MAV’s Dangerous Inflection

This post concerns a question on the 2019 VCE Specialist Mathematics Exam 2 and, in particular, the solution and commentary for that question available through the Mathematical Association of Victoria. As we document below, a significant part of what MAV has written on this question is confused, self-contradictory and tendentious. Thus, noting the semi-official status of MAV solutions, that these solutions play a significant role in MAV’s Meet the Assessors events, and are quite possibly written by VCE assessors, there are some troubling implications. Question 3, Section B on Exam 2 is a differential equations problem, with two independent parts. Part (a) is a routine (and pretty nice) question on exponential growth and decay.* Part (b), which is our concern, considers the differential equation

    \[\boldsymbol{\color{blue}\frac{{\rm d}Q}{{\rm d}t\ } = e^{t-Q}}\,,\]

for t ≥ 0, along with the initial condition

    \[\boldsymbol{\color{blue}Q(0) =1}\,.\]

The differential equation is separable, and parts (i) and (ii) of the question, worth a total of 3 marks, asks to set up the separation and use this to show the solution of the initial value problem is

    \[\boldsymbol{\color{blue}Q =\log_e\hspace{-1pt} \left(e^t + e -1\right)}\,.\]

Part (iii), worth 2 marks, then asks to show that “the graph of Q as a function of t” has no inflection points.** Question 3(b) is contrived and bitsy and hand-holding, but not incoherent or wrong. So, pretty good by VCE standards. Unfortunately, the MAV solution and commentary to this problem is deeply problematic. The first MAV misstep, in (i), is to invert the derivative, giving

    \[\boldsymbol{\color{red}\frac{{\rm d}t\ }{{\rm d}Q } = e^{Q-t}}\,,\]

prior to separating variables. This is a very weird extra step to include since, not only is the step not required here, it is never required or helpful in solving separable equations. Its appearance here suggests a weak understanding of this standard technique. Worse is to come in (iii). Before considering MAV’s solution, however, it is perhaps worth indicating an approach to (iii) that may be unfamiliar to many teachers and students and, possibly, the assessors. If we are interested in the inflection points of Q,*** then we are interested in the second derivative of Q. The thing to note is we can naturally obtain an expression for Q” directly from the differential equation: we differentiate the equation using the chain rule, giving

    \[\boldsymbol{\color{magenta}Q'' = e^{t-Q}\left(1 - Q'\right)}\,.\]

Now, the exponential is never zero, and so if we can show Q’ < 1 then we’d have Q” > 0, ruling out inflection points. Such conclusions can sometimes be read off easily from the differential equation, but it does not seem to be the case here. However, an easy differentiation of the expression for Q derived in part (ii) gives

    \[\boldsymbol{\color{magenta}Q' =\frac{e^t}{e^t + e -1}}\,.\]

The numerator is clearly smaller than the denominator, proving that Q’ < 1, and we’re done. For a similar but distinct proof, one can use the differential equation to replace the Q’ in the expression for Q”, giving

    \[\boldsymbol{\color{magenta}Q'' = e^{t-Q}\left(1 - e^{t-Q}\right)}\,.\]

Again we want to show the second factor is positive, which amounts to showing Q > t. But that is easy to see from the expression for Q above (because the stuff in the log is greater than \boldsymbol{e^t}), and again we can conclude that Q has no inflection points. One might reasonably consider the details in the above proofs to be overly subtle for many or most VCE students. Nonetheless the approaches are natural, are typically more efficient (and are CAS-free), and any comprehensive solutions to the problem should at least mention the possibility. The MAV solutions make no mention of any such approach, simply making a CAS-driven beeline for Q” as an explicit function of t. Here are the contents of the MAV solution:

Part 1: A restatement of the equation for Q from part (ii), which is then followed by 

.˙.  \boldsymbol{ \color{red}\  \frac{{\rm d}^2Q }{{\rm d}t^2\ } = \frac{e^{t+1} -e^t}{\left(e^t + e -1\right)^2} } 

Part 2: A screenshot of the CAS input-output used to obtain the conclusion of Part 1.

Part 3: The statement   

Solving  .˙.  \boldsymbol{\color{red} \  \frac{{\rm d}^2Q }{{\rm d}t^2\ } = 0} gives no solution  

Part 4: A screenshot of the CAS input-output used to obtain the conclusion of Part 3.

Part 5: The half-sentence

We can see that \boldsymbol{\color{red}\frac{{\rm d}^2Q }{{\rm d}t^2\ } > 0} for all t,

Part 6: A labelled screenshot of a CAS-produced graph of Q”.

Part 7: The second half of the sentence,

so Q(t) has no points of inflection

This is a mess. The ordering of the information is poor and unexplained, making the unpunctuated sentences and part-sentences extremely difficult to read. Part 3 is so clumsy it’s funny. Much more important, the MAV “solution” makes little or no mathematical sense and is utterly useless as a guide to what the VCE might consider acceptable on an exam. True, the MAV solution is followed by a commentary specifically on the acceptability question. As we shall see, however, this commentary makes things worse. But before considering that commentary, let’s itemise the obvious questions raised by the MAV solution:
  • Is using CAS to calculate a second derivative on a “show that” exam question acceptable for VCE purposes?
  • Can a stated use of CAS to “show” there are no solutions to Q” = 0 suffice for VCE purposes? If not, what is the purpose of Parts 3 and 4 of the MAV solutions?
  • Does copying a CAS-produced graph of Q” suffice to “show” that Q” > 0 for VCE purposes?
  • If the answers to the above three questions differ, why do they differ?
Yes, of course these questions are primarily for the VCAA, but first things first. The MAV solution is followed by what is intended to be a clarifying comment:

Note that any reference to CAS producing ‘no solution’ to the second derivative equalling zero would NOT qualify for a mark in this ‘show that’ question. This is not sufficient. A sketch would also be required as would stating \boldsymbol{\color{red}e^t (e - 1) \neq 0} for all t.

These definitive-sounding statements are confusing and interesting, not least for their simple existence. Do these statements purport to be bankable pronouncements of VCAA assessors? If not, what is their status? In any case, given that pretty much every exam question demands that students and teachers read inscrutable VCAA tea leaves, why is it solely the solution to question 3(b) that is followed by such statements? The MAV commentary at least makes clear their answer to our second question above: quoting CAS is not sufficient to “show” that Q” = 0 has no solutions.  Unfortunately, the commentary raises more questions than it answers:
  • Parts 3 and 4 are “not sufficient”, but are they worth anything? If so, what are they worth and, in particular, what is the import of the word “also”? If not, then why not simply declare the parts irrelevant, in which case why include those parts in the solutions at all?
  • If, as claimed, it is “required” to state \boldsymbol{e^t(e-1)\neq 0} (which is indeed the key point of this approach and should be required), then why does the MAV solution not contain any such statement, nor even the factorisation that would naturally precede this statement?
  • Why is a solution “required” to include a sketch of Q”? If, in particular, a statement such as \boldsymbol{e^t(e-1)\neq 0} is “required”, or in any case is included, why would the latter not in and of itself suffice?
We wouldn’t begin to suggest answers to these questions, or our four earlier questions, and they are also not the main point here. The main point is that under no circumstances should such shoddy material be the basis of VCAA assessor presentations. If the material was also written by VCAA assessors, all the worse. Of course the underlying problem is not the quality or accuracy of solutions but, rather, the fundamental idiocy of incorporating CAS into proof questions. And for that the central villain is not the MAV but the VCAA, which has permitted their glorification of technology to completely destroy the appreciation of and the teaching of proof and reason. The MAV is not primarily responsible for this nonsense. The MAV is, however, responsible for publishing it, promoting it and profiting from it, none of which should be considered acceptable. The MAV needs to put serious thought into its unhealthily close relationship with the VCAA.   *) We might ask, however, who refers to “The growth and decay” of an exponential function? **) One might simply have referred to Q, but VCAA loves them their words. ***) Or, if preferred, the points of inflection of the graph of Q as a function of t.

Update (26/06/20)

The Examination Report is out and is basically ok; none of the nonsense and non sequiturs of the MAV solutions are included. The solution to (b)(iii) correctly focuses upon the factoring of Q”, although it needlessly worries about the sign of the denominator. There is no mention of the more natural approach to obtaining and analysing Q” but, given the question is treated by the VCAA and pretty much everyone as just another mindless exercise in pushing buttons, this is no surprise.

10 Replies to “MAV’s Dangerous Inflection”

  1. The most disheartening aspect of all this is that I don’t think the MAV will ever acknowledge that its inbred solutions are substandard (NB: This is my opinion, based on the evidence).

    ‘Written by VCAA Assessors’ is a marketing tool intended to imply quality and reliability, regardless of the reality. Unfortunately, many teachers buying this (and the Meet the Assessors) products are generally not interested in critically evaluating these products, being happy instead with near enough is good enough. (If this was not the case, then surely Meet the Assessor would not be as popular as it apparently is, and there would be a much stronger demand by teachers for better quality).

    It’s little wonder that many teachers (including myself) much prefer the FREE products readily available on-line. They may have similar issues but at least you’re not paying hundred of dollars, and the clear implication of semi-official status is not trumpeted. The free solutions are also willing to acknowledge mistakes and publish errata ….

    1. Thanks, JF. Yes, it’s difficult to imagine the MAV or VCAA ever publicly admitting they stuffed up. Whether this behaviour stems from arrogance or cluelessness, I’m not sure. (Yeah, yeah, the taco commercial …)

      I’m not sure the MAV or the VCAA will get away with this so easily, however. These solutions are written evidence of the basis of meetings with the assessors. If there’s clear error or blatant nonsense, it is difficult to deny or to ignore. The question above is particularly difficult to ignore because, not only is the solution a complete mess, there is explicit and totally confused advice on what is supposedly a VCE-acceptable solution.

  2. I cannot wait to see the VCAA “report” on this exam, at the very least to see what students, on the whole, made of this mess.

    But specifically to this question, it will be really interesting to see VCAA’s “answer” since we have all heard at one point or another that “show that and verify are not the same thing”

    1. Jesus. I’m not going to ask for the difference between “show that” and “verify”. What a tiny, idiotic world.

      But yes, the reports on all the exams will be interesting. There were plenty of dodgy questions on all four MM/SM exams from last year, and the MAV/assessor solutions to both Specialist exams have clear flaws. (I haven’t see the MM solutions, but I’m willing to bet what they’re like.)

      1. The MAV MM Exam 2 ‘solutions’ could not possibly be worse than the 2016 ‘solutions’, that make no mention of the erroneous probability density function but just go ahead and blithely calculate medians and means.

        Provable fact 1: Those 2016 solutions were written by a member of a VCAA exam setting panel.
        Provable fact 2: The same person is quoted as saying: “…. comments about VCAA have to be removed [from MAV solutions]. We are supposed to be writing solutions for MAV, not referring to VCAA.”

        Solutions to VCAA exams (and presentations purporting to discuss VCAA exams) will NEVER be accurate when written and/or influenced by VCAA stooges. Irrespective of advertising to the contrary. There is an insurmountable conflict of interest.

        It’s long overdue for mathematics teachers to demand BETTER products for their time and money and not betrayed by rampant conflicts of interest and misleading advertising.

        1. I think there is more here than just conflict of interest. The VCAA is in and of itself not in conflict: it is simply incompetent or dishonest, or both, depending upon the particular issue. Similarly, the MAV has done plenty that is inept, simply through being inept. Of course the type of MAV-VCAA conflict of interest you mention, and which has reared its head on more than one occasion, is shameful and makes matters much worse.

          There was also blatant dishonesty before 2016, at least from the VCAA. The 2014 MM2 Exam Report had, and has, a blatant falsehood, and it is almost certainly a falsehood of which the VCAA was and is aware. That is, it’s a blatant fucking lie. I’ll bet London to a brick that deliberate dishonesty has been long-standing practice.

          However, I still think the issue of current MAV solutions is different, and cannot be swept away in the same manner. This time, I think the MAV has bitten off more conflict than they can chew.

  3. Marty, for those of us with the memory of a goldfish, can you remind me what the falsehood is in the 2014 MM Exam 2 Report.

    And does anyone know whether this falsehood was/is perpetuated elsewhere (commercial exam solutions, presentations etc), and where it’s been called out (apart from here)?

    1. JF, It was MCQ14. See here. Note that the error in the question was not as own-goalish or as significant as other VCAA exam errors and idiocies, although it certainly was a bad error. The main point is that the dishonesty in (and, hilariously, insufficiency of) the examination report is as blatant as it gets.

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