WitCH 81: Pointed Criticism

This is another one courtesy of Mysterious Michael. We’re not quite sure whether it’s a WitCH, or a PoSWW, or one for the error list, or simply a roll-your-eyes-and-ignore. In any case, it’s a question from the notorious 2016 Methods Exam 2 (report here), and we’ve expanded slightly from MM’s stated concern. (We could’ve expanded to include the entire question, as a Road to Nowhere, but decided not to lose too much of MM’s original focus.)

Have fun.

34 Replies to “WitCH 81: Pointed Criticism”

  1. i) is routine i.e. expand f(x), determine its anti-derivative, apply g(0) = 1 to get g(x). Boring but reasonable.
    ii) should read “Find the values of x at which g(x) has stationary points”. Plural, because there are two.
    Referring to g(x) is sufficient, why mention “the graph of y = g(x)”? Maybe a reminder to get the CAS out?

  2. Would VCAA find it acceptable to just differentiate g(x) and show that you get f(x), and also evaluate g(0) to show it is 1? With some argument about uniqueness? Or is that not showy enough?

      1. The answer to this question is that VCAA would NOT find it acceptable. A secondary reason for this answer (the primary reason is obvious) comes directly from the 2021 Maths Methods Examination 1 Report:

        “A reminder that ‘show that’ questions require a reasoned argument. The answer is given and students are required to provide a detailed progression to the answer.”

        (But it’s only a matter of time before VCAA contradicts itself on this statement).

        1. Thanks John. It seems to me that it would be so much easier to answer this question if they just wrote “find a function g(x) that satisfies g'(x) = f(x) and g(0) = 1.” Then students wouldn’t have to worry about what steps to write down. They could just demonstrate their understanding by finding the answer. Including the answer in the question makes it harder to know what they want.

          I don’t understand what students are meant to write, if apparently they are supposed to use CAS to do it. Do they really write “I used CAS to find the anti-derivative of f(x) and it said it is the same as what’s here”. I don’t understand why they would even need to use CAS because the answer is right there.

          1. Sorry, I just had a look at the examination report. It does seem that it’s just a case of copying the functions down like a fill-the-blanks exercise, making sure to add the appropriate integration symbols and “+c”. Marks off for transcription errors.

            1. Given that the examiners report effectively admits that students will just use CAS to do the integration, I wonder what they might have made of a student writing down \int_0^x g'(t)dt+1=g(x).

              1. If you’re gave this as the answer, VCAA would give it zero. VCAA would say it wasn’t sufficiently simplified and then launch into some sanctimonious crap about what students should be doing.

          2. No. It would be so much easier to answer

            exam questions in general if VCAA had a competent vetting panel that included a mathematician and someone with sufficient intelligence to understand why this is such an idiotic question in a CAS-enabled Exam. Unfortunately, this is impossible under current VCAA policy.

  3. What is the point of defining f(x) separately when g'(x) does better in less? It is never “considered” as the actual function of interest anywhere in the entirety of Question 2.

      1. In some schools… not naming names… some teachers (also not naming names) instruct students to define the first function they see in a question as f(x) and then define g(x) as its derivative.

        Perhaps (although I doubt it) this is the examiner attempting to push back…?

        1. RF:
          Sounds like pushing back, or could just be a preference of letter choices.
          Defining functions with f, g, h, etc seems like just one “convenient way” to inform the students that there will be different functions throughout one extended question with multiple parts.
          For some MM exam 2 in certain years, students are expected to use distinct letters correctly for different functions within one question. If a function f is already given at the beginning but student used f again to stand for something else in later parts, this can be contradictory to the existing information and *could be* penalized.
          Curious what would happen if student used dy/dx in place of f’(x) for the max/min type of questions. Based on the advices from the MTA sessions in recent years, it feels like exam 1 are more strict with the notation things than exam 2.

          As to the domain, with polynomials, R should be the default domains, despite being understated. As to other functions, appropriate constraints and considerations must be taken. So in this instance, though less ideal and rigorous, it is not as concerning as other issues raised. The subtle change in the order of function letters seemed ringing a bell to teachers and students to read the questions and check the conditions more closely.

          I can still remember six year ago when I first saw this paper I was a bit surprised with this “show that” question because it involved three or four lines working just for one mark (should it be an exam 1 question, students would need to expand the function first, integrate and add a constant, substitute the point to find the constant, and restate the given function). Later I realized it was set for exam 2 so students would have just picked up their CAS to integrate the function, put a +c, and to state the condition g(0)=1 while asserting c = 1. This seems to be the expectation. Since then, be it either Methods or Spesh, the “one-marker show that” questions have become increasingly popular, where students often need to demonstrate numerous correct steps to get the mark.

          WST has pointed out above whether differentiating the given function and substituting x=0 to show g(0) = 1 would be a valid response or not.
          Although it looks like “backward engineering” and it is more of the “verify” approach, I am personally not against this approach but who knows if VCAA would accept it? The “applied maths” part of mine tells me this should be ok…That said, it is always in the best interest of students to start working from the given condition towards the result, not the other way around to jeopardize their mark(s)…

          As for the potential grammar issue in this question, it is open to interpretations.
          The answers could be:
          “For x = -2, there is a stationary point”
          “For x = 1, there is also a stationary point”
          So I would accept the preamble, despite it using single rather than plural.
          The second part could also be phrased as
          “State the values of x at which the stationary points of g occur”.
          “State the values of x where the stationary points of g occur”
          “State the values of x for the stationary points of g”, etc.

          1. On the topic of “show that” questions… I do constantly wonder how the (small group, but not zero) students doing Methods CBE are meant to deal with this and are their lines of calculation treated as “showing” in a way paper-based exam students are not…?

            Perhaps it doesn’t matter.

            1. Very good question. You get some inkling of an answer from this post. Also, my memory is that Sai had a lot to say about this, and which didn’t make the cut of his Vinculum article. He may wish to comment (if he’s not too busy teaching new kids how to game VCE with Mathematica).

              1. I’ve been summoned! JF is correct, the exams were slightly modified in the case of exam 2. Some of my records indicate that the Methods CBE were printed out and then marked, but that was via word of a teacher at my school. It’s likely that they would have required you to use the text box/computation cell to “show” that you’ve determined the value of c manually.

                On the topic of data, I uncovered an old document a long time ago looking at CBE performance in Methods. Unfortunately, there’s no raw data, but beggars can’t be choosers. The link is here: https://www.researchgate.net/publication/276207224_Comparing_the_Score_Distribution_of_a_Trial_Computer-Based_Examination_Cohort_with_that_of_the_Standard_Paper-Based_Examination_Cohort

                1. Thanks very much, Sai. Re the paper, my memory is that this paper was used by DLL to argue that Methods and Methods-CBE needn’t be distinguished.

                  I also spent a fair amount of time a couple years ago, clawing some data from a knee-jerk reluctant VCAA, on students’ performance on Methods-CBE. It would appear that in general students did not do as well as regular Methods students. That was surprising, but presumably has mostly to do with the manner in which schools selected themselves to offer CBE, and the manner in which they then implemented it.

                  1. That would undoubtedly be a factor. Another factor might be that many of the students thought (and still think) that they can abrogate their understanding and thinking to Mathematica. It is used as a substitute/crutch for thinking. Despite everything to the contrary said by their teachers. This is the seductive trap that many students fall into, despite assessments providing constant evidence to the contrary.

            2. Students would have to have their 3 cells of Mathematica code. Mathematica doesn’t give the ‘+c’, so the question is a golden opportunity for students to really demonstrate their mathematical knowledge and understanding.

              It’s possible – but I can’t remember offhand and I can’t be bothered checking – that the Methods CBE Exam* might be a modified version of the ‘write on paper’ exam. When the CAS-calculator was being piloted, Exam 2 was modified.

              *Yes, I know Methods Computer Based Exam Exam. Ha ha ha. By the way, I don’t say PIN number.

              1. The only CBE I had experience with as a teacher/supervisor/needed a job and this filled in the time nicely was the 2013 paper and as far as I could tell is was not modified in any way.

                The pink cells for working and blue cells for text were useful in a way but again, an examiner can SEE a CBE student’s working in a way that cannot be seen for other students unless they hand in their CAS with the exam… so… yeah. Fairness?

                When CAS was being piloted (2002-2005?) there was a CAS and a non-CAS paper 2. There was also a slightly modified study design from memory. There was a lot of overlap between the papers (again, from memory) and the examiner’s report made mention of how students didn’t perform much differently on the two different papers. Of course, publishing the raw data is another question (I am unable to answer for now but remain hopeful)

                1. We have the same memory. (And what a stroll down memory lane it’s been!)

                  Fairness? None. Fortunately the CBE has gone the way of the dodo… for now.

                  Raw data? Forget it. We have to be content with vague statements like “didn’t perform much differently on the two different papers.”

  4. Does it strike anyone else as odd that VCAA, who are/is normally obsessed with domains did not give a domain for function f?

    Is it fair to assume the domain is R?

    Something tells me this is not a safe assumption.

  5. Since it’s only one mark, VCAA’s expected working out is probably:

    find g(x)= indefinite integral of f(x) using cas, with +c
    solve g(0)=1 to find c (using the cas)
    write g(x)

    Of course that doesn’t test anything. The question wouldn’t be too bad if it were in the tech-free exam and worth three or so marks, but it’s really pointless in the tech-active exam.

    1. VCAA have the explicit instruction
      “For questions worth more than one mark, appropriate working must be shown.”
      A reasonable implication is that if the question is not worth more than 1 mark, appropriate working is not required. Only the answer is required. There has been a plague of 1 mark VCAA exam questions in recent years where clearly only the (conjectured) answer is required. To justify the (conjectured) answer to these questions requires significant working.

      All bets are off when it comes to the idiotic 1 mark “Show that” question. I recall the 2019 Specialist Maths Exam 2 Question 2 (a)(i) that required students to “Show that” the solutions to a given quadratic equation were …. To get the 1 mark required five steps, two of which were so trivial that a decent student wouldn’t have bothered with them. The fact that 32% of the state got zero for this question is a damning indictment on VCAA. Such questions on a CAS-assisted exam are truly stupid and suggest that, deep down, VCAA knows that having a 2 hour CAS-assisted exam is a joke.

      The question in this blog is a classic example of this idiocy. I agree with what A says, the Examination Report confirms the required working suggested by A and implicitly confirms that it doesn’t test anything (except whether a student can satisfy the VCAA Notation Police). It takes a special kind of idiot to write a question like this.

      1. Not only the person who wrote the question, but the panel who approved its inclusion in the paper…

          1. No point panel beating when the Vcar’s a write off. What we need is a new Vcar.

          2. Marty – this is your best comment EVER.

            And yes, there are lots of witty headlines and one-liners to choose from, but I suggest this is currently in the gold medal position.

  6. The only real issue I had with this was the spinal fluid leaking out of my every cranial orifice as I read it. Otherwise, good to go.

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