WitCH 39: Field of Dreams

This WitCH comes courtesy of frequent commenter, SRK. It comes from the 2016 Specialist Mathematics Exam 2. The examination report does not comment, except to indicate that about 2/3 of students gave the intended answer of B.

22 Replies to “WitCH 39: Field of Dreams”

  1. Good to see you posting new thread, Marty. Welcome back to this tough term 3.

    This is a very typical question which requires student using either by-hand estimation approach, or CAS approach.

    I was wondering what issue was noticed by Mr SRK.


  2. Oh joy, a direction field question, a question that barely appears except as a MCQ. Look at the pretty picture they say and draw a line and make sure your eyes don’t fail you and you have your answer…..Except the fact that none of those points lie on the graph. Should you choose to solve the differential equation with the boundary condition, well you’d be in for a nasty surprise, that none of the points do in fact lie on the graph. Unless we live in some absurd world where -2.56 = -2.5? By that very logic, I guess -2 = -1 or -1.66 = -1 if you squint hard enough/have a questionable graph? Ah but silly me, it asks which point could lie on the graph. The very fact that they had to give the differential equation made the entire thing completely and utterly false. What is their definition of “could”?

    1. M.K:

      Is my picture illustrating your intended point?

      I definitely see your argument on wrong logic in rounding numbers, such as “-2=-1 or -1.66=-1”.

      Well, strictly speaking, if I were a rigorous mathematician who was sitting this paper, I’d rather leave the answer blank because no option was a perfect option. But, that will be in no one’s interest. So let’s just compare which option is closer…

      1. P.N., it’s not about speaking strictly, it’s about speaking truly. Yes, we know what the examiners want, and 2/3 of students knew it as well. But what the examiners asked was meaningless.

    2. Thanks, MK. That is of course the problem. Once they include the differential equation, there is no “could” about it: it’s either an “is” or an “isn’t”, and what we have to choose from are five isn’ts.

  3. At first glance (I usually miss something, so be nice everyone…) this is a typical solve with CAS, draw graph, trace graph, find an acceptable answer, move on after wasting way too much time question.

    I find it interesting that sometimes the paper gives the differential equation (as it did here), sometimes it does not.

    Of course, when one has the DE, one can work out the coordinates much more accurately.

    Would answer B be better if it was -2.6 and not -2.5? Perhaps.

    1. RF, no the question would be no better with -2.6, and it would still be a stupid question even if the correct answer were an option. The fundamental problem is the use of “could” for something that is either true or false.

      But you raise another point. Even interpreting “could be” as “closest to” (FFS), what the hell were students expected to do? Run their finger on the paper? CAS the problem? Solve the DE? Given the finger works and is quickest, what is the point of giving the DE?

  4. Well, digging my old files allows me to see what I was thinking when I obtained a copy of 2016 Exam 2.

    I will just extract my sketching here.

    Even though the question could have been phrased as
    “which of the following point could be the closest to the solution to the differential equation with given condition (0, -1)”, still, by sketching and inspecting by eye, I cannot say option B is a perfect answer, but closest answer.

  5. I wonder if the reason for including the DE is in response to 2015 Exam 2 Q13, where options C and D are difficult to choose between just by trying to draw the solution curve on the slope field. Only 47% chose the correct answer D, and the examination report just states “the solution doesn’t cross any gradient segments” – as though most of the 26% who chose C didn’t already know that.

    1. Well-said.

      Without a DE, guessing the answer or guessing the DE really needs some mathematical dexterity or imagination.

      Now I can see why Marty named this thread as “Field of Dreams”!

      1. Well, without the DE, one has no option but to guess/approximate. That would still leave the question as idiotic, but it at least it would have been coherent.

    2. Jesus, what a gratuitously evil question. But what is the point of any of these questions? Wasn’t connect-the-dots properly covered in about Grade 1?

      1. “ Wasn’t connect-the-dots properly covered in about Grade 1?”

        My Dear Dr Marty., you probably overestimated our Vic-curriculum at primary level too much LOL!

  6. I think you can kind of make it work, if you assume everything is approximate, correct to the first decimal place.

    Solving the equation by hand gives y(x) = 1 - x + C e^{-x}.
    If you apply the constraint y(0) = -1 exactly, then that means you get C=-2 and y(3.5)  = -2.6 (to the first decimal place) so it doesn’t work.

    But if you assume the first point is also approximate, that means it could be, for instance, (-0.02, -1.01), giving the constant C = -1.99, and if “3.5” is only approximate, it could be say 3.46, which allows the second point to fit.
    ETA: actually you only have to be approximate about “3.5” to make it work!

    1. student-teacher, you’re like the defense lawyer for Ghislaine Maxwell: “Your honour, you can kind of trust my client to be out on bail, if you’re willing to assume she’ll approximately stay put”.

      Nope, not having it. Especially not with these VCAA assholes. They love nothing better than whacking a kid if she dares give an approximate answer when an exact answer was “required”. Sauce for the goose.

  7. The coarseness of the diagram suggests that an answer to an accuracy of about \pm 0.1 is reasonable. Within that context, Option B is a correct choice – the other options lie outside of the \pm 0.1 uncertainty range and are incorrect. This question would advantage Chemistry and/or Physics students who would be accustomed to making estimates from given data.

    But the choice of wording “could also include” is very poor and obviously not written by someone with much experience in calculating estimated values from given data. Better wording would have been to NOT include the DE and then ask something like:

    “A solution to the differential equation includes the point (0, -1). Another point included in this solution can be estimated to be”

    (Yep, not perfect but I can only work with what’s given)

    But then I guess this opens up the whole can of worms of estimation from given data. In which case, you keep the lid on the can by not asking this sort of question. Incidentally, a reliable source told me that for the 2017 Specialist Maths Exam 1 Q8, VCAA had an interval of acceptable answers ….

    When calculating an estimate of a value from given data, awareness of the maximum error in the estimate is fundamental. In this question VCAA is sending the message that estimates calculated from inexact data can be considered as exact …. (2016 was a vintage VCAA year – in Methods functions that are ‘almost a pdf’ were considered genuine pdf’s).

    1. JF, I agree with all you write, although I would reorder it. The first thing to say is that the question as written is meaningless. As for that 2017 question, it’s pretty weird. I can’t see the sense of any of these vector field questions, since, at best, they must be either trivial or incomprehensible.

  8. Seems like your kids learn a lot. In the US, that kind of direction field would not be addressed until the ODEs course sophomore year. Beginning of it granted. But still. The AP program (and regular college calculus) has a stripped down ODE module towards the end of second semester. But it doesn’t include direction fields.

    There is a growing tendency for kids to take BC calculus in junior year of HS (or before), but it is still a very small minority. And then there’s no organized coverage for them. They have to run to a local college to get 3rd semester calc and ODEs, if they want to keep going.

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