Thank you all for coming to the funeral that Mommy decidedly didn’t want. I apologise in advance for this mess of a tribute. I’m used to public speaking, but it usually involves assigning homework, and doesn’t include Mommy watching from above and judging. Luckily, she’s probably not listening, and is just staring, shocked that I’m finally wearing shoes. It’s a convenient distraction from a eulogy Mommy didn’t want, delivered by someone with no clue what a eulogy is.

I think I’m supposed to say who Mommy was, and I guess that’s easy. She was Mommy. For decades Jeff and I have endured the bemusement of people hearing overgrown men referring to their mother in such a childlike manner. But that’s who she was, a loving, securing, motherly presence, from our births until her final day.

And probably long before our births. Mommy was born in 1928, just in time for the great depression, which helped make a mess of her childhood and the childhood of her brother Lee and sister Louise. From her early teens, Mommy took on the duties of a parent, working to protect them all from a loving but drunken and inept father, and a cold and irresponsible mother, and mercenary foster parents. That Mommy could emerge from this swamp so determined and strong, and still so loving says exactly who she was.

At 22, after having trained as a nurse, Mommy escaped her family, to marry the wrong guy. Then, she met David, our father, the right guy, sort of; true love discovered in one of America’s most notorious asylums. Mommy and Daddy soon moved from Pennsylvania to California, had Jeff and then me. And since California was not far enough from their families, we moved to Melbourne, which in 1961 was like moving to the Moon.

Once in Melbourne, the marriage almost immediately failed. Mommy found herself on a Victorian era moon, a single mother with two young kids. She devoted the next decade and a half to me and Jeff, to keeping us happy and healthy and safe. She became a teacher and later a nurse again, worked herself ragged to be able to buy us a house in Macleod.

In the late 70s, once Jeff and I had left home for our studies, Mommy looked hard for new social outlets. She joined the Hawthorn football club, a betrayal I’ll now forgive. She met Diane and Steve, and they remained the closest of friends for forty years. You were so, so important to her, and she loved you both so much.

In the early 90s Mommy found herself in Echuca, alone but settled and comfortable, tending her garden and reading about nazis. In 2004, Mommy wrote her memoirs: that’s your homework there. Mommy wrote of her extraordinarily tough life in a clear, honest and heartfelt manner. Mommy closed her memoirs by coming to terms with her past, her loves and her mistakes, and summing up. She wrote:

A psychiatrist once asked me what was the most important thing that ever happened to me in my life and I did not hesitate a second before answering, ‘having my sons’. They gave my life meaning and purpose as nothing else ever could.

Mommy thought of her memoirs as ending things, but there were sixteen more years to go, and they were sixteen of Mommy’s most content and happiest years. She was hugely comforted to have Ying and Jackie appear, sensible and loving souls to watch over her error-prone sons. And then there was Eva and Lillian. Grammy adored you both beyond words. You made her last years, as a proud, doting Grammy so happy and so rich.

As Mommy grew older, it became harder for her to take care of herself, but she refused to do otherwise, refused countless pleadings to live with us or just to move to Melbourne. She figured that it would be the death of her, and she was probably right. Three weeks ago, in Echuca hospital, I told Mommy she would be coming home to Melbourne with me and she immediately replied “Oh, I can’t do that.” It was heart-breaking to explain to her that there was now no choice. But Mommy nodded, and she understood. Better than me, she understood.

Yesterday, I finally got the courage to look at Mommy’s box of important documents. I found the instructions Mommy left for taking care of matters after her death. And, along with a reminder to cancel the newspaper, and the strategically placed advertisement for a cut-rate crematorium service, Mommy left a note for me and Jeff. The note is undated, but here it is:

For Jeff + Marty

My life has been like most other people – a series of ups and downs. I want both of you to know that my life was a happy one only because of you two. From the time when you were small boys right up to the present your love, consideration and basic goodness makes me very proud. You are so good. I am grateful for the love you have shown me over the years. You made my life a happy one.

Love, Mommy

There is plenty there with which to argue. There was plenty more that made her happy, including very much those here today. But it shows who she was. She was Mommy. Thank you for being here to say goodbye to Mommy, to Marian, to Grammy. Mommy didn’t want this, she didn’t want us here. But it is good and it is right that we are, to remember her and to honour her. Thanks.

WitCH 44: Estimated Worth

This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, Estimating the area under a graph. (It is followed by 21B, Finding the exact area: the definite integral.)

We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.

MAV’s Trials and Tribulations

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

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

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

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

\color{blue}\boldsymbol{f(x) =\frac{2}{(x-1)^2}- \frac{20}{9}}

on the domain \boldsymbol{(-\infty,1)}. Part (a) asks students to find \boldsymbol{f^{-1}} and its domain, and part (b) then asks,

Find the coordinates of the point(s) of intersection of the graphs of \color{blue}\boldsymbol{f} and \color{blue}\boldsymbol{f^{-1}}.

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

Solve  \color{blue}\boldsymbol{\frac{2}{(x-1)^2}- \frac{20}{9} =x}  for  \color{blue}\boldsymbol{x},

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

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

It is simply not legitimate to swap the equation \boldsymbol{f(x) = f^{-1}(x)} for \boldsymbol{f(x) = x}, unless a specific argument is provided for the specific function. When valid, that can usually be done. Easily. We laid it all out, and if anybody in power gave a damn then this type of problem could be taught properly and tested properly. But, no.

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

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

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

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

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

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


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

Q2. There are infinitely many choices of integers \boldsymbol{a} and \boldsymbol{b} with \boldsymbol{a/\sqrt{b}} equal to the indicated answer of \boldsymbol{-2/\sqrt{3}}.

Q3. This is not, or at least should not be, a Methods question. Integrals of the form \boldsymbol{\int\!\frac{f'}{f}\ }  with \boldsymbol{f} non-linear are not, or at least are not supposed to be, examinable.

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

Q5. “Appropriate mathematical reasoning” is a pretty fancy title for the trivial application of a (stupid) definition. The choice of the subscripted \boldsymbol{g_1} is needlessly ugly and confusing. Part (c) is fundamentally independent of the boring nitpicking of parts (a) and (b). The writers still don’t appear to know what “hence” means.

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

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

Q8. The endpoints of the domain for \boldsymbol{\sin x} are needlessly and confusingly excluded. The sole purpose of the question is to provide a painful, Magrittesque method of solving \boldsymbol{\sin x = \tan x}, which can be solved simply and directly.

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

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

UPDATE (28/09/20)

John (no) Friend has located an excellent paper by two Singaporean maths ed guys, Ng Wee Leng and Ho Foo Him. Their paper investigates (and justifies) various aspects of solving \boldsymbol{f(x) = f^{-1}(x)}.

Bernoulli Trials and Tribulations

This one feels relatively minor to us. It is, however, a clear own goal from the VCAA, and it is one that has annoyed many Mathematical Methods teachers. So, as a public service, we’re offering a place for teachers to bitch about it.*

One of the standard topics in Methods is the binomial distribution: the probabilities you get when repeatedly performing a hit-or-miss trial. Binomial probability was once a valuable and elegant VCE topic, before it was destroyed by CAS. That, however, is a story is for another time; here, we have smaller fish to fry.

The hits-or-misses of a Binomial distribution are sometimes called Bernoulli trials, and this is how they are referred to in VCE. That is just jargon, and it doesn’t strike us as particularly useful jargon, but it’s ok.** There is also what is referred to as the Bernoulli distribution, where the hit-or-miss is performed exactly once. That is, the Bernoulli distribution is just the n = 1 case of the binomial distribution. Again, just jargon, and close to useless jargon, but still sort of ok. Except it’s not ok.

Neither the VCE study design nor, we’re guessing, any of the VCE textbooks, makes any reference to the Bernoulli distribution. Which is why the special, Plague Year formula sheet listing the Bernoulli distribution has caused such confusion and annoyance:

Now, to be fair, the VCAA were trying to be helpful. It’s a crazy year, with big adjustments on the run, and the formula sheet*** was heavily adapted for the pruned syllabus. But still, why would one think to add a distribution, even a gratuitous one? What the Hell were they thinking?

Does it really matter? Well, yes. If “Bernoulli distribution” is a thing, then students must be prepared for that thing to appear in exam questions; they must be familiar with that jargon. But then, a few weeks after the Plague Year formula sheet appeared, schools were alerted and VCAA’s Plague Year FAQ sheet**** was updated:

This very wordy weaseling is VCAA-speak for “We stuffed up but, in line with long-standing VCAA policy, we refuse to acknowledge we stuffed up”. The story of the big-name teachers who failed to have this issue addressed, and of the little-name teacher who succeeded, is also very interesting. But, it is not our story to tell.


*) We extend our standard apology to all precious statisticians for our language.

**) Not close to ok is the studied and foot-shooting refusal of the VCAA and textbooks to use the standard and very useful notation q = 1 – p.

***) Why on Earth do the exams have a formula sheet?

****) The most frequently asked question is, “Why do you guys keep stuffing up?”, but VCAA haven’t gotten around to answering that one yet.

One FEL Swoop: The Further Error List

This is the home for Further Mathematics exam errors. The guidelines are given on the Methods error post, and there is also a Specialist error post.


2019, Exam 1 (Here, and report here)

QB6 (added 21/09/20) The solution requires that a Markov process is involved, although this is not stated, either in the question or in the report.

2018 NHT, Exam 1 (Here, and report here)

MCQ4 (added 23/09/20) The question provides a histogram for a continuous distribution (bird beak sizes), and asks for the “closest” of five listed values to the interquartile range. As the examination report almost acknowledges (presumably in time for the grading), this cannot be determined from the histogram; three of the listed values may be closest, depending upon the precise distribution. The report suggests one of these values as the “best” estimate, but does not rely upon this suggestion. See the comments below.

2015 Exam 1 (Here, and report here)

MCQ9 Module 2 (added 30/09/20) The question refers to cutting a wedge of cheese to make a “similar” wedge of cheese, but the new wedge is not (mathematically) similar. The exam report states that the word “similar” was intended “in its everyday sense” but noted the confusion, albeit in a weasely, “who woulda thought?” manner. A second answer was marked correct, although only after a fight over the issue.

Hard SEL: The Specialist Error List

This is the home for Specialist Mathematics exam errors. The guidelines are given on the Methods error post, and there is also a Further error post.


2006 EXAM 2 (Here, and report here)

MCQ20 (added 24/09/20) The notation F_1, F_2, F_3 refers to the forces in the question being asked, and seemingly also in the diagram for the question, but to the magnitudes of these forces in the suggested answers. The examination report doesn’t acknowledge the error.

MELting Pot: The Methods Error List

We’re not really ready to embark upon this post, but it seems best to get it underway ASAP, and have commenters begin making suggestions.

It seems worthwhile to have all the Mathematical Methods exam errors collected in one place: this is to be the place.*

Our plan is to update this post as commenters point out the exam errors, and so slowly (or quickly) we will compile a comprehensive list.

To be as clear as possible, by “error”, we mean a definite mistake, something more directly wrong than pointlessness or poor wording or stupid modelling. The mistake can be intrinsic to the question, or in the solution as indicated in the examination report; examples of the latter could include an insufficient or incomplete solution, or a solution that goes beyond the curriculum. Minor errors are still errors and will be listed.

With each error, we shall also indicate whether the error is (in our opinion) major or minor, and we’ll indicate whether the examination report acknowledges the error, updating as appropriate. Of course there will be judgment calls, and we’re the boss. But, we’ll happily argue the tosses in the comments.

Get to work!

*) Yes, there are also homes for Specialist Mathematics and Further Mathematics errors.


2016 EXAM 2 (Here, and report here)

Q3(h), Section B (added 06/10/20)- discussed here. This is the error that convinced us to start this blog. The question concerns a “probability density function”, but with integral unequal to 1. As a consequence, the requested “mean” (part (i)) and “median” (part (ii)) make no definite sense.

There are three natural approaches to defining the “median” for part (ii), leading to three different answers to the requested two decimal places. Initially, the examination report acknowledged the issue, while weasely avoiding direct admission of the fundamental screw-up; answers to the nearest integer were accepted. A subsequent amendment, made over two years later, made the report slightly more honest, although the term “screw-up” still does not appear.

As noted in the comment and update to this post, the “mean” in part (i) is most naturally defined in a manner different to that suggested in the examination report, leading to a different answer. The examination report still fails to acknowledge any issue with part (i).

Q4(c), Section B (added 25/09/20) The solution in the examination report sets up (but doesn’t use) the equation dy/dx = stuff = 0, instead of the correct d/dx(stuff) = 0.

2016 EXAM 1 (Here, and report here)

Q5(b)(i) (added 24/09/20) The solution in the examination report gives the incorrect expression \pm\sqrt{e^x} - 1 in the working, rather than the correct \pm\sqrt{e^x -1}.

2014 EXAM 2 (Here, report here)

MCQ4 (added 21/09/20) – discussed here. The described function need not satisfy any of the suggested conditions, as discussed here. The underlying issue is the notion of “inflection point”, which was (and is) undefined in the syllabus material. The examination report ignores the issue.

2011 EXAM 2 (Here, and report here)

Q4, Section 2 (added 23/09/20) The vertex of the parabola is incorrectly labelled (-1,0), instead of (0,-1). The error is not acknowledged in the examination report.

2011 EXAM 1 (Here, and report here)

Q7(b) (added 23/09/20) The question asks students to “find p“, where \boldsymbol{p} is the probability that a biased coin comes up heads, and where it turns out that \boldsymbol{p^2(4p-3)=0}. The question is fatally ambiguous, since there is no definitive answer to whether \boldsymbol{p=0} is possible for a “biased coin”.

The examination report answer includes both values of \boldsymbol{p}, while also noting “The cancelling out of p was rarely supported; many students incorrectly [sic] assumed that p could not be 0.”  The implication, but not the certainty, is that although 0 was intended as a correct answer, students who left out or excluded 0 could receive full marks IF they explicitly “supported” this exclusion.

This is an archetypal example of the examiners stuffing up, refusing to acknowledge their stuff up, and refusing to attempt any proper repair of their stuff up. Entirely unprofessional and utterly disgraceful.

2010 EXAM 2 (Here, report here)

MCQ17 (added 28/09/20) – discussed here. Same as in the 2014 Exam 2, above: the described function need not satisfy any of the suggested conditions, as discussed here.

2007 EXAM 2 (Here, report here and discussed here)

MCQ12 (added 26/09/20) Same as in the 2014 Exam 2, above: the described function need not satisfy any of the suggested conditions, as discussed here.

MitPY 9: Team Games

This MitPY is from commenter HollyBolly, who asked on the previous MitPY for some advice on diplomacy.*

Can you guys after all the serious business give me some advice for this situation: on a middle school Pythagoras and trig test, for a not very strong group of students. Questions are to be different from routine ones provided with the textbook subscription. I try “Verify that the triangle with sides (here: some triple, different from 3 4 5) is right, then find all its angles”. After reviewing, the question comes back: “Verify by drawing that a triangle with sides…”

How do you respond if that review has come from:

A. The HoD;

B. A teacher with more years at the school than me but equal in responsibilities in the maths department;

C. A teacher fresh from uni, in their20s.


*) Yeah, yeah. We’ll stay right out of the discussion on this one.

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?