## Right-Angles and Wrong Angles

It hit the news last week, that two high school kids had come up with a new proof of Pythagoras’s Theorem:

Two New Orleans high school seniors who say they have proven Pythagoras’s theorem by using trigonometry – which academics for two millennia have thought to be impossible – are being encouraged by a prominent US mathematical research organization to submit their work to a peer-reviewed journal.

I had decided to leave it alone. I had figured out enough of the story, that the two kids had done something very cool, which had then been way over-egged by the predictably clueless, Homerically lazy media. I’ve done these stories, and they can be tiring and tricky.

My friend Grant Cairns,* however, tipped it over the line, by pointing out an MAV tweet:

Good ‘ol MAV.

How a right-angled triangle is any more about “the world around us” than the number 3, God only knows. Grant summarised it as the MAV being “totally grounded in reality”. More accurately, the MAV is totally grounded by reality. To an earthbound hammer, everything is an earthbound nail. Continue reading “Right-Angles and Wrong Angles”

## Witch 91: Engaging With Reality

Following on from Adam Carey’s report on the decline of enrolments in Specialist Mathematics, Radio National held an interview-discussionHow to re-engage students in mathematics. The interview was with the president of the MAV and the Chief Engineer of Engineers Australia. We haven’t the energy to document or analyse their various diagnoses and caveats and solutions. Go for it. Continue reading “Witch 91: Engaging With Reality”

## WitCH 85: The Continuation of MAV’s Trials

With Methods exams next week, this one’s kinda important.

We try to avoid critiquing, or even being in the same room as, third party VCE practice exams. They are invariably clunky and weird, with plenty to criticise, but they matter infinitely less than the yearly screw-ups of the official exams.

Even MAV trial exams we do our best to ignore. Yes, the MAV is (too) closely aligned with the VCAA (with a number of people in conflicted, dual roles), and so MAV has a significantly greater professional and moral obligation to maintain high standards. But still, third party is third party, and we try our best to just ignore MAV’s nonsense. On occasion, however, MAV’s nonsense matters sufficiently, or is simply sufficiently annoying, to warrant a whack.

## MAV’s Valuing of Mathematics

The MAV have started the engines for their 2022 Annual Conference. The organisers have already lined up an impressive list of speakers and, as always, the MAV have put out the call for anyone and everyone to present, at a cost of only \$500 or so for the two-day extravaganza.

Of course, if planning to attend or present, one should keep in mind the theme and sub-themes of the conference: Continue reading “MAV’s Valuing of Mathematics”

## MAV’s Trials and Tribulations

Yeah, it’s the same joke, but it’s not our fault: if people keep screwing up trials, we’ll keep making “trial” jokes. In this case the trial is MAV‘s Trial Exam 1 for Mathematical Methods. The exam is, indeed, a trial.

Regular readers of this blog will be aware that we’re not exactly a fan of the MAV (and vice versa). The Association has, on occasion, been arrogant, inept, censorious, and demeaningly subservient to the VCAA. The MAV is also regularly extended red carpet invitations to VCAA committees and reviews, and they have somehow weaseled their way into being a member of AMSI. Acting thusly, and treated thusly, the MAV is a legitimate and important target. Nonetheless, we generally prefer to leave the MAV to their silly games and to focus upon the official screwer upperers. But, on occasion, someone throws some of MAV’s nonsense our way, and it is pretty much impossible to ignore; that is the situation here.

As we detail below, MAV’s Methods Trial Exam 1 is shoddy. Most of the questions are unimaginative, unmotivated and poorly written. The overwhelming emphasis is not on testing insight but, rather, on tedious computation towards a who-cares goal, with droning solutions to match. Still, we wouldn’t bother critiquing the exam, except for one question. This question simply must be slammed for the anti-mathematical crap that it is.

The final question, Question 10, of the trial exam concerns the function

on the domain . Part (a) asks students to find and its domain, and part (b) then asks,

Find the coordinates of the point(s) of intersection of the graphs of and .

Regular readers will know exactly the Hellhole to which this is heading. The solutions begin,

Solve    for  ,

which is suggested without a single accompanying comment, nor even a Magrittesque diagram. It is nonsense.

It was nonsense in 2010 when it appeared on the Methods exam and report, and it was nonsense again in 2011. It was nonsense in 2012 when we slammed it, and it was nonsense again when it reappeared in 2017 and we slammed it again. It is still nonsense, it will always be nonsense and, at this stage, the appearance of the nonsense is jaw-dropping and inexcusable.

It is simply not legitimate to swap the equation for , unless a specific argument is provided for the specific function. When valid, that can usually be done. Easily. We laid it all out, and if anybody in power gave a damn then this type of problem could be taught properly and tested properly. But, no.

What were the exam writers thinking? We can only see three possibilities:

a) The writers are too dumb or too ignorant to recognise the problem;

b) The writers recognise the problem but don’t give a damn;

c) The writers recognise the problem and give a damn, but presume that VCAA don’t give a damn.

We have no idea which it is, but we can see no fourth option. Whatever the reason, there is no longer any excuse for this crap. Even if one presumes or knows that VCAA will continue with the moronic, ritualistic testing of this type of problem, there is absolutely no excuse for not also including a clear and proper justification for the solution. None.

What of the rest of the MAV, what of the vetters and the reviewers? Did no one who checked the trial exam flag this nonsense? Or, were they simply overruled by others who were worse-informed but better-connected? What about the MAV Board? Is there anyone at all at the MAV who gives a damn?

*********************

Postscript: For the record, here, briefly, are other irritants from the exam:

Q2. There are infinitely many choices of integers and with equal to the indicated answer of .

Q3. This is not, or at least should not be, a Methods question. Integrals of the form   with non-linear are not, or at least are not supposed to be, examinable.

Q4. The writers do not appear to know what “hence” means. There are, once again, infinitely many choices of and .

Q5. “Appropriate mathematical reasoning” is a pretty fancy title for the trivial application of a (stupid) definition. The choice of the subscripted  is needlessly ugly and confusing. Part (c) is fundamentally independent of the boring nitpicking of parts (a) and (b). The writers still don’t appear to know what “hence” means.

Q6. An ugly question, guided by a poorly drawn graph. It is ridiculous to ask for “a rule” in part (a), since one can more directly ask for the coefficients , and .

Q7. A tedious question, which tests very little other than arithmetic. There are, once again, infinitely many forms of the answer.

Q8. The endpoints of the domain for are needlessly and confusingly excluded. The sole purpose of the question is to provide a painful, Magrittesque method of solving , which can be solved simply and directly.

Q9. A tedious question with little purpose. The factorisation of the cubic can easily be done without resorting to fractions.

Q10. Above. The waste of a precious opportunity to present and to teach mathematical thought.

UPDATE (28/09/20)

John (no) Friend has located an excellent paper by two Singaporean maths ed guys, Ng Wee Leng and Ho Foo Him. Their paper investigates (and justifies) various aspects of solving .

## MAV’s Mathematica Games

I’ve already posted Sai kumar Murali krishan’s Mathematica-VCE article, and fleshed it out a little. Here, I’ll give some of the back story, including a statement from Vinculum‘s editor, Roger Walter, and ending with a summary and a list of questions I sent to the MAV regarding the reviewing of Sai’s article, and to which I never expect to receive an answer. Throughout, I was curious whether the MAV would, once again, act in a gratuitously censorious manner, or whether they would now be wiser and publish Sai’s article; impressively, they accomplished both.

Last year, Sai was a student in my Monash Extension class. (It is irrelevant but ironic that Mathematica was used in this class in a limited but intelligent manner, for computing powers of large matrices, row reduction and the like.) The class was small and friendly, and fun, at least for me. I knew Sai well, and it’ll come as no surprise that he did very well in my class, but I had no sense of Sai’s Mathematica superpowers until the year had ended. Then John Kermond, Sai’s Specialist teacher, suggested I talk to Sai about Mathematica in VCE. Sai and I emailed back and forth a bit, and it became clear Sai had a very interesting story to tell.

I encouraged Sai to write up his gaming of VCE with Mathematica, with the goal of publishing in MAV’s journal Vinculum. Given the MAV’s previous conduct and general obsequiousness towards VCAA, some may suspect that goal was foolish or deliberate possum-stirring. With hindsight, it may have been the former but it was in no sense the latter.

There were a number of strong reasons to aim for Sai to write for Vinculum. First and foremost, Vinculum is the main senior school mathematics journal read by Victorian teachers, and was thus the natural home for an article such as Sai’s. Secondly, although Sai will clearly go far and needs absolutely no assistance from me, I thought it would be very good for Sai to have such a publication on his CV. Thirdly, I respected and continue to respect Vinculum‘s editor, Roger Walter, and I trusted he would see the importance of Sai’s article and would work hard to publish it. Finally, given the MAV had acted censoriously in the past and had been publicly called on it, I expected the MAV to be more circumspect in considering Sai’s article. On this last point, I was very wrong. Which is why we’re here now.

Sai quickly put together a very good draft, which I helped Sai tighten and polish. And, amusingly, I got Sai to tone down his language; Sai’s contempt for VCAA’s Mathematica crusade is significantly stronger than is indicated by the published article. Unfortunately, giving a clearer focus to Sai’s article also meant cutting out some very good material, on other clumsy and silly aspects of Mathematica in VCE. In particular, Sai put a lot of work into critiquing the sample Mathematica solutions provided by the VCAA, aspects of which Sai variously described, with supporting argument, as “contrived”, “incomplete”, “silly”, “bloated”, “obnoxious”, “abysmal” and “ludicrous”. (If he didn’t have better things to do, I’d retire and let Sai take over the blog.) I’m hoping to have Sai write a guest post on these solutions, and on other aspects of the Mathematica trial, in the near future.

In February, with a solid draft in hand, Sai submitted his article to Roger Walter, Vinculum‘s editor; I stayed in the loop, for the obvious reasons. The back and forth with Roger was sensible, efficient and amicable. Then, however, the MAV Publications Committee kicked in. What follows is a statement from Roger Walter, followed by my letter to the MAV (Publications Committee and CEO and President); that letter, as well as asking a number of questions, outlines the reviewing process and the frustration that it entailed.

And, failing any response from the MAV, that will end the story. I have heard nothing to indicate that the MAV is anything but satisfied with the manner in which Sai’s article was reviewed, which, if true, I find astonishing.

Statement from the Editor of Vinculum, Roger Walter (13/7)

I was very insistent that Sai’s article be published in Vinculum. This was partly my desire to publish, as far as is possible, all material that contributors have spent time and effort to put together, and partly because I was pleased to see a contribution from a student resourced from his experiences at secondary school. However, the main reason I pushed for publication was that the article itself had merit. This was for two reasons. Firstly, it was true, i.e. it clearly and accurately described the situation. Secondly, it was, at least in my mind, relevant – to both secondary teachers and to those who are responsible for planning our curriculum.

To me, two important statements were made (among others). One was that Mathematica was extremely powerful: too powerful for VCE, in fact, as it meant that students could answer questions without understanding the mathematics involved – an understanding which would be important for their future studies. The other is that Mathematica, being so much more powerful than the CAS calculators used by the majority of students, has the potential to create a non-level playing field. It is important that both teachers and those responsible for our curriculum are aware of this, if they aren’t already.

One of the things I try to do as editor, particularly in the editorials I write, and the material that is published, is to make educators think about what they are doing in their classes. I hope that if nothing else, this article achieves that. Also, as editor, I need to be impartial, and publish according to relevance and reality, regardless of my personal opinions, and the opinions and policies of other organisations. This impartiality is not always easy, but is relevant in the case of this article, and many others.

Letter From Marty to the MAV (13/7)

Dear Publications Committee, I am writing to you in regard to the article Mathematica and the Potential Gaming of VCE, by Sai kumar Murali krishnan (cc-ed) and just now published in the Term 3 issue of Vinculum

By way of background, it was at my suggestion that Sai wrote his article and submitted it to Vinculum. I also consulted with Sai during what turned out to be the lengthy and erratic reviewing process. Now, with Sai’s interest and agreement, there are a number of questions I wish to ask about that reviewing process. I am willing to publish any response by the Publications Committee or the MAV on my blog. I will interpret a lack of reply by Friday, 17 July as a decision to not comment.

Kind Regards, Marty Ross

******************************************

Sai initially submitted his article to Roger Walter, the editor of Vinculum, in mid-February of this year. After some back and forth, by early March Roger and Sai considered the article polished and ready for the Term 2 issue of Vinculum. The Publications Committee, however, objected. In late March the Publications Committee demanded that the following two paragraphs, the final paragraphs of Sai’s article, be cut:

To the extent VCAA is aware of these issues, there is reason to doubt that they are sufficiently aware, or at least sufficiently concerned. VCAA, after all, has created and continues to maintain this strange and uneven playing field. As further evidence, VCAA provides sample Mathematica solutions, and it is telling that these solutions are clumsy, uninventive and calculator-mimicking, suggesting a limited understanding of Mathematica’s capabilities.

Whatever naivety may exist, I believe it is unlikely to last. Nothing precludes the marketing of Mathematica packages designed specifically for VCE testing and, if Mathematica becomes widely available in VCE, I believe this commercialisation is inevitable. Such a development would turn VCAA’s implementation of Mathematica, which is already very problematic, into an obvious farce.

Q1. Was there any reason for these cuts, beyond members of the Publications Committee being “not happy with comments about the VCAA”? Does the Publications Committee generally regard such unhappiness as sufficient reason to censor an author?

Q2. Given that the criticism of VCAA was objectively valid and directly relevant, and given the potential commercialisation of Mathematica in VCE is an obvious and significant concern, will the Publications Committee now acknowledge there was no editorial or policy reason for demanding these paragraphs be cut? If not, will the Publications Committee now, finally, state any such reason?

Reluctantly, Sai then accepted these unjustified cuts, together with a new conclusion, with the understanding that publication could go ahead in Term 2, and with no further requests for substantial changes. Almost immediately, however, the Publications Committee demanded this second version of Sai’s article be held over until the Term 3 issue of Vinculum. The reason given to Sai for this delay was the Publications Committee “wanted time to consider the rest of the article and the conclusion”.

Q3. Will the Publications Committee now acknowledge that demanding a substantial and unjustified cut, and then subsequently demanding further time for review was a flawed and unfair process? Will the Publications Committee now acknowledge that in these circumstances, and in any circumstances, such a demand for further time should be accompanied by clear and substantive reasons, reasons that were entirely absent in this instance? Will the Publications Committee now indicate what specific parts of the article needed to be considered further, and why?

In mid-April the Publications Committee contacted Sai about further revising the second version of his article. The Publication Committee failed to indicate, much less argue for, a single flaw in this version. Rather, the Publications Committee requested that Sai add to his article, that the article also indicate what teachers could do in “using calculators and technology to support rather than bypass technology [sic]”. To this end, the Publications Committee also indicated they had contacted an MAV consultant familiar with Mathematica “to help [Sai] complete the article”.

Q4. Was the intent of the Publication Committee at that stage simply to dilute the clear content and message of Sai’s article? Will the Publications Committee now acknowledge that the suggested expansion of Sai’s article was unnecessary and unhelpful, at best orthogonal to the clear content and message of his article? Given this orthogonality and the absence of any claim of error in Sai’s article, will the Publications committee now acknowledge that at that stage they simply should have apologised to Sai for the needless delay and have accepted the second version of Sai’s article?

Q5. Does the Publication Committee understand the distinction between offering “help” and attempting to impose it, and will the Publications Committee now acknowledge the extraordinary presumptuousness of initiating “help” before having even canvassed the idea with Sai? Sai quickly replied to the Publications Committee, rejecting this proposal, and making it clear that his article should be accepted or rejected as is. Sai also clearly and carefully detailed the flaws and frustrations of the review process to that stage.

Q6. Why did the Publications Committee not respond to the concerns raised in Sai’s email? Why did the Publications Committee still decline to publish Sai’s article, still without providing a single reason beyond a vague and unjustified “too negative”?

Over Roger’s objections, the Publications Committee continued to refuse to publish the second version of Sai’s article. In an attempt to placate a member of the Publications Committee, Roger suggested “a possible insertion which … doesn’t need to be at the end”:

Technology, including Mathematica, calculators, spreadsheets and the many online programs, have tremendous potential to assist students with learning, understanding and applying mathematics. What is important for educators is to be careful that students are not using this technology to bypass learning and understanding mathematics.

Although Roger’s proposal was clearly well-intentioned, Sai considered, and considers, Roger’s paragraph to be clumsy, unnecessary and forced, particularly as a concluding paragraph. He also didn’t believe for a minute the inclusion of this paragraph would placate the objecting member.  Nonetheless, Sai was willing to consider it, and asked Roger: IF Sai agreed to this third version, with the original conclusion cut and this new conclusion properly incorporated, would that THEN be acceptable to the Publications Committee? Sai never received an answer.

Q7. Why did Sai never receive an answer to this question, on a proposal originating from discussion within the Publications Committee? Does the Publications Committee now acknowledge that this failure to respond was rude and unprofessional? In early May, Sai received the following communication from the Publications Committee:

The MAV are continuing internal discussions regarding the publication of the Mathematica article in alignment with MAV’s publication policy. It is expected that a decision may be provided by the end of Term 3.

Q8. Why, after months of failing to indicate a single flaw in Sai’s article, did the Publications Committee consciously and pointedly fail to tell Sai anything about any further “internal discussions”? What, precisely, in the “publication policy” necessitated that Sai was given no opportunity to comment on these “internal discussions” and, in particular, why was Sai given no opportunity to confirm or correct the version of his article then being considered?

Sai responded, indicating his frustration with the further delay and lack of communication. The Publication Committee responded:

1. Mathematica article is not to be included in Term 3. Pending subcommittee decision, it will be published in Term 4.

2. MAV are ‘continuing internal discussions regarding the publication of the Mathematica article in alignment with MAV’s publication policy. It is expected that a decision may be provided by the end of Term 3”.

Q9. Why was there a loud and definitive, and subsequently false, statement that Sai’s article would be further delayed until Term 4? Why was this further delay left unexplained?

Q10. Why did the Publications Committee not inform Sai of this “subcommittee” directly and immediately upon its formation? Who were the members on the subcommittee, what was the role of the subcommittee, and who determined this membership and role? On what formal basis and with what justification did the Publications Committee deprive Sai of this information?

Q11. Was the subcommittee properly informed that Sai had never agreed to Roger’s inserted paragraph being the conclusion to Sai’s article, and if not then why not? If, as appears to be the case, the subcommittee was not informed of this, will the Publication Committee now acknowledge that this lapse was a very serious error, and will the Publication Committee now apologise to Sai for this error?

Q12. What summary and/or advice and/or opinion did the Publications Committee provide to the subcommittee, and why did Sai not also receive any such material?  In particular, if the Publications Committee indicated substantive objections, after having failed for months to do so to Sai directly, why did the Publications Committee not then inform Sai of these objections?

Finally, in early June, the Publications Committee presented Sai with a fourth version of his article, presumably the work of the “subcommittee”.  The Publications Committee indicated they had agreed to publish this fourth version in Vinculum. It was made clear that this version of the article, which still included Roger’s inserted paragraph as conclusion, was not open to any further discussion, and that Sai had to either accept or decline. It was also indicated that the “aim” was still to publish in Term 4. Given the changes from the third to the fourth version were few and very minor, and swallowing his annoyance with the demand to conclude with Roger’s paragraph, Sai quickly agreed to this fourth version of his article.

Sai was relieved when, presumably due to the wise counsel of the subcommittee, the reviewing ordeal finally ended with an agreement to publish. He is also very pleased to see his article appear in the Term 3 issue of Vinculum. The article as published is identical to the fourth version, except for a new title and the inclusion of a clarifying footnote, both agreed upon without dispute. Which raises the final questions.

Q13. Given that the changes from the third version of Sai’s article to the fourth version were very few in number and were all very minor, does the Publications Committee accept that the decision of the “subcommittee” repudiates the months of secretive stonewalling of the Publications Committee?

Q14. Given there are only minor differences between the second, March, version of Sai’s article and the final, July, published version of Sai’s article, and given Sai was never presented with a single substantive criticism of his article, will the Publications Committee now acknowledge that this whole review process could have been handled in a significantly more efficient, more thoughtful, more open and more respectful manner

Q15. Will the Publications Committee now extend a formal apology to Sai?

## How to Play VCE with Mathematica

An article titled Mathematica and the Potential Gaming of VCE has just appeared in the MAV’s journal Vinculum (and we have posted it here). By Sai kumar Murali krishnan, who completed VCE last year and who we previously mentioned in this post, the article delivers what the title promises (noting the “Potential” is redundant): Sai demonstrates how Mathematica’s huge library of functions and extremely powerful programming can be used, and has been used, to trivialise VCE maths exams. We believe Sai’s article is very interesting and very important. (For anyone interested to do so, Sai can be contacted by email here.)

Also likely to be of interest, at least to readers of this blog, is the story of the long and weird battle to have Sai’s article appear. Roger Walter, Vinculum’s editor, deserves a hell of a lot of credit for seeing that battle through and ensuring Sai’s article survived, largely unscathed. And a disclaimer: we played a role in Sai choosing to write the article, and we were also involved in the subsequent battle. We intend to write on all of this in the near future.

With Sai’s permission, we’ve posted his article here. In this post we’ll give a few more examples and we’ll provide some concluding paragraphs, which didn’t make it into Sai’s published article. By way of background, Mathematica memory need not be cleared before taking an exam or SAC. Secondly, in computer-based (CBE) Methods, a student enters their answers directly into the Mathematica notebook; this means that Mathematica code and output in and of itself constitutes acceptable working, and is very close to sufficient as answer.

First, here is a multiple choice question from the 2019 Mathematical Methods exam, which we also discussed here:

The problem is to determine Pr(X > 0). Here is Sai’s solution, utilising standard Mathematica functions:

The point is, of course, that the application of functions such as Area and Polygon requires very little sense of the mathematics involved. For an example requiring no mathematical sense whatsoever, consider the following multiple choice question, which appeared on the 2017 Mathematical Methods exam:

The question is of a standard type, and for these questions Sai created the Mathematica function FTest. The following is Sai’s complete Mathematica working to solve the question above:

A final example, again from the 2019 Mathematical Methods exam:

Here is Sai’s Mathematica working for this question, using two functions he created, FInfo and TangentLine:

Sai’s Vinculum paper contains a number of other examples, and Sai has created a huge library of incredibly sophisticated functions to tackle VCE questions, a library which he shared with his fellow VCE students. Sai’s work raises obvious issues, not least of which is the grossly unfair competition between the majority handheld-CAS students and the few Mathematica-powered students. The original version of Sai’s article ended with two paragraphs, which the MAV Publications Committee demanded be cut:

To the extent VCAA is aware of these issues, there is reason to doubt that they are sufficiently aware, or at least sufficiently concerned. VCAA, after all, has created and continues to maintain this strange and uneven playing field. As further evidence, VCAA provides sample Mathematica solutions, and it is telling that these solutions are clumsy, uninventive and calculator-mimicking, suggesting a limited understanding of Mathematica’s capabilities.

Whatever naivete may exist, I believe it is unlikely to last. Nothing precludes the marketing of Mathematica packages designed specifically for VCE testing and, if Mathematica becomes widely available in VCE, I believe this commercialisation is inevitable. Such a development would turn VCAA’s implementation of Mathematica, which is already very problematic, into an obvious farce.

Of course the MAV having cut these paragraphs, along with every single reference to the VCAA, doesn’t make their content any less true, any less obvious or any less important.

We intend to write more later in the week.

## MAV’s Sense and Censor Ability

We’ve written about MAV’s censorship previously. It seems, unfortunately, that we may have another such incident to write about in the near future. We’ll see.

There is also a third incident that we’ve long planned to write about, but have never gotten around to. It is rather involved, and we won’t give the full story here, but one specific aspect is perhaps worth telling now.

In 2016, we accepted an invitation from the MAV to give a keynote address at their Annual Conference. We chose as our keynote title Same Sermon, New Jokes. We also submitted a “bio pic” – the graphic above – and an abstract. The abstract indicated our contempt for twenty or so organisations and facets of Australian mathematics education.

A couple months later, the Conference organisers emailed to indicate their objection to our abstract. One can argue the merits of and the propriety of this objection, and we will write generally on this at a later date, but one aspect of the objection was particularly notable. The email included the following:

“While we welcome all points of view, we do need to be respectful of the organisations we work with, and with whom we need to maintain good relations … We would like you to re visit the text … without the criticism of formal organisations.”

We pushed back against the criticism, and ended our reply with what we intended as a rhetorical question:

“You wrote that you (plural) welcome all points of view, which I was very reassured to read. Given that, which formal organisations do you consider to be above criticism?”

The email reply from the organisers included a response:

“In regards to the formal organisations with which the MAV has relations, you have stated some of them, e.g. ACARA, VCAA.”

No one at the MAV, including the then President, indicated to us any problem with this request or its clarification.

For now, we’ll leave it there.

## 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

for t ≥ 0, along with the initial condition

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

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

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

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

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

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 ), 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

.˙.

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

Part 3: The statement

Solving  .˙.  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 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 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 (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 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.

## 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 should be noted. In part (c), it is preferable to begin with the chain rule in the form , 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.

UPDATE (26/06/20)

The Specialist Maths examination reports are finally, finally out (here and here), so it seems worth revisiting the MAV “Assessor” solutions. In summary, the clumsiness of and errors in the MAV solutions as indicated above (and see also here and here) do not appear in the reports; in the main this is because the reports are pretty much silent on any aspect involving some subtlety. Sigh.

EXAM 1

Q5 Yes, Magritte-ish. Justifying that the critical points are extrema was not expected, meaning conscientious students wasted their time.

Q6 The error in the MAV solutions is ducked in the report.

Q7 The error in the MAV solutions is ducked in the report.

EXAM 2 (Section A)

MCQ6   The error in the MAV solutions is ducked in the report.

MCQ11 The report is silent.

MCQ12 A huge screw-up of a question, to which the report hemidemisemi confesses: see here.

MCQ14 The report suggests the better method for solving this problem.

EXAM 2 (Section B)

Q2 Jesus. This question was intrinsically confusing and very badly worded, with the students inevitably doing poorly. So, why the hell is the examination report almost completely silent? The MAV solutions were a mess, but the absence of comment in the report is disgraceful.

Q3 The solution in the report is ok, although more could have been written. But, it’s not the garbled nonsense of the MAV solution, as detailed here.