22 Replies to “PoSWW 21: A Model Answer”

      1. Slope fields are a new concept in NSW senior maths so I don’t have much experience with them and the kind of questions that get asked at school.
        But it seem the slope field doesn’t make sense for the situation it is modelling. There are solutions to the slope field that do not intersect the y-axis but then that would mean there is unbounded capacity to this damn.

  1. Which model? Heidi Klum? Oh … the model from which this slope field has been drawn …? “this model” is \displaystyle that model. OK.

    Wrong options can be rejected:

    All slopes < 0 therefore options B and D are rejected.
    All slopes \displaystyle appear to approach zero as \displaystyle t \rightarrow \infty therefore option A is rejected.
    All slopes \displaystyle appear to approach zero as \displaystyle V \rightarrow 0 therefore option E is rejected.

    So option C is the winner. We have the answer, now let's move on. Actually … let's NOT move on.

    From option C: \displaystyle \frac{dV}{dt} = -k V^2.

    It follows that \displaystyle V = \frac{\alpha}{t + \beta} where \displaystyle \alpha, \beta \in R^{+}. \displaystyle \alpha = \frac{1}{k} and \displaystyle \beta is determined from the initial condition. That's one lousy model for the volume of water in a reservoir as time goes by, but that's the collateral damage we expect from these moronic 'real-life' contexts.

    But, wait a moment … From what I can see, it looks like there's a constant solution if the initial volume is small enough. That's a bit awkward.

    And correct me if I'm wrong, but does the slope field suggest that if the initial volume is large enough, then the volume approaches a limiting \displaystyle non-zero value …? That's awkward too.

    And if you stare at the slope field for long enough, I swear you can see someone flipping the bird at you. Right back at you, VCAA.

    1. Thanks, John. Of course the (eventual) model is ridiculous, but what first pissed me off was referring to a bunch of stupid dashes as a “model”. Then, supposedly, “according to this model” to derive a precise equation is insane. Plus, the placement of the k is really, really stupid.

  2. The thing that bugs me about this one is the pseudo-context. Just ask which derivative could have produced the slope field and be done with it.

  3. I’m sure nobody will agree with me again, but I see this one as also incorrect. There is no right answer.

    The reason is similar to last time. When we write in mathematics “for k > 0, blah involving k is true” we mean that k can take a range of values greater than zero. If we mean a specific k, then we write “there is a k>0 such that blah involving k is true”. (Sometimes you will see “for some k>0, blah involving k is true” which is IMO bad writing, because the word “some” here makes it ambiguous.)

    You might think that this is pedantic, and that we should go with the interpretation that gives a plausible answer. But these words are *qualifiers* and being able to understand them is absolutely crucial in learning how to properly read and understand mathematics.

    Again, we might want to rewrite the question to say “Let k>0 be a constant. Which of the following equations could be consistent with the slope field for V above?”. Or how about we just remove k from the question entirely? Hey, maybe we should just ask another question? What are we talking about again? Oh that’s right, we are making excuses for idiotic wrong exam questions that should not have excuses made for them. Don’t mind me.

    1. Glen, I absolutely agree with you! I hadn’t thought of that.

      But I must ask: Sitting the exam, what would you do? Would you leave this question blank, claiming there’s no correct answer, and doggedly pursue VCAA through the legal system? I see on the Examination Report that 1% of students left the question blank. Perhaps for that reason … Or would you leave your principles at the doorway to VCAA World, swear under your breath, walk through the door and shade C?

      1. If you mean Glen the 90’s era High School Student, then I would have likely answered randomly because I didn’t have any clue what I was doing in high school.

        If you mean Glen the 2020 era Cynic then I would have written in large bold letters “THIS QUESTION HAS NO CORRECT ANSWER” and coloured none of them. Then after the exam I would have written to whoever ran the exam and complained. I’d shout it from the rooftops, if I was put in a situation like that.

        I think your question is really very relevant however. Because often I see a lot of teachers — very well-meaning teachers — make this decision *for the student*. I don’t think that’s right. Your job as a teacher is to teach, and not to make personal decisions related to exams on the fly.

        I’m willing to concede this point as impractical but that’s my opinion. I know, your colleagues, parents of students, and students themselves might think you are nuts. Just teach the kids how to get the highest mark possible! That’s the mantra, right? I reject that. Teachers have a truly important job and it isn’t to game a broken exam.

        I find it so sad that disciplines like mathematics are reduced to the maximisation of marks on an exam. Mathematics is so much more than this.

        1. I think we can do both.

          I think we can teach mathematics AND I think we can teach students to game a broken exam. And, morally and ethically, I think we must try our best to do both.

          There are teachers with students far smarter than the VCAA idiots who write this shit who face the genuine difficulty of helping that student to NOT get penalised for being smarter than those idiots.

          To get on our high horse and deny that as teachers part of our job is to help students maximise their results is to do those students a huge injustice.

          We can do both and we must do both.

          @Glen 20: My question was for ‘Glen the 90’s era High School Student’, who I think adroitly avoided answering the question.

          1. Maybe *you* can do both, but I’m not really referring to the exceptional few teachers at the top end of the bell curve that will teach mathematics well no matter what garbage curriculum or exams are thrown at them. It is missing the point to say “we can do both” — no matter what, there is a priority and I think it is a grave mistake (and as you say, an injustice, although I would say an injustice in the complete opposite direction) to choose “gaming the exam” as that priority. But, here we are.

            What I think is closer to the truth, is that most teachers teach how to game the exam, many of them think that’s it for math, and that’s the end of the story. Some teachers try harder to teach things beyond just gaming the exam, and very very few of them do so well. It is completely backwards in my view.

            I like our side conversation. Honestly, I thought that I answered your question. Glen the 90’s era HS student would have answered randomly because he had NFI.

  4. John, you’d figure there’s something wrong with you and you just can’t do this stuff. You’d never do maths again and spend the rest of your life telling people “I’m no good at maths”.

    1. I like to think that I’ve taught my students to have a healthy disrespect and contempt towards VCAA so that they know NOT to jump to that conclusion when they see this sort of shit on VCAA exams.

      What’s needed is for ALL teachers to make their students (and parents) aware of how contemptible, loathsome and deceitful VCAA is by presenting the evidence and discussing it. When that happens, there’s a good chance of forcing cultural change at VCAA.

    2. Dear Sigma (what the hell do I call you?), I strongly agree. I think the specific errors and the general absurdity is way, way more damaging than the loss of a few marks here and there, although that’s bad enough.

      The clear message from VCE (and earlier) is that “maths is arbitrary and ritualistic and, often, plain nuts”. The preponderance of nonsense must be convincing tons of students to not study mathematics any more than is absolutely necessary.

      1. Marty, it’s obvious. Given that you moonlight in abnormal psychology (for the purposes of knowing your enemy), it’s clear that you’ve caught the eye of Sigma Freud.

  5. 😂 “Steve” is fine!

    It pains me how many fine minds are turned away from the discipline in high school. We’re all the poorer for it.

    1. It infuriates me, and it infuriates me that it is the work of the sanctimonious clowns who never shut up about getting students to do and to enjoy mathematics.

    2. Absolutely CORRECT. People fall in love with mathematics when they see how it all hangs together, beautifully connected as an infinitely intricate tapestry.

      It has the most impact when students are allowed to observe this for themselves. They will become convinced, and start to have an idea of what mathematics is all about. For this, they need to know how to do mathematics, how to work with its most basic objects easily and how to reason properly about them.

  6. I left high school swearing I’d never do maths again. I hated it. I knew it was powerful and important, but it was disjointed and confusing and impossible to solidly grasp. My teachers thought mathematics was a magical machine that does stuff when you turn some handle (but nobody really knows how … other than maybe some distant, mythical unicorns/geniuses). It would be years before I’d realize that mathematics is actually just the slow, careful, sober investigation of patterns.

    After a few years of McJobs, I went back to uni to do a science degree (Latrobe). The plan was physics and I was just going to do the maths I had to. At the start of the first maths lecture, my lecturer Grant Cairns announced “the first thing we need to do is unteach you everything you were taught in high school”. And he did. And I have never looked back.

    Throughout my degree, I was convinced my issues in high school were down to my being a really shitty student (which I probably was too BTW). Then in honours, I volunteered at a high school to help in a mathematics class and my jaw dropped. By this stage I was a full 10 years out of high school and guess Cairns had done a really good job of unteaching me. My. God. The crap those poor students were being subjected to!

    It took me all that time to come full circle and re-realize how truly shitty high school mathematics is. And the teacher in that class had zero understanding of the state of affairs.

    I think one of the most fundamental issues is that there is nothing confusing or difficult in the machinery of mathematics if you’re doing it right, the individual steps are almost embarrassingly simplistic … you just need the time and discipline to step. Very. Very. Slowly. (… and build up to the big picture.) But mathematics teachers wear the purported difficulty of mathematics as a badge of honour. A key ingredient of their egos. They don’t possess the humility to accept the simplicity of what they’re doing. Maybe some only learnt it as a mysterious machine themselves and genuinely believe it’s difficult and confusing.

    Unfortunately, I think the only solution is making sure the workforce of mathematics teachers by-and-large have mathematics degrees. I can’t see a path to that state of affairs.

    For me, there’s no curriculum of sufficient quality to make high school mathematics work when high school mathematics teachers haven’t done much beyond high school mathematics!

    (Present company excluded. 😂)

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