The Complex Roots of VCAA’s Defence

1. Introduction

Sometimes VCAA is their own worst enemy. Well, no: we all know the identity of VCAA’s worst enemy. But on occasion VCAA places runner up. Maybe third.

I’ve been pondering VCAA’s major 2022 errors, how they could have occurred and how VCAA could continue to defend the flawed questions for so long, against all reason. Yes, “VCAA is nuts” springs to mind. But that’s not enough. VCAA being nuts is necessary but not sufficient to explain this extended episode of madness.

I’ve now thought long and hard about VCAA’s defence of the five questions, and I think there is a little more there than the nuttiness that meets the eye. I think that with a better lawyer VCAA might have fared better. To be clear, VCAA is guilty: all five questions are stuffed; all five defences are stuffed. But VCAA could have argued their case better, and they perhaps might have received a more merciful judgment.* I’ll consider the five questions in turn, one of which is worth more consideration than the others.

*) Not really.


2. The Four Quick Questions

VCAA has not provided any defence of the Methods question, centred upon and garbled by the empty regions. Their likely defence, however, seems easy to predict: there’s no region, therefore no area, therefore no function. In brief, the problem here is simply that VCAA’s writers and vetters and reviewers are mathematically illiterate. They don’t understand mathematical definition and argument well enough, and they cannot write well enough, to be able to see the glaring flaw in the question.

The explanation for the coffee question, beyond the nuttiness, appears to be VCAA’s pure arrogance. Having written something other than what was intended, VCAA is clutching at the straw of overall plausibility in order to pretend that they didn’t write the words they wrote.

The population confidence interval problem similarly appears to have been a product of arrogance, but along with a thin curriculum, and along with some unknown man not knowing his limitations. VCAA’s claim regarding the intent of the question is presumably true. The writers, knowing that every confidence interval question is like every other confidence interval question, decided to spice things up with a “novel” scenario. Then, when the reality was pointed out, that the “novel” scenario had no connection to statistical reality, VCAA’s trademark arrogance kicked in, with VCAA subsequently unwilling to admit the error.

The can question is similar to the confidence interval question, but is more interesting, a little trickier and a hell of a lot weirder. The key, as VCAA referred to in their defence, appears to be the line in the question preamble:

“The [soft drink] company sources empty cans from an external supplier”.

According to VCAA, the phrase “external supplier” was intended to trigger “insight into the physical situation” and thereby the required independence assumption. Of course such an assumption had not previously been required of the students, so why require it now? I’m guessing exactly because it had not been required previously. Similar to the population-confidence scenario, all these sums of random variables questions are essentially identical, an identity forced upon the examiners by a thin and foundationless topic. So, some clever man, who doesn’t know his limitations, spices up the question.

Then, the nuttiness kicks in. VCAA decides that a phrase uttered sotto voce a page and a half before the question is a fair trigger for the required independence assumption. And, VCAA decides that because they cannot imagine a different manufacturing process, their independence assumption can be the only reasonable assumption.

Then the arrogance and the weirdness kick in. VCAA refuses to consider that there might be a different and reasonable approach to the question, even though their own exam report takes a different and reasonable approach to the question.


3. The Long Question

Finally, we have the indefensible complex numbers question. For which I will offer a defence.

Well, not really. The question is appalling and wrong. And I think, in their big hearts and smaller brains, the VCAA people know it is appalling and wrong. But I think VCAA may genuinely be puzzled by the outrage that this question has provoked. I think I finally figured out, with no thanks to VCAA, what VCAA is on about.

As a reminder, the preamble to the question is,

The polynomial p(z) = (z – a)(z – b)(z – c) has complex roots a, b and c, where Re(a) ≠ 0, Re(b) ≠ 0, Re(c) ≠ 0 and Im(b) = 0. When expanded, the polynomial is a cubic with real coefficients.

From this, we are supposed to be able to conclude that a and c are conjugate, and thus |a| = |c|. The loud objection is that this argument only works if we somehow know a and c are not real. VCAA’s loud defence is that students were supposed to realise from the information that a and c are not real. The loud re-objection has been “You guys are nuts”.

I will not try to dissuade the reader. They are nuts. But I will try to explain VCAA’s thinking, at least as I understand it, and as VCAA utterly failed to do.

VCAA’s defence begins,

The case where both a and c are strictly real is a specific case, and may be considered a degenerative case reducing all three roots to real values;

Ignoring the bad punctuation and bad writing, this is reasonable, but it doesn’t flag anything about interpreting the question. (Just as, for example, the possibility of a polynomial having a repeated root doesn’t automatically flag anything.)

a and c are declared to be complex roots and the general case where each has both real and imaginary parts should be considered.

Remember when I suggested VCAA could use a better lawyer?

Yes, a and c are “declared to be complex roots”. So what? There is a “so what”, which we’ll come to, but one cannot possibly glean it from the remainder of the sentence. The generic case comma where each has both non-zero real and imaginary parts comma is of course being “considered”. The issue is why VCAA seems to regard the generic case as the only case to be considered.

Any condition for their imaginary parts to be zero would have been mentioned just as it was declared that Im(b)=0.

This reverses the onus of declaration. It was declared that Im(b) = 0 because the question required that Im(b) = 0. But no objector to this question is trying to declare that a and c are real: they are noting the possibility that a and c are real, unless and until it is declared or clearly implied otherwise.

The question prompts students to work with conjugates in the general case.

If so, nothing in the defence so far indicates how this generic case has been prompted.

Questions regarding complex polynomials in general form in the subject of Specialist Mathematics must consider the roots of the polynomial to have non-zero real and imaginary components, unless otherwise stated.

At least this time they remembered to include the “non-zero”, but the statement is of course ludicrous and utter fantasy. It also undermines the purpose of the question declaring that the real parts of a, b and c are non-zero, although of course there was never any purpose anyway.

This is consistent with theory descriptions of conjugate root theorem developed in texts.


And, finally,

The constraints in the question did not preclude imaginary elements making option B correct.

Once again, reversing the onus. The question must do more than not preclude non-zero imaginary parts; the question must somehow ensure non-zero imaginary parts.

That is the sum of VCAA’s defence. So where was the defence? You have to look very closely.


4. Complex Roots in VCE

The only aspects of VCAA’s defence that suggests any substance at all are the references to complex roots and [the] conjugate root theorem. So, I took a look at the complex numbers chapters in the three Specialist Textbooks: Cambridge (bad), Jacaranda (worse) and Nelson (off the charts). Here, for example, is how Cambridge first states the conjugate root theorem:

Let P(z) be a polynomial with real coefficients. If a + bi is a solution of the equation P(z) = 0, with a and b real numbers, then the complex conjugate a − bi is also a solution.

Which is perfect. Then, in the chapter summary, Cambridge states the theorem as follows:

If a polynomial has real coefficients, then the complex roots occur in conjugate pairs.

Which is very far from perfect.

The problem, of course, is that in the second statement Cambridge is using “complex roots” to refer only to non-real complex roots. Exactly as VCAA is claiming for a and c in the exam question.

Cambridge makes other slips on this as well, but in general the language is more careful (or luckier). Jacaranda is similar but less accurate, typically employing the expression “pair of complex conjugate roots”, and occasionally “complex roots”, to refer to non-real roots. Nelson is diabolical. Nelson often and routinely, although not universally, uses “complex roots” to mean non-real roots (and there is much worse). It is also fair to note, without excusing these texts, that “complex roots” is used often enough in casual classroom language to refer to non-real roots.

Similarly, VCE exams have slipped up on occasion in the past, although not often. MCQ6 on 2022 NHT Exam 2 refers to a polynomial having

… one real and two complex solutions …

NHT is never a strong precedent, however, and otherwise it seems we have to go back to MCQ8 on 2010 Exam 2:

The polynomial equation P(z) = 0 has one complex coefficient.

That is obviously not good, if only for declaring that an equation has a coefficient, but it is a while ago. There are a couple other, more benign occurrences. I could find nothing else.


5. Back to VCAA’s Defence

That, then, is the core of VCAA’s defence of the complex number question: the precedent in VCE is that “complex root” means non-zero root, and thus students should have so interpreted a and c being declared to be complex roots. How does this defence stand up? Like a flounder.

First of all, there is simply not the precedent that VCAA is claiming. If “complex root” has previously been used to mean non-real root, such usage has hardly been sufficiently common or uniform to expect and require students in an exam to interpret the term in this manner. More importantly, not all past actions act as precedents. My older daughter having swiped five cookies is not a precedent for my younger daughter to now swipe five more. Using “complex” to mean “non-real” is, purely and simply, wrong. A teacher might slip it in on a suitable casual occasion, but they should know they’re abusing the term, and in a dangerous manner. There is zero argument for an exam abusing the language in this manner, no matter how many exams have done so previously.

Secondly, even accepting that “complex root” means non-real root, the exam question is stuffed. In their defence, VCAA note that a and c are declared to be complex roots, while pretending not to notice that b has also been declared to be a complex root. Which means of course that b cannot be real, which the question requires.

Now VCAA are aware of this weakness, of course. So, VCAA does not define “complex root” to mean non-real root. They effectively define “complex root” to mean non-real root “unless otherwise stated“. The legal term for this argument is “bullshit”. Before the 2022 exam, we doubt that anyone in the history of the planet has used “complex root” with such a meaning.

Sorry, VCAA. I defended you as well as I could, but you’re still screwed.


6. Conclusion

Was there a purpose to this long, long post, which ended up pretty much where it began? Yes.

The main purpose was to get all the “what the hell were they thinking” thoughts out of my head and safely onto electronic paper. I can now forget them. But the other purpose was, genuinely, to try to understand VCAA’s thinking. I think I do now, in a way I hadn’t previously.

It now seems clear to me that VCAA intended the complex number question to be written exactly as it appeared. They expected the question to be interpreted, and naturally interpreted, exactly as they intended. They believe, at least at some level, that they have mounted a proper defence of their question. Freud might query whether they really believe this defence, but they somehow believe it.

I think there are two conclusions to be drawn from this. The first conclusion is that VCAA is, to their very core, mathematically incompetent. VCAA’s defence is so disorganised, so incoherent, so plainly non-mathematical, it is in ways madder, more damning, than the question being defended.

The second conclusion is that VCAA is nuts. Simply, they are nuts.*


*) Perhaps I should have written “were nuts”. As I wrote, this defence from VCAA is from months ago, from before Burkard and I went nuclear. Still, as long as VCAA refuses to correct the record, and as long as the Deloitte report is treated as if it weren’t garbage, it’s fair enough to stick with “are nuts”, and I probably will.

31 Replies to “The Complex Roots of VCAA’s Defence”

  1. I’m not even sure why I teach Number Sets and Subsets… Maybe I am confusing my students by teaching them the correct mathematical understanding.

    The funniest thing to me is that they say:

    The polynomial p(z) = (z – a)(z – b)(z – c) has complex roots a, b and c, where Re(a) ≠ 0, Re(b) ≠ 0, Re(c) ≠ 0 and Im(b) = 0. When expanded, the polynomial is a cubic with real coefficients.

    So b can be a real solution, whilst still being a complex root. (ie. They seem to understand this…) They say it’s complex, but then define it to be real (Im(b)=0), so they understand real numbers are a subset of complex numbers..

    They effectively define “complex root” to mean non-real root “unless otherwise stated“. The legal term for this argument is “bullshit”. Before the 2022 exam, we doubt that anyone in the history of the planet has used “complex root” with such a meaning.

    Anyone wondering why we didn’t just get:

    The polynomial p(z) = (z – a)(z – b)(z – c) has complex roots a, b and c, where Im(a) ≠ 0, Im(b) ≠ 0, and Im(c) = 0. When expanded, the polynomial is a cubic with real coefficients.

    This just makes me feel like screaming…

    The case where both a and c are strictly real is a specific case, and may be considered a degenerative case reducing all three roots to real values;

    So, we only need to consider the solutions that they deem appropriate, however, we are supposed to just know which ones they want? Are they going to start giving us rational functions with parameters in the denominator and ask us what happens to the graph when we change the values of the parameters, however, they only wan’t specific cases… But we have to guess which ones they are considering? (Of course, it may just be a “degenerative case”)

  2. This question has taught me how little I understand about ordinary cubic equations; so what follows may well display my ignorance. But the whole point of VCAA is to be boringly precise so that ordinary students and teachers are mostly spared from having to guess, and so display their ignorance.

    1. Marty quotes (presumably from some official syllabus booklet):
    “Questions regarding complex polynomials in general form in the subject of Specialist Mathematics must consider the roots of the polynomial to have non-zero real and imaginary components, unless otherwise stated.”

    In mathematics, the “generic” case applies whenever we are simply given an object (in this case a cubic polynomial) *without further conditions*. The generic “triangle ABC” cannot be assumed to have a right angle, or be isosceles.

    But neither can “triangle ABC” be assumed *not* to have a right angle!

    VCAA may wish to include “z” and “z^2” as bona fide polynomials. So the official note needs to change – perhaps to read:
    “Questions regarding complex polynomials in general form in the subject of Specialist Mathematics must consider the roots of the polynomial to have *possibly* non-zero real and imaginary components, unless otherwise stated.”

    2. Moreover, as soon as one asks students to work with given conditions on the roots (as here), the cubic is no longer “generic”! Instead the imposed conditions restrict it to some specific subfamily of possible polynomials.

    3. Marty wrote: “VCAA’s defence begins,
    “The case where both a and c are strictly real is a specific case, and may be considered a degenerative case reducing all three roots to real values””

    The more they wriggle, the worse it gets. There appear to be two (very different) fresh errors here.

    (a) At a higher level, one can consider the family of all possible polynomials, or all possible polynomials of degree at most n. Each such polynomial can then be parameterised (e.g. by the set of coefficients; or by the leading coefficient and the set of roots). One can then consider subsets, or sequences, of polynomials, and ask all sorts of questions about what happens to the polynomials as one varies the sequence of parameters.
    It may then become clear that there is some notion of “degenerate”: for example, a cubic polynomial with a repeated root is sometimes said to be “degenerate” [see (b) below] – though I can find no standard mathematical justification for using the word “degenerative”.

    (b) I *can* imagine considering a sequence of cubic polynomials with roots a, b, c, in which (i) b is real and fixed, and (ii) Re(a) is fixed, and Im(a) tends to 0. But this is then going to affect the other root “c” (the complex conjugate of a!). In other words: the “limit” polynomial will then have a repeated root a = c (and so might quite properly be called “degenerate”, but never “degenerative”!).
    However, even if the language were used correctly, it cannot obviously be used to incorporate the interesting family of cubic polynomials with three *distinct* real roots (which was the original objection).

    1. Thanks, Tony.

      What I’m quoting from is VCAA’s reply to the “unwavering teacher”, on why the four 2022 Specialist questions are hunky-dory. So, it’s not exactly VCAA policy. But it is, at least until VCAA says otherwise, VCAA’s offical stance on these questions.

      Of course what VCAA has written is completely nuts. What I was trying to do here was to figure out why, in VCAA’s nutty minds, it’s not nuts.

    2. One minor disagreement, Tony. I used, and use, “generic” in a slightly different manner to you. For me “generic” means somehow the typical, so what’s true on dense set, or except for a set of measure zero, or similar. So, for me a generic triangle would *not* be right-angled.

      Not that it matters much. It’s more a colloquial term. If “generic”, or it’s “general” alternative, is being used to justify the wording of an exam question then there are much larger fish to fry.

  3. 2017 Specialist Mathematics Exam 1 Question 3:

    “Let \displaystyle z^3 + a z^2 + 6z + a=0, \displaystyle z \in C, where a is a real constant. Given that \displaystyle z = 1 - i is a solution to the equation, find all other solutions.”

    In other words, all other complex roots of \displaystyle z^3 + a z^2 + 6z + a are required.
    The Exam Report gives the (correct) answer z = 1 + i and z = 2.
    VCAA’s attempt to try and define “complex root” to mean non-real root “unless otherwise stated“ is doomed, if only by it’s own answers (and marking scheme) to this question.

    VCAA clearly understands (or at least, used to understand) that real numbers are a subset of complex numbers and complex roots can be real or non-real. “Oh, what a tangled web we weave when first we practice to deceive.”

    1. BiB, I don’t understand your point. The fundamental issue with the 2022 question is the meaning of “complex root”. How does either the 2017 question or the report solution shed any light on VCAA’s understanding of the meaning of “complex root”.

      1. Solutions to the equation \displaystyle z^3 + a z^2 + 6z + a=0 are roots of the polynomial \displaystyle z^3 + a z^2 + 6z + a.

        We are told without any qualification to find all other solutions to \displaystyle z^3 + a z^2 + 6z + a=0 where \displaystyle z \in C.
        That is, we must find all other complex solutions to \displaystyle z^3 + a z^2 + 6z + a=0.
        That is, we must find all other complex roots of \displaystyle z^3 + a z^2 + 6z + a.
        Without any further qualification. Without any “otherwise stated”.

        The Examination Report says the answer is z = 1 + i and z = 2. So VCAA considers z = 2 a complex root.

        But … in its defence of the 2023 MCQ4, VCAA seems to be saying that unless otherwise stated, complex roots mean non-real roots. VCAA’s defence is inconsistent with its answer to the 2017 question.

  4. Mea culpa: I took Marty’s word “generic” and interpreted it in the only sense I could think of at school level. In other words I confused “generic” and “general”.

    The correct use of “generic” is not at the heart of this discussion. It is not fixed. Marty is clearly in good company in taking it to mean “true on a dense subset”; however, this seems distinctly unhelpful. (I can swallow “a generic real number is irrational”, but not “a generic real number is rational”.)

    I suspect the idea behind “generic” is better captured by “almost all” – which then carries with it a clear responsibility to explain how this is being interpreted.

    1. Yes, you’re right Tony: “dense” is wrong. That one came up in a quick search, after I was puzzled by your reply, and so I threw it in, but it clearly doesn’t correspond to “typical”. My own (rare) use of “generic” would correspond to “almost all”.

  5. This is somewhat off-topic from the well deserved opprobrium towards VCAA, but one has to wonder why exactly we even bother teaching complex numbers at the high-school level. What was striking to me when I took the UMEP course was that it was one of the first cases where I had seen complex numbers being used in a purposeful way to solve seemingly unrelated problems (in that case, it was to do exponential-trigonometric integrals). As far as I know, the VCE course, and most other secondary mathematics courses, do not do anything of this sort. If you asked the average VCE specialist maths student what an application of complex numbers would be, my bet is that they would not have a clue. Of course, this is not in and of itself a reason to exclude complex numbers from the curriculum, but it is symptomatic of the fact that this topic is taught with a remarkable level of purposelessness. This, coupled with the clear evidence that the curriculum designers do not have an adequate understanding of the topic, suggest to me that removing it from the syllabus may be the best course of action.

    1. Hear, hear. And why stop with complex numbers. For starters I’d definitely throw rational functions onto the scrap heap. Show me a student that can tell you the purpose of learning about those useless buggers. And what use is a point of inflection to any secondary school student?

      They’ve already made a good start and gotten rid of dynamics and statics – useless stuff for secondary school students when you think about it. And don’t get me started on all the other irrelevant stuff that gets taught in every other subject.

      By the way, are you a madman? What possible purpose can you conceive for doing exponential-trigonometric integrals as a Yr 12 student? And haven’t you heard of a CAS?

      There’s so much useless stuff (well, useless at secondary school level) that gets taught. Get rid of it all, I say. Let’s properly teach kids only the ‘important’ stuff that can be used with a clear purpose and then kick them out of school at the end of Yr 9 unless they want to be engineers etc.

        1. Then I’ll limit it to the following (byo tone):

          What possible purpose can you (the OP) conceive for doing exponential-trigonometric integrals as a Yr 12 student? And haven’t you heard of a CAS?

    2. Thanks, Anand. It’s slightly off-topic, and I might now drag the discussion further off-topic, but I think it’s worth it.

      First of all, in principle I disagree with you about getting rid of C from the curriculum. It’s a natural topic and naturally extends previous topics. Yes, it could go further or deeper, but I don’t think it has to do so to be meaningful. You seem to have been a strong student (UMEP, AMT), and so you would have benefited from a more extensive and/or deeper coverage, as would other stronger students. But that’s not properly an argument for the topic as is.

      Secondly, the C topic is of course currently pretty awful, and not simply because of the grotesque exam errors: they are much more a symptom than the disease. The fundamental disease is that the “curriculum designers” – for this discussion let’s put the curriculum writers and exam writers and textbook writers in the same stupid boat – have no proper appreciation of or respect for mathematical thought. They simply don’t get the way mathematics should be structured, so the topic as taught and examined simply ends up being a conglomeration of poorly organised and poorly presented facts. This pretty much guarantees that errors will occur but, even before the errors, the presentation makes the topic fundamentally worthless.

      This was not always the case, at least not in Victoria. The subject used to be a little deeper, with a better coverage of polynomial roots, à la Vieta, and it would be a better topic if this material were reintroduced; deeper is preferable to larger. But more importantly, the topic used to be presented in the texts in a properly logical order. It was nice.

      Why is C not so presented now? Because nobody cares. Yes, the writers-writers-writers are incompetent, but no one cares that they are incompetent. No one has cared for decades. The solution is not to change topics: these idiots will, and do, screw up anything and everything. All SM and MM topics are bad as now presented. The solution is to re-instil a culture of mathematical thought, the first step of which is putting the current writers-writers-writers on an iceberg and wishing them farewell.

      I will be writing more on this soon. People often remark that the Cambridge texts are good, which only proves how painfully low standards have sunk. Yes, Cambridge is way better than the alternatives, and Cambridge is getting better. But Cambridge still really, really sucks. Two of my very first WitCHes were on the awfulness of Cambridge‘s chapter on complex numbers; that was five years ago and the chapter is seemingly unchanged, just as bad. And, again, the other texts are much worse. I’ll soon have a bunch of posts on Nelson‘s complex numbers chapter, which defies belief.

      Finally, UMEP is not that great a subject. You presumably only think so because you did MM and SM at the same time.

      1. I will add that the history of complex numbers is very much more interesting than most students realise or will ever realise (they were NOT invented to solve quadratic equations or to deal with some physical problem). Their history alone makes them worthy of inclusion in any secondary school advanced mathematics subject. Hopefully teachers of Specialist Mathematics units 1&2 find the time to make this history known.

      2. Thanks for your very thoughtful points Marty. What you are saying makes a lot of sense and you are right that a lot of the problems are due to the poorly conceived nature of the curriculum and its presentation in the textbooks. Interesting to hear your thoughts about UMEP: I took the course in 2015 during the last year of its previous iteration, when it was a cognate subject for Accelerated Mathematics 1, before they basically changed it to match Calculus 2 and Linear Algebra. I thought at the time that it had quite a lot of proof-based content and that it provided good preparation for further subjects at Melbourne.

        1. Thanks again, Anand. The topics within UMEP are reasonable, but the presentation hasn’t ever been great, and the assessment was often arbitrary and unfair. The people in charge of UMEP have often been opaque and arrogant.

  6. In NSW from around 1900 (my evidence from Victoria is more sparse, but it seems to be similar) the main “leaving” examinations were set by a major university (Sydney, Melbourne) and they were long papers.

    Students could take papers in British History, Mathematics, Greek, Latin, English Literature and that was about it (Drawing and Bookkeeping were added a bit later).

    The Mathematics papers were largely (if not entirely) built around the Oxford and Cambridge (mostly Cambridge in the case of NSW) entry examinations and covered Arithmetic, Algebra, Calculus, Mechanics and Statistics.

    Complex numbers came and went a bit over the years, the papers were all around 10 questions long and the papers were 3 hours in length (although students wanting to sit the advanced paper would first need to pass the ordinary paper, so that would have been six hours total).

    Finding marking guides for pre WW2 papers has proven impossible, and most of the early papers had no mark allocations written on the paper, so I’m assuming the universities made their own judgements (as would be their right at the time to decide which students were to be admitted).

    What is noticeably absent until much later (certainly not in anything pre 1966) is “modelling” – genuine or imagined.

    Part of me wonders if this obsession with mathematical models (especially in SACs but also in Paper 2 Methods and Specialist exams) was the catalyst for everything heading south…

    1. Thanks, RF. That sounds similar to Vic/Matric. I think by the 40’s mechanics and complex numbers were well established. But I don’t have details.

      1. It is difficult to know without looking at specific papers because for a long time pre 1970s the exam papers basically were the syllabus.

        It seems like Complex Numbers came and went a bit for a while, similar patterns with vectors.

        Algebra, Trigonometry and Calculus are the common themes in all the papers I’ve seen from Victoria but the process of investigating is rather slow because very few of these papers are easy to find online, which slows the process quite a bit.

        The UK papers from decades ago are oddly enough easier to find and seem to be the main influence for the 1850-1970 NSW and VIC papers/contents, especially the Cambridge/Oxford examination boards.

        But I digress.

        Complex numbers good. Modelling bad. Algebra skills good. CAS skills bad. Old exams OK. Modern exams… choose your expletive in most states of Australia. NSW still OK.

  7. At the risk of hitting a possibly inappropriate note, may I urge caution when going on a “chucking things out” spree?

    I can’t comment on Victoria – where syllabus constraints may make the following comments largely relevant to first year undergrad (for scientists, engineers, and mathematicians): if so, the spirit of my comment may still apply to other topics.

    In the UK many students in their last two years of high school report that they first had their eyes opened by meeting complex numbers. My impression is that (like having a crush on some totally inaccessible film star) this often had painfully little to do with what one could *do* with them: more the fact that they forced one to rethink familiar slogans like “you can’t take the square root of -1”; and perhaps “tidying up consequences”, like the fact that “a polynomial equation of degree n now had exactly n roots”. (Long before one gets to “exponential-trigonometric integrals”, the key idea one should perhaps aim for is the fascinating interplay of Cartesian/additive and polar/multiplicative form, leading to e^(i*theta), etc..)

    Rational functions (from sketching [*not* plotting], to integrating – after partial fractions) also deserves a little respect at some stage – if the syllabus has room. A key implicit idea is that of *elementary functions*, and whether mathematics is a closed calculational world, where one can get away with using just these. Unfortunately, syllabuses and texts tend to grind through lots of increasingly complicated examples without ever getting kids to reflect on what is really going on: namely: that (i) we start with *y [or f(x)] =x*; then (ii) extend to powers and simple polynomials; then (iii) extend further to simple rational functions, trig functions, and exponentials; then (iv) introduce their their inverses. Through this extended process, one can get a notion of *generality* without writing out the “most general” polynomial or rational function, or the most general composite trig function, or trying to capture their inverses. When differentiation then comes along, we notice (thanks to the product rule, the quotient rule, and the chain rule) that the derivative of anything familiar (i.e. built up from *elementary functions*) seems to be still an *elementary function*! There is then the question: “to what extent is the same true for integration?” Rational functions (and partial fractions) offer a partial answer, leaving the surprises until later (except that one can hardly avoid noticing that some simple-looking integrals remain tantalizingly excluded from those cheat-lists of standard integrals).

    So complex numbers and rational functions are both examples of that (largely invisible and unacknowledged) mathematical urge to achieve “closedness” or “completeness” – and the associated perennial challenge: What can happen when we are forced to open the door to some imaginary New World?

    1. Thanks, Tony. I think Anand accepted the argument. I also think BiB was being sarcastic, but they made it impossible to be sure.

      1. Yes, I was engaging in the lowest form of wit. Although I think the point of my question might have been missed (no big deal).

    2. Thanks again, Tony. I agree, that it is the coherence and depth of topics that matters. As it stands in Victoria, complex numbers are a Godawful mess, not least because Cambridge, which amounts to the paper of record, does such a disastrous job of it.

      As for integration, I’ve been yelling for a good decade that it is utter madness to not teach substitution in the Methods treatment. Anybody with any sense realises this; any sensible MM teacher will teach substituion anyway. But VCAA hasn’t budged. The reader can complete the syllogism.

      1. It’s not uncommon to see questions of the form \displaystyle \int \frac{f'(x)}{f(x)} dx on the MAV Maths Methods Trial exam 1 (*). The most recent example is 2020 Question 3: Solve \displaystyle \int_0^a \frac{x}{x^2 + 4} dx = 3 for \displaystyle a, where \displaystyle a is a real constant.

        I queried this sort of question with the MAV back in 2011 (**) and got an unsatisfactory response. I queried VCAA and was eventually told that it could be considered to be part of the course but it would be very unlikely to ever appear on an exam.

        * NOT as part of an ‘integration by recognition’.

        ** The 2011 MAV Maths Methods Trial Exam 1 Question 1 part (b) required finding an integral of the form \displaystyle \int \frac{f'(x)}{f(x)} dx.

        1. I am prepared to be convinced otherwise, but I have yet to be convinced that integration “by recognition” is a thing, outside of Methods exams of course.

          Saying that it appears in reduction formulas in Specialist is a bridge too far for me – that is integration by parts, which is well beyond the Methods syllabus.

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