WitCH 64: Decreasing Intelligence

Just in case anybody got the wrong impression and hoped or feared we’d turned over a new leaf, we’ll be posting a number of WitCHes in the next few days. We’ve finally had a chance to look at the 2021 NHT exams (although the exam reports have still not appeared). As usual, the exams are clunky and eccentric, and we’ll be posting a brief question-by-question overview of the exams. But, first, some highlights.

The following question is from the 2021 NHT Methods Exam 2. It seems to us more of a PoSWW than a WitCH, but we have seen the exact same issue arise in a recent SAC. So, presumably, this issue is more widely spread.

UPDATE (30/08/21)

As commenters have pointed out, none of the options are correct. A function f is defined to be strictly decreasing on its domain if f(b) < f(a) whenever b > a, and this is false for all the given restricted domains. It is also standard in VCE to only apply the definition on intervals, which also kills options A, D and E. (There is no problem applying the definition on more general domains but, for the first reason given, this does not save the question.)

There is also a “local” notion of decreasing function. So, for example, one might describe a function as decreasing at x = -4, indicated, say, by the derivative at that point. But, as illustrated by the failure of option D, the global definition does not equal the sum of the local notions.

UPDATE (22/09/21) Surprising no one, the examination report simply pretends that VCAA didn’t screw up, and gives the answer as D.

41 Replies to “WitCH 64: Decreasing Intelligence”

  1. Well there may be a turning point at the origin to think about too, no? And given that the function changes behaviour there, well that fact may rate a mention perhaps?

    1. Thanks, Simon. Yes, for pedants, leaving out discussion of the behaviour at the origin is mighty weird. But the more important issue is the one Glen hammered.

      1. I’m not sure what “discussion” about the turning point at (0, 0) has been omitted …? The ‘interesting’ thing about this point is that x = 0 is common to an interval where f is increasing and an interval where f is decreasing. Some students struggle to accept this, despite it being obvious from the definition (in their minds they equate strictly decreasing/increasing with decreasing/increasing and create a paradox for themselves).

        On a related note, I have no idea why VCAA introduced the concept of strictly increasing/decreasing. Particularly since it doesn’t apply the concept anywhere (for example, in its favourite question – finding the intersection points of a function and its inverse). As stated in the Stupid Design and assessed in exams, it’s completely pointless.

        1. Thanks, John.

          Strictly monotonic is used at least implicitly in the existence of inverse functions. If there’s an error, it is not making this condition explicit.

          The fact that there is a turning point at (0,0) is relevant, and nothing is stated about it. Which is weird. Of course the graph makes it obvious, but then why not mention the others either, rather than simply labelling the graph?

          1. Call me stupid, but I don’t see what should be stated about the turning point at (0, 0) …?

            Re: “Strictly monotonic is used at least implicitly in the existence of inverse functions. If there’s an error, it is not making this condition explicit.”

            Yes, the context of inverse functions is another missed opportunity to apply the concept (when f is continuous). All we get in the Stupid Design and examinations is “one-to-one” rather than “strictly incrasing/decreasing”.

  2. So they are just saying things without any meaning now, or what? f strictly decreasing on I means:

    For all a, b in I, a < b implies f(b) < f(a).

    There is no correct answer.

    1. The correct interval should be (-∞, -2√2) or (0, 2√2).
      Option C includes the stationary point but that is not a decreasing part of f.

    2. Absolutely. Increasing/decreasing (strictly or not) is defined on an interval, not on a union of two separate (non-intersecting) intervals. The function is indeed decreasing on each of the two intervals given in option D, but not on their union.

      This is possibly a subtle point; but then, mathematics is about getting the subtleties correct.

      I’ve just looked up the Study Plan, which includes: “identification of intervals over which a function is constant, stationary, strictly increasing or strictly decreasing,” – but the set shown in option D is not an interval (nor are those in A and E).

      I think what might annoy many of us here is the frustration of seeing these things done wrong, when a little more sense and thought would have them done right. Are these things never proof-read?

      1. The general impression I’ve gotten from working for various publishers is that these things aren’t proof-read by anyone at tertiary level (and hence lack that attention to rigour that would otherwise be picked up by those more experienced e.g. us).

        1. Matt, what do you mean by “tertiary level”? One would also hope for more rigorous checking of a formal exam.

          1. By ‘anyone at tertiary level’ I refer to anyone who works in tertiary education (i.e. universities). Like how one distinguishes between primary/secondary education and their respective primary/secondary schools.

            1. Thanks, Matt. Unfortunately, I’m not sure “tertiary level” is the guarantor of expertise that one might expect or wish for.

              1. The necessary attributes of the competent vettor (and writer) include:

                1) Strong mathematics background,

                2) Intimate knowledge of the current Stupid Design and past examinations, and

                3) Extensive teaching experience in the subject including current/recent experience.

                Clearly there are some teachers that do actually have these attributes, but there are also a lot of ‘fakers’. And I know of ‘fakers’ that have been asked to be vettors.

                The problem is that teachers with the necessary attributes probably don’t want anything to do with VCAA. And the feeling is probably mutual. ‘Fakers’ on the other hand …

                All of which leads to the production of exams that contain errors, both obvious and less obvious.

  3. As with many of these, you can guess what the intended answer is; whether or not any of the answers are actually correct of course I will leave to the numerous superior intellects who comment here.

    (I still get confused by the VCAA definition of strictly decreasing and when the interval is a union of sets I just give up!)

    Thanks Glen, your answer makes sense. If only a Cambridge author had mentioned it somewhere…

    1. Does VCAA have a definition? I couldn’t find one. In any case, the definition on Planet Earth is the one Glen provided, and implies that the endpoints are included.

      1. VCAA published an entire Bulletin (Supplement 2) on it in April 2011 (No. 87) – attached for the historians.

        Funnily, the correct definition (as stated by Glen above) is explicitly given, with examples consistent with Option D of the NHT question.

        As an aside, if VCAA did this on a semi-regular basis for FAQ’s and archived the bloody things in an intuitively logical place (near the Stupid Design, for example!), life would be a lot clearer.
        For example, a similarly styled Bulletin on points of inflection.
        And surely it’s wouldn’t be too hard to include the content of such Bulletins in the Study Design.

        PS – @Marty. The NHT Examination Reports are as useful as an inflatable dartboard (as I’m sure you know). They have answers, and if we’re lucky there might be some brief discussion. The fact that only a couple of dozen students sit the VCAA NHT Methods Exams might have something to do with this.

        VCAA MMCAS notice-strictly increasing

        1. Thanks, John. Yes, VCAA could make the information *slightly* more accessible. As I wrote, I say the same error on a recent SAC, at a good school. My guess is the misunderstanding is common.

          As for the NHT reports, of course I know they are useless. I was just stating that to indicate why I didn’t link a report, or to indicate the intended answer.

          1. Marty, this raises an ‘interesting’ dilemma:

            Many teachers write SACs with questions that model what students will see on the VCAA exams. To do otherwise is to set the student up for an ambush. I will (generously) assume that this is what happened at the “good school”.

            Now, if the SAC question is written so that it’s totally mathematically correct, the student is trained on a question that is worded differently to what s/he will see on the VCAA exams. This will be off-putting for many students and will disadvantage his/her performance.

            Hence the dilemma: The trade-off between the technically correct question (that the student will never see on the exam) and the less-technically correct question (that the student will very likely see on the exam).

            The dilemma is particularly dilemmarish with respect to the NHT question, where it’s a couple of technical details that makes the question incorrect – details that most teachers and certainly VCAA don’t know, understand or appreciate and hence there’s no harm (?)

            I know what you’re reaction will be: No compromise! (just like Rorschach). Unfortunately, things aren’t that black-and-white … (And I know what you’re going to say: “Of course it’s fucking black-and-white!)

            I’m not advocating either way – what I’m saying is that teachers are damned either way … (Of course, you can always say damn the torpedoes and have no dilemma at all https://en.wikipedia.org/wiki/David_Farragut)

            The blame for all this lies with VCAA – VCAA is inconsistent, often wrong, lacks transparency and produces Stupid Designs that lack clarity and hence foster uncertainty.

            1. I am so misunderstood! Of course VCE teachers should “compromise”. But, to whatever extent teachers “compromise”, they should also be fully aware of the “compromises” they are making, and the effect of those “compromises”. And, the reality here is the vast majority of VCE teachers are just as guilty as the VCAA: most simply do not give a toss whether what they are teaching is correct or not, let alone whether it has any worthwhile mathematical sense.

              The whole thing is ritual. No one cares.

              1. lol! I think you’re better understood than you think.

                “Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” – Bertrand Russell.

                Fuck VCAA for screwing with this! It should be VCAA that says no compromise. It should be VCAA that produces a Study Design (NOT a Stupid Design) that makes this possible. It should be VCAA that writes exams that makes this possible.

                Most teachers do “give a toss whether what they are teaching is correct or not”. Plenty of teachers care, when they have the time to care! Fuck the system for making sure that teachers don't get enough time to care *all the time*.

                1. Of course VCAA are villains here, and always. But, no, I’m not willing to give teachers a pass. I don’t buy the “they’re so busy” excuse. I simply don’t believe that the vast majority of VCE teachers care if what they teach is right or wrong, meaningful or not.

                  What type of PD do teachers choose? What does that tell you?

                  1. Teachers choose PD for a number of reasons, but lately, quite a lot do it to satisfy VIT requirements.

                    If you get a brief glimpse into the workings of the mind of a VCAA assessor and somehow manage to determine how they are interpreting their own questions – bonus!

                    SACs on the other hand add nothing of value, encourage a gaming of the system and should be scrapped entirely.

                    So should calculators.

                    Use the time saved by these measures to perhaps teach integration by parts.

        2. But even that supplemental note is dodgy. On page three they define a function as being decreasing “over an interval”, and then immediately given an example of a function which is “not decreasing” – but without saying what interval it is not decreasing over. They could have said: not creasing over (-\infty,\infty), but they didn’t. It’s as if they don’t even fully understand their own definition.

          1. I’m fine with not specifying the interval each time, if the interval is obvious. More generally, there is no particular need to restrict “increasing” to an interval, although that is natural.

            One difficulty is that we naturally talk about “increasing” more locally, so “the function is increasing at x = 5”, and so on. This casualness and semi-contradiction in language is fine as long as there are clear underlying definitions and a careful, intelligent culture. However, …

  4. Could have just asked: “f'(x) < 0 for …" and then adjust the brackets as required.

    Although the point that Simon makes plagues this version as well.

    1. It’s not hard for VCAA to get the wording right …!

      1) Define the function over a finite domain, perhaps [-5, 5].

      2) Write the last sentence of the preamble as:

      “f(x) is strictly decreasing on the intervals”

      3) Dump the union notation in the options:

      D. \displaystyle \left [-5, - 2\sqrt{2}\right] and \displaystyle \left[0, 2\sqrt{2\right]

      It’s not rocket science (fortunately for the astronauts).

        1. In JF’s re-wording I think “and” is taken to mean there are two intervals over which the function is strictly decreasing, whereas the union of these sets makes them one, over which the function is not strictly decreasing.

          If I read the comment correctly, VCAA’s attempt to use set notation has made response D incorrect, because the author did not fully comprehend the UNION of two sets makes one set.

          An understandable but not forgivable error on their part.

          1. The idea of strictly monotonic functions (at least from what I’ve been taught) only applies to intervals, and the issue here is that the union of two intervals is not necessarily an interval, since, well the definition supplied breaks down. I suppose that’s also another oversight, from their end. The usage of and makes it distinct that there are two intervals on which the function has the above property.

            1. Hi, Sai. Hope things are going well. Thanks for advising Mr. X.

              When introduced for everyday real-valued functions, just defining monotonicity for interval domains is natural. But there’s obviously nothing in the definition that requires an interval. All that is required is that the domain and range be ordered. Eventually, and pretty soon, it’s natural to consider monotonic functions in this more general setting.

  5. In fact they could alleviate all confusion and problems with not understanding how sets work by simply giving a list of intervals, and asking on which of those intervals the function is decreasing. So it would be a multiple selection question rather than a multiple choice question.

  6. Has anybody considered the idea that all this is intentional? Not necessarily in the sense that a particular person willfully decided to make an error, but rather in a kind of “systemic” sense?

    Think of a company that wants/needs to hire mathematically skilled stuff to do some not very exciting, but necessary every-day calculations to keep the business running. But the last thing they need are real geniusses who can think for themselves, understand what they are doing, or even have their own ideas how to improve the carefully designed workflow. As such a company, you need to create a test that filters out both mathematically completely unskilled applicants (who wouldn’t be able to do the job) and really skilled and clever people (who might become dangerous to the not-so-intelligent management).

    The present kind of test does exactly this. You can’t solve it if you have no idea about math, but you also can’t solve it if you really and deeply understand math, including its logic and formalism. The only student that passes the test is the one who reels off learned algorithms without thinking about them.

    And maybe [now comes the conspiracy theory part] this is exactly the kind of working class people (the elite of) a modern, digitalized country needs.

    1. You had me going there for a minute, Hans. There’s no doubt in my mind that many organisations do this – and that’s a good thing. You don’t want bored geniuses and you don’t want challenged idiots, you want someone ‘in between’. So you need a ‘test’ that ‘filters’ out both extremes.

      Where I thought you were going with this is that VCAA don’t want geniuses writing the exam, but they also don’t want idiots. So they apply a ‘filter’ to get writers that are ‘in between’. I can *almost* believe this. Particularly since you have the filtered doing the filtering etc. so you end up with regression to mediocrity.

      But I don’t buy a conspiracy of social engineering via VCAA exams – smart students (and certainly the very smartest) will nearly always give the answer that’s expected, even when the question is defective. They might raise an eyebrow at the question but they will nevertheless answer it ‘correctly’.

Leave a Reply to Alasdair Cancel reply

Your email address will not be published.

The maximum upload file size: 128 MB. You can upload: image, audio, video, document, spreadsheet, interactive, text, archive, code, other. Links to YouTube, Facebook, Twitter and other services inserted in the comment text will be automatically embedded. Drop file here

This site uses Akismet to reduce spam. Learn how your comment data is processed.