Witch 113: Smoothing Over the Cracks

This one is a combo WitCH. The main concern is a multiple choice question from last week’s Methods Exam 2. The question may not be an “error” in the newspaper sense, but it is bad. To appreciate some (but far from all) of its badness, however, we need to see VCAA’s solution. We won’t likely see that solution, however however, for months, if ever; transparency is not VCAA’s strong suit (Section 7).

To deal with this, we’ve teamed up last week’s MCQ with a similar MCQ from the 2021 Exam 2, together with VCAA’s solution to that earlier question from the exam report. Last week’s question appears first.

UPDATE (05/11/23)

Some clarifying remarks.

First of all, I am guessing the majority of students and teachers (and 99.9% of the general public) who read this blog have no proper understanding of the mathematical issues. If only for this reason, it’s not gonna make it into the Herald Sun. But, trust me, it’s bad. It’s the kind of thing that makes mathematicians start blogs.

Secondly, I’ve already written about this VCE nonsense many times, beginning in 2012. See also here and here (Q7) and here and here and here (MCQ19). It will not die.

Thirdly, the underlying material is all explained well enough, but at a superficial level, in a reasonable first year uni subject. Because the explanation is superficial, however, the ideas don’t stick well and people forget. (Few teachers do, or at least do well enough, the solid second year subject in “real analysis” you need to really get this stuff.)

It is perhaps reasonable to include this material in VCE, even if there’s no compelling reason to do so. If it is to be included, however, the syllabus and textbooks and exams have to be written and vetted by people who understand it. That ain’t happening.

UPDATE (06/11/23)

This stuff is gonna drive me nuts, I know. I will try to kill it one more time, with a proper update soon.

For now, let me just make clear what a student should do in practice with an MCQ like the above: autopilot. Just match the function values, match the derivative values and do whatever the two resulting equations suggest that you do. That’s all there is.

23 Replies to “Witch 113: Smoothing Over the Cracks”

  1. I’d just left a related comment on the Methods discussion, which I’ll partially reproduce here:

    I’ve always taken [‘smooth’] to mean infinitely differentiable without a second thought, but it seems VCAA uses it to mean continuously differentiable (e.g. the function here is only C^2, and Google led me to the MM21 exam report’s discussion MCQ19). I checked both the Cambridge and Jacaranda textbooks for clarification; the only thing I could find was Jacaranda’s statement that “The gradient of a function only exists where the graph is smooth and continuous.”

    Neither function is smooth in the ‘all derivatives’ sense, but are C^1, so are smooth in the VCAA sense. Some comments on the report’s solution to Q19:

    1. The fact that lim_{x -> 0} f(x) exists doesn’t mean that f is continuous at 0.
    2. Strictly speaking, it is f that is continuous/smooth and not the graph of f.
    3. The step from

        \[f(x) = \begin{cases} 4x+1 &x<0 \\ (2x+1)^2 &x \geq 0 \end{cases}\]

    to

        \[f'(x) = \begin{cases} 4 & x<0 \\ 4x+1 & x\geq 0 \end{cases}\]

    deserves much more explanation. Even if f is continuous, we don't a priori know that f'(x) exists at x=0, and in general, it may not — so the form given here presumes that f'(x) exists everywhere… which was the question itself. Without further explanation, the implicit assertion here is that any continuous piecewise function is differentiable.
    4. If I want to check that a function is differentiable, there's no need for me to check that the derivative is continuous: it just has to exist. So I don't know what the f'(x) = 4 discussion is, and even then, we get the same error as in 1.
    5. Since it's a local condition, I'd prefer saying "f is differentiable at all real x" rather than "for all real x". (Incidentally, specifying 'all real' matters here: option B is also differentiable everywhere in its domain).

    1. Thanks, Javert. Of course these are exactly the ideas to get straight. And, yes, I’ve written about this stuff a number of times. It won’t die.

      I’ll update the post with some links.

      1. Sorry, I’m too busy to follow closely. Javert, if there’s an edit you want me to make, just let me know in a comment or an email.

  2. Putting the issue of Mathematical accuracy aside for a moment (yes, I know that is a rather dangerous way to start a comment…)

    IF there is consistency from year to year on how VCAA assesses this concept, does that make the problem less of a problem? The answer may not be correct. There may not be a correct answer. But IF the answer marked as correct is done so in a manner that is consistent with previous years’ examinations… does that make the question “fair” in some sense?

    I see what you are both saying here and agree that it is far from good. I’m less upset about these questions though than some of the others for reasons of consistency.

    1. The question explicitly disadvantages people who are more knowledgeable about mathematics, and also explicitly advantages people who are more knowledgeable about VCAA-speak. Even if it’s fair (which it isn’t, because it favours those who went to expensive private schools and those who have tutors), it is unacceptable.

      Of course the actual (dis)advantage in this case is probably not that big.

      1. Thanks, Joe. I agree with all that. But the main thing here is the students are learning Not Maths.

        The exam report is not close to correct for the 2021 question, and I’ll bet London to a brick that the 2023 question is not asking what was intended to be asked, if there is even any clear sense to any of the writers what was intended. So, sure, students will match the endpoints and the derivatives. But only a tiny few will have any clue of the meaning of what they are doing, and why it doesn’t properly answer the question(s) as written.

        All three textbooks get this way wrong.

      2. Speaking of VCAA-speak now as a graduate student it’s now strange to me that VCAA insists on using the term “vector resolute” rather than “projection” when it’s a much more intuitive term. I have never seen the term vector resolute since year 12 while vector projection is a much more widely used term and again just makes more sense with what it is as well. Not having standard language just makes it harder to use other resources to learn with.

        1. Yes. It’s a small thing but it’s important. And totally unnecessary. Not nearly as bad as the corruption of “smooth” and “show that”, however.

    2. Hi, RF. The problem here isn’t one of consistency. As Joe suggests, these are autopilot MCQs, students will do their autopilot and arrive where VCAA intends. The problem is, VCAA and the overwhelming majority of students (and teachers) have no idea where they’ve flown.

      Now, let’s deal to your first sentence …

      1. OK. Thanks everyone – when put that way it is obvious. What VCAA is doing may be regarded by some as not encouraging students to think. What they are actually doing though is encouraging students to not think.

        The two sound similar, but are in fact wildly different.

        Understood.

        Another issue to consider is that, as well as the “casual understanding of limits”, continuity, differentiability and “smoothness” in these questions, one only has to look at commercial SACs produced by a certain organization to see that this runs much deeper than just the exams.

        1. Thanks, RF. Yes, the third party SAC/Exam producers of course take their lead from the VCAA exams and reports. Even if these guys recognise that something from VCAA is nonsense, they have little choice but to reproduce the nonsense. Which amplifies the nonsense.

          1. Why though? Why do they have to include questions pretty much each year (I gave up reading them many years ago) in Methods SAC 1s (and the old SAC 3 investigation tasks) about “smooth joins” between two curves that each formed part of a (hybrid/piecewise) function…?

            Is it because it leaves scope to play with sliders on the CAS machines to find the best “model” or is it because it has been assessed on SACs in the past so much, it feels right to do so?

            Not sure I care so much at the moment, but just wait until the 2024 audit season…

            1. I would assume that the only selling point for these third party products is that they are VCAA-like. If so, of course they will look to follow VCAA’s lead, on anything and everything.

              1. The only selling point, and it’s a good one, is that teachers are time poor and writing SACs and solutions takes a lot of time (particularly if you’re the only teacher and there is no division of labour!).

                (Some may think that another selling point is that they are VCAA-compliant and will therefore pass an audit. One of several arguments against this is that the VCAA is aware of these products and has made a clear statement that commercial SACs need to be significantly modified in order to be compliant. Which takes some gas out of the above. But if you’re not being audited … ).

  3. (I’m trying to re-learn a few bits of my undergraduate degree here, so be gentle…)

    Can the question be fixed by asking it this way?

    For what values of x are y=f(x) and y=f'(x) continuous for all x \in R?

    1. Unless I’m wrong (highly likely), that latter is VCAA’s implied definition of ‘smooth’.
      I have attached amended calculations (the ones I posted here

      Secret 2023 Methods Business: Exam 2 Discussion


      go wrong after a while – I blame a combination of tiredness and then being led down the garden path by someone I thought knew better. The amended calculations may still go wrong somewhere and I am open to constructive criticism).

      When you assume f is smooth at 2pi, then a = -1/2 is the only possible solution. However, to \displaystyle verify that
      if a = -1/2, then f is smooth (ie. that a = -1/2 is a solution)
      requires knowledge outside the scope of the course.

      To appropriate the opinion of someone else, I believe the word ‘smooth’ should be replaced with the word ‘differentiable’ (and I also believe that this is what VCAA actually intended). To \displaystyle verify that
      if a = -1/2, then f is differentiable at 2pi
      does not require knowledge outside the scope of the course (although maybe it does given the amount of limit stuff that has been deleted from the new Study Design).

      Anyway, that’s my nine cents (cost of living pressures).

      2023Mathematical Methods Exam 2 MCQ 9

      1. Thanks, BiB. I’ve taken a quick look and your calculations seem correct. But I think you’re being misled somewhat by the exam report. There’s no need to go the MVT path. I’ll add a comment or an update soon.

      2. Hi Back in Black!

        It’s nice that you gave your working for this. One comment: once you’ve shown it to exist, f':(4,8)\rightarrow\mathbb{R} is a function, just like any other. So to show that it is continuous you do not need to do anything more special than usual. In particular, it is enough to check that

            \[\lim_{x\rightarrow c} f'(x) = f'(c)\]

        for all c\in(4,8). Since f' is piecewise smooth, we only need to check the transition, which corresponds to c=2\pi. But this is straightforward since we can calculate the limits from the left and the right.

        It’s also worth pointing out (you may know this already) that when calculating f'(x) (to show it exists), it is not necessary that f' is defined at x a-priori. In general, when calculating a limit, the function does not need to be defined at the limiting value in its domain. That’s why we take a punctured neighbourhood in the definition. (In fact this is an important point to resolve some later logic about constructing functions *from* limits.)

        Analysis of this kind is poorly understood at university, so it would shock me if the situation is better in schools!

        I haven’t been able to keep up with all of the posts, so if this was already noted, my apologies.

  4. Thanks. That is an awful lot of working for 1 mark and I’m guessing it took you significantly longer than the 90 seconds available per mark…

    Seriously though, your efforts here are much appreciated.

    1. Because it’s an MCQ, the question in practice is not much work: make sure the function values match and make sure the derivative values match. That’s all.

      That doesn’t indicate at all what you’re proving, or what it has to do with “smooth” or the exam report’s nonsense. But that’s the computational essence behind the question.

      Of course if VCAA asks something on this as an Exam 1 question or an Exam 2B question, all bets, and gloves, are off.

    2. Thanks, RF. I played the game (pretty well if I say so myself!) to get what I think is the intended answer (option C has been confirmed by several independent sources).
      But getting the VCAA intended answer doesn’t necessarily mean that the answer is correct. So I wanted to see what happened when a = -1/2 and pretend I was checking for continuity of f(x), continuity of f'(x) (‘smoothness’) and existence of f'(x) (differentiability) at x = 2pi.

      I wanted to vet the question and check that there was a correct option (the question had been brought to my attention for checking by an independent source when I was too tired to think I was too tired). And it bugged me. It all checks out as far as I can see (and now I’m checking out!)

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