VCAA Puts the “Con” into Consultation

As we have written, the Victorian Curriculum and Assessment Authority is “reviewing” Victoria’s senior secondary maths, which amounts to the VCAA attempting to ram through a vague and tendentious computer-based curriculum, presented with no evidence of its benefit apart from change for the sake of change. Readers can and should respond to the VCAA’s manipulative questionnaire before May 10. In this post we shall point out the farcical nature of VCAA’s “consultation”, as evidenced by VCAA’s overview and questionnaire.

The overview begins by framing VCAA’s review with the following question:

What could a senior secondary mathematics curriculum for a liberal democratic society in a developed country for 2020–2030 look like?

This is peculiar framing, since it is difficult to imagine how a society being “liberal” or “democratic” or otherwise has any bearing on the suitability of a mathematics curriculum. Why would a good curriculum for China not also be good for Victoria?

One could easily write off this framing as just jingoistic puffery; neither word reappears in VCAA’s overview. It is, however, more insidious than that. The framing is, except for the odd omission of the word “suitable”, identical to the title of the Wolfram-CBM paper promoting “computer-based mathematics” in general and Wolfram-CBM in particular. This paper is the heavy propaganda gun VCAA has procured in furtherance of its struggle to liberate us all from the horrors of mathematical calculation. Though the Wolfram-CBM paper never states it explicitly, this makes clear the purpose of the framing:

“[L]iberal” and “democratic” and “developed” amounts to “rich enough to assume, demand and forever more have us beholden to the omnipresence of computers”.

The VCAA overview continues by noting the VCAA’s previous review in 2013-2014 and then notes the preliminary work undertaken in 2018 as part of the current review:

… the VCAA convened an expert panel to make recommendations in preparation for broad consultation in 2019.

Really? On whose authority does this anonymous panel consist of experts? Expert in what? How was this “expert panel” chosen, and by whom? Were there any potential or actual conflicts of interest on the “expert panel” that were or should have been disclosed? How or how not was this “expert panel” directed to conduct its review? Were there any dissenters on this “expert panel”?

The only thing clear in all this is the opacity.

The overview provides no evidence that VCAA’s “expert panel” consists of appropriately qualified or sufficiently varied or sufficiently independent persons, nor that these persons were selected in an objective manner, nor that these persons were able to and encouraged to conduct the VCAA review in an objective manner. 

Indeed, any claim to breadth, independence or expertise is undermined by the constrained formulation of the questionnaire, the poverty of and the bias in the proposed curriculum structures and the overt slanting of the overview towards one particular structure. Which brings us to the issue of consultation:

There is no value in “broad consultation” if discussion has already been constrained to the consideration of three extremely poor options.

But, “consult” the VCAA will:

The VCAA will consult with key stakeholders and interested parties to ensure that feedback is gained from organisations, groups and individuals.

Well, great. The writer of this blog is a keenly interested stakeholder, and an individual well known to the VCAA. Should we be waiting by the phone? Probably not, but it hardly matters:

The VCAA has provided no indication that the consultation with “key stakeholders” and “interested parties” will be conducted in a manner to encourage full and proper critique. There is very good reason to doubt that any feedback thus gained will be evaluated in a fair or objective manner.

The overview then outlines three “key background papers” (links here). Then:

… stakeholders are invited to consider and respond to the consultation questionnaire for each structure.

Simply, this is false. Question 1 of VCAA’s questionnaire asks

Which of the proposed structures would you prefer to be implemented for VCE Mathematics?

Questions 2-8 then refer to, and only to, “this structure”. It is only in the final, catch-all Question 9 that a respondent is requested to provide “additional comments or feedback with respect to these structures”. Nowhere is it possible to record in a proper, voting, manner that one wishes to rank the Wolfram-CBM Structure C last, and preferably lower. Nowhere is there a dedicated question to indicate what is bad about a bad structure.

The VCAA questionnaire explicitly funnels respondents away from stating which structures the respondents believe are inferior, and why.

The good news is that the manipulativeness of the questionnaire probably doesn’t matter, since the responses will be presumably just be considered by another VCAA “expert panel”.

The VCAA overview gives no indication how the responses to the questionnaire will be considered and provides no commitment that the responses will be made public.

The VCAA overview goes on to provides outlines of the three structures being considered, which we’ll write upon in future posts. We’ll just comment here that, whereas Structures A and (to a lesser extent) B are laid out in some reasonable detail, Structure C looks to be the work of Chauncey Gardiner:

What is written about Structure C in the VCAA overview could mean anything and thus means nothing. 

True, for a “detailed overview” the reader is directed to the Wolfram-CBM paper. That, however, only makes matters worse:

A 28-page sales pitch that promotes particular software and particular commercial links is much more and much less than a clear, factual and dispassionate curriculum structure, and such a pitch has absolutely no place in what VCAA describes as a “blue-sky” review. By giving prominence to such material, the VCAA fails to treat the three proposed structures in anything close to a comparable or fair manner. 

If there were any doubt, the overview ends with the overt promotion of Structure C:

The distinctive proposal … contain[s] aspects which the Expert Panel found valuable … There was support for these aspects, indeed, many of the invited paper respondents [to the 2018 paper] independently included elements of them in their considerations, within more familiar structures and models.

Nothing like putting your thumb on the scales.

It is entirely inappropriate for a VCAA overview purportedly encouraging consultation to campaign for a particular structure. A respondent having “included elements” of an extreme proposal is a country mile short of supporting that proposal lock, stock and barrel. In any case, the cherry-picked opinions of unknown respondents selected in an unknown manner have zero value. 

Though woefully short of good administrative practice, we still might let some of the above slide if we had trust in the VCAA. But, we do not. Nothing in VCAA’s recent history or current process gives us any reason to do so. We can also see no reason why trust should be required. We can see no reason why the process lacks the fundamental transparency essential for such a radical review.

In summary, the VCAA review is unprofessional and the consultation process a sham. The review should be discarded. Plans can then be made for a new review, to be conducted in the professional and transparent manner that Victoria has every right to expect.

Reviewing the VCAA Review – Open Discussion

The VCAA is currently conducting a “review” of VCE mathematics. We’ve made our opinion clear, and we plan to post further in some detail. (We’ll update this post with links when and as seems appropriate.) We would also appreciate, however, as much input as possible from readers of (especially critics of) this blog.

This post is to permit and to encourage as much discussion as possible about the various structures the VCAA is considering. People are free to comment generally (but carefully) about the VCAA and the review process, but the intention here is to consider the details of the proposed structures and the arguments for and against them. We’re interested in anything and everything people have to say. Except for specific questions addressed to us, we’ll be pretty much hands-off in the comments section. The relevant links are

Please, go to it.

The Wolfram at the Door

(Note added 20/4: A VCAA questionnaire open until May 10 is discussed at the end of this post. Anyone is permitted to respond to this questionnaire, and anyone who cares about mathematics education should do so. It would be appreciated if those who have responded to the questionnaire indicate so in the comments below.)

Victoria’s math education is so awful and aimless that it’s easy to imagine it couldn’t get much worse. The VCAA, however, is in the process of proving otherwise. It begins, and it will almost certainly end, with Conrad Wolfram.

We’ve long hoped to write about Wolfram, the slick salesman for Big Brother‘s Church. Conrad Wolfram is the most visible and most powerful proponent of computer-based maths education; his Trumpian sales pitch can be viewed here and here. Wolfram is the kind of ideologue who can talk for an hour about mathematics and the teaching of mathematics without a single use of the word “proof”. And, this ideologue is the current poster boy for the computer zealots at the VCAA.

The VCAA is currently conducting a “review” of VCE mathematics, and is inviting “consultation”. There is an anonymous overview of the “review”, and responses to a questionnaire can be submitted until May 10. (Below, we give some advice on responding to this questionnaire. Update 25/4: Here is a post on the overview and the questionnaire.) There is also a new slanted (and anonymous) background paper, a 2017 slanted (and anonymous) background paper, a 2014 slanted (and anonymous) background paper, and some propaganda by Wolfram-CBM.

In the next few weeks we will try to forego shooting Cambridge fish in the barrel (after a few final shots …), and to give some overview and critique of the VCAA overview and the slanted (and anonymous) background papers. (We hope some readers will assist us in this.) Here, we’ll summarise the VCAA’s proposals.

The VCAA has stated that it is considering three possible structures for a new VCE mathematics study design:

  • Structure A.1 – the same warmed over swill currently offered;
  • Structure A.2 – tweaking the warmed over swill currently offered;
  • Structure B – compactifying the warmed over swill currently offered, making room for “options”;
  • Structure C – A “problem-centred computer-based mathematics incorporating data science”.

What a wealth of choice.

There is way, way too much to write about all this, but here’s the summary:

1. Structure C amounts to an untested and unscripted revolution that would almost certainly be a disaster.

2. The VCAA are Hell-bent on Structure C, and their consultation process is a sham. 

So, what can we all do about it? Pretty much bugger all. The VCAA doesn’t give a stuff what people think, and so it’s up to the mathematical heavy hitters to hit heavily. Perhaps, for example, AMSI will stop whining about unqualified teachers and other second order trivia, and will confront these mathematical and cultural vandals.

But, the one thing we all can do and we all should do is fill in the VCAA’s questionnaire. The questionnaire is calculatedly handcuffing but there are two ways to attempt to circumvent VCAA’s push-polling. One approach is to choose Structure C in Q1 as the “prefer[red]” option, and then to use the subsequent questions to critique Structure C. (Update 25/4: this was obviously a poor strategy, since the VCAA could simply count the response to Q1 as a vote for Structure C.) The second approach is to write pretty much anything until the catch-all Q9, and then go to town. (20/4 addition: It would be appreciated if those who have responded to the questionnaire indicate so below with a comment.)

We shall have much more to write, and hopefully sooner rather than later. As always, readers are free to and encouraged to comment, but see also this post, devoted to general discussion.

The VCAA Dies Another Death

A while back we pointed out two issues with the 2018 Specialist Mathematics Exams. The Exam Reports (though, strangely, not Exam 1) are now online (here and here). (Update 27/02/19: Exam 1 is now also online.) Ignoring some fresh Hell suggested by the Exam 2 Report (B2(b), B3(c)(i), B6(e)), how did the VCAA address these issues?

Question 3(f) on Section B of Exam 2 was a clumsy and eccentrically worded question that covered material outside the curriculum. Unsurprisingly the Report made no mention of these issues. But, what about a blatant error by the Examiners? Would they remain silent in the face of such an error? Again?

Question 6 on Exam 1 (not online) required students to find the “change in momentum” of an accelerating particle. Unfortunately, the students were required to express this change in kg m s-2. The Exam had included the wrong units, just a careless typo, but a blatant error. The Report addressed this blatant error with the following:

Students who interpreted this question as asking for the average rate of change of momentum to be dimensionally consistent with the units and did this correctly were awarded marks accordingly.

That’s it. Not an honest word of having stuffed up. Not a hint of regret or apology. Just some weasely no-harm-no-foul bullshit.

 

 

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 

    \[\boldsymbol{\frac{{\rm d}h}{{\rm d}t} = \frac{4 - 5\sqrt{h}}{25\pi\left(4h^2 + 1\right)}\,.}\]

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.

Untried Methods

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. Update: Now online.). 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?

 

Inferiority Complex

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 {-2-2\sqrt{3}i} 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 \color{red}\boldsymbol{z^2 + 4z + 16 = 0}} are {\color{red} \boldsymbol{z=-2-2\sqrt{3}i}} and \boldsymbol{\color{red}{z=-2+2\sqrt{3}i}}.

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:

\color{blue} \boldsymbol{z = \frac{-4\pm \sqrt{4^2 \ - \ 4 \times 1 \times 16}}{2}=\frac{-4\pm \sqrt{-48}}{2}=\frac{-4\pm 4\sqrt{3}i}{2} = -2\pm2\sqrt{3}i\, .}

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”.

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Onwards we trek to 4(c):

Express the roots of \color{red} \boldsymbol{z^2 + 4z + 16 = 0}} in terms of  \boldsymbol{{\color{red}   2 -2\sqrt{3}i}}.

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  {\color{blue} \boldsymbol{2 -2\sqrt{3}i}} 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 w = 2 - 2\sqrt{3}i, as the examiners could have and should have and did not. Then, the intended answers were -w and -\overline{w}.

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 w-4 and \overline{w-4} and w^2/4 and w^4/|w|^3 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.

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The pain now temporarily lessens (though the worst is yet to come). 4(d) asks for students to show that the relation |z| = |z - (2-2\sqrt{3}i)| has the cartesian form x-\sqrt{3}y - 4 = 0, 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).

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Finally, the debacle of 4(f):

The equation of the line passing through the two roots of {\color{red} \boldsymbol{z^2 + 4z + 16 = 0}} can be expressed as {\color{red} \boldsymbol{|z-a| = |z-b|}}, where \color{red}\boldsymbol{a, b \in C}. Find \color{red}\boldsymbol{b} in terms of \color{red}\boldsymbol{a}.

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 \operatorname{Re} z = x = -2. 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 a = (-2+c) + di. Since b must be symmetrically placed about the line \operatorname{Re} z = -2, it follows that b = (-2-c) + di. Then b+2 = -c + di = -\overline{(a+2)}. This gives b = -2 - \overline{(a + 2)}, and finally

\color{blue}\boldsymbol{b = -4 -\overline{a}\, ,}

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.

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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.

The Arc Enemy

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 \boldsymbol{f(x) = 10\arccos(2-2x)} 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 \boldsymbol{5\pi} 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

    \[\boldsymbol{{\Huge \int\limits_{\frac12}^{\frac32}}\sqrt{1+\left[\tfrac{20}{1 - (2-2x)^2}\right]^2}}\ {\bf d}\boldsymbol{x}\]

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.

Inverted Logic

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 –1 is a perfectly fine and familiar name for the inverse of f. It takes a special cluelessness to imagine that renaming –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, 

    \[\boldsymbol {\left(f^{-1}\right)'(b) = \frac1{f'\left(f^{-1}(b)\right)}}.}\]

IVT is typically stated, and in particular the differentiability of –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 –1 and and then differentiates –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.

VCAA Plays Dumb and Dumber

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?