Inferiority Complex

This one is long, a real Gish gallop. Question 4, Part 2 from the 2017 VCE Specialist Mathematics Exam 2 is a mess. The Examiners’ Report is, predictably, worse.

Part (a) of Question 4 is routine, requiring students to express {-2-2\sqrt{3}i} in polar form. One wonders how a quarter of the students could muck up this easy 1-mark question, but the question is fine.

The issues begin with 4(b), for which students are required to

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

The question can be answered with an easy application of completing the square or the quadratic formula. So, why did almost half of the students get it wrong? Were so many students really so clueless? Perhaps, but there is good reason to suspect a different source of the cluelessness.

The Examiners’ Report indicates three general issues with students’ answers. First,

students confused factors with solutions or did not proceed beyond factorising the quadratic.

Maybe the students were confused, but maybe not. Maybe some students simply thought that, once having factorised the quadratic, the microstep to then write “Therefore z = …”, to note the roots written on the exam in front of them, was too trivial in response to a 1 mark question.

Second, some students reportedly erred by

not showing key steps in their solution.

Really? The Report includes the following calculation as a sample solution:

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

Was this whole tedious, snail-paced computation required for one measly mark? It’s impossible to tell, but the Report remarks generally on ‘show that’ questions that

all steps that led to the given result needed to be clearly and logically set out.

As we have noted previously, demanding “all steps” is both meaningless and utterly mad. For a year 12 advanced mathematics student the identification of the roots is pretty much immediate and a single written step should suffice. True, in 4(b) students are instructed to “show” stuff, but it’s hardly the students’ fault that what they were instructed to show is pretty trivial.

Third, and by far the most ridiculous,

some students did not correctly follow the ‘show that’ instruction … by [instead] solely verifying the solutions given by substitution.

Bullshit.

VCAA examiners love to worry that word “show”. In true Princess Bride fashion, however, the word does not mean what they think it means.

There is nothing in standard English usage nor in standard mathematical usage, nor in at least occasional VCE usage (see Q2(a)), that would distinguish “show” from “prove” in this context. And, for 4(b) above, substitution of the given values into the quadratic is a perfectly valid method of proving that the roots are as indicated.

It appears that VCE has a special non-English code, in which “show” has a narrower meaning, akin to “derive“. This cannot alter the fact that the VCE examiners’ use of the word is linguistic and mathematical crap. It also cannot alter the fact that students being penalised for not following this linguistic and mathematical crap is pedagogical and mathematical crap.

Of course all the nonsense of 4(b) could have been avoided simply by asking the students to find the roots. The examiners declined to do so, however, probably because this would have violated VCAA’s policy of avoiding asking any mathematical question with some depth or difficulty or further consequence. The result is a question amounting to no more than an infantile and infantilising ritual, penalising any student with the mathematical common sense to answer with the appropriate “well, duh”.

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Onwards we trek to 4(c):

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

Less than a third of students scored the mark for this question, and the Report notes that

Misunderstanding of the question was apparent in student responses. Many attempts at solutions were not expressed in terms of  {\color{blue} \boldsymbol{2 -2\sqrt{3}i}} as required.

Funny that. The examiners pose a question that borders on the meaningless and somehow this creates a sea of misunderstanding. Who would’ve guessed?

4(c) makes little more sense than to ask someone to write 3 in terms of 7. Given any two numbers there’s a zillion ways to “express” one number “in terms of” the other, as in 3 = 7 – 4 or whatever. Without further qualification or some accepted convention, without some agreed upon definition of “expressed in terms of”, any expression is just as valid as any other.

What was expected in 4(c)? To approach the question cleanly we can first set w = 2 - 2\sqrt{3}i, as the examiners could have and should have and did not. Then, the intended answers were -w and -\overline{w}.

These expressions for the roots are simple and natural, but even if one accepts a waffly interpretation of 4(c) that somehow requires “simple” solutions, there are plenty of other possible answers. The expressions w-4 and \overline{w-4} and w^2/4 and w^4/|w|^3 are all reasonable and natural, but nothing in the Examiners’ Report suggests that these or similar answers were accepted. If not, that is a very nasty cherry on top of an incredibly silly question.

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The pain now temporarily lessens (though the worst is yet to come). 4(d) asks for students to show that the relation |z| = |z - (2-2\sqrt{3}i)| has the cartesian form x-\sqrt{3}y - 4 = 0, and in 4(e) students are asked to draw this line on an Argand diagram, together with the roots of the above quadratic.

These questions are routine and ok, though 4(d) is weirdly aimless, the line obtained playing no role in the final parts of Q4. The Examiners’ Report also notes condescendingly that “the ‘show that’ instruction was generally followed”. Yes, people do tend to follow the intended road if there’s only one road.

The final part, 4(g), is also standard, requiring students to find the area of the major segment of the circle |z| = 4 cut off by the line through the roots of the quadratic. The question is straight-forward, the only real trick being to ignore the weird line from 4(d) and 4(e).

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Finally, the debacle of 4(f):

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

The Report notes that

This question caused significant difficulty for students.

That’s hilarious understatement given that 99% of students scored 0/1 on the question. The further statements acknowledging and explaining and apologising for the stuff-up are unfortunately non-existent.

So, what went wrong? The answer is both obvious and depressingly familiar: the exam question is essentially meaningless. Students failed to comprehend the question because it is close to incomprehensible.

The students are asked to write b in terms of a. However, similar to 4(c) above, there are many ways to do that and how one is able to do it depends upon the initial number a chosen. The line through the two roots has equation \operatorname{Re} z = x = -2. So then, for example, with a = -4 we have b = 0 and we can write b = a + 4 or b = 0 x a or whatever. If a = -5 then b = 1 and we can write b = -a – 4, and so on.

Anything of this nature is a reasonable response to the exam question as written and none of it resembles the answer in the Report. Instead, what was expected was for students to consider all complex numbers a – except those on the line itself – and to consider all associated complex b. That is, in appropriate but non-Specialist terminology, we want to determine b as a function f(a) of a, with the domain of f being most but not all of the complex plane.

With the question suitably clarified we can get down to work (none of which is indicated in the Report). Easiest is to write a = (-2+c) + di. Since b must be symmetrically placed about the line \operatorname{Re} z = -2, it follows that b = (-2-c) + di. Then b+2 = -c + di = -\overline{(a+2)}. This gives b = -2 - \overline{(a + 2)}, and finally

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

which is the answer indicated in the Examiners’ Report.

In principle 4(f) is a nice question, though 1 mark is pretty chintzy for the thought required. More importantly, the exam question as written bears only the slightest resemblance to the intended question, or to anything coherent, with only the slightest, inaccurate hint of the intended generality of a and b.

99% of 2017 Specialist students have a right to be pissed off.

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That’s it, we’re done. One more ridiculous VCE exam question, and one more ridiculously arrogant Report, unsullied by an ounce of self-reflection or remorse.

4 Replies to “Inferiority Complex”

  1. Couple of things Marty (adding to your case, not speaking against it):

    1. The 1% of students who got the correct answer (according to “meet the examiners” which is always hilarious if you leave any expectation of a sensible answer at the door) was apparently much closer to 0%, quite possibly below 0.1%.

    2. The double negative in the complex number is what would have thrown students who would have found arctan(sqrt(3)) and failed to account for the correct quadrant (for which you can thank calculators, at some length).

    3. one always wonders why “verify” is considered different to “show” in the minds of these examiners… I’m no expert, but I have studied logic at tertiary level (outside my Mathematics degree) and am in no doubt that “show that” as a phrase does imply a direction that the argument must follow. On this point though, there is enough precedent to not be caught-out as much. One just needs to remember that VCAA (as a whole, not just in this one subject) rewards memory over logic.

    1. Thanks, Number 8. In brief response:

      1) I’m not at all surprised.

      1)’ Did the “examiners” you “met” make any acknowledgment that the (possibly different) “examiners” totally stuffed things up? (Stop laughing.)

      2) Double negative or not, 25% is pretty woeful for such a simple question. Yes, I’m sure the calculators are involved. But, to steal from the NRA, calculators don’t rot people’s brains, people rot people’s brains.

      3) Sure, in principle Victorian teachers and students should know to interpret “show that” according to VCAA’s non-English code. In practice, it is poison, as evidenced by 4(b) above. And it’s not English.

      3)’ I have no idea what the examiners think “verify” means, but on Planet Earth it means to “check/demonstrate the truth of”, making it another synonym for “prove”. There’s some convention in mathematics, where “verify” tends to be used to refer to checking cases (a la substitution), but it’s definitely not a strict rule.

  2. To speak plainly about (1) I have met many examiners. Some avoided the questions put to them, others answered after giving the “this is not the official VCAA position” and others avoided the question, laughing it off and blaming others. An example of this would be the explanation provided to the Methods PDF which wasn’t a PDF (I believe your post was titled “the median is the message” or similar). In short, my experience with examiners falls into a few broad categories, with some overlap:

    1. They blame students for not knowing things or for being sloppy in their working (see just about every Specialist Paper 1 examiners report ever written).
    2. They blame students for not following the instructions given in the question.
    3. They divert blame from themselves by saying things such as “a variety of responses were accepted” without giving any indication of what this really means.

    The sessions ARE worth going to, at worst it is a day away from school, reunions with other teachers, both of which are nice; but you really need to leave your expectations at the door. Apparently the March sessions are better than the December sessions (even though the examiners reports are not always published by March… which I really don’t understand).

    The MAV does their best to make these sessions valuable, but to imagine for one moment that this satisfies the need for transparency of the VCAA is misguided.

    1. I have been to a number of Meet the Examiners sessions. They are clearly as valuable for a VCE teacher as they are nauseating for a mathematician.

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