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.

An Untrustworthy T-Shirt

Among the many Australian mathematics organisations that are making matters worse rather than better, the Australian Mathematics Trust must not be included. AMT is great, a rare beacon of hope. A beacon somewhat dimmed, it is true, by the fact that the AMT guys have an average age of about 95. Still, any beacon in a storm, or whatever.

Apart from their wonderful work on the Australian Mathematics Competition and their associated endeavours, AMT sells excellent books on problem-solving, as well as some very cool (and some very uncool) mathematical t-shirts. One shirt, however, is particularly eye-catching:

This shirt celebrates Norwegian mathematician Niels Henrik Abel and his 1824 proof of the non-existence of a solution in radicals of the general quintic equation. That is, in contrast to the quadratic formula, and to the cubic and the quartic analogues, there does not exist a quintic formula. It’s a pretty shirt.

It’s also a pretty weird shirt. AMT’s blurb reads

This T Shirt features Abel’s proof that polynomials of order five or higher cannot be solved algebraically.

Stylewise, it is probably a good thing that Abel’s “proof” doesn’t actually appear on the shirt. What is not so good is the sloppy statement of what Abel supposedly proved.

Abel didn’t prove that “[polynomial equations] of order five or higher cannot be solved algebraically”. What he proved was that such equations could not generally be solved, that there’s no general quintic formula. In particular, Abel’s theorem does not automatically rule out any particular equation from being solved in terms of radicals. As a very simple example, the quintic equation

    \[\boldsymbol{x^5-2x=0}\]

is easily shown to have the solutions \boldsymbol{0,\pm\sqrt[4]2,\pm\sqrt[4]2i}.

Which brings us back to AMT’s t-shirt. Why on Earth would one choose to illustrate the general unsolvability of the quintic with a specific equation that is solvable, and very obviously so?

Even good guys can screw up, of course. It’s preferable, however, not to emblazon one’s screw-up on a t-shirt.

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.

WitCH 4

Well, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle are still there to ponder. A reminder, it’s up to you, Dear Readers, to identify the crap. There’s so much crap, however, and so little time. So, it’s onwards and downwards we go.

Our new WitCH, courtesy of New Century Mathematics, Year 10 (2014), is inspired by the Evil Mathologer‘s latest video. The video and the accompanying articles took the Evil Mathologer (and his evil sidekickhundreds of hours to complete. By comparison, one can ponder how many minutes were spent on the following diagram:

OK, Dear Readers, time to get to work. Grab yourself a coffee and see if you can itemise all that is wrong with the above.

Update

Well done, craphunters. Here’s a summary, with a couple craps not raised in the comments below:

  • In the ratio a/b, the nature of a and b is left unspecified.
  • The disconnected bubbles within the diagram misleadingly suggest the existence of other, unspecified real numbers.
  • The rational bubbles overlap, since any integer can also be represented as a terminating decimal and as a recurring decimal. For example, 1 = 1.0 = 0.999… (See here and here and here for semi-standard definitions.) Similarly, any terminating decimal can also be represented as a recurring decimal.
  • A percentage need not be terminating, or even rational. For example, π% is a perfectly fine percentage.
  • Whatever “surd” means, the listed examples suggest way too restrictive a definition. Even if surd is intended to refer to “all rooty things”, this will not include all algebraic numbers, which is what is required here.
  • The expression “have no pattern and are non-recurring” is largely meaningless. To the extent it is meaningful it should be attached to all irrational numbers, not just transcendentals.
  • The decimal examples of transcendentals are meaningless.

VTACKY

It’s been a long, long time. Alas, we’ve been kept way too busy by the Evil Mathologer, as well as some edu-idiots, who shall remain nameless but not unknown. Anyway, with luck normal transmission has now resumed. There’s a big, big backlog of mathematical crap to get through.

To begin, there’s a shocking news story that has just appeared, about schools posting “wrong Year 12 test scores” and being ordered to remove them by the Victorian Tertiary Admissions Centre. Naughty, naughty schools!

Perhaps.

The report tells of how two Victorian private schools had conflated Victoria’s VCE subject scores and International Baccalaureate subject scores. The schools had equated the locally lesser known IB scores of 6 or 7 to the more familiar VCE ATAR of 40+, to then arrive at a combined percentage of such scores. Reportedly, this raised the percentage of “40+” student scores at the one school from around 10% for VCE alone to around 25% for combined VCE-IB, with a comparable raise for the other school. More generally, it was reported that about a third of IB students score a 6 or 7, whereas only about one in eleven VCE scores are 40+.

On the face of it, it seems likely that the local IB organisation that had suggested Victorian schools use the 6+ = 40+ equation got it wrong. That organisation is supposedly reviewing the comparison and the two schools have removed the combined percentages from their websites.

There are, however, a few pertinent observations to be made:

None of the sense or substance of the above is hinted at in the schools-bad/VTAC-good news report.

Of course the underlying issue is tricky. Though the IBO tries very hard to compare IB scores, it is obviously very difficult to compare IB apples to VCE oranges. We have no idea whether or how one could create a fair and useful comparison. We do know, however, that accepting VTAC’s cocky sanctimony as the last word on this subject, or any subject, would be foolish.

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?

 

WitCH 3

First, a quick note about these WitCHes. Any reasonable mathematician looking at such text extracts would immediately see the mathematical flaw(s) and would wonder how such half-baked nonsense could be published. We are aware, however, that for teachers and students, or at least Australian teachers and students, it is not nearly so easy. Since school mathematics is completely immersed in semi-sense, it is difficult to know the rules of the game. It is also perhaps difficult to know how a tentative suggestion might be received on a snarky blog such as this. We’ll just say, though we have little time for don’t-know-as-much-as-they-think textbook writers, we’re very patient with teachers and students who are honestly trying to figure out what’s what.

Now onto WitCH 3, which follows on from WitCH 2, coming from the same chapter of Cambridge’s Specialist Mathematics VCE Units 3 & 4 (2018).* The extract is below, and please post your thoughts in the comments. Also a reminder, WitCH 1 and WitCH 2 are still there, awaiting proper resolution. Enjoy.

* Cambridge is a good target, since they are the most respected of standard Australian school texts. We will, however, be whacking other publishers, and we’re always open to suggestion. Just email if you have a good WitCH candidate, or crap of any kind you wish to be attacked.

Update (06/02/19)

The above excerpt is indicative of the text’s entire chapter on complex numbers. It is such remarkably poor exposition, the foundations so understated and the direction so aimless, it is almost impossible to find one’s way back to sensible discussion.

Here is a natural framework for a Year 12 topic on complex numbers:

  • First, one introduces a new number \boldsymbol i for which \boldsymbol i^2=-1.
  • One then defines complex numbers, and introduces the fundamental operations of addition and multiplication.
  • One then at least states, and hopefully proves, the familiar algebraic properties for complex numbers, i.e. the field laws, \boldsymbol {u(z + w) = uz + uw} and so forth. All these properties are obvious or straight-forward to prove, except for the existence of multiplicative inverses; one has to prove that given any non-zero complex \boldsymbol z there is another complex \boldsymbol w with \boldsymbol {zw = 1}.
  • That is the basic complex algebra sorted, and then one can tidy up. This includes the definition of division \boldsymbol {\frac{z}{w} = zw^{-1} = w^{-1}z}, noting the essential role played by commutativity of multiplication.
  • Then, comes the geometry of complex numbers, beginning with the definition and algebraic properties of the conjugate \boldsymbol {\overline{z}} and modulus \boldsymbol {|\boldsymbol z|},  the interpretation of these quantities in terms of the complex plane, and polar form.
  • Finally, the algebra and geometry of complex numbers are related: the parallelogram interpretation of addition, the trigonometric-polar interpretation of multiplication, roots of complex numbers and so forth.

Must complex numbers be taught in this manner and in this order? No and yes. One obvious variation is to include a formal definition of a complex number \boldsymbol {z = a + bi} as an ordered pair \boldsymbol {(a,b)}; as Damo remarks below, this is done as an asterisked section in Fitzpatrick and Galbraith. Though unnervingly abstract, the formal definition has the non-trivial advantage of reinforcing, almost demanding, the interpretation of complex numbers as points in the complex plane. More generally, one can emphasise more or less of the theoretical underpinnings and, to an extent, change the ordering.

But, one can only change the ordering and discard the theory so much, and no more. Complex numbers are new algebraic objects, and defining and clarifying the algebra is critical, and this fundamentally precedes the geometry.

What is the Cambridge order? The text starts off well, introducing \boldsymbol i with \boldsymbol {i^2  = -1},  and then immediately goes off the rails by declaring that \boldsymbol {i  = \sqrt{-1}}. Then, in brief, the text includes:

(a) an invalid treatment of the square roots of negative numbers;

(b) complex addition stated, presumably defined, with the inverse \boldsymbol {-z} introduced but not named;

(c) complex subtraction, followed by an almost invisible statement of the relevant field laws, none of which are proved or assigned as exercises;

(d) scalar multiplication;

(d) the complex plane and “the representation of the basic operations on complex numbers”;

(e) complex multiplication defined, with an almost invisible statement of field laws, none of which are proved or assigned as exercises, and with no mention of the question of multiplicative inverses;

(f) the geometry of multiplication by \boldsymbol i;

(g) the modulus of a complex number defined, with algebraic properties (including {\boldsymbol {|\frac{z}{w}| = \frac{|z|}{|w|}}) stated and assigned as exercises;

(h) the conjugate of a complex number defined, with algebraic properties stated and either proved or assigned as exercises.

(h) Finally, as excerpted above, it’s on to reciprocals of complex numbers, multiplicative inverses in terms of modulus and conjugate, and division.

(i) This is followed by sections on polar form, de Moivre’s theorem and so forth;

(j) CAS garbage is, of course, interspersed throughout. (Which is not all Cambridge’s fault, but the text is no less ugly for that.)

At no stage in the text’s exposition is there any visible concern for emphasising or clarifying foundations, or for following a natural mathematical progression. There is too seldom an indication of what is being defined or assumed or proved.

What is the point? Yes, one can easily be overly theoretical on this topic, but this is Year 12 Specialist Mathematics. It is supposed to be special. The students have already been introduced to complex numbers in Year 11 Specialist. Indeed, much of the complex material in the Year 11 Cambridge text is repeated verbatim in the Year 12 text. Why bother? The students have already been exposed to the nuts and bolts, so why not approach the subject with some mathematical integrity, rather than just cutting and pasting aimless, half-baked nonsense?

Now, finally and briefly, some specific comments on the specific nonsense excerpted above.

  • division of complex numbers has already appeared in the text, in the list of (unproved) properties of the modulus.
  • the algebraic manipulation of \boldsymbol {\frac1{a+bi}} is unfamiliar and unmotivated and, as is admitted way too late, is undefined. There is a place for such “let’s see” calculations – what mathematicians refer to as formal calculations –  but they have to be framed and be motivated much more carefully.
  • There is no need here for a “let’s see” calculation. The critical and simple observation is that \boldsymbol {(a + bi)(a-bi) = a^2 + b^2} is real. It is then a short step to realise and to prove that \boldsymbol {\frac{a}{a^2 + b^2} - \frac{bi}{a^2 + b^2}} acts as, and thus is, the multiplicative inverse of \boldsymbol {a + bi}.
  • Having finally admitted that \boldsymbol {\frac1{a + bi}} has not been defined, the text goes on to not define it again. The text states the multiplicative inverse of \boldsymbol z, but it is not clear whether this statement amounts to a definition or a conclusion.
  • Division of complex numbers is then defined with needless subscripts and, more importantly, with no mention of the fundamental role of commutativity of multiplication.
  • Throughout, the use of conjugate and modulus is muddying rather than clarifying.
  • At no stage is it made clear why \boldsymbol {\frac1{a + bi}} makes sense in contrast to, for example, the non-sense of \boldsymbol {\frac1{M}} for a matrix.