WitCH 29: Bad Roots

This one is double-barrelled. A strange multiple choice question appeared in the 2019 NHT Mathematical Methods Exam 2 (CAS). We had thought to let it pass, but a similar question appeared in last’s weeks Methods exam (no link yet, but the Study Design is here). So, here we go.

First, the NHT question:

The examination report indicates the correct answer, C, and provides a suggested solution:

\Large\color{blue} \boldsymbol{ g(x)=f^{-1}(x)=\frac{x^{\frac15}-b}{a},\ g'(x) = \frac{x^{-\frac45}}{5a},\ g'(1) = \frac1{5a}}

And, here’s last week’s question (with no examination report yet available):

WitCH 27: Uncomposed

Ah, so much crap …

Tons of nonsense to post on, and the Evil Mathologer is breathing down our neck. We’ll have (at least) three posts on last week’s Mathematical Methods exams. This one is by no means the worst to come, but it fits in with our previous WitCH, so let’s quickly get it going. It is from Exam 1. (No link yet, but the Study Design is here.)

WitCH 26: Imminent Domain

The following WitCH is pretty old, but it came up in a tutorial yesterday, so what the Hell. (It’s also a good warm-up for another WitCH, to appear in the next day or so.) It comes from the 2011 Mathematical Methods Exam 1:

For part (a), the Examination Report indicates that f(g)(x) =([x+2][x+8]), leading to c = 2 and d = 8, or vice versa. The Report indicates that three quarters of students scored 2/2, “However, many [students] did not state a value for c and d”.

For Part (b), the Report indicates that 84% of students scored 0/2. After indicating the intended answer, (-∞,-8) U (-2,∞) (-∞,-8] U [-2,∞) or R\backslash(-8,-2), the Report goes on to comment:

“This question was very poorly done. Common incorrect responses included [-3,3] (the domain of  f(x); x ≥ -2 (as the ‘intersection’ of  x ≥ -8 with x ≥ -2); or x ≥ -8 (as the ‘union’ of x ≥ -8 with x ≥ -2). Those who attempted to use the properties of composite functions tended to get confused. Students needed to look for a domain that would make the square root function work.”

The Report does not indicate how students got “confused”, although the composition of functions is briefly discussed in the Study Design (page 72).

WitCH 24: The Fix is In

We’ve finally found some time to take a look at VCAA’s 2019 NHT exams. They’re generally bad in the predictable ways, and they include some specific and seemingly now standard weirdness that we’ll try to address soon in a more systematic manner. WitCHwise, we were tempted by a number of questions, but we’ve decided to keep it to two or three.

Our first NHT WitCH is from the final question on Exam 2 (CAS) of Mathematical Methods:

As usual, the NHT “Report” indicates nothing of how students went, and little of what was expected. In regard to part f, the Report writes,

p(x) = q(x) = x, p'(x) = q'(x) = 1, k = 1/e

For part g, all that the Report provides is the answer, k = 1.

The VCAA also provides sample Mathematica solutions to schools trialling Methods CBE. For the questions above, these solutions are as follows:

Make of it what you will.

The MAV and a Matter of Opinion

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

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

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

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


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

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

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

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

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

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

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

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

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

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

e) The author has not contested the rejection.

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

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

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

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

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


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

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

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

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

VCAA’s SAC of Roaming

One type of educational horror that we haven’t yet written about are SACs, those internal assignmenty-examinationy things that make every second week of Year 12 studies a living hell. It is a tricky topic since SACs are school-based, often teacher-specific, and our primary goal is to attack inept authority. In that regard, schools and beleaguered teachers are in a weird middle ground, part victim and part villain, and they already have plenty of critics. Nonetheless, SACs are the sea in which students and teachers swim (or sink), and mathematics SACs are typically appalling; the overwhelming majority of mathematics SACs that we see are pointless, anti-mathematical, error-strewn blivits. So, something has to be written about such SACs of shit. And, we have a plan.

Our hand has been forced a little, however, by an email we received from a VCE student. The student is taking Mathematical Methods CBE, the trial version of Methods that uses Mathematica instead of CAS, and the student wrote about a recent Mathematica-based SAC at their school. We then asked a teacher at the school about the SAC, and they confirmed the students’ report.

This then is the students’ story, exactly as written to me.

I’m not sure of how many other students you know that are doing CBE methods but my sac today served pretty well to show how awful things can go, so a new perspective is always welcome right?

Starting from the top, we have school-provided laptops with Mathematica preinstalled. So we go in, and we have to utilise this thing called a palette which takes control of Mathematica (I have many complaints against the palette) and downloads the SAC from some remote server. No problems here right? Well, I’d imagine 200 people simultaneously downloading an item from a server would MAYBE just MAYBE cause some congestion in the network. Hell breaks loose here, a class has their file downloaded and enters reading time while the other 4 or so classes are in utter chaos. The downloader is failing over and over while also saying it succeeded. This goes on for a good half an hour before the teachers collectively decided that the sac would be rescheduled to Thursday. The class that began reading? Oh, they just stop. God knows what they could have done, taken photos of the sac with a snipping tool, copied the files over, the possibilities are endless. This whole thing is almost appalling yet terrifying because this is what I’ll have to do at the end of the year. I’ve got a load of other concerns, among the unending variety of methods to do a question with Mathematica and how an assessors report would assign marks, to which our official VCAA quiz provided pissant “solutions” that were often wrong.

Anyway, my slightly irritated take on the abhorrent state of the CBE system, which I thought may be interesting to you.

To that, the teacher at the school added the following:

[The former Head of mathematics at the school] made a deal with the devil and agreed to the school doing Methods CBE. I don’t feel the consultation process valued the feedback from myself and other teachers – it was always going to happen despite the misgivings of other teachers. I can’t help but think that my feelings on the issue is the reason I’m not teaching Maths Methods this year (for the first time in 9 years). There are many problems with this deal – it makes my blood boil. The current head of maths is a very decent guy and has done a fantastic job dealing with the mess he inherited.

I heard that the SAC was a disaster and actually saw events unfolding from afar – like watching a car crash in slow motion. Blind Freddy could have seen what was going to happen. As I left school, I saw the VCE coordinator and the current Head of Maths running around grim faced.

All of the student’s concerns are legitimate. Furthermore, the SAC was meant to run until 4.45 pm, so many students will have made alternative and inconvenient arrangements to accommodate this and now they have to do it all over again. Not to mention what it’s done to the stress levels of many students. Not to mention the time and resources that had to be expended re-writing the SAC. At every stage VCAA have washed their hands of CBE problems and left the school to do its dirty work, using the students as the guinea pigs.


1. The palette provided by VCAA had a bug.

2. The VCAA server failed. VCAA are trying to blame the school for both errors and no apology has been given. Re: The server fail. VCAA said that the school should have downloaded from the server prior to the SAC starting (which is not practical). VCAA are saying everything worked fine at the other CBE schools (which all have small student cohorts as opposed to our school’s cohort of over 200, which makes a big difference).

That’s it. Our own point of view is that SACs are all but guaranteed to be awful and Mathematica in the classroom is all but guaranteed to be awful. Here, however, those predictable awfulnesses are beside the point. The point here is VCAA’s Trumplike level of incompetence combined with VCAA’s Trumplike unwillingness to accept responsibility.

WitCH 23: Speed Bump

Our second WitCH of the day also comes from the 2017 VCE Specialist Mathematics Exam 2. (Clearly an impressive exam, and we haven’t even gotten to the bit about using inverse trig functions to design a brooch.) It is courtesy of the mysterious SRK, who raised it in the discussion of an earlier WitCH.

Question 5 of Section B of the (CAS) exam concerns a boat and a jet ski. Though SRK was concerned with one particular aspect, the entire question is worth pondering:

The  Examiner’s Report indicates an average student score of 1.4 on part a, and comments,

Students plotted the initial positions correctly but significant numbers of students did not label the direction of motion or clearly identify the jet ski and the boat. Both requirements were explicitly stated in the question.

For part i, the Report indicates an average score of 1.3, and comments,

Most students found correct expressions for velocity vectors. The most common error was to equate these velocity vectors rather than equating speeds. 

For part ii, the Report gives the intended answer as (3,3). The Report indicates that slightly under half of students were awarded the mark, and comments,

Some answers were not given in coordinate form.

For part i, the Report suggests the answer {\sqrt{(\sin t - 2\cos t)^2 + (1 + \sin t + \cos t)^2}} (with the displayed answer adorned by a weird, extra root sign). The report indicates that a little over half of the students were awarded the mark, and comments,

A variety of correct forms was given by students; many of these were likely produced by CAS technology, including expressions involving double angles. Students should take care when transcribing expressions from technology output as errors frequently occur, particularly regarding the number and placement of brackets. Some incorrect answers retained vectors in the expression.

For Part ii, the Report indicates the intended answer of 0.33, and that 15% of students were awarded the mark for this question. The Report comments,

Many students found this question difficult. Incorrect answers involving other locally minimum values were frequent.

The Report indicates an average score of 1.3 on part d, and comments;

Most students correctly equated the vector components and solved for t . Many went on to give decimal approximations rather than supplying the exact forms. Students are reminded of the instruction saying that an exact answer is required unless otherwise specified.

Lots there. Get hunting.