An Offer: Checking For SACking Offences

Pauline Baynes

This is an open offer to review Methods and Specialist SACs. Here are the conditions:

0) The review is free. (You can consider donating to Tenderfeet.)

1) You may email me any Methods or Specialist SAC, by anyone.

2) You should indicate whether or not you are the writer of the SAC.

3) If you are the writer of the SAC, I will be diplomatic.*

4) It’s on your head, in particular for future SACs, if you’re breaking confidentiality rules or conventions. This is not my concern.

5) I will keep all SACs confidential, except to the extent there is explicit agreement otherwise. (See 12-14, below.)

6) Future SACs should, at minimum, be close to a final draft.

7) All SACs should include solutions and a grading scheme.

8) I may decline to review a SAC for being too old, or for other reasons.

9) I will review only for mathematical sense and mathematical correctness.

10) In particular, I will not check for, and do not give a stuff about, VCAA compliance.

11) I will not check all arithmetic and a review should not be taken as a guarantee that the SAC is error-free.

12) Each time I review a SAC I will record so below, with brief and, modulo points 13 and 14, anonymity-preserving comments.

13) I will identify commercial SACs as such, possibly indicating the commercial entity.

14) If you are the author of the SAC and you agree, I will consider making a separate post, to review the SAC in detail and to allow for comment.

 

I will be interested to see who is brave enough to enter (and who is tossed into) the lion’s den.

 

*) Yes, I am capable of diplomacy. I just prefer to do without.

UPDATE (26/7)

We have our first taker: a brave soul has entered the den. I’ll look at the proffered SAC asap. I was also asked what I am after, in making this offer, which is a fair question. The answer is two-fold:

a) (Jekyll) I’m making a genuine offer to provide a critique of a SAC from a mathematical perspective, for any writer who wants it. I’m hoping that by providing such a critique, the writer will become more attuned to any mathematical shortcomings in their (and all) SACs, and in VCE generally. Hopefully then, to the limited extent that VCAA’s idiot curriculum permits it, this will help the writer produce more mathematically coherent and rich SACs in the future.

b) (Hyde) I’m looking to see as much as I can of the nonsense the SAC system is producing. This will allow me to confirm for any teacher or student who has been served swill that they have indeed been served swill. It will also allow me to write upon such SACs, even if in very oblique terms.

UPDATE (27/7)

OK, this post is being steered away from what I intended, but I’m happy to let others steer.

First, a clarification. By “SAC”, I mean any school-based Year 12 assessment that counts towards the final VCE grade. I don’t care if the assessment takes five minutes or five days.

Now, the question is what to do with SACs offered to me by authors? I have two currently. I can either

a) Make the SACs into posts on this blog. The SACs would then be a basis for discussion, and a model for future SACs, but the SACs themselves would presumably not be usable. (Again, I don’t give a stuff about protocol, but obviously teachers must.)

or

b) Keep the SACs off the site, except for brief comments below, and set up a Free SACs to Good Home post. Teachers can then contact me to obtain copies.

Readers can suggest to me what they prefer. They can also suggest how (b) might work in practice.

A Request: Auditory Lapses

It’s SAC time and, as indicated below, I have a request for people to contact me.

A couple days ago I was talking to a Head of Maths, who suggested/requested/pleaded that I take a whack at the auditing of SACs. In principle, I’d love nothing more. (Well, I’d love a bottle of Laphroaig more, but you get the point.) Maths SACs are a soul-drowning swampland and consequently, and independently, the auditing of SACs is a Kafkaesque nightmare. That is currently amplified to 11, with the VCAA making its astonishingly stupid and ineptly delayed decision to maintain SACs during a plague year.

The difficulty with me writing on SAC audits is that, although I am generally aware of the brain-drilling arbitrariness in SAC auditing, I seldom see the specific idiocies. And, this is a case where the idiot devil is most clearly evident in the idiot details. Hence my request:

If anyone has a SAC audit horror story, please DO NOT provide the details as a comment below, but please feel free to email me.

Then, if you wish, we can chat about your horror story. Of course, I will maintain all confidences, and I will not use any information, even in an anonymised manner, without clear and specific agreement.

Even with information in hand, natural concerns for confidentially make this a very difficult topic upon which to write. I have no idea what I might be able to do. But, the first step is to see what there is to see.

Again, please don’t include specifics in the comments below, although general bitching is appropriate, welcome and to be expected.

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.

Guest Post: Mathematica and the Potential Gaming of VCE

What follows is the article Mathematica and the Potential Gaming of VCE, by Sai kumar Murali krishnan, which has just appeared in Vinculum and which we have written about here. Sai’s article is reproduced here with Sai’s permission. Sai can be contacted by email here

 

INTRODUCTION

Last year I completed VCE, including Mathematical Methods (CBE) and Specialist Mathematics. At my school these subjects employed the computer system Mathematica in place of handheld CAS calculators. The CBE (Computer-Based Examination) version of Methods also entailed the direct submission of SACs and the second (tech-active) exam on the Mathematica platform.1

Mathematica is extraordinarily powerful and, as it happens, I consider myself a decent programmer. During VCE, I entertained myself by creating custom functions to automate tedious computations, which I then shared with my fellow students. We were able to store these functions in a paclet (package), ready for use on the SACs and the exam. While handheld CAS calculators can also store (less complex) custom-made functions, Mathematica’s vast in-built library and ease of use moves it into a different class. Mathematica enables the creation of exam-ready functions to perform any computation a student might require.

I have witnessed, and experienced, many problems with the implementation of Mathematica, but in this article I will focus upon the two most glaring and most important issues. First and foremost, Mathematica is so powerful that it can trivialise the testing of the mathematics for which it is purported to be a tool. It enables any student who can program in Mathematica or, more perversely, who has a friend, teacher or tutor who can program in Mathematica, to perform well in VCE mathematics. Secondly, and as an inevitable consequence of this trivialisation, the current partial implementation of Mathematica could create a grossly unfair competition, an unfairness enhanced in Methods CBE by effectively permitting Mathematica code to be submitted as an answer. The students equipped with handheld CAS calculators are the victims. Armed with toys sporting 70s Nintendo displays, they are being outgunned by students deploying full-screen guided missiles.

In this article I will illustrate how Mathematica can trivialise exam questions in Mathematical Methods. In Part 2, I provide an example of the use of Mathematica’s in-built functions. In Part 3, I consider the application of custom-built functions. In Part 4, I summarise, and I indicate why I believe the problems with the implementation of Mathematica are only likely to worsen.

 

IN-BUILT FUNCTIONS

We begin by looking at Question 5, Section B of 2019 Exam 2, which concerns the cubic \boldsymbol{f(x) = 1 - x^3}.

The question first prompts us to find the tangent at x = a, which we perform in one step with the function TangentLine.2 We then find the intersection points Q and P with two applications of the function Solve. Next, the area of the shaded region as a function of a is found by subtracting the area under the cubic from a triangular area: the former is found using the function Integrate, and the latter is found directly from the coordinates using the functions Polygon and Area. Finally, we are required to find the value of a that minimises the area, which is found in one step with the function ArgMin.

What follows is the complete Mathematica code to answer this five-part question:

This solution requires little mathematical understanding beyond being able to make sense of the questions. In particular, the standard CAS approach of setting up integrals and differentiating is entirely circumvented, as is the transcription. In Methods CBE, the above input and output would be considered sufficient answers.

 

CREATED FUNCTIONS

We’ll now venture into the world of custom Mathematica functions, where programmers can really go to town. We’ll first look at the topic of functions and the features of their graphs. Mathematica does not have a built-in function to give all the desired features, so I created the function DetailPlot. To begin, I use a module to gather data about a function, including endpoints, axial intercepts, stationary points, inflection points and, if required, asymptotes. I then turn the module into an image to place over the graph.

Let’s fire this new weapon at Q2(c), Section B of 2016 Exam 2, which concerns the pictured quartic. We are given the equation of the graph and the point A, and we are told that the tangents at A and D are parallel. We are then required to find the point D and the length of AE.

And, here we are:

With very little input, DetailPlot has provided a rich graph, with every feature one might require within easy reach. The intersection points are ‘callouts’, which means that the points are labelled with their coordinates. In particular the coordinates of D and E have been revealed by DetailPlot, without any explicit calculation. We can then press forward and finish by finding the length of AE, a trivial calculation with the in-built function EuclideanDistance.

In the next example I demonstrate that a multi-stage question can still be trivialised by a single piece of pre-arranged code. In the multiple choice question MCQ10 from 2017 Exam 2, the function \boldsymbol{f(x) = 3\sin \left(2\left[x+\frac{\pi}{4}\right]\right)} undergoes the transformation \boldsymbol{T\left(\Big[\begin{smallmatrix}x \\ y\end{smallmatrix}\Big]\right) = \Big[\begin{smallmatrix}2 & 0 \\ 0&\frac13\end{smallmatrix}\Big]\Big[\begin{smallmatrix}x \\ y\end{smallmatrix}\Big]} and we are required to identify the resulting function. For such questions I created the function Transform, and then the in-built function FullSimplify polishes off the question:

My last example is on functional equations, for which I created two functions, FTest and RFTest. I will illustrate the use of the latter function. For MCQ11 on 2016 Exam 2, the equation f(x) – f(y) = (yx) f(xy) is given and it is required to determine which of the given functions satisfies the equation. Here is my entire solution:

 

CONCLUSION

It is impossible to have a proper sense of the power of Mathematica unless one is a programmer familiar with the package. This article presents just a few examples from the vast library of functions I created for Mathematical Methods and I found even more so for Specialist Mathematics. My libraries for both subjects barely scratch the surface of what is possible.

Creating such packages requires skill in both programming and mathematics, but the salient point is that any subsequent application of those programs by another student requires no comparable skill. The programs I have written may improve the performance of mathematically weaker students. Conversely, any student without access to such programs or, worse, is required to use handheld CAS instead of Mathematica, will be at a significant disadvantage.

This demonstrates the potential power of Mathematica to change the focus of VCE mathematics and, consequently, to debase its teaching and its assessment. True, the same issues had already arisen with the introduction of handheld CAS; clever teachers and cleverer students have always engaged in creating and sharing push-a-button CAS programs. Mathematica, however, has massively elevated the seriousness of these issues, all the more so since only a fraction of students have access to the platform.3

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.

1. All non-CBE students take the same tech-active exams and are considered in the same cohort for ATAR purposes. The Methods (CBE) exam appears to differ in only a superficial manner, and it appears that CBE students have not been considered a separate cohort since 2016.

2. The examination diagrams have been redrawn for greater clarity.

3. Although the Victorian Government offers Mathematica to all schools, to date many schools have not implemented it.

© Sai kumar Murali krishnan 2020

Letter from a Concerned Student

A few days ago we received an email from “Concerned Student”, someone we don’t know, requesting advice on how to approach VCE mathematics. We have thoughts on this and intend to reply, but the email also seemed generally relevant and of likely interest. The email also raises interesting questions for teachers, and for the writer of this blog. With Concerned Student’s permission, we’ve reproduced their email below. We’ll hold off commenting until others, who actually know what they’re talking about, have had a go. Here is CS’s email:

It seems clear from reading this blog that a significant proportion of the VCE Methods & Specialist curricula are in direct conflict with good mathematical education. As someone entering these subjects next year, what’s the recommended approach to make it through all the content of the study design while also *learning maths*? Should I largely ignore the (… Cambridge) textbook and overall course and focus on self-teaching content along the same lines from better sources, stopping only to learn specifically from the curriculum whatever button mashing is necessary for an exam; or should I instead focus on fighting through the curriculum, and learn some proper maths on the side – I guess the productive question there is “is it easy enough to apply properly learnt maths to the arcane rituals found in VCE course assessments?”

It’s probably worth noting that, as far as I’m aware, the Methods & Specialist teachers at my school are known for being quite good, but they’re obviously still bound by the curriculum they teach.

SACs of Shit

SACs may not be the greatest problem with VCE mathematics, but they’re right up there. SACs are torture for teachers and torture for students. They teach nothing. As assessment, they are unnecessary, unreliable and phenomenally inefficient. They are a license for VCAA’s unaswerable auditors applying Kafkaesque rules to act either as favour-givers or as little Hitlers, as the mood takes them. These problems are currently amplified to eleven by VCAA’s “We’ll give you some kind of guidance in, oh, a little while” plan for the plague year.

For all of the awfulness of the above, that’s not the worst of it. The worst is that the majority of SACs are monumentally stupid. Literally. A SAC has the imposing presence of a monument, its towering stupidity casting a shadow over everything.

How are SACs so bad? Many contain errors, often subtle although too often not, but, as irritating as that is, that is not the main problem. The main problem is that they are mathematical nonsense. Typically they will present the student with a ridiculous model of a contrived problem, which is then all redone in greater, brain-bludgeoning generality by throwing in a needless parameter in a randomly chosen location. All of this is undertaken, of course, in the nihilistic world of CAS. Finally, somewhere near the end, the poor beleaguered student, who by this stage just wants to escape with their life, will be required to “comment on the model”, to which the usual response is “It’s really nice, please let me go” and to which the only reasonable response is “It’s fucking insane”.

How do we know SACs are this bad? Because we see them. We see the commercial SACs, and the sample SACs, and the past SACs, and the current SACs. Are they all as bad as we suggest? No, of course not. Specialist SACs are typically nowhere near as bad as Methods SACs, and even many Methods SACs will fall short of truly idiotic torture, rising only to the level of being dumb and painful. Then there are the rare few SACs we see that are good, resulting in an exchange:

“This actually makes sense. Who’s your teacher?”

“Oh, it’s Mr. ….”

“Ah. Yes.”

So, yes, the quality and worth of SACs varies widely, but the average is squarely in the neighbourhood of monumental, tortuous stupidity. Which bring us to the “why”. Why are SACs in general so awful? There are two reasons.

The first and fundamental reason is the VCAA and their view of what they imagine is a curriculum. VCE mathematics subjects are so shallow and so lacking in a foundation of solid reason, that almost any attempt at depth and substance in a SAC is destined to be farce. The VCAA has replaced foundation and depth with CAS, which reaches peak awfulness in SACs. The VCAA promotes the fantasy that CAS magically transforms students into mathematical explorers, clever little Lewises and Clarks skilfully navigating the conceptual wilderness. The reality, of course, is much less Lewis and Clark than it is Burke and Wills. To top it off, SACs must follow guidelines that Terry Gilliam would be proud of, giving us Burke and Wills’ Bogus Brazilian Journey. Or, just Eraserhead. Something like that.

The second reason is the teachers. Sort of. Even if the subjects were coherent, even if they were unpoisoned by CAS and were unconstrained by vague and ridiculous conditions, even then writing a good SAC would be a very difficult and massively time-consuming task. Most teachers just don’t have the mathematical background, or the literary skill, to write a coherent, correct and mathematically rich SAC; many cannot even recognise one. And, that’s writing a good SAC for this imaginary good subject; writing a good SAC for these fundamentally flawed subjects with their ridiculous constraints is close to impossible, even for a strong teacher. And which teachers, particularly weaker teachers, have the time to compose such a good SAC? Why bother trying? And so, with the greatest common sense, most teachers do not. Most teachers stick to the audit-proof and meaningless formulaic SAC bullshit that the VCAA expects and effectively demands.

The VCAA’s SAC system is a crime against mathematical humanity.

UPDATE (15/5)

We received the following from a student acquaintance (who hadn’t read this post):

Hi Marty, given the upcoming math SACs approaching soon, the pressure is on to practice and practice. Attached below is last year’s Methods SAC1 (Unit 3/4) for [the student’s school]. I remember many talented friends of mine who were stumped, and didn’t do very well on this SAC. Personally, I thought this SAC was horrifying. In contrast to Specialist, (I actually quite enjoy Specialist!), Methods seems to be a huge prick because of frustrating, ambiguous SACs containing questions seemingly cooked from the pits of hell itself. Are these sort of SACs common across the state?

The student is, of course, correct. The SAC, which comes from a highly respected school, is a nightmare in all of the ways canvassed above. From start to end it is idiotic CAS-driven pseudo-modelling, complete with Magritte nonsense and a pointlessly prissy grading scheme. And, yes, the SAC contains an error.

Of course we won’t reveal the school, much less any teachers involved, which means that we are also unable to critique the SAC in detail. But that is one of the insidious aspects of the SAC system; an entirely proper concern for privacy means that SAC nonsense, although endemic, fails to be exposed to the public critique that is so very much needed.

UPDATE (26/7)

Once again, this time in response to this post, a student from a “good” school has contacted us in regard to their SAC. This was a Specialist SAC, and the student had contacted us because the teachers/writers had screwed up: some tech aspects of the SAC were a mess, and the subsequent clean-up of the mess was clearly disingenuous and clearly insufficient. But, as always, the situation was much worse that the student suggested.

The student’s SAC was ridiculous. From beginning to end it was pointless, CAS-driven pseudo-modelling. It had the idiotic parameters thrown in. It was poorly written. It displayed poor mathematical understanding, leading to ridiculous own goals. It. Was. Not. Mathematics.

And, we can’t write the details of any of this.

The Methods Intelligence Test

Psychologist Daniel Kahneman dedicates his book Thinking Fast and Slow  to the memory of Amos Tversky, his long-time collaborator. Tversky was considered so brilliant by his colleagues that they came up with the Tversky Intelligence Test:

The faster you realise that Amos Tversky is smarter than you, the smarter you are.

It has occurred to us that there is a similar Methods Intelligence Test:

The slower you realise that Methods is stupider than you, the stupider you are.

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.