Signs of the Times

Our second sabbatical post concerns, well, the reader can decide what it concerns.

Last year, diagnostic quizzes were given to a large class of first year mathematics students at a Victorian tertiary institution. The majority of these students had completed Specialist Mathematics or an equivalent. On average, these would not have been the top Specialist students, nor would they have been the weakest. The results of these quizzes were, let’s say, interesting.

It was notable, for example, that around 2/5 of these students failed to simplify the likes of 81-3/4. And, around 2/3 of the students failed to solve an inequality such as 2 + 4x ≥ x2 + 5. And, around 3/5 of the students failed to correctly evaluate \boldsymbol {\int_0^{\pi} \sin 5x \,{\rm d}x}\, or similar. There were many such notable outcomes.

Most striking for us, however, were questions concerning lists of numbers, such as those displayed above. Students were asked to write the listed numbers in ascending order. And, though a majority of the students answered correctly, about 1/4 of the students did not.

What, then, does it tell us if a quarter of post-Specialist students cannot order a list of common numbers? Is this acceptable? If not, what or whom are we to blame? Will the outcome of the current VCAA review improve things, or will it make matters worse?

Tricky, tricky questions.

Feynman on Modernity

We plan to have more posts on VCAA’s ridiculous curriculum review. Unfortunately.

Now, however, we’ll take a semi-break with three related posts. The nonsensical nature of VCAA’s review stems largely from its cloaking of all discussion in a slavish devotion to “modernity”, from the self-fulfilling prediction of the inevitability of “technology”, and from the presumption that teachers will genuflect to black box authority. We’ll have a post on each of these corrupting influences.

Our first such post is on a quote by Richard Feynman. For another project, and as an antidote to VCAA poison, we’ve been reading The Character of Physical Law, Feynman’s brilliant public lectures on physical truth and its discovery. Videos of the lectures are easy to find, and the first lecture is embedded above. Feynman’s purpose in the lectures is to talk very generally about laws in physics, but in order to ground the discussion he devotes his first lecture to just one specific law. Feynman begins this lecture by discussing his possibly surprising choice:

Now I’ve chosen for my special example of physical law to tell you about the theory of gravitation, the phenomena of gravity. Why I chose gravity, I don’t know. Whatever I chose you would’ve asked the same question. Actually it was one of the first great laws to be discovered and it has an interesting history. You might say ‘Yes, but then it’s old hat. I would like to hear something about more modern science’. More recent perhaps, but not more modern. Modern science is exactly in the same tradition as the discoveries of the law of gravitation. It is only more recent discoveries that we would be talking about. And so I do not feel at all bad about telling you of the law of gravitation, because in describing its history and the methods, the character of its discovery and its quality, I am talking about modern science. Completely modern.

Newer does not mean more modern. Moreover, there can be compelling arguments for focussing upon the old rather than the new. Feynman was perfectly aware of those arguments, of course. Notwithstanding his humorous claim of ignorance, Feynman knew exactly why he chose the law of gravitation.

This could, but will not, lead us into a discussion of VCE physics. It suffices to point out the irony that the clumsy attempts to modernise this subject have shifted it towards the medieval. But the conflation of “recent” with “modern” is of course endemic in modern recent education. We shall just point out one specific effect of this disease on VCE mathematics.

Once upon a time, Victoria had a beautiful Year 12 subject called Applied Mathematics. One learned this subject from properly trained teachers and from a beautiful textbook, written by the legendary J. B. “Bernie” Fitzpatrick and the deserves-to-be-legendary Peter Galbraith. Perhaps we’ll devote some future posts on Applied and its Pure companion. It is enough to note that simply throwing out VCE’s Methods and Specialist in their entirety and replacing them with dusty old Pure and Applied would result in a vastly superior, and more modern, curriculum.

Here, we just want to mention one extended topic in that curriculum: dynamics. As it was once taught, dynamics was a deep and incredibly rich topic, a strong and natural reinforcement of calculus and trigonometry and vector algebra, and a stunning demonstration of their power. Such dynamics is “old”, however, and is thus a ready-made target for modernising zealots. And so, over the years this beautiful, coherent and cohering topic has been cut and carved and trivialised, so that in VCE’s Specialist all that remains are a few disconnected, meat-free bones.

But, whatever is bad the VCAA can strive to make worse. It is clear that, failing the unlikely event that the current curriculum structure is kept, VCAA’s review will result in dynamics disappearing from VCE mathematics entirely. Forever.

Welcome to the Dark Ages.

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.

 

 

WitCH 6: Parallel Reality

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.

Have fun.

Update

Below, we go through the passage line by line, but that fails to capture the passage’s intrinsic awfulness. The passage is, as John put it pithily below, a total fatberg. The passage is worse than wrong; it is clumsy, pompous, circuitous, barely comprehensible and utterly pointless.

Why do this? Why write like this? Sure, ideas, particularly mathematical ideas, can be tricky and difficult to convey; dependence/independence isn’t particularly easy to explain. And sure, we all write less clearly than we might wish on occasion. But, if you write/proofread/edit something that the intended “readers” will obviously struggle to understand, then all you’re doing is either showing off or engaging in a meaningless ritual.

An underlying problem is that the entire VCE topic is pointless. Yes, this is the fault of the idiotic VCAA, not the text, but it has to be said, if only as a partial defence of the text. No purpose is served by including in the curriculum a subtle definition that is not then reinforced in some meaningful manner. Consequently, it’s close to impossible to cover this aspect of the curriculum in an efficient, clear and motivated manner. The text could have been one hell of a lot better, but it probably never could have been good.

OK, to the details. Grab a whisky and let’s go.

  • First, a clarification. The definition of “parallel vectors” appears in a slightly earlier part of the text. We included it because it is clearly relevant to the main excerpt. We didn’t intend, however, to suggest that the discussion of dependence began with the “parallel” definition.
  • For the given definition of “parallel vectors” it is redundant and distracting to specify that the scalar k not be 0.
  • As discussed by Number 8, the definition of “parallel vectors” should not exclude the zero vector. The exclusion may be natural in the context of geometric proofs, but here it is a needless source of fussiness, distraction and error.  As an example of a blatant error, immediately following the above passage the text begins a proposition with “Let a and b be two linearly independent (i.e. not parallel) vectors.” A second and entirely predictable error occurs when the text later goes on to “resolve” an arbitrary vector a into components “parallel” and “perpendicular” to a second vector b.
  • The definition of “linear combination” involves a clumsy and needless use of subscripts. Thankfully, though weirdly, subscripts aren’t used in the subsequent discussion. The letters used for the vector variables are changed, however, which is the kind of minor but needless, own-goal distraction that shouldn’t occur.
  • No concrete example of linear combination is provided. (The more abstract the ideas, the more critical it is that they be anchored immediately with very specific illustration.)
  • It is a bad choice to begin with “linear combination”. That idea is difficult enough, but it also leads to a poor and difficult definition of linear dependence, an unswallowable mouthful: “… at least one of its members [elements? vectors?] can be expressed as a linear combination of [the] other vectors [members? elements?] …” Ugh! What really kills this sentence is the “at least one”which stems from the asymmetry hiccup in the definition. (The hiccup is illustrated, for example, by the three vectors a = 3 + 2j + k, b = 9i + 6j + 3k, c = 2i + 4j + 3k. These vectors are dependent, since b = 3a + 0c is a combination of a and c. Note, however, that c cannot be written as a combination of a and b.)
  • As was appropriately done for “linear combination”, the definition of linear dependence should be framed in terms of two or three vectors staring at the reader, not for “a set of vectors”. 
  • The language of sets is obscure and unnecessary.
  • No concrete example of linear dependence is provided. There is not even the specialisation to the case of two and/or three vectors (which, again, is how they should have begun).
  • If you’re going to begin with “linear combination” then don’t. But, if you are, then the definition of linear independence should precede linear dependence, since linear independence doesn’t have the asymmetry hiccup: no vector can be written as a combination of the other vectors. Then, “dependent” is defined as not independent.
  • No concrete example of linear independence is provided. 
  • The properly symmetric “examples” are the much preferred definition(s) of dependence. 
  • The “For example” is weird. It is more accurate to label what follows as special cases. They are not just special cases, however, since they also incorporate non-obvious reworking of the definition of dependence.
  • No proof or discussion is provided that the “example[s]”  are equivalent to the definition. 
  • No genuine example is provided to illustrate the “example[s]”.
  • The simple identification of two vectors being parallel/non-parallel if and only if they are dependent/independent is destroyed by the exclusion of the zero vector.
  • There is no indication why any set of vectors including the zero vector must be dependent. 
  • The expression “two-dimensional vector” is lazy and wrong: spaces have dimension, not vectors. (Ditto “three-dimensional vectors”.)
  • No proof or discussion is provided that any set of three “two dimensional vectors” is dependent. (Ditto “four three-dimensional vectors”.)
  • The “method” for checking the dependence of three vectors is close to unreadable. They could have begun “Let a and b be linearly independent vectors”. (Or, with the correct definition, “Let a and b be non-parallel vectors”.)
  • There is no indication of or clarification of or illustration of the subtle distinction between the original “definition” of linear dependence and the subsequent “method”.

What a TARDIS of 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.

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

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”.

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

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.