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)}.

112 Replies to “MAV’s Trials and Tribulations”

      1. I only attempted the fixed point question. Is it indicative of the quality of the rest of the exam…? Ugh, keep it away from students.

        1. It depends what you mean by “quality”. But the fixed point thing is way beyond “lazy”. This crap has been going on for at least ten years.

  1. “Interchange variables, then…” set of steps of this commonly recognised routine may not make much sense to students trying to understand why they are doing this. Say we are to find the inverse of Fahrenheits to Celsius formula, or the other way around, F(C) to C(F). All clear, no “interchanging”. Plotting both on the same set of axes may cause some problems – how to label axes. Dual labelling? Two graphs might be a better solution, perhaps?
    I think pedagogically instructing students to find x in terms of y would work better, like with F and C. Only then, when both graphs are required, on one Cart plane interchanging is needed.
    Can you please inspire me, why f and f-1 need to be presented together at all?
    Re. intersections of f and f-1, changing “one to one” to “strictly increasing” would be sufficient for the points of intersection to lie on y=x then it is the way to go. And yes, proving that some f is strictly increasing should be in the course. Is the voice of the teachers not strong enough?

    1. Banacek, the voice of teachers is not strong enough, but I doubt that that is the issue here. I have met very few teachers who give a stuff about the idiocy of this f = f-inv ritual.

    2. The way I hear it (from a little-name), at least one ‘complaint’ has been made to The Bizarro Twilight Zone (TBTZ) (NB: *The Twilight Zone* represents *quality*) and been brushed off:

      1) Allegedly the fact that the question does not ask students to mathematically justify why f(x) = x can be used is a good enough reason not to give a mathematical justification in the ‘solutions’.

      2) Allegedly eyeballing the curvature of the graph is sufficient to prove that there are two points of intersection and they occur on the line y = x. However …

      3) Allegedly the ‘complaint’ encouraged TBTZ to add a graph to its *cough* ‘mathematical explanation’ (is anyone aware that this actually occurred): Apparently solving f(x) = x is a very common approach that is taught to finding the points of intersection of the graph of a function and its inverse, and the added graph will help to confirm or add to that understanding.

      I’m not sure what the evidence is for assertion 3). I certainly do NOT teach this without also teaching the conditions under which it is valid. Perhaps by ‘very common approach’ TBTZ means that it’s the approach always used by VCAA without any justification.

      So the way I hear it, TBTZ was given the chance to amend its solution by adding some mathematical rigour and decided that adding a graph was sufficient – rigour mortis, one might say.

      And as for finding the inverse function by “interchanging the variables” …. This is the epitome of sloppy, lazy and negligent teaching – and TBTZ is guilty of encouraging and promoting it.

      It is clear that TBTZ solutions for this exam are not intended to be exemplary or instructional. I suggest that the writers are not capable (by competence and/or inclination) of writing such solutions to more than a very superficial level.

      1. JF, I don’t want to get into third hand aspects, which is not necessary, I’ll be briefly respond to the hypothetical arguments later.

      2. Hi, JF. On the possible arguments (not your arguments) that you raise:

        1) The claim that 10(b) doesn’t ask for a “reason” would not be an argument, but a confession. It is inexcusable to ask that question without the presumption or explicit statement that justification is also to be expected. It was inexcusable in 2010, and it is fucking inexcusable now. Furthermore, even if one is stupid enough to set such a no-reason question, that is absolutely no justification for not including an argument as part of the solutions; there is no reason, other than rank laziness, for solutions to be nothing other than expected response.

        In summary, Q10 and its “solution” are screwed for two related but distinct reasons.

        2) There was no graph included at that point in the solutions I saw, but it doesn’t matter, for a bad reason and a worse reason. The bad reason is that, graph or no graph, one should give the argument. The worse reason is that perhaps the writers or reviewers were incapable of giving the argument: if someone actually referred to “curvature” as part of the (Magritte or proper) argument, then it is pretty clear that person doesn’t understand the argument.

        3) This is too stupid to be worthy of a response. “Common” is not the measure of sense or correctness. Not anywhere, and definitely not in Methods.

        1. Nicely put, Marty. I’ll disagree on only one thing:
          Re: “there is no reason, other than rank laziness, for solutions to be nothing other than expected response.”
          There’s another reason: Rank mathematical incompetence.

          Linked to both of the above reasons, I always try to achieve four things with the solutions I write (whether the question be on a commercial trial exam, a VCAA exam, a SAC, or a pissant test):
          1) The solutions provide a clear marking scheme.
          2) The solutions are exemplary.
          3) The solutions are instructional.
          4) The solutions show different methods (when more than one method is available) and the relative merits/demerits of each method are briefly discussed.

          Related to 3), I will include ‘interesting’ relevant background or generalisations. And it goes without saying that I try to ensure that the mathematics is correct.

  2. Hi,
    I wonder how many schools actually fork out the $200+ for these trial exams and sketch solutions when so much similar past exam material is freely available elsewhere?
    Steve R

    1. Quite a few, actually. They’re in high demand by schools to provide a “breadth” of exam material practice to pass on to students.
      Our school purchases these each year, as well as from other companies.
      As to the usefulness of these, as well as their mathematical fluency, that’s something this post is covering.
      Yes I agree though, charging such ridiculous amounts is daylight robbery, but schools don’t have much choice!

      1. Hi Steve.

        There’s plenty of choice. As of 2020, every school has:

        14 Past VCAA exams
        3 Past NHT exams
        However many number of trial exams that have been bought over the years.
        However many *free* exams straight off the internet (at least 14 itute exams etc.)

        I’d estimate every student in every school has access to at least 40 pairs of exam 1 and exam 2. Seriously, is any student going to need more than that? Insanity for any student to think they might.

        As for wanting an October practice exam, how hard to use a commercial exam from a few years earlier?

        Schools buy these trial exams:
        1) out of stupid habit (“But we *always* buy trial exams!”),
        2) out of an insecure misconception that they need to (“But what will our students practice on?”),
        3) because of stupid pressure from students that only the very latest trial exams are of any value and they’ll be disadvantaged if they don’t have them. (“My friend at Thirdrate Secondary College got four 2020 practice exams. How come we didn’t get any!?”)

        Commercial organisations prey on this force of habit, insecurity and stupidity. It is an absolute gravy train.

    2. Yeah, they’re not cheap! I assume that regular commercial organisations can’t charge as much, because they don’t have the Twilight Zone Stamp of Officialness. I’m not sure what I would consider a fair price, in either category, and although it’s not really for me to judge the general usefulness of non-VCAA exams, I’m skeptical.

      Any exam with the TZSoO, however, should be up to some minimum standard. I don’t see how anyone can argue that the Methods 1 trial exam meets any such standard.

    3. Hi SR, it’s easy to do a Fermi calculation. I’d conservatively estimate the costs of writing both trial exam 1 and 2 for any commercial organisation to be as follows:

      1) Writing fees: 3000.
      2) Vettors fees: 1500.
      3) Reviewers fees: 1000.

      So call it 5000 – 6000 for both exams.

      Let’s say 200 for both exams is charged. Then to break even you’d need roughly 25 – 30 schools buying both exams. Furthermore:

      Number of secondary schools in Victoria: Roughly 600.
      Number of commercial organisations selling exams: Roughly 10.
      Assume half the schools buy both exams from a random organisation: 300/10 = 30.

      So at a minimum every organisation is at least breaking even, and I’d bet The Bizarro Twilight Zone is doing much better that that …. Trial exams are big business.

      1. Yes, you hit the nail on the head, JF.
        Trial exam writing is serious business, in fact most “Education” companies are (to name a few, there’s MAV, TSFX, Neap, Access, the list goes on and on; don’t get me started on how many there are).
        Many schools want as many exams as possible, in fact I’m aware that it becomes a point of boasting for some students.
        Student 1: “Our school gave us 7 practice exams from 2020”
        Student 2: “Oh really, we got only 3. Our school’s always so stingy”
        You get the gist – it’s a self fulfilling prophecy. And per your previous comment, it’s an absolute gravy train – not necessarily for the writers but for the organisations that sell these.
        The writers will be lucky to take home 50% of the revenues earnt from exam sales.

  3. I wrote a trial SM exam once for a company that I will not name. The experience was interesting. I did it mainly because (1) I felt I could and (2) I thought it would be a really useful, collaborative experience.

    What actually happened was: I wrote the exam, sent it in and was paid for the hours I worked. A few months later a teacher who had paid the company for the exam queried one of my solutions (the solution was correct, but I did see the teacher’s point and sent a slightly modified solution with the same answers). A teacher who had BOUGHT the paper. Not someone at the company proof reading, checking solutions, having a discussion about appropriate scope of questions… So I never did it again.

    In short: I’m a bit sympathetic towards the exam authors and not so much for the companies that sell them.

    1. Thanks, RF. An interesting and illuminating story. I generally agree with your concluding point. It is hard as hell to write a good exam, even for a good subject and even without the kind of formulaic constraints that are part and parcel with VCE. If a 3rd party exam has an element of clumsiness/error, which is pretty much always the case, I tend to be forgiving, of both the authors and the company.

      There are two reasons why I am not forgiving with the above exam. The first reason is because MAV products have the Twilight Zone Stamp of Officialness, and if you live by the TZSoO then you die by the TZSoO. If the MAV is going to cloak itself with the authority of VCAA, and now AMSI, then they get judged by an appropriately higher standard. Do you want to distinguish the authors from the MAV? It’s possible, but I don’t buy it. I don’t for a minute think MAV works in a way that warrants that distinction.

      The second reason is that, even by general commercial standards, the exam is shoddy. Yes, a couple of the points I made are nitpicks; I didn’t have to pick on, for example, the writers’ misunderstanding of “hence” (but boy, it gets up my nose). But most are very far from nitpicks. A number of the questions simply should have been rejected as not up to standard, or worse.

      1. Re: “If the MAV is going to cloak itself with the authority of VCAA, and now AMSI, then they get judged by an appropriately higher standard.”

        You’re in fine form, Marty. That is exactly right.

    2. Hi RF. It sounds to me like either:
      1) The company knew you were a star and so didn’t need to bother with vetting and reviewing your exams, or
      2) The company decided to increase its profit margin by deliberately not paying for vetting and reviewing (by my estimate this would save a couple of thousand dollars) and hoped that your were a star.

      Every company I’ve done work for has had a robust vetting process. In fact, some of those companies probably had a process superior to VCAA’s. But any vetting process only works if:
      1) the people who do the vetting are mathematically competent (VCAA has illustrated how essential this is on numerous occasions),
      2) the writer is open-minded and intelligent – the writer has to be able to accept feedback when it’s valid and provide good reasons if it’s rejected, and
      3) the vettors are ‘fearless’ – they call it as they see it and don’t get ‘bullied’ into submission.

      In the case of the TBTZ, which I’m sure has a robust vetting process, the process has obviously failed. It’s hard to know where … But in my experience with several companies, it’s the writer that gets the final say (actually the editor gets the final say, but unless the editor is mathematically competent, the writer will get the final say) …

      1. I suspect there are a few issues at play here:

        1. It is hard enough to find *good* VCE Mathematics teachers, so I cannot imagine how hard it is to find *good* exam setters. Part of me wonders how many of the truly good non-VCAA exams were written by teachers either while on leave or after their retirement (I know of one company started by a teacher after their “retirement” – read: got sick of their school leaders and went their own way, successfully)

        2. Assuming (1), finding good proof-readers is likely to be just as difficult, because anyone vetting a paper has to (a) know the curriculum, arguably better than the paper setter, (b) be able to communicate very clearly what is wrong with a question or solution and (c) be able to assist the author to fix the problem without re-writing the question. I suspect that properly proof-reading a paper in this way may take significantly more time than a company may be prepared to pay for…

        3. The pressure to produce good exams just isn’t there. I’ve had really poor examples of trial exams and yet my HOD keeps purchasing them because, (a) the FM teachers like that company, so we may as well buy them all, (b) we have room in the budget, (c) the various “showing off” mentioned earlier that private schools seem under invisible pressure to do (but I’ve never really felt as a teacher) and/or (d) it is what we have always done, why would we change?

        1. RF, re 1 and 2, I’m not sure if you’re talking general commercial or MAV; I don’t think such excuses are valid for the MAV. Re 3, have you considered the possibility that your HoD is an idiot?

          1. Re (1) and (2); I’m not offering an excuse, far from it, but I believe this may be part of the reason.

            Re (3); don’t need to consider. I have the M Ross: “Proof by Contradiction” (unpublished) paper as reference.

        2. Hi RF. You’ve hit the bullseye with every point.

          Re: “I suspect that properly proof-reading a paper in this way may take significantly more time than a company may be prepared to pay for…”

          From experience, I will confirm your suspicion. I’ve been burnt several times now vetting exams that I’ve been assured were close to final draft, when in fact nothing could be further from the truth. After a spectacular burning (*) earlier this year, I now ask who the writer is when asked to vet an exam. I have a list of names whose exams I refuse to vet, for exactly the reason you mention.

          * The exam was VERY poorly formatted, the grammar and punctuation were poor, two of the section B questions were based on material no longer on the course, solutions were poorly set out, incomplete and in some instances conceptually wrong. Had this been a normal year and hence statistics included on this exam, I hate to think how much more terrible the exam would have been.

  4. Okay, but just supposing you were teaching this, what do you tell students that isn’t a lie and doesn’t scare them?

    Would a typical methods student be okay with graphing this function to show that the solutions are on the line y=x? (I’m guessing they’d find it hard?)

    Or would it be better to say if f(x) = x, then f^{-1} (x) = x = f (x), so a solution to the second equation will necessarily be a solution to the first (but not necessarily the other way around)
    and in this case either way you get a cubic, so if you get three solutions of f(x)=x then you know you’ve got them all? Or something like that? To be honest, I can’t see this going well at all.

    So what do you say? Also, is it just me or was that kind of a gross cubic to have to solve?

    1. s-t, that’s an excellent question, both of the f = f-inv nonsense and of plenty of other Methods nonsense. It is one reason why you have previously been advised, and everyone is advised, to avoid teaching Methods if it is at all possible. It’s also not my job to suggest how people cope with Methods’ lunacies, but here are a few notes, and I know the approach of at least a few teachers is along these lines.

      1) Before anything, there is the Mathematic Oath: if something is screwed and you know it’s screwed then it is your professional and moral obligation to tell your students that it is screwed. The f = f-inv stuff is screwed and teachers must say so. That doesn’t necessarily meaning going into great depth. But, at minimum, it means throwing something like y = 1/x at them.

      2) Secondly, I believe that Year 12 maths is not, and has never been, primarily about learning mathematics; it is about playing the game to get as high a grade as possible.* What does that imply here? As it stands, every indication is that VCAA expects students to mindlessly replace f = f-inv with f = x and to solve the latter.** So, that is the main thing you instruct as a “technique”.

      3) How can you somewhat justify things simply and non-scarily, either for the pure goal of teaching mathematics, or for the practical goal of protecting the students from VCAA unpredictability? Yes, pointing out (and proving) that f = x solutions are also solutions of f = f-inv is easy and natural. I don’t think graphing is a good guide, not only because of the Magritte aspect; it leads too directly to thinking that searching for f = x solutions suffices. (See again y = 1/x.) As you suggest, counting solutions is a recipe for disaster. (also f = f-inv will in general amount to a higher order polynomial or whatever, with the possibility of more solutions.)

      The natural condition to require is that the function is increasing, and it is easy to show that this precludes “other” solutions appearing. But again, I don’t expect teachers to go into this in great detail. Time is precious.

      4) Yes, the actual cubic on the above trial exam was needlessly gross. Again, it’s not just the dumb elephants on the exam; overall, it was shoddy.

      *) Others here, who are less cynical, will disagree. For me, the telling difference is that in the past you couldn’t get a high grade without learning the maths.

      **) It is difficult to see how VCAA can alter this without causing a riot. For example, just imagine the result and the reaction if, this year, they asked students to solve f = f-inv for the function f(x) = 1/x.

      1. Thank you. Okay, I can see how the monotonically increasing or decreasing case would work and I think I could explain that. And I’ll keep the 1/x example up my sleeve.

        (And actually I think on reflection I made a mistake counting solutions so yeah, silly me, and not a good strategy.)

        I totally agree with your point (2). I think that’s the attitude I’d take if I did teach it – because in any case, the students don’t have the option to opt out if they want to study mathematics or anything related at university, do they?

        (Long aside: I watched this movie called Lagaan about a mean colonial guy challenging a village to a cricket match – if they won they wouldn’t have to pay tax during a drought year, but if they lost they would pay triple. While the rules of cricket can be quite arbitrary and so forth, in that case it would be helpful to learn them. I think the VCE is somewhat analogous, though less dramatic and with less singing.)

      2. Point (2) is on the mark. Some may say dangerously so, allowing for the *.

        Results at Year 12 VCE level invite comparison between teachers (yes, even of different subjects and at times I have been asked to examine English, English Language, English Literature results statistically) and even though my answer is the same each year: “the sample size is not big enough to reject H0”, very few principals/school leaders understand statistics.

        MM34 is a nasty subject in this regard, and the scaling up does not compensate enough (my opinion). SM34 is a little bit less nasty, but still horrible enough, although Year 11 Specialist allows a teacher to at least cover some genuine mathematics, unlike Year 11 Methods in many schools.

        But again, Marty’s point (2) is so true it is scary.

        1. Of all the things which bug me about the way schools use Year 12 results (I’m sure this varies between schools), the “adjusted study scores” is probably the worst. It’s just a black box for innumerate and gullible principals / school leaders to use to unfairly malign teachers or subjects.

          Is there any information available on how these scores are calculated? I’m aware that they involve a “predicted study score” which apparently involves the GAT, and perhaps some other things, but other than that I’m clueless.

          1. SRK, I’m not quite sure of the issue, or why you brought it up on this post. If you want me to set up a “study score” MitPY, or whatever, I’m happy to.

            1. Marty, it’s in response to RF’s comment about the pernicious ways that Year 12 results are used.

              The “adjusted study scores” aren’t specific to this year. Rather every year, each student, for each student is given an adjusted study score which is the difference between their actual score and their predicted score (based on the GAT and perhaps other things). I think the (dubious) idea is that schools can use these adjusted study scores as an assessment of how well a student has performed because it “controls” for general ability.

                1. I went to a VTAC briefing on this, and it was quite interesting, but unless you either have a degree in statistics (or in my case a relative with a degree in statistics) it is difficult to pull-apart and get to the core of it all.

                  SAC moderation is another briefing I went to 6 or so years ago and again, it was very interesting but I doubt many of the participants took much away from it because it was so heavy on the statistics.

                  At the end of the day, some of the choices made are quite arbitrary, but I will give them (the moderators) that it is quite fairly applied within a subject. Between subjects is a totally different case and one that I agree, some school leaders think they understand a lot more deeply than is actually the case.

    1. Hi, Terry. I’ll just give the same reply as on the other post. I think the original idea, lost in the mists of time, was to be teaching and testing the symmetry of f and f-inv around y = x, and if properly taught that could work. But, the question has ossified into a ritualistic, meaningless and unvalidated “trick”.

    2. I’ll bet London to a brick that the only purpose TBTZ writers had for writing this question was to copy similar past VCAA questions. And so of course they also copy the flawed VCAA ‘solution’.

      Interlude: The brief given to all trial exam writers for all the companies I’ve had experience with is to write an exam similar in style to the VCAA exams. In fact, some of these companies flatly reject exam questions that are ‘different’. This is why you will often see the ‘same’ questions on a trial exam that were on the previous year’s VCAA exam. The reason given is “This is what the teachers want. It is our selling point.” The fact that these trial exams continue to sell so well is strong evidence that when it comes to trial exams, (most) teachers are gullible fools.

      As for VCAA’s purpose, I agree with Marty. There probably was a good assessment purpose once upon a time, but that purpose has been corrupted by lazy and stupid writers who copied the question without understanding or caring about the curriculum context of the question. These writers are too lazy and stupid to frame the question within the current curriculum context. And then you get the copier of the copiers …

      Then again, it might actually have been nothing more than a stupid question when it first appeared, but it just kept getting copied again and again by stupid and insipid writers because it ‘looked good’.

      BTW I’ve attached an interesting paper – a good paper.

      atm-mt228-39-42

      1. Jf, to be clear, I’ve never seen this question outside of VCE, and I’ve never seen it asked properly inside of VCE. When I first noticed this shit, on the 2011 exam, somewhat point out to me that the Cambridge text had some half-sense idiot description for what was going on. My assumption has always been that (a) Cambridge was the origin of this stuff in VCE; (b) Cambridge included it because of the symmetry message, but had always stuffed it up.

            1. I’ve done a bit more digging.

              That VCAB sample question was on the 1967 VUSEB Pure Mathematics Matriculation Exam, almost word-for-word. It was Q7(a) (all one part, no part (i) and part(ii) business).

              The MAV solution (yes, you read correctly) states (and I quote):

              The graphs intersect at (x, y) where
              2 + \sqrt{x} = (x - 2)^2, ………….. (1)
              which is also where the line {(x, y) : y = x} intersects either graph.
              \therefore x = (x - 2)^2
              = x^2 - 4x + 4,
              \therefore x^2 - 5x + 4 = 0,
              (x - 4) (x - 1) = 0,
              x = 4 or 1.
              But x = 1 is not on the domain of f^{-1}. And x = 4 satisfies equation (1).
              Hence the graphs intersect at the point (4, 4).

              Unfortunately I don’t have the ‘Report of Examiners’, so we don’t know what sort of solution the VUSEB had in mind. (The 1967 report appears to be missing from the State Library – https://guides.slv.vic.gov.au/education/exams. Maybe available at Melb Uni …?)

              This sort of question doesn’t re-appear until the VCAB sample question. So the shit we see today is inherited from at least 1967.

              (For those of you with scorecards trying to keep track of all the players:

              VUSEB was the predecessor of VISE (the re-brand occurred 1979), which was the predecessor of VCAB (the re-brand occurred 1986 – see https://researchdata.edu.au/victorian-institute-secondary-education/490147).

              More succintly: VUSEB ~1900-1977, VISE 1978-1986, VCAB 1986-1991, BoS and VBoS 1991-2003, VCAA 2003-2020.

              Matriculation Exams were the predecessor of HSC Exams (the re-brand occurred in 1970), which were the predecessor of VCE Exams (the re-brand occurred in 1992 – see https://en.wikipedia.org/wiki/Victorian_Certificate_of_Education)

              1. Thanks very much, John. My guess is that the MAV solution is what was expected on the exam, which is very surprising. You’re better at this game, but I’ll do my own investigation.

              2. A correction to the VUSEB epoch: {\large 1964} – 1977. History lesson:

                “From 1857 to 1964 the University of Melbourne set the Matriculation Examinations for mathematics. In 1912 a Schools Board consisting of representatives of the Education Department, independent schools and the university replaced the university’s Board of Public Examinations. The Board relinquished control of the matriculation from 1945, and all other examinations from 1965.

                In July 1964 the Victorian Universities and Examinations Board (VUSEB) took over prescribing the courses and setting the
                examinations. The newly created Monash University joined with Melbourne in this structure.

                This changed in 1979 when a new state structure for curriculum and assessment was established which the universities no longer controlled. This was the Victorian Institute of Secondary Education (VISE). VISE was replaced by the Victorian Curriculum and Assessment Board (VCAB) in 1986 and this in turn by the Board of Studies (BOS) and Victorian Board of Studies (VBOS) in 1993.

                BOS was abolished in 2002 and replaced by the current Victorian Curriculum and Assessment Authority (VCAA) in 2002.”

                (Source: https://www.vcaa.vic.edu.au/Documents/vce/mathematics/consultation/Paper-1-NewDirections2014-VCEMathematics.pdf)

                pp 55 onwards are historically interesting.

  5. I am not an expert on CAS calculators … however, if students are expected to solve the cubic equation that results from f(x)=x with a CAS calculator, why not just solve the original equation f(x) = f^{-1}(x) with the calculator?

    1. In this case, it’s an Exam 1 question, so there is no calculator. In general, I’m not sure how grungy an equation can be before a handheld CAS will resort to numerical solving.

    2. And there again, is an interesting point. A CAS calculator, such as a TI NSpire tends to struggle a bit with the algebraic solution to these but Wolfram Mathematica is much more successful.

      See Marty’s other posts about the inherent unfairness of this.

      1. Interestingly enough, I have found that there is a significant disparity even between the various CASes–the Casio Classpad can solve some algebraic solutions that the TI-Nspire cannot, and a TI-Nspire with a port of Giac/Xcas installed can solve algebraic expressions that both of them cannot.

      2. If I may make an objective comparison (from a programmer perspective that is) then you’re looking at the following:

        The Casio Classpad has 512kb of working RAM available to the user. (There is actually 2mb internally but 512kb allocated for userland stuff). A side comment is that Casio uses a BASIC-like language (I say this primarily because I suspect the internal workings of functions are seldom updated).

        With TI, you can compare either first or second iteration of calculators. The new iteration features 64mb of working RAM as does the original (if I’ve interpreted correctly). So ~128 times the working power relative to the classpad. They also suffer less compared to the classpad, but you can give it terrible equations to suffer.

        Then there’s Mathematica….Not much else to be said except the RAM is limited to each laptop, typically above 8gb for school-like laptops. Now you can choose to select the 8gb as a whole, but a computer program typically doesn’t use that much RAM. Even at a conservative 2GB, it should be obvious that one calculator has evolved with time, whereas the others are fragments of the past. It should also be noted that functions within Mathematica are also updated since they’re used in Wolfram Alpha. That means the solving functions are (typically unless for some reason it gets confused solving tan(x)+sec(x)=0) are likely to be state of the art relative to the handhelds. It is well known that the handhelds have firmware updates, but its typically bUgFixEs or trying to patch ndless (a program for emulating games).

        Why am I mentioning RAM? It’s actually possible for the classpad to run out of it on certain integrals. We really can’t tell whether if its primative programming on the calculator or a lack of memory (or both) but from what I’ve been doing a bit of research into, it would be entirely possible to create a malicious exam 2 that would be undoable on one, or potentially both handhelds. In the case of the casio classpad, such a “malicious exam” would most certainly exist, or at least the calculations it could not handle. See 2010 MM, which was in the news a long while back and had students unable to do certain operations. That’s not the end however, one could easily create equations that are designed to be taxing on the two calculators which is a common occurence with the casio classpad. Of course, you’ve wandered entirely off the course of teaching mathematics and into the realm of screwing one calculator over, which is um, hilarious but pointless.

        1. https://www.heraldsun.com.au/news/victoria/answers-just-didnt-add-up-on-year-12-maths-methods-exam/news-story/fcdf73d3cb1f17e82978573a6208b116?sv=644530f908f30c274e59b2ec6adaf9af

          I remember it well. Obviously none of those kids had been taught Plan B – get a solution by plotting graphs and find the x-coordinate of the intersection point.

          However, I do agree that it should be a level playing field for all allowed CAS technology in every Exam 2 question, and VCAA screwed up. I’ve heard anecdotally that in those pre-Mathematica days, Exam 2 was independently ‘blind reviewed’ using the three common CAS calculators – TI-Nsipid, Crappio, and Spewlett-Hackard – to ensure there WAS a level playing field. So just another story of a VCAA vettor asleep at the wheel.

          Two random observations whilst re-reading the 2010 Exam 2 Examination Report (no mention of the above controversy in the Report):

          1) The non-exemplary specimen solution given for Q1(a)(i) is:

          g(x) = …” let y = …”, inverse swap x and y …
          x = ..., g^{-1}(x) = …”

          2) The non-exemplary specimen solution given for Q1(a)(iii) is:

          “Solve g^{-1}(x) = g(x), g^{-1}(x) = x or g(x) = x“.
          Now remember, this is a CAS-Active exam. It goes on to say:
          “This question was done quite well. Most students solved g^{-1}(x) = x or g(x) = x. A small number of students solved g^{-1}(x) = g(x), …”
          So in a CAS-ACTIVE exam, only a small number of students solved g^{-1}(x) = g(x) …!!

          I think the accumulation of evidence is incontrovertible: Either some TBTZ writers are copying this shit directly from VCAA, or they’re the actual authors of this VCAA shit and it bleeds into all the other shit they write.

          1. It’s pretty hilarious that you mention 2010 which I skimmed through to find… you guessed it, another stationary point of inflexion question. Q17 MCQ, the comment reads “The graph has a stationary point of inflection at x = 4 because f'(4) = 0 and f'(x) > 0 for 2<x4”. @Marty Unless this example is different in a way I have not accounted for, it should be added to the list…?

            1. Hi Sai. There’s no error here. Not this time! The answer follows directly from the sign test (and there’s only one correct answer).

                1. OK, yes. I see the counter-example. My bad. So the problem with writing these ‘clever’ questions is they unconsciously consistently assume polynomial functions.

                  1. It’s not actually clear how this question was born, but I think there’s a few things going on. I’ll add a comment on the error list post.

  6. “Fitzpatrick and Galbraith”, the standard, and generally excellent, pure maths textbook from the 70s makes the same mistake (Example 17, Chapter 2, 1971 and editions). There is just one related exercise that I could find (Revision A(7) Q3); that’s a trapped area question, beginning with \boldsymbol{f(x) = (x-2)^2}, and just looking at it now, it’s clear that the exercise is just the 1967 exam question rejigged (and stuffed up, as it happens). The question by its nature requires a diagram, and the function is simple enough that the Magritte element is not so annoying. But, it’s still poor.

    Amusingly, the 1970 Year 11 text by Lucas and James briefly introduces the notion of a “self-inverse” function, which of course gives the lie to all this autopilot stuff. I don’t have the Year 12 Lucas and James books, but I’m sure John will check what’s there.

    1. Re: Lucas and James. I quote from the 1976 edition of ‘Sixth Form Pure Math’ (pp 28-29):

      “The peculiar examples in Example 19(c) and (d) are now seen to be self-inverse, since ff = I. Let us look at their graphs [graphs of y = 4 - x and y = \frac{2}{x} on separate axes with the line y = x drawn on both are given]. Notice that both graphs are symmetrical about y = x so that the reflection in this line maps the graph onto itself, and hence f^{-1} = f.”

      There are no f(x) = f^{-1}(x) questions in the Exercises.

      Self-inverse functions are known as involutions. There are several interesting papers to be found on the internet using search strings such as “involutions classroom”. I’ve attached one such paper.

      A Glimpse Into the Wonderland of Involutions

      1. Thanks very much, John. That’s very interesting. It raises a couple of questions, at least for the history buffs here:

        1) Did Fitzpatrick and Galbraith, and the HSC world more generally, acknowledge and correct their error after 1976? HSC in the 70s was a very small world, so it seems difficult to imagine that the matter wasn’t discussed by the big shots of the time.

        2) When, precisely, did the autopilot thing resurface?

        Regarding (2), I’ve roughly checked exams up to 1990 and couldn’t find any sign of it. Indeed, there was a small anti-sign in the 1976 exam (QB2(b)), when students were given the function \boldsymbol{f(x) = (x+3)/(2x-1)}. They were instructed to find \boldsymbol{f\circ f}, sketch it’s graph, and to “write down the inverse function of \boldsymbol{f}“.

        The only other (slightly) related questions I could see were in 1981 and 1985. The 1981 question concerned the function \boldsymbol{f(x) = (x-2)^2} yet again, but this time with the inverse of the left branch. The 1985 question was very involved, but the final part required the graphing of the right branch of \boldsymbol{h(x) = (x+1)^2-1} and it’s inverse; pretty obvious once you look.

        1. Re: Fitzpatrick and Galbraith. There was a second edition in 1974 and I’ve looked at my 1980 copy (third edition) and the Example 17 error you mentioned is still there. I quote from the third edition (p 53):

          “The simplest method of finding the coordinates of the point of intersection in this case is to make use of the fact that any [my emphasis] intersections of the graphs of f and f^{-1} will take place along the line y = x. Hence we must solve …. This is a much simpler process than solving [f(x) = f^{-1}(x)] …. However, the relevant solution must be within the domain of both f and f^{-1} and hence …”

          So there were two opportunities for this error to be fixed by Fitzpatrick and Galbraith but it wasn’t.

          Rough summary (for the history buffs):

          Examination history:

          1) f(x) = f^{-1}(x) is on the 1967 exam and a dodgy solution is given by the MAV. At the moment we don’t know what solution the VUSEB intended.

          2) It’s not on any exam between 1968 and 1990 (checked by Marty and I) or 1997 – 2009 (checked by me). Exams for 1991 – 1996 need to be checked. (My guess is that it doesn’t appear).

          3) It appears on the 2010 Methods Exam 2 (Section 2 Q1 (a)(iii)), then 2011 Exam 1, and then continues making regular appearances.

          Curriculum history:

          1) 1986 – 1990: It appears explicitly as a sample question (Q55 (ii) on p22) in the VCAB syllabus. Virtually word-for-word from the 1967 exam. No solution is given.

          Secondary school textbook history:

          1) 1971 – 1980: It appears as an example (Ch 2 Eg. 19) in the first three editions of Fitzpatrick and Galbraith. It does not appear in Lucas and James. These were the two most common textbooks of the time.

          2) 1995(?) – 2015: It appears in Cambridge Essential Mathematical Methods (the predecessor of ‘Cambridge’ Mathematical Methods).

          3) 2015 – : It appears in ‘Cambridge’ Mathematical Methods.

          It is noted that large sections of ‘Cambridge’ (particularly the questions) appear to be heavily ‘influenced’ by Fitzpatrick and Galbraith, so its appearance in Cambridge is not a big surprise.

          If my guess is correct, before 2010 the question last appeared as a sample question (copied from the 1967 VUSEB exam) in the 1986-1990 syllabus – a 20 year gap.

          1. Thanks very much, John. I’m surprised that it didn’t eventually get fixed in F & G.

            Regarding your (2), do you actually know of any texts from 1981 – 2010 that refer to the \boldsymbol{f(x) = f^{-1}(x)} question, either correctly or incorrectly? (I consider the killing of Pure and Applied in 1985, as the death of Victorian school mathematics, and so have little knowledge of the wasteland that followed.)

            The 2011 Cambridge text (Chapter 3I) has a hilarious mix of second-guessing and autopilot. They remark that the equation \boldsymbol{f(x) = f^{-1}(x)} is “usually” solved by the dodgy method, while noting solutions off of the line y = x “sometimes exist”. Then they have a number of autopilot exercises, while also having a number of self-inverse/involution exercises, including the 1976 exam question.

            Cambridge 2020 (Chapter 1F) is a lot better (Example 25), although still not correct (subsequent discussion and Example 27). The authors clearly know they should give up (or suitably justify) the dodgy approach, but they can’t quite get themselves to do it. I may have missed them, but I didn’t see any exercises pushing or presuming the dodgy method. The 1976 exam question is there, as well as some very nice review questions (with clunky solutions) generalising the exam question.

            1. I don’t know of any texts from 1981 – 2010 that refer to the \boldsymbol{f(x) = f^{-1}(x)} question, either correctly or incorrectly. Once the restrictions lift I can check a few old edition textbooks in a couple of libraries (assuming they haven’t been thrown onto the scrap heap).

              Re: I consider the killing of Pure and Applied in 1985, as the death of Victorian school mathematics. Indeed, the evidence from both the syllabi and the exams is inarguable.

              Re: Cambridge. I don’t know why they can’t simply hold off on those examples and questions until after the strictly increasing stuff is introduced. Then they’d get the best of both worlds. It astounds me that one of the authors – a Brobdingnagian-name – has let it fester like this for 20 years or more.

              1. Just a minor update and narrowing of the window: the 2005 edition of “Cambridge” appears to be identical to the 2011 edition in regard to the f = f-inv question.

    1. Hi, Terry, there are there options:

      1) Say “This is a 3rd party exam, so who gives a shit?” Then get on with more rewarding study.

      2) Ponder why the exam writers didn’t even bother to get rid of the fractions before plucking a factor out of thin air.

      3) Plug and pray. (x = -2 works.)

        1. Since the question mentions “point(s)” of intersection, would candidates be expected to justify why there is only one answer?

          1. But play this game with VCAA and it would seem that you’re dicing with trouble.

            Because saying P(a) = 0 therefore a is a root of P is NOT a VCAA-accepted method for showing that a is a root of P (and therefore not acceptable for constructing the factor x - a and therefore not acceptable for factorising P).

            1. Oh really? But surely an equality like

                  \[P(x) = (x-a)Q(x)\]

              where Q is a polynomial of degree one less than P should be enough to then claim that a is a root of P?

              After all, we can do our dicing on the back of a napkin, using it to inspire our formal response.

              1. For those who came in late … I’m obviously trying to highlight the insanity of VCAA’s position by being an agent-provocateur. To continue:

                Sorry Glen, but no dice.

                VCAA has stated that:

                1) correct answers obtained from invalid mathematics receive zero credit, and

                2) P(a) = 0 is an invalid way of showing that a is a root of P.

                Therefore by VCAA-logic, P(x) = (x - a) Q(x) is based on invalid mathematics and so you’re penalised. And if you want to Get Smart and say that P(x) = (x - a) Q(x) by inspection, sorry 86, but you haven’t shown all the steps and so you’re penalised. It’s a classic Catch-22.

                But I suppose if you expanded it and simplified and then showed it was the same … I’ve proposed this method earlier – VCAA has not yet commented on the validity or otherwise of this method (I’m not sure if that’s good or not) …

                These are the VCAA rules of the game laid out in the Examination Reports. Maybe Marty could start a new blog – the most insane conclusion(s) one can arrive at using VCAA-logic. What a marvellous topic for an MAV Conference option.

  7. Let me compare the original issue that Marty raised concerning Q10 this trial exam with the first issue raised in “One FEL Swoop: The Further Error List” concerning the Markov condition; https://mathematicalcrap.com/2020/09/20/one-fel-swoop-the-further-error-list/

    Marty points out that you cannot simply replace the equation f(x) = f^{-1}(x) with the equation f(x)=x without some mathematical justification. At least, in the case in question, this could be justified.

    Similarly, in the Markov question, one cannot multiply the transition matrices without some mathematical justification. The difference is that, in this case, there is no mathematical justification.

    1. Terry, you’re not getting an argument from anyone here. (True, you’re not getting much response at all, but it *is* Further.) Perhaps you should write an article for Vinculum.

      1. I thought about writing an article for Vinculum. Maybe I will. Thanks for the suggestion.

        Further Mathematics is the most popular mathematics subject in VCE, and its relative popularity is increasing.

        Also, when I was working in a large school recently, I was told that enrolments in VCAL are increasing more quickly than enrolments in VCE.

        1. Sheesh. The argument that will not die.

          For the Nth time, McDonalds’ popularity doesn’t mean that their food isn’t shithouse.

          1. The other way of looking at it is that it’s precisely because of Further’s large cohort that it’s so important for it to be a good subject.

            Of course, given how completely FUBAR it currently is, no amount of tinkering will fix it. (0.01% of Further teachers and 0.0001% of Further students will care about Terry’s point). It needs a complete rewrite, removal of CAS, new assessment structures, etc.

            1. SRK, I think both of your points are valid. It’s because of your second point that I just can’t get myself to care about your first point.

    2. I’ll bite. I’d contend that this particular sin — asking this kind of question and expecting this kind of robotic “solution strategy” — is of a similar level. It isn’t exactly the same, I agree, but they are both just awful.

      The question doesn’t say “solve f(x) = x“; it says:

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

      If a given function f:I\rightarrow J is invertible, then f^{-1}:J\rightarrow I (here I,J\subset\mathbb{R}). What does \emph{intersection of the graphs} mean? Does it have any meaning, without more information? These aren’t a pair of functions f,g with common domain, for which you can ask “For which x does f(x) = g(x)?”. But that’s what the question pretends is happening.

      I think that’s pretty darn terrible.

      1. Hah! Maybe I should start a “What’s the Worst Thing in VCE?” competition.

        I’m not quite sure how to judge these things in the school context. I think both the maybe-fixed-point questions and the maybe-Markov questions are lazily, needlessly dumb, and practically flawed; whether one is more intrinsically flawed is of third order importance.

        I’m not so fussed about the intricacies of functions and their domains; indeed there’s too much of that in VCE already. (Example: What is the composition \boldsymbol{f\circ g} if \boldsymbol{f(x) = \sqrt{x}} and \boldsymbol{g(x) = x}?)

        1. I’m pretty fussy about domains and codomains. Because they really do make a difference later down the track. I don’t think you need to worry students about them too much until much later. But *teachers* and *examiners* should definitely have issues like the one I mentioned in their minds, since their understanding of the materials should be at a higher level. In particular, the point I’m trying to make is that the very act of asking this question about intersection of graphs confuses the notion of domain with codomain, which can cause quite significant issues in these kinds of questions.

          About your question, supposing that f:[0,\infty)\rightarrow\mathbb{R} and g:[0,\infty)\rightarrow[0,\infty), the composition h = f\circ g : [0,\infty)\rightarrow\mathbb{R} is given by

              \[h(x) = \sqrt{x}\,.\]

          If g is given a larger domain, then the question would fall into a similar “sinful” category, and without teaching students about determining the right choice for domains etc, you’re just setting them up for confusion.

          While students should not need to tweak the domains and codomains of their functions (at this stage), their teachers and examiners should definitely worry about doing so. It’s pretty easy to cook up questions that should be avoided because of these considerations.

          I guess I should also add that when we test students at uni on domains etc, lecturers often fall into the annoying trap of trying to trick students by giving them ridiculous functions to determine the range of. I really hate that.

          1. Amen to your remarks about tricking students.

            In setting an examination question, one should always ask “What is the purpose of this question?”

          2. Thanks, Glen. We agree on the nastiness and pointlessness of trickery. We mostly disagree on the other aspects, at least for school.

  8. I’m surprised I didn’t ask earlier but for the non-linear substitution integration question of the form f'(x)/f(x), what was did the solution use? A straight u-substitution or some wishy-washy explanation? I’d definitely begin crucifying if a solution that used content outside the Methods content was used….

    1. No argument was given, the implication being that students should just recognise integrals of that form. I have absolutely no idea why anyone would think that is fair game in Methods.

      1. I know it’s pointless but reading off the study design, the dot point for Methods on anti-derivatives state “find anti-derivatives of polynomial functions and power functions, functions of the form f(ax+b) where f is x^n, for n \in Q, sine, cosine and simple linear combinations of these, using pattern recognition or by hand”. Nowhere could one in any sense come to the conclusion the form \int \frac{f'}{f}. Only SM states the use of the substitution is there…. I know there was a discussion in the other post on whether these integrals could come up, but the wording cannot reasonably suggest that a student would know how to integrate such a thing. The formula sheet does not suggest anything either, especially not anything in the table of integrals. Maybe this irks me a lot but given where the MAV exams end up, it would be irresponsible to carelessly shove in artifically difficult questions that tread outside the study design that would end up confusing students, and perhaps even teachers (hypothetically I would say).

        1. Sai, any time you want to take over the blog, I know I can be at peace in my retirement.

          I agree entirely. I cannot see any way of reading the study design that remotely suggests an integral of the form \boldsymbol{\int f'/f} for non-linear \boldsymbol{f} is kosher. So, what’s with Q3 on the trial exam? You’d have to ask the MAV. Good luck with that.

  9. Let me rush to defend the original MAV question. If the purpose of the trial exam is to prepare students for what they can expect to find on the VCE examination, then the question seems appropriate.

    (Flaming me will take the number of comments over the ton.)

    1. TM, I agree with you that the main purpose of trial exams is to prepare students for the VCAA exams. Therefore your rushed defence fails:

      There are questions on this exam you will NEVER see on a VCAA exam! How does including content NOT on the Methods course (like asking for the anti-derivative of \frac{f'(x)}{f(x)} where f(x) is non-linear) prepare a student for the VCAA Methods exam?

      And the fact that VCAA continually get their ‘f(x) = f^{-1}(x)‘ questions wrong doesn’t mean that they might not eventually get it right – maybe even this year … Slavishly copying defective questions from the past under the guise of preparing students for the VCAA exams in the present just doesn’t wash.

      Are you suggesting that trial exams should purposefully include defective questions similar to ones that have appeared on past VCAA exams (like the one here https://mathematicalcrap.com/2020/02/14/witch-35-overly-resolute/) simply to best ‘prepare’ students? Should trial exams in a non-COVID year claim a function is a pdf when it fails to integrate exactly to 1 (see 2016 Methods Exam 2)?

      Where do you want to draw the line on including defective questions to ‘prepare’ students?

      Finally but no less importantly: The ‘main purpose’ can never and should never be used to justify the shitful, poxy, crap being passed off as ‘solutions’: The ‘solution’ for finding the inverse function is shit, the justification for using f(x) = x is shit etc. At least VCAA add the qualifier that the ‘solutions’ in the Examination Reports are not meant to be exemplary or complete (they should also add ‘correct’ to this disclaimer). Trial exam solutions often carry the disclaimer of not being representative of VCAA views or marking schemes. However, I would expect they ARE meant to be exemplary.

      1. I was implying that it is reasonable to expect a question asking students to solve f(x) = f^{-1}(x) on the VCE paper. There is nothing defective in the question for an appropriate f. Hence it is fair enough to prepare students for such a question.

        1. TM, unfortunately you are correct in saying it’s reasonable (and, I’ll add, responsible) to prepare students for this love affair VCAA keeps inflicting on students and hapless teachers.

          However, the point we are arguing is that within this love affair, students should be required to provide a valid mathematical argument as to why it’s reasonable to solve the alternative equation f(x) = x (or f^{-1}(x) = x if it’s simpler) for an ‘appropriate’ f. And it should be DEMANDED that the solutions provide a valid mathematical argument for this.

          All of which can be easily done within the scope of the course (and as a bonus, give some relevance to the inclusion of ‘strictly increasing’ in the Study Design). But neither VCAA or TBTZ do this in their questions or ‘solutions’/reports – both are guilty of promoting poor mathematics. And TBTZ have double downed on this (and several other defects noted in this blog and brought to its attention). There is no conceivable defence, rushed or otherwise, for TBTZ (and VCAA) exam writers. However, the Dunning-Kruger Effect provides an explanation.

        2. John has responded but let me re-hammer two points. These points have been made, but you keep ignoring them.

          1) You claim that there is nothing defective in the question for an appropriate f, but that is simply false. It may be theoretically true, but we’re not discussing a theoretical situation: we’re discussing the actual situation, which includes a long history of VCAA asking the question in a screwed, autopilot form.

          What that implies is that, until VCAA establishes some theory of the question and flags this theory as “expected”, it is absolutely impossible for VCAA to ask a question on this material that is both mathematically sound and properly fair.

          2) You are correct that “it is reasonable to expect a question asking students to solve f(x) = f^{-1}(x)“, although “reasonable” is the last word I would have used in that context. To that extent, it is “reasonable” for the trial exam to include such a question, complete with autopilot proof, although it’s hardly mandatory: there’s plenty of other question types from which to choose. It is not remotely “reasonable”, however, to include such a question without the solutions also containing a clear discussion of the proper mathematical justification.

          The Mathematic Oath: First, tell no lies.

    2. Flaming is usually synonymous of personal attack, and I would expect to see attacks only directed at arguments and ideas on this blog, not at people. I think you’re pretty safe.

  10. Another thought I’ve had for a while was why VCAA would have any reason to test the particular feature related to inverse functions. Logarithmic/exponential functions have an inverse that can be found by hand, whereas for cubics and polynomials of degree greater than 3, trying to find a rule for the inverse would boil down to using the cubic/quartic formula. When you use such a formula, you introduce complex numbers unless the function is crafted to have a nice inverse (say f(x) = (x-3)^3+5 and the like).

    Now it would be extremely foolish to use the CAS calculator to rearrange for such an expression since you can have up to 3 or more branches for an inverse function and such a function would not be useable in an exam (for a variety of reasons, I’m not sure a CAS calculator could compute the inverse although it will have to be tested). So, getting students to solve f(x) = f^{-1}(x) is out of the question.

    All of this may be conjecture but the “hack” for finding f(x) = f^{-1}(x) happened to work (by luck or stupidity) for VCAA/Cambridge etc which gave them “a method” for solving such intersections and having a method for testing a students “understanding” of inverse functions. Everything I’ve said so far is conjecture but it’s what I have a feeling is going on, especially with the emphasis on symmetry between the line y=x when talking about functions and their inverses.

    1. Thanks, Sai. I think that’s basically what is going on. The equation f = f-inv has not intrinsic interest, and I don’t think is interests the VCE guys. It’s the inverse and the symmetry they’re interested, together with a poor understanding on that symmetry.

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