WitCH 27: Uncomposed

Ah, so much crap …

Tons of nonsense to post on, and the Evil Mathologer is breathing down our neck. We’ll have (at least) three posts on last week’s Mathematical Methods exams. This one is by no means the worst to come, but it fits in with our previous WitCH, so let’s quickly get it going. It is from Exam 1. (No link yet, but the Study Design is here.)

Update (15/06/20)

The examination report (and exam) is out, so it’s time to wade into this swamp. Before doing so, we’ll note the number of students who sank; according to the examination report, the average score on this question was 0.14 + 0.09 + 0.14 ≈ 0.4 marks out of 4. Justified or not, students had absolutely no clue what to do. Now, into the swamp.

The main wrongness is in Part (b), but we’ll begin at the beginning: the very first sentence of Part (a) is a mess. Who on Earth writes

“The function f: R \to R, f(x)  is a polynomial function …”?

It’s like writing

“The Prime Minister Scott Morrison of Australia, Scott Morrison is a crap Prime Minister”.

Yes, you may properly want to emphasise that Scott Morrison is the Prime Minister of Australia, and he is crap, but that’s not the way to do it. This is nitpicking, of course, but there are two reasons to do so. The first reason is there is no reason not to: why forgive the gratuitously muddled wording of the very first sentence of an exam question? From these guys? Forget it. The second reason is that the only possible excuse for this ridiculous wording is to emphasise that the domain of f is all of R, which turns out to be entirely pointless.

Now, to Part (a) proper. This may come as a surprise to the VCAA overlords, but functions do not have “rules”, at least not unique ones.  The functions f(x) = -4x^2\left(x^2 - 1\right) and h(x) = 4x^2-4x^4, for example, are the exact same function. Yes, this is annoying, but we’re sorry, that’s the, um, rule. Again this is nitpicking and, again, we have no sympathy for the overlords. If they insist that a function should be regarded as a suitable set of ordered pairs then they have to live with that choice. Yes, eventually ordered pairs are the precise and useful way to define functions, but in school it’s pretty much just a pedantic pain in the ass.

To be fair, we’re not convinced that the clumsiness in the wording of Part (a) contributed significantly to students doing poorly. That is presumably much more do to with the corruption of students’ arithmetic and algebraic skills, the inevitable consequence of VCAA and ACARA calculatoring the curriculum to death.

On to Part (b), where, having found f(x) = -4x^2\left(x^2 - 1\right) or whatever, we’re told that g is “a function with the same rule as f”. This is ridiculous and meaningless. It is ridiculous because we never did anything with f in the first place, and so it would have been a hell of lot clearer to have simply begun the damn question with g on some unknown domain E. It is meaningless because we cannot determine anything about the domain E from the information provided. The point is, in VCE the composition \log(g(x)) is either defined (if the range g(E) is wholly contained in the positive reals), or it isn’t (otherwise). End of story.  Which means that in VCE the concept of “maximal domain” makes no sense for a composition. Which means Part (b) makes no sense whatsoever. Yes, this is annoying, but we’re sorry, that’s the, um, rule.

Finally, to Part (c). Taking (b) as intended rather than written, Part (c) is ok, just some who-really-cares domain trickery.

In summary, the question is attempting and failing to test little more than a pedantic attention to boring detail, a test that the examiners themselves are demonstrably incapable of passing.

8 Replies to “WitCH 27: Uncomposed”

  1. “Find the rule of f”. Why does this annoy me so much?

    I suppose it’s because I’d much rather them just ask for the roots with multiplicity.

    What is the point of g? Am I missing something with this “rule” idea? Are f and g not the same thing?

    What is the point of subtracting the second logarithm? The method to solve the problem is the same.

    Is “maximal domain” ever actually defined for these students? I really doubt it.

    Why is the last part worth 2 marks? If they just want a statement, it is going to be all or nothing.

    Argh these questions are infuriating!

  2. Seconded. Does “the rule” include the domain or not??? (picture me yelling at a cloud similar to Abe Simpson).

    Because the wording of part b implies “the rule” does not include the domain. In which case why didn’t they say “the equation”??? (keeps yelling at cloud)

    And the 2 marks for the final part is usually (not always) a clue that there are two parts to the answer…

    1. Right, that’s why I’m confused with the two marks. Because how could you mark it wrong if a student just wrote “(-\infty, \log 4]”? (Assuming my working is right.)

  3. Sloppy in the extreme. The question already says that f is defined by a fourth degree polynomial. So why not, in (a), simply ask for the polynomial? And what do they mean by g “having the same rule as f”? This makes no sense at all. If what they mean is that f will from now on have a restricted domain, that’s fine. I don’t see the point of introducing a new function g. The only extra variable needed here is the polynomial p(x) – which is the one thing missing.

    But wait – there’s more! Given that f = x^2(1-x^2), an astute student could try to simplify the logarithms in the definition of h as log(x^2(1-x^2)/(x^3+x^2)) = log(1-x) which is defined for x less than 1. But the examiners will leap onto this with a cry of delight like starving hyenas attacking a baby gazelle and say that log(f(x)) is only defined for x strictly between -1 and 1, except for x = 0. So as far as I can see this question is designed more to trick students – the learning outcomes seem pretty damned blurred to me.

    It’s hardly a wonder students are turned off mathematics.

    1. Yes. Another “designed to trick students” aspect of this is that actually f(x) = 4x^2(1-x^2), that coefficient of 4 seems quite nasty given that the plot in the question itself is likely for x -> x^2(1-x^2), and the “trick” that the coefficient of 4 is present only becomes apparent if you pay close attention to the coordinates of the local maxima. (Note that for the simpler function the height of these humps should be 1/4, and not 1. Very underhanded how they changed the scale on the axes.)

      Why do the examiners hate mathematics?

  4. Thanks everyone for your comments. The underlying issues, of course, are what a “function” is, and what it means to “compose” two functions. One can be very formal about this, or very informal. The task here is to figure out what these two concepts mean in VCE (which means consulting the study design and exam reports and authoritative (?) texts), whether that meaning is sensible, and to what extent the exam question is clear and faithful to that (non)sense.

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