WitCH 28: Tone Deaf

We haven’t yet had a chance to go through the 2019 VCE exams, but this question was flagged to me independently by two colleagues: let’s call them Dr. Death and Simon the Likeable. It’s from Mathematical Methods Exam 2 (CAS). (No link yet.)

UPDATE (05/07/20)

Even ignoring the stuff-ups, this question is ugly and pointless; the pseudo-applied framing is ugly and pointless; the CASification is ugly and pointless; the back-to-front integral is ugly and pointless; the matrix equation is ugly and pointless; the transformation is really ugly and really pointless. Part (f) is the pinnacle of ugliness and pointlessness, but the entire question is swill, from beginning to end.

And then there’s Part (e). “This question was not answered well” the examiners solemnly intone. Gee, really? Do you think your question being completely stuffed might have had something to do with it? Do you think maybe having a transformation of x when there’s not an x in sight may have been just a tad confusing? Do you think that the transformation then resulting in a function of t was maybe not the smartest move? Do you think writing an integral backwards was perhaps just a little too cute? Do you think possibly referring to the area of, rather than to the value of, an integral was slightly clunky? And, most importantly, do you think perhaps asking a question for which there is an infinite and impenetrable jungle of answers may have been an exercise in canyon-sized incompetence?

But, sure, those troublesome students didn’t answer your question well.

Part (e) was intended to have students find a transformation of the function f that effectively switches the behaviour on the intervals [0,4] and [4,6] to the intervals [2,6] and [0,2].  Ignoring the fact that the intended question was asked in an absurdly opaque manner, and ignoring the fact that no motivation for the intended question was either provided or is imaginable, the question asked was entirely different, and was ridiculous.

Writing the transformation out,

    \[\boldsymbol{\left\{\aligned &X = ax + c \\ &Y = by + d, \right.\endaligned}\]

we then have

    \[\boldsymbol{\left\{\aligned &x = \frac{X - c}{a} \\ & y = \frac{Y - d}{b}. \right.\endaligned}\]

So, the function y = f(t) y = f(x) can be written

    \[\boldsymbol{\dfrac{Y - d}{b} = f\!\left(\dfrac{X - c}{a}\right).}\]

Solving for Y, that means our transformed function Y = g(X) can be written

    \[\boldsymbol{g(X) = b\, f\!\left(\dfrac{X - c}{a}\right) + d.}\]

Well, this is our function g unless a = 0, in which case g doesn’t exist. Whatever. Back to the swill.

Using the result from Part (d), we have Part (e) asking for a, b, c and d such that

    \[\boldsymbol{\int\limits_2^0 + \int\limits_2^6 \ \left[ b\, f\! \left(\dfrac{X - c}{a}\right) + d\right]  \, {\rm d}X \ = \ \dfrac{15}{\pi}.}\]

What then are the solutions to this equation? The examination report lists a couple of families and then blithely remarks “There are other solutions”. Really? Then why didn’t you list them, you clowns? 

We’ll tell you why. Because the complete solution to this monster is a God Almighty multi-infinite mess. As a starting idea, pick any three of the variables, say a and b and c, to be whatever you want, and then try to adjust the fourth variable, d, to solve the equation. We’ll offer a prize for anyone who can give a complete solution. 

This question is as good an example as there can be of the pointlessness, the ugliness and the monumental klutziness of VCAA’s swamp mathematics.

55 Replies to “WitCH 28: Tone Deaf”

  1. 3e: I don’t understand how there’s a transformation in terms of x applied to a function in terms of t. And wouldn’t there also be an infinite number of solutions for a, b, c, d?

  2. Also, isn’t 3e grammatically senseless? An integral doesn’t have an area, it has a value. Whether or not that value represents an area is another matter entirely. Maybe?

    1. Those were my thoughts too, Red Five. And even if you admit a charitable reading of the question, then you’re confronted with what Tau says above.

    1. RF, I’m not really concerned about EAL students for now. I don’t think EFL students have a hope of making sense of this garbage.

  3. Coming back to this question there are a few big problems (and probably a really big, fundamental one that I’ve somehow missed…):

    The variable x is never defined since the original function has dependent variable t. OK, perhaps students will just miss this issue in their rush to push buttons and solve the equations. VCAA would not be forgiving of students if they wrote x instead of t however!
    Why do the integrals need to be 2 to 0 and 2 to 6, why can’t they be 0 to 2 and 6 to 2? If absolute value functions hadn’t been (for some reason known only to VCAA) removed from the methods course, the translation would be simpler, surely?
    Finally, surely it is just a case of a=b=1, d=0 and c=some general solution with a +8k in the answer?

        1. Hopefully some time this year. The delay is an absolute disgrace – the NSW reports get published within weeks of the exam.

          I think VCAA is still trying to figure out where the word ‘the’ should go in its gutless “All students given mark” comment – this time for MCQ 12.

          1. Most other states publish their reports within days. NSW is interesting, they publish two sets of reports from what I can see, the “solutions” and the “marking guide”. Different way of marking, too. VCAA could learn a lot… but won’t.

            1. VCAA will never publish their marking guide. VCAA has too much to hide, publishing the marking guide would expose it to too much risk. What I don’t understand is why teachers think that things like Meet the Asses are useful. None of those asses will ever say anything worthwhile – they certainly won’t comment on the marking guide. Any meaningful question is met with “I can’t tell you that”, “I don’t know that” etc. All they give is a preview of the Examination Report – maybe that’s why the reports come out so late, it’s a cosy arrangement to make gravy trains like Meet the Asses look worthwhile …?

              1. Meh. Its a day off work to catch up with other teachers I only see at these things or MAVCON (which I’ve stopped going to/presenting at anyway). Its PD hours and a few laughs (privately, to myself of course, although at a recent SAC workshop I will admit to not being able to contain myself during the prob and stats presentation…)

                1. Yes, that explains meeting the asses during the day (and has been said in other posts). However, many of these sessions run in the evening – who wants to waste their time meeting asses in the evening (unless they’ve been suckered into it??)

                  1. I don’t go to those. At my school it seems to be the first time teachers of a subject that ask to go… Some teachers seem to think it is good for SAC 1 ideas in MM and SM. I’m a lot less certain about the merits.

              2. JF, I’m in two minds about the assessors. They’re not responsible for the idiot culture of the VCAA, and they don’t make the idiot rules that result. On the other hand, when they sign up to be assessors they either know the VCAA culture is idiotic with idiot rules, or they don’t; either assumption is then a basis for criticism.

                1. I’d posit that every assessor knows the culture they’ve implicitly signed up to. That’s on their own conscience. What really shits me are the ones who then present at things like Meet the Asses, duping attendees into thinking they’re going to get the inside scoop on how the exams were marked.

                  Maybe this is why the MAV now provides solutions written by those asses, so that attendees think they’re leaving with something snort valuable (from the solutions I’ve seen, I’m tempted to retract this sentence. The freely available itute solutions are superior and more reliable – they consistently flag when a VCAA question is defective).

                  (If the MAV had integrity, it would include a clear disclaimer stating that the asses who present are bound by confidentiality agreements with VCAA not to discuss marking schemes or anything else that won’t be in the Examination Report. But that’s not good for business).

                  1. Well, JF, I still think you’re probably being a little (but only a little) too hard on the assessors.

                    There’s no question that every assessor, and every teacher and every student, is fully aware of the Little Hitler culture of VCAA grading. It is not at all obvious, however, that the assessors realise how pointless and how damaging is this Little Hitlerism. Such pettiness is extremely common in mathematics teaching, in both schools and universities.

                    I think it’s worth a whole blog post but, in brief, I think such (lack of) thinking stems from a fundamentally weak sense of mathematics. Without a clear understanding of the way mathematics works, one clings desperately to the “rules”. Any good teacher recognises that kind of fear in some (way too many) students, but the exact same fear is evident in some (way too many) teachers and administrators and textbook writers and university lecturers.

                    For similar reasons, it is also not clear to me that the assessors realise just how much is wrong in VCE mathematics, both in specific fact and in general sense. The assessor solutions and the few assessor meetings I’ve attended suggest to me they are simply not sufficiently strong in mathematics to realise the fundamental third-ratedness of VCE mathematics.

                    In summary, forgive them a little, for they know not what they do. But, as for the super-cosy MAV-VCAA “Meet the Assessors” partnership, yes, that is repugnant.

                  2. JF:
                    Thanks for indicating the over-due published Exam 2 Assessor’s report.
                    We all have seen the “adjusted exam specification” as well as “Adjusted Formula Sheet”.
                    Looks like there would be less ill-post questions in the end of this year, hopefully.

                    On the other hand, I think this time the 2019 Exam 2 Report is “pretty much” the marking scheme itself, except for specific marking allocations and the way to mark the transformation question with wrong letter x instead of t.

                    An extra question to ask you: why do you think MCQ12 in MA094 should have been given a free mark for all students? I think the answer acceptable was A. Some students sent me their mark statement for spesh last year, which indicates the “correct answer” as A.

                    P. N.

                    1. Hi P.N. What exam is MA094? Do you mean Specialist 2019 Exam 2? If so, in Q12 none of the equations can be solved because the given vector resolute is a geometrical impossibility (it is longer than the original vector). The question is defective. Can the equations in Option A actually be solved ….?

                    2. P.N. If you are referring to MCQ12 from last year’s SM Exam 2, that question is completely stuffed: it is stuffed in design, in execution and in practical fact. Anyone suggesting otherwise is being moronic or deceitful. I understand if the examiners chose A as the “correct” answer but, as is so annoyingly frequent, they are wrong.

        2. Hello Red Five:

          I think the author in charge had already finished this SM report long ago (in the middle of March), but it then becomes mysterious for examination operation units to then publish the document… If I see the updates, I would try to flag it in this post ASAP.

          The setting panel has already done this year’s Spesh paper (originally), but due to the Covid-19, they have to delete the stat questions and fill it up or expand existing questions to a satisfactory level.

          Well, let’s pray for the accuracy, rigor, depths and broadness for the MA093 and MA094 paper this year, as well as MA113 and MA114…in such a special 2020!


          1. PN, I’m also confused by your acronyms. As for the SM report having been completed “long ago”, March is already inexcusable.

            1. JF and Marty,

              Sorry for any confusion I made.

              MA07=Further maths
              MA09=Specialist maths
              MA11=Mathematical Methods

              An extra “3” represents Exam 1, and an extra “4” for Exam 2.

          2. Thanks PN, I still don’t understand why the document hasn’t been published when other reports have been. Sure, some modifications to exams are needed, but I find it quite odd when the other reports are out.

            1. My understanding (based on things said at MAV Conference sessions discussing the exams) was that the Chief Assessor is not on the Exam writing panel. So modification of the 2020 exam should have no bearing on the writing and publishing of the 2019 Examination Report.

              RF, I’ve previously commented (see below) on what I think are the answers to your oddity. I would speculate that an additional reason is debate/concern over how to report on Q12 given the blowtorch applied to it both here (https://mathematicalcrap.com/2020/02/14/witch-35-overly-resolute/) and elsewhere.

              1. JF and Marty,

                Thanks for reminding me the old post.

                Yes, I now see where the mistake is.

                Another pick-up wakes up in my memory, once the MCQ12 in 2019 Exam being mentioned.

                Strangely, every time we have a vector MCQ on Spesh exam, it more or less lacks rigorous ground to hold true, theoretically.

                See 2018, Specialist Maths Exam 2, MCQ12. The same position. Similar flaw.

                The ‘correct’ answer again, is A: vector a is parallel to b….

                Honestly speaking, I’d rather say vector a is in the same direction of b, rather than saying parallel….

                By the way, one more mistake to mention (maybe Marty would be interested in picking it up at some stage), 2019 NHT Exam 2, Question 2, part b: “for what values of k will f_k(x) has no stationary points?”

                The official answer suggested ‘-2<k<0’. However, they ignored the fact that, when k = – 2 or k = 0, there will be one cancellable factor for each case of k, and then, the function f_k(x) will reduce to two rectangular hyperbolae respectively…none of which has any stationary point at all…

                1. Jesus. P.N., is the other 2019 NHT error commonly known? I’ll post on it soon.

                  As for MCQ12 on SM 2018, your suggested answer is more precise but I think “parallel” is ok. There is an underlying issue that “parallel” doesn’t work very well for vectors, as I wrote about in this WitCH.

                  1. Unless incentivised in some way to carefully consider VCAA exam questions, I think many teachers (myself included) take a brief glance at them, see nothing but vanilla and move on. So I think errors will often go unnoticed for some time.

                    And in this particular case the error is in the so-called Report, not the exam, and so it will go unnoticed for even longer because:

                    1) It’s not a report, it’s just a list of answers so there’s no useful insight and so no real point in giving it more than glance (just to check if it’s suffered a case of mistaken identity and is actually a report).

                    2) The report comes out so late that it’s become mostly irrelevant anyway.

                    The troubling aspect of this error is that VCAA has published detailed solutions to this exam and gets the same wrong answer for Q2(b). So the error is clearly not a typo in the report. So in all likelihood the marking scheme has the same wrong answer (and therefore incomplete solution) and therefore every single student who sat that exam has been incorrectly marked.

                    One could say meh it’s just another mathematical error among a laundry list of errors. True, but this latest example of mathematical incompetence by VCAA is very special because there’s empirical evidence showing the thinking that led to the error.

                    Does anyone know of alternative solutions available in the public domain (itute don’t do NHT solutions, unfortunately)?

                1. Well spotted P.N.

                  Regarding those interesting implications: VCAA could almost get away with this by saying that it was a typo. Except for one small thing …. So now there are some different interesting implications ….

      1. Thanks, JF. You’re referring to the expression “reflect in the x-axis” in the answer to Part (e)? For me, the funniest part was the deadpan “There are other solutions.” As, to why it takes them half a year, and counting, to come out with the reports, God only knows.

        I’ll look carefully at the report, and try to put up some new posts in the next few days.

        1. Yes, that’s the expression I’m referring to.

          Indeed, and maybe some of those other solutions involve t. At some stage some quiet revisionism will occur.

          Re: Half a year and counting. Surely the answer, in the ratio of 1 : 1: 1, is because

          they can : they don’t give a flying Philadelphian : they’re a gang of idiots.

          You can say what you like about NESA, but at least they get their reports out for teachers in a timely manner.

    1. Thanks, RF. Yes, to everything (once your correction is included). It’s pretty much what Tau wrote above. I’ll try to update this WitCH soon. (So much crap …).

      1. Sure. I completely agree with what Tau wrote, and would add the question of why matrices ONLY occur in VCAA Methods questions for transformation questions such as these… (which are surely more easily done without matrices at this level???)

  4. Thanks for the post- I had read it earlier and one of my conchy students had been doing MM 2019 Exam 2 today and said she had one question she didnt understand ie 3e. I told her not to fret, its nonsense, recalling this discussion.

    1. Thanks, davraus. I’m glad the post was of use, beyond the psychological satisfaction of raking VCAA over the coals. And of course if the VCAA just had the balls to directly admit in the examination report that they fucked up, then students and teachers wouldn’t have to hunt in underworld blogs to find the truth. They really are assholes.

      My broader plan is to have my recent “error list” posts be a complete list of all such screw-ups. I’m just working to find time to add all the errors, such as the one above, of which I’m aware.

  5. In fact, the two ‘solutions’ given in the Examination Report for part (e) are described incorrectly: The Report has mixed up its lefts and rights … The answers are correct, however (as far as they go).

    It’s totally dishonest to ask for “\displaystyle the values” (my emphasis) when there are an infinite number of possible values from an infinite set of families of solutions and VCAA, by its own admission, doesn’t/can’t even give them all. The Report never makes it clear whether or not students were required to include the “\displaystyle + 12n” (multiples of the period of \displaystyle f) as part of the value of \displaystyle c.

    VCAA can try and pretend that all this is no big deal. But it is a big deal. It’s a BIG, BIG fucking deal. Not the least because of the amount of time and effort that has to be spent giving clarifications and explanations to every student that attempts the question and then asks for help because they can’t make sense of it or the Report. So as always, teachers have to clean up VCAA’s shit. I’m writing up a solution – well, a solution that explains the VCAA solution. Mainly for my own convenience (I invest several hours of time to save even more hours of time in the long run …) I’ll post the solution so that others don’t have to waste their valuable time doing the same thing. Thanks a lot – yet again – VCAA. clowns. I hope one day to return the favour. With interest.

    Fortunately, I can refer students to this blog when they ask about this question (and most do). So students don’t just have to take my word that VCAA are a bunch of By the way, I’m also unimpressed with the solutions I’ve seen for this question. They are no better than the swill VCAA gives in the Report. Has anyone seen solutions that clearly explain part (e), as well as mentioning the errors in the change of variable from t to x …?

    1. Furthermore, the restrictions to positive and negative integers in each ‘solution’ for \displaystyle c given in the Examination Report is totally unnecessary since 12 is the period and so \displaystyle f(t + 6 + 12n) = f(t + 6) and \displaystyle f(-t + 6 + 12n) = f(-t + 6) for all integer values of \displaystyle n.

      It’s very clear that VCAA does not understand its own ‘solutions’, let alone the question itself.

        1. If only we have more strong maths warriors like you…JF…

          Must say…what a nice solution document and in-depth discussions!

      1. Yeah, that’s pretty dumb. At that stage, VCAA was probably just trying to rush out of there and pretend the question never existed.

      1. Yes, I know. The deal is much bigger than even a BIG, BIG … deal. But the word seemed strangely apt given the huge amount of unnecessary work VCAA creates due to its incompetence, dishonesty, arrogant disregard and lack of professionalism.

    2. John, your first paragraph makes no sense. I can also see no reason to assume or imagine that students were required to give more than one solution.

      1. OK, first paragraph elaboration:

        For part (e) the Examination Report says:

        “Reflect f in the y-axis and translate 6 units left …
        Alternatively, reflect f in the x-axis and translate 6 units right …”

        I said the Report has mixed up its lefts and rights. And it has. It should have said:

        Reflect f in the y-axis and translate 6 units RIGHT …
        Alternatively, reflect f in the x-axis and translate 6 units LEFT …

        The Reports description of the transformation is incorrect. The answers for a, b, c and d are correct, however (as far as they go).

        Re: “I can also see no reason to assume or imagine that students were required to give more than one solution.”

        Can you see any reason to assume or imagine that students were NOT required to give more than one solution?

        The question is inconclusive. It asks for “the values” … Does that mean all the values or some values? Who knows? The Report gives two solutions (within which there are an infinite numbers of values given for \displaystyle c) but never says how many solutions students needed to find. If only one solution was required, the question should have said “find a set of values …” And since a single solution contains an infinite number of values for \displaystyle c, how many values of \displaystyle c should be given. One, or infinite (in the form 6 + 12n)?

        I was going to say all this earlier but thought I might get misunderstood as thinking the question was OK apart from this wording.

        By the way, I’ll bet London to a brick that:
        1) both solutions were intended, possibly to include the infinite number of values of \displaystyke c until someone outside of VCAA pointed out that “There are other solutions”, a comment in the Report that reeks of a panicked inclusion at the last minute.
        2) The writer had no clue that “There are other solutions”, nor did the vetters, nor did the assesors.

        1. John, the question is obviously a disaster, and I think it is such a disaster that it’s not worth thinking about further. You’re the teacher, but I cannot see the benefit to a Methods student to spending more than two minutes thinking about the question. But, in brief reply:

          0) Since the thing has period of 12, a translation 6 to the left is the same as translation 6 to the right. Nonetheless, I think you have a point, that the translations opposite to the report’s are easier to see.

          1) I think you win your brick. Rereading the exam question, the plain reading is that it is asking for *all* transformations of the given form that work. Even without the screw-up that would have been a killing field, and I wonder if the implied “all” was intended or just another case of idiotic wording. In any case, it’s pretty clear that the eventual grading accepted any individual solution, as evidenced by the very specific solutions indicated in the report, plus the obvious need for VCAA to try salvage something from the disaster. But, the report cannot even state directly and honestly what was eventually accepted as a solution. Utterly disgraceful.

          2) Yes, it is clear that the writers and vetters and assessors had absolutely no idea what they were doing. This question and its handling is as clear an example as there is of VCAA’s incompetence, dishonesty, arrogance and cowardice.

          1. Re: You’re the teacher, but I cannot see the benefit to a Methods student to spending more than two minutes thinking about the question.

            Obviously you’re correct. The problem is convincing the student of this. They think they have to understand every minute element of a VCAA question, regardless of how dumb the question is. It is no easy thing trying to teach students the good advice of Kenny Rogers: “You gotta know when to hold ‘em, know when to fold ‘em, know when to walk away”

            1. When they’re lined up against the wall (one day), I don’t want the pistol with the blanks.

          2. Re 0): I agree that f(t + 6) = f(t – 6). But, and I might stand corrected on this, to set up the correct terminals on the integrals involving g(t), you’ve got to go left to begin with. So translation to the left by 6 and \displaystyle then you can translate \displaystyle f(t = 6) (left or right) by integer multiples of periods.

            Yeah, I know, we’re all beyond caring by now. Except for some present and future Methods students …

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