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.

4 Replies to “Inverted Logic”

  1. Ignoring for the moment the sheer crap of even thinking to find the derivative of y=log(x) using an inverse function and not the actual definition of the logarithm function as the area underneath a y=1/x graph (oops, guess I failed to ignore something, don’t tell VCAA…) this type of multiple choice question feels just a bit too familiar for VCAA Methods exams.

    Although the sample size (n=2) is too small to draw a meaningful conclusion, the NHT exams do seem to require more leaps of logic (thankfully mostly valid) than what Victorian Methods students have become accustomed to. This is perhaps a reflection on the superiority of Mathematical education of the intended candidates more than anything else, but I suspect continuing this rant would involve preaching to the choir.

    1. Thanks, Number 8. Certainly, defining log(x) as an integral is easier, although I can understand the reasons to first (pseudo)define ex, with log(x) as the inverse. What I can’t understand is the failure to be more explicit and systematic about inverses.

      I haven’t looked carefully at the NHT exams, though I can believe they’re at least somewhat deeper than the Victorian VCE exams. If true, it would certainly be interesting to know why.

  2. What I really want to know is who gives the guidance to the setters of the NHT papers and if this is a different person to the November papers. In the IB world there was one chief examiner for the subject but several deputies. The exams (2 May and 1 November) were all comparable in difficulty but there was quite a difference between timezone 1 and timezone 2 for the May exams. I will leave it to others to ponder why.

    I suspect the opposite is happening with VCE NHT, namely that the papers are deliberately different.

    While on the subject of neing systematic about inverses (which I wasn’t but you were) I do sometimes wonder if the need doesn’t run much deeper, into functions more generally and their treatment, not just inverse functions. The jump from Year 10 to 11 seems wide in this regard and perhaps without good reason?

    1. VCAA’s issues with quality, consistency and transparency know no bounds.

      Regarding functions, inverses and so forth, I’d actually prefer *less* focus on functions and function notation. Most of the function content in VCE is fetishistic, and pointless and needlessly confusing for secondary students. I may do a post on this. But the main issue for this post is that the inverse material, and all material, should be presented in a clear and considered manner, which is not the case. And, the material should be examined in the form in it is presented and learned, which is also not the case.

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