{1-3x}{2\sqrt{x}}\,.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-58125fb7c89d0faca5b8c4f1d7ce8b4d_l3.png "Rendered by QuickLaTeX.com")
Easy, and done.
So, how is it that 65% of Methods students scored 0 on this contrived but routine 1-point question? Did they choke on “the gradient of the tangent to the graph of f” and go on to hunt for a question written in English?
The Examiners’ Report pinpoints the issue, noting that the exam question “required a step-by-step demonstration …“. And, “[w]hen answering ‘show that’ questions, students should include all steps to demonstrate exactly what was done“ (emphasis added). So the Report implies, for example, that our calculation above would have scored 0 because we didn’t explicitly include the step of obtaining a common denominator.
Jesus H. Christ.
Any suggestion that our calculation is an insufficient answer for a student in a senior maths class is pedagogical and mathematical lunacy. This is obvious, even ignoring the fact that Methods questions way too often are flawed and/or require the most fantastic of logical leaps. And, of course, the instruction that “all steps” be included is both meaningless and utterly mad, and the solution in the Examiners’ Report does nothing of the sort. (Exercise: Try to include all steps in the computation and simplification of f’.)
This is just one 1-point question, but such infantilising nonsense is endemic in Methods. The subject is saturated with pointlessly prissy language and infuriating, nano-step nitpicking, none of which bears the remotest resemblance to real mathematical thought or expression.
What is the message of such garbage? For the vast majority of students, who naively presume that an educational authority would have some expertise in education, the message is that mathematics is nothing but soulless bookkeeping, which should be avoided at all costs. For anyone who knows mathematics, however, the message is that Victorian maths education is in the clutches of a heartless and entirely clueless antimathematical institution.
The Victorian Minister for Education has announced that the state’s senior school curriculum will undergo a review. The stated focus of the review is to consider whether “there should be a more explicit requirement for students to meet minimum standards of literacy and numeracy …“. The review appears to be strongly supported by industry, with a representative of the Australian Industry Group noting that “many companies complained school leavers made mistakes in spelling and grammar, and could not do basic maths“.
Dumb and dumber.
First, let’s note that Victorian schools have 12 years (plus prep) to teach the 3 Rs. That works out to 4 years (plus prep/3) per R, yet somehow it’s not working. Somehow the standards are sufficiently low that senior students can scale an exhausting mountain of assignments and exams, and still too many students come out lacking basic skills.
Secondly, the Minister has determined that the review will be conducted by the VCAA, the body already responsible for Victorian education.
If the definition of insanity is doing the same thing over and over and expecting different results, then the definition of insane governance is expecting the arrogant clown factory responsible for years of educational idiocy to have any willingness or ability to fix it.