MitPY 8: Verification Code

It’s a long time since we’ve had a MitPY. But, the plague goes on (including the plague of right-wing Creightons).

This one comes from frequent commenter Red Five, and we apologise for the huge delay in posting. It is targeted at those familiar with and, more likely, struggling with Victoria’s VCE rituals:

VCAA uses some pretty strange words in exam questions, and the more exam papers I read, especially for Specialist Mathematics 34, the more I can’t get a firm idea of how they distinguish between the meanings of “show that“, “verify that” and “prove that“.

Verify” seems to mean “by substitution”, “show that” seems to mean “given these very specific parameters” and “prove that” seems to be more general, but is it really this simple?

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.

WitCH 38: A Deep Hole

This one is due to commenter P.N., who raised it on another post, and the glaring issue has been discussed there. Still, for the record it should be WitCHed, and we’ve also decided to expand the WitCHiness slightly (and could have expanded it further).

The following questions appeared on 2019 Specialist Mathematics NHT, Exam 2 (CAS). The questions are followed by sample Mathematica solutions (screenshot corrected, to include final comment) provided by VCAA (presumably in the main for VCE students doing the Mathematica version of Methods). The examination report provides answers, identical to those in the Mathematica solutions, but indicates nothing further.

UPDATE (05/07/20)

The obvious problem here, of course, is that the answer for Part (b), in both the examination report and VCAA’s Mathematica solutions, is flat out wrong: the function fk will also fail to have a stationary point if k = -2 or k = 0. Nearly as bad, and plenty bad, the method in VCAA’s Mathematica solutions to Part (c) is fundamentally incomplete: for a (twice-differentiable) function f to have an inflection point at some a, it is necessary but not sufficient to have f’’(a) = 0.

That’s all pretty awful, but we believe there is worse here. The question is, how did the VCAA get it wrong? Errors can always occur, but why specifically did the error in Part (b) occur, and why, for a year and counting, wasn’t it caught? Why was a half-method suggested for Part (c), and why was this half-method presumably considered reasonable strategy for the exam? Partly, the explanation can go down to this being a question from NHT, about which, as far as we can tell, no one really gives a stuff. This VCAA screw-up, however, points to a deeper, systemic and much more important issue.

The first thing to note is that Mathematica got it wrong: the Solve function did not return the solution to the equation fk‘ = 0. What does that imply for using Mathematica and other CAS software? It implies the user should be aware that the machine is not necessarily doing what the user might reasonably think it is doing. Which is a very, very stupid property of a black box: if Solve doesn’t mean “solve”, then what the hell does it mean? Now, as it happens, Mathematica’s/VCAA’s screw-up could have been avoided by using the function Reduce instead of Solve.* That would have saved VCAA’s solutions from being wrong, but not from being garbage.

Ask yourself, what is missing from VCAA’s solutions? Yes, yes, correct answers, but what else? This is it: there are no functions. There are no equations. There is nothing, nothing at all but an unreliable black box. Here we have a question about the derivatives of a function, but nowhere are those derivatives computed, displayed or contemplated in even the smallest sense.

For the NHT problem above, the massive elephant not in the room is an expression for the derivative function:

    \[\color{red} \boldsymbol{f'_k(x) = -\frac{x^2 + 2(k+1)x +1}{(x^2-1)^2}}\]

What do you see? Yep, if your algebraic sense hasn’t been totally destroyed by CAS, you see immediately that the values k = 0 and k = -2 are special, and that special behaviour is likely to occur. You’re aware of the function, alert to its properties, and you’re led back to the simplification of fk for these special values. Then, either way or both, you are much, much less likely to screw up in the way the VCAA did.

And that always happens. A mathematician always gets a sense of solutions not just from the solution values, but also from the structure of the equations being solved. And all of this is invisible, is impossible, all of it is obliterated by VCAA’s nuclear weapon approach.

And that is insane. To expect, to effectively demand that students “solve” equations without ever seeing those equations, without an iota of concern for what the equations look like, what the equations might tell us, is mathematical and pedagogical insanity.

 

*) Thanks to our ex-student and friend and colleague Sai for explaining some of Mathematica’s subtleties. Readers will be learning more about Sai in the very near future.