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.)
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 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 is all of , 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 and , 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 or whatever, we’re told that is “a function with the same rule as ”. This is ridiculous and meaningless. It is ridiculous because we never did anything with in the first place, and so it would have been a hell of lot clearer to have simply begun the damn question with on some unknown domain . It is meaningless because we cannot determine anything about the domain from the information provided. The point is, in VCE the composition is either defined (if the range 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.