VCAA’s Mathematical Flea Circus, Part One

To people not involved in Mathematical Methods it is difficult to convey the awfulness of the subject. Even regular readers of this blog, who will be somewhat aware of our million or so posts, cannot properly appreciate Methods’ unrelenting shallowness and pedantry and clunkiness and CASiness. The subject, to the extent there is one, is infested with fleas.

For non-Methods people to appreciate this they would need to be shackled to a desk, to work through the exams and the SACs and the textbook exercises. This would take time and it would violate several human rights laws. As a quicker, legal and somewhat less tortuous approximation, people can read the exam reports.

The general advice in the Methods exam reports indicate to teachers and students the systemic issues, the things to watch out for in the coming year. (Well, in the current year, given it’s April, but whatever.) I had initially intended this post to quickly highlight a few irritants in the 2023 Exam 1 report (Word, idiots), after which I would have focussed upon the more annoying Exam 2 report (Word, idiots). The more that I read the Exam 1 report, however, the more irritating became the irritants. So, I’ve decided to split the discussion into two; this post will be on the Exam 1 Report, with the following post on the Exam 2 Report.

The Exam 1 Advice to Students contains some sensible, and sensibly worded, advice. The irritation begins in the third paragraph:

Students need to familiarise themselves with the language of mathematics. Terms such as ‘verify’, ‘show that’ and ‘hence, show’ have particular meanings, and students should ensure that they understand what approach is required to respond appropriately to the question.

Well, no. What is of concern is not the language of mathematics but the language of VCE mathematics, which is its own weird thing. Of course students need to know VCE language, whatever it is, but it is not students’ fault that the VCE meaning of “show that” bears little resemblance to either regular English usage or general mathematical usage. It is somewhat reasonable to whine if students have been told repeatedly and still don’t know what you mean, but the whining loses a lot of its force if it lacks any proper foundation.

Continuing with such concerns,

Mathematical notation is a precise language and students should pay attention to its use

Mathematical notation is not a language; it is the collection of symbols used to assist in conveying messages composed in a language. Not a big deal, but there is no little irony in the use of such imprecise language to whine about the need for precise language.

In any case, the report then highlights four specific and general examples of students’ imprecise use of notation:

… students should use the names of functions as given in the stem of the question. If the question required f'(x), it was not acceptable to write ‘y = \boldsymbol{\lq}; nor, when asked to determine the maximum area, was it appropriate to name the derivative function A'(x) when the variable being used was k. The proper representation of a logarithm has the base as a subscript, for example logex. An integral or integration statement must be accompanied by a ‘dx’, as in ∫f(x) dx.

None of these complaints is properly a big deal and all are best thought of as invalid. Let’s consider them one by one.

To begin, it is indeed inappropriate to rename f'(x), or any function, as y without announcing the fact. But it is so common and typically so useful to label the function of interest as y, the lack of announcement will almost never cause confusion. It is also unclear to which question(s) this y = f'(x) objection is referring. The only question explicitly referring to f’ in a required answer is Q1(b), which asks for f'(π/4) and which seems an unlikely candidate for such an objection. Q9(b) requires f'(x) along the way, but it also appears to be an odd basis for the objection, particularly given the larger issues with the that question and its reporting, to which we shall return. Finally, since we’re here and we’re nitpicking, the quotation marks on the y = f'(x) whine are stuffed up.

Secondly, let’s deal with the dx in integrals, a perennial source of examiners’ whining. Yes, as a matter of style an integral should either be written as ∫f(x) dx or (better) ∫f, but it really really really doesn’t matter. The dx is nothing more than an historical artefact, the inclusion of which (in single variable calculus) adds absolutely nothing to the meaning. The report’s demand that an integral “must” be accompanied by a dx is precious and silly. It will still be precious and silly next year, when VCAA whine about it again.

Thirdly, the natural logarithm is so overwhelmingly the most important and most frequently employed base, it is very standard for it to be written as a subscript-free log x. VCAA can choose to operate differently, perhaps for questionable “we also have other logs” reasons, but VCE VCAA should be aware that in doing so they are, as is too often the case, the odd man out.

Finally, let’s consider the most plausibly valid whine, students writing A'(x) for the derivative of A, a function of k. Which I would argue generally is not a big deal. Specifically here, this whine should be viewed in context, which requires a lengthy digression.

Question 9 is a Methods Special, a thoroughly idiotic and badly flawed question about two walking tracks, given as the graphs of functions f(x) and g(x), and which are drawn as tangent at a point, P. The point P, however, is not defined in the question prompt. Consequently, when asked in Part (b) to “verify” that f(x) and g(x) have a “turning point” at P, and to also determine the “co-ordinates” of P, it is understandable if students were plenty confused. The exam report whines,

It was not sufficient to assume by inspection that the tracks met at P.

Yes it was. If VCAA is too incompetent to define P then students have every right to define the point as best they can on the information provided, which includes that it is (one point) where the tracks intersect.

Some students misinterpreted the question and solved f(x) = g(x).

Gee, I wonder why.

So, how were students to know where P is, or what P is? The report solution never determines that the graphs of f and g ever meet, that “P” even exists. The report first computes f'(x), and then determines f'(2) = 0, but without a single clue of how they chose to substitute x = 2. Later, when the simpler function g is considered, we obtain some way-too-late insight. That is, P is effectively defined as the common stationary point, and the problem is to determine that such a point exists and where it is. Which is not properly stated or remotely clear in either the exam report or, more importantly, the exam question.

Then, to top it off,

… they stopped short of showing that the point of intersection was a turning point for both curves.

As did the solution in the exam report, which, as well as being ass-backwards and incomplete, appears to conflate “turning point” with “stationary point”.

Question 9(b) was clearly a debacle, salted to be much more painful by the reports’ self-righteous twaddle. But I digress.

We are here for 9(c), which concerns a triangular shaped amusement park. The coordinates of the park are to be (0,0), (k,0) and (k, g(k)), for God knows what reason, and the problem is to maximise the area of the park. So the area A(k) is half base times height and away you go. Except, some students then wrote A'(x) = 0 and hence VCAA’s whine.

Yes, students should not define A(k) and then write A'(x) for the derivative, and they deserve about 10% of the blame for doing so. Why? Because the variable k serves absolutely no purpose whatsoever. The question would be identical if the coordinates were written as (0,0), (x,0) and (x, g(x)). The solution in the exam report itself does not once refer to k. It is hardly difficult to understand if students are distracted and confused by the introduction of an entirely gratuitous variable, and they should not then be blamed for doing so.

This post is meant to be about the nitpicking, about the fleas, but the point has to be made. There are many reprehensible aspects of VCAA’s conduct but by far the most reprehensible is when VCAA stuffs up exam questions and then blames the students for the resulting confusion.

Returning to the fleas, I will make the point as clearly as I can: if I were still lecturing first year university classes I wouldn’t lose a moment’s sleep over any of the four highlighted notational issues above, even in clearer and more egregious cases. On an assignment, I might take off a meaningless micro-mark, as a flag to be concerned with accuracy and proper style. On an exam, I would not even consider deducting marks for such matters.* I would be much more concerned with whether the students got the maths right than whether they had used the preferred pronouns. The latter being the evident depth of VCAA’s concerns.

*) This is a common but not universal policy. Mathematicians tend to be more reasonably focussed on substance, but plenty of university lecturers are anal retentive assholes. 

UPDATE (25/04/24)

I’ve added the graphics for Q9.

UPDATE (25/04/24)

For those unclear on how bad Q9 and the report “solution” are, consider the functions \boldsymbol{f(x) = 2x^3-4x^2+2x+1} and \boldsymbol{g(x) = -x^2 + 2x+1}. Do the graphs of these two functions contain a “point P”? If so, where? If not, why not? Why does or does not the report “solution” also apply for these two functions?

5 Replies to “VCAA’s Mathematical Flea Circus, Part One”

  1. Agree with all of the above and want to add the following: given the nature of VCE, where individual marks can be REALLY IMPORTANT, these things REALLY MATTER to students.

    If the VCE wants to reward the students who are the best at guessing answers to multiple choice or who have teachers who know how to game the SAC system to their advantage, then fine. Something tells me this is not the case though.

    If the VCE wants to reward students who have LEARNED a lot, then a few tweaks here and there are not going to cut it.

  2. VCAA’s whining about 9b is quite ironic, considering they gave me full marks when I failed to verify that the two functions’ turning points had the same y coordinate — all I showed was that the two functions had a turning point with the same x coordinate, and I was kicking myself almost immediately after the exam because I was certain I would lose marks for it.

    Alas, VCAA’s incompetence swings both ways on occasion.

    1. Thanks, Bugle. The entire thing is nuts. I should have done Q9 and the report as a WitCH, but didn’t think to change direction.

  3. “…but VCE should be aware that in doing so they are, as is too often the case, the odd man out.”

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