PoSWW 15: Slippery Slope

The question below is from the second 2020 Specialist exam (not online), and was raised by commenter Red Five in the discussion here. This’ll probably turn into a WitCH but, really, the question is so damn stupid, it doesn’t deserve the honour.

11 Replies to “PoSWW 15: Slippery Slope”

  1. It looks like the ‘clever’ person who wrote MCQ Q7 wrote this one too. I don’t know what’s meant to be tested here … Setting up a DE or interpreting a slope field? It’s stupid to try and test both in a MCQ. This is the sort of question that drives TM nuts.

    Again, I get the feeling that the motivation behind the question is one or more of the following:

    1. The writer was trying to test slope fields in a ‘different’ way. Probably because students have very demonstrated the ability to interpret typical types of slope field questions very successfully in previous exams. So why be a wanker and try to screw them over? Probably because …

    2. The writer thought that s/he was being clever. Wrong. You’re being *stupid* and a *wanker*. Not clever.

    3. The writer was trying to test slope fields in a way that thwarted directly using a CAS. Stupid. If you don’t want students using a CAS, test the skill in Exam 1. Don’t muddy the waters by adding extra skills.

    Memo to VCAA: You don’t try to test skills that are trivialised using CAS technology in Exam 2. Because it leads either to trivialisation or to stupid questions. Additional memo: Test one skill in a MCQ and be clear what that skill is. Don’t test multiple skills in a MCQ.

    Verdict: There is no reasonable justification for this question. It’s stupid to test two very separate skills in a single multiple choice question. What conclusion do we draw from the performance of the cohort in this question? That students can’t construct DE’s, or they suddenly can’t get the slope field for a DE?

  2. Who in their right minds believes a SLOPE FIELD represents a CURVE? What the fuck. A slope field is a vector field, represented e.g. by a map V:\mathbb{R}^2\rightarrow\mathbb{R}^2. A CURVE is represented by a (continuous) map p:[a,b]\rightarrow\mathbb{R}^2.

    There are things called “integral curves” for a vector field, but they aren’t unique. Anyway, I don’t want to guess at what they meant. The question is presented as written to students and it should be condemned for that.

  3. Thanks, JF and Glen. JF it is definitely an awful question, but I think it’s much worse than that. As Glen, suggests, the question makes no sense. In fact, I think the question makes even less sense than the senselessness that Glen suggests.

    1. Well slope fields don’t represent curves. Slope fields are a tool to graphically obtain the solutions to a first order differential equation.

      The question *might* have been improved by asking
      “Which of the following slope fields best represents [the solutions to the differential equation that defines] this curve?”

      Overlooking the nonsense wording, even if we assume the *intent* of the question is for students to construct a DE and then choose an appropriate slope field for this DE, the question is DUMB. I can’t get past how utterly stupid it is to test both those things in a single MCQ question. What was going through the unicellular brain of the writer? What will it mean that 31% of the state got it right and 69% of the state got it wrong?

      The big worry for me is that teachers, including myself, subconsciously translate nonsense like this into something that makes more sense. So in my mind I see
      “Which of the following slope fields best represents [the solutions to the differential equation that defines] this curve?”
      And then my outrage neurons fire at the stupidity of asking this and the ineptness of the ‘asleep at the wheel’ vettors.

      1. Hi John!

        Genuine question: What do you mean when you say solutions to a differential equation to define a curve? I can see how they define many curves (they are the integral curves I referred to earlier), or with given initial data, a single curve. But I don’t understand how the whole solution set to a given DE (not that this question has given any DE) defines a \textit{single} curve?

        1. Hi Glen.

          I’ve understood “a curve” to mean any generic curve that satisfies the given property. I’m probably guilty on multiple counts of subconsciously translating this question.

          1. Something like this?

            Consider a vector field X:\mathbb{R}^2\rightarrow\mathbb{R}^2.
            We wish to impose the following condition on X.
            Let \gamma:[0,1]\rightarrow\mathbb{R}^2 be an integral curve of X; that is, for each u\in[0,1], we have

                \[\gamma'(u) = X(\gamma(u))\,.\]

            Set \gamma(u) = (x(u),y(u)). Our condition on X is that for all such \gamma we require that \gamma has the property that, for each fixed u\in[0,1], the line starting at \gamma(u) with tangent vector \gamma'(u) meets the x-axis at y(u).

            Which of the following plots is a plausible visual representation of the vector field X?

            1. What I had in mind is the following:

              “P(x, y) is a point on a curve. The curve is such that the x-intercept ….
              Which of the following slope fields best represents [the solutions to the differential equation that defines] such curves?”

              My discussion is a hypothetical exercise in how the question might be rehabilitated so that it makes mathematical sense and is appropriate at the Specialist Maths level. This thread is an example of what the interaction between writer and vettor(s) *should* have looked like.

              But the exercise is moot because you could word the question perfectly and I’d still hate it. For the simple reason I’ve previously given.

              Moral of the story: The cuter you try to be, the more of a wanker you look. There would have been no issue with the wording if the writer hadn’t tried to be such a smart-ass. Just write a straight multiple choice question that tests a single skill without all the bullshit.

              1. JF, there’s nothing hypothetical about it. Students in the exam have no choice but to somehow make sense of the question, as then do students and teachers concerned for 2021. The requirement to determine what the hell VCAA *intended* to ask, however, doesn’t preclude, and preferably comes after, critics such this blog noting that *in fact* VCAA fucked up and asked nothing close to making sense.

              2. I just don’t like conflating so many ideas in such a small space. Writing everything out explicitly reduces confusion in my view. I also don’t like the reference to DEs — not all vector fields come from DEs. I understand somehow that this was the intent of the question, but you don’t need to know anything about DEs to work out the right answer. So long as you know what the restriction on the vector field is supposed to be, you can just look at the plots and use a ruler to eliminate the incorrect choices.

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