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.

The Troubling Cosiness of the VCAA and the MAV

It seems that what amounts to VCE exam marking schemes may be available for purchase through the Mathematical Association of Victoria. This seems very strange, and we’re not really sure what is going on, but we shall give our current sense of it. (It should be noted at the outset that we are no fan of the MAV in its current form, nor of the VCAA in any form: though we are trying hard here to be straightly factual, our distaste for these organisations should be kept in mind.)

Each year, the MAV sells VCE exam solutions for the previous year’s exams. It is our understanding that it is now the MAV’s strong preference that these solutions will be written by VCAA assessors. Further, the MAV is now advertising that these solutions are “including marking allocations“. We assume that the writers are paid by the MAV for this work, and we assume that the MAV are profiting from the selling of the product, which is not cheap. Moreover, the MAV also hosts Meet the Assessors events which, again, are not cheap and are less cheap for non-members of the MAV. Again, it is reasonable to assume that the assessors and/or the MAV profit from these events.

We do not understand any of this. One would think that simple equity requires that any official information regarding VCE exams and solutions should be freely available. What we understand to be so available are very brief solutions as part of VCAA’s examiners’ reports, and that’s it. In particular, it is our understanding that VCAA marking schemes have been closely guarded secrets. If the VCAA is loosening up on that, then that’s great. If, however, VCAA assessors and/or the MAV are profiting from such otherwise unavailable information, we do not understand why anyone should regard that as acceptable. If, on the other hand, the MAV and/or the assessors are not so profiting, we do not understand the product and the access that the MAV is offering for sale.

We have written previously of the worrying relationship between the VCAA and the MAV, and there is plenty more to write. On more than one occasion the MAV has censored valid criticism of the VCAA, conduct which makes it difficult to view the MAV as a strong or objective or independent voice for Victorian maths teachers. The current, seemingly very cosy relationship over exam solutions, would only appear to make matters worse. When the VCAA stuffs up an exam question, as they do on a depressingly regular basis, why should anyone trust the MAV solutions to provide an honest summary or evaluation of that stuff up?

Again, we are not sure what is happening here. We shall do our best to find out, and commenters, who may have a better sense of MAV and VCAA workings, may comment (carefully) below.

UPDATE (13/02/20)

As John Friend has indicated in his comment, the “marking allocations” appears to be nothing but the trivial annotation of solutions with the allotted marks, not a break-down of what is required to achieve those marks. So, simply a matter of the MAV over-puffing their product. As for the appropriateness of the MAV being able to charge to “meet” VCAA assessors, and for solutions produced by assessors, those issues remain open.

We’ve also had a chance to look at the MAV 2019 Specialist solutions (not courtesy of JF, for those who like to guess such things.) More pertinent would be the Methods solutions (because of this, this, this and, especially, this.) Still, the Specialist solutions were interesting to read (quickly), and some comments are in order. In general, we thought the solutions were pretty good: well laid out with usually, though not always, the seemingly best approach indicated. There were a few important theoretical errors (see below), although not errors that affected the specific solutions. The main general and practical shortcoming is the lack of diagrams for certain questions, which would have made those solutions significantly clearer and, for the same reason, should be encouraged as standard practice.

For the benefit of those with access to the Specialist solutions (and possibly minor benefit to others), the following are brief comments on the solutions to particular questions (with section B of Exam 2 still to come); feel free to ask for elaboration in the comments. The exams are here and here.

Exam 1

Q5. There is a Magritte element to the solution and, presumably, the question.

Q6. The stated definition of linear dependence is simply wrong. The problem is much more easily done using a 3 x 3 determinant.

Q7. Part (a) is poorly set out and employs a generally invalid relationship between Arg and arctan. Parts (c) and (d) are very poorly set out, not relying upon the much clearer geometry.

Q8. A diagram, even if generic, is always helpful for volumes of revolution.

Q9. The solution to part (b) is correct, but there is an incorrect reference to the forces on the mass, rather than the ring. The expression  “… the tension T is the same on both sides …” is hopelessly confused.

Q10. The question is stupid, but the solutions are probably as good as one can do.

Exam 2 (Section A)

MCQ5. The answer is clear, and much more easily obtained, from a rough diagram.

MCQ6. The formula Arg(a/b) = Arg(a) – Arg(b) is used, which is not in general true.

MCQ11. A very easy question for which two very long and poorly expressed solutions are given.

MCQ12. An (always) poor choice of formula for the vector resolute leads to a solution that is longer and significantly more prone to error. (UPDATE 14/2: For more on this question, go here.)

MCQ13. A diagram is mandatory, and the cosine rule alternative should be mentioned.

MCQ14. It is easier to first solve for the acceleration, by treating the system as a whole.

MCQ19. A slow, pointless use of CAS to check (not solve) the solution of simultaneous equations.

UPDATE (14/02/20)

For more on MCQ12, go here.

UPDATE (14/02/20)

Exam 2 (Section B)

Q1. In Part (a), the graphs are pointless, or at least a distant second choice; the choice of root is trivial, since y = tan(t) > 0. For part (b), the factorisation \boldsymbol{x^2-2x =x(x-2)} should be noted. In part (c), it is preferable to begin with the chain rule in the form \boldsymbol{dy/dt = dy/dx \times dx/dt}, since no inverses are then required. Part (d) is one of those annoyingly vague VCE questions, where it is impossible to know how much computation is required for full marks; the solutions include a couple of simplifications after the definite integral is established, but God knows whether these extra steps are required.

Q2. The solution to Part (c) is very poorly written. The question is (pointlessly) difficult, which means clear signposts are required in the solution; the key point is that the zeroes of the polynomial will be symmetric around (-1,0), the centre of the circle from part (b). The output of the quadratic formula is neccessarily a mess, and may be real or imaginary, but is manipulated in a clumsy manner. In particular, a factor of -1 is needlessly taken out of the root, and the expression “we expect” is used in a manner that makes no sense. The solution to the (appallingly written) Part (d) is ok, though the centre of the circle is clear just from symmetry, and we have no idea what “ve(z)” means.

Q3. There is an aspect to the solution of this question that is so bad, we’ll make it a separate post. (So, hold your fire.)

Q4. Part (a) is much easier than the notation-filled solution makes it appear.

Q5. Part (c)(i) is weird. It is a 1-point question, and so presumably just writing down the intuitive answer, as is done in the solutions, is what was expected and is perhaps reasonable. But the intuitive answer is not that intuitive, and an easy argument from considering the system as a whole (see MCQ14) seems (mathematically) preferable. For Part (c)(ii), it is more straight-forward to consider the system as a whole, making the tension redundant (see MCQ14). The first (and less preferable) solution to Part (d) is very confusing, because the two stages of computation required are not clearly separated.

Q6. It’s statistical inference: we just can’t get ourselves to care.

 

The MAV and a Matter of Opinion

This post is tricky. It is not about us, but there is context, and that context should be kept in mind.

Many readers of this blog will be aware of the long relationship we have had with the Mathematical Association of Victoria. It dates back to 2001, when we first came up with the weird idea that mathematics teachers may be interested in learning some maths beyond the thin gruel they were typically served while at university. That idea morphed into 15+ years of teaming up with the Evil Mathologer, of presenting under the banner of and as a consequence of the MAV, of spreading ideas and rousing the rabble. It was quixotically stupid and exhausting and incredibly rewarding. The prehistory of this blog is an interesting story, which is probably of interest to no one.

Fewer readers of this blog will be aware that our association with the MAV ended a few years ago, when the MAV threatened to (and arguably did) censor the abstract of our (invited) keynote. That story may be of more interest, and we hope to write on it in the near future.

In summary, and notwithstanding our long association with and our gratitude to the MAV, we have no love for the MAV in its current form. That is the context. Now for the post.

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A few months ago we heard that an article was rejected for publication in the MAV’s teachers’ journal Vinculum. The manner of and the reason for that rejection sounded very strange, and so we began to ask questions. As indicated below, the MAV has not been particularly forthcoming, but this is our current understanding of the story:

1) An opinion piece was submitted to Vinculum. In the piece, the author argued that all VCE mathematics exams in Year 12 should be calculator-free.

2) Roger Walter, the editor of Vinculum, accepted the piece for publication and included it to be published in the next issue.

3) Peter Saffin, the CEO of the MAV, overruled the editor, instructing Walter to retroactively reject the piece.

4) Saffin’s stated reason for the rejection was that the author’s position was in conflict with the VCAA’s strong advocacy of calculator use.

That is the bare bones of the story. Here is a little flesh (once again, as we understand it):

a) The author of the article is a long-standing member of the MAV, a respected gentleman who has devoted decades to Australian mathematics education generally and to the MAV specifically.

b) The author’s piece was topical, well-written and not flame-throwing.

c) In early September we contacted Michael O’Connor, the President of the MAV, seeking information and clarification. After a back and forth, the President declined to confirm or deny point 3, declaring that as a member of the public we had “no need to know”, and that “even MAV members would have to show sufficient reason”. O’Connor citied his “duty of care towards MAV staff and volunteers”.  Similarly, O’Connor declined to confirm or deny point 4.

d) To our knowledge, no MAV editor has ever previously been overruled in such a manner, by anyone.

e) The author has not contested the rejection.

f) Notwithstanding (d), O’Connor indicated that “proper processes have been followed”.

g) O’Connor indicated that he is “expecting there to be a policy discussion at the next publications meeting”.

h) At this stage, the rejection of the article has not been rescinded.

i) At this stage, no one at the MAV, nor the MAV as a body, has apologised to the author for the rejection of the article or the manner of that rejection,

j) In late September we replied to O’Connor, critiquing various aspects of this incident and his characterisation of it. O’Connor indicated his intention to respond.

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That then is the post. O’Connor and Saffin were invited to comment on a close version of the above. O’Connor reiterated his intention to reply and suggested out our posting now was “premature”, arguing that the MAV had not had “sufficient time to perform due diligence”. Saffin did not reply as of the time of posting.

We will update the post if and when any new information comes to hand.

UPDATE (05/12/19):  In response to a query in the comments of another post, here is a brief and empty update:

  • Michael O’Connor has not replied further, and, written indication notwithstanding, presumably has no intention of doing so.
  • We do not know of any officer of the MAV having expressed, formally or publicly, the view that unilateral censorship of the type above is inappropriate.
  • We are not aware of any formal or informal steps the MAV may have taken to preclude such censorship in the future.
  • We are not aware of any officer of the MAV, nor the MAV as a body, having apologised to the author of the Vinculum article.