WitCH 76: Cos and Effect

Yeah, yeah, we should have done this one a week ago. Feel free to transcribe your comments from the Methods Exam 2 discussion post. You can also check out the discussion on stackexchange, courtesy of Stog the Stirrer.

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UPDATE (15/11/21)

The question is awful, with the final part the pinnacle of awfulness. We’ll consider that part in some detail, but first the other awfulness.

  • Part (a) Part (b) could be interesting since finding the critical points amounts to solving a (nasty) cubic, but it is not interesting here. Here, it is meaningless CAS garbage.
  • The wording in Part (b) Part (c) is atrocious. If you mean the functions f(h-x) and f(x) are equal then damn well use the word “function”. And, why not define a new function, Fh or whatnot? You guys are forever, painfully, defining functions for no good purpose. Why not here, when there is actually a purpose? Of course, it also never occurs to anyone that one might prove that the two functions are equal for whatever h. Nope, just look at the damn picture.
  • When does ga equal f? Seriously?

 

  • Part (e)(i) is fine, but Part (e)(ii) contains possibly the worst sentence in human history. The question itself could be a good test of knowledge of trig symmetry, but of course here it is just a test of pushing buttons.
  • Part (f) is good, although we wonder what students will make of it.

 

And now, Part (g), which is, at best, Magritte garbage. Is it worse than that? Yes, it is.

The question asks students to

Find the greatest possible minimum of ga.

There are (at least) six plausible interpretations of this question:

Interpretation 1 For each a, let Ma be the (absolute) minimum of ga. Find the maximum value of Ma over all possible a.

Interpretation 2 As for Interpretation 1, but find the minimum value of Ma over all possible a.

Interpretation 3 For each a, let La be the set of local minima of ga. Find, for each a, the maximal element of La.

Interpretation 4 As for Interpretation 3, but find, for each a, the minimal element of La.

Interpretation 5 As for Interpretation 3, but find the maximal element over (the union of) all La.

Interpretation 6 As for Interpretation 4, but find the minimal element over (the union of) all La.

Now, for the kind of reasons that commenter Tungsten suggests, it is likely that Interpretation 1 was intended, but it’s no gimme. In particular, Interpretation 2 is quite plausible; it takes a special born-that-way stupidity to use the term “greatest” when optimising negative quantities. Moreover, as John Friend and Glen have suggested, below and on the discussion post, Interpretation 3 is also very natural. Then, Interpretation 4 is not far behind. In any case, this is insane. Students shouldn’t need to engage in an idiotic guessing game at the end of the exam, for 1 mark, simply because the writers cannot write.

Anyway, guessing over, let’s assume Interpretation 1 is correct. What then do students do? Yep, as Tungsten suggests, they just fiddle with their buttons, note that Ma gives a minimum of -√2, and that Ma appears to be decreasing. That’s all they can reasonably do. Well, they can also reasonably scream out “This is meaningless garbage”, but that probably won’t score them the mark.

Note that there is a very nice and natural and easy proof that Ma has a minimum of -√2. See the stackexchange reply. But this is Methods. No one gives a damn.

A stupid, hateful question to end a stupid, hateful exam for a stupid, hateful subject. Utter lunacy.

 

UPDATE (23/04/22)

Not a meaningful word in the exam report. A complete disgrace, and entirely predictable.

19 Replies to “WitCH 76: Cos and Effect”

  1. e ii)
    ” … the area bounded … is equal above and below the x-axis …”
    I’m guessing it is meant to describe two separate areas that are equal.

    g) “Find the greatest possible minimum value of g_a.”
    I assumed that this meant the greatest global minimum, greatest taken over all possible a. But after looking at comments in the Methods Exam 2 discussion post, I guess it could mean the greatest (or least) minimum for the various local minima for a fixed a. Either way it is too difficult.

    When I was a Year 12 student the exams were tightly controlled by E. R. Love who had an international reputation for the rigour and clarity of his writing. The early eighties saw a move to give examining power to secondary teachers. This is the result.

    1. I will attempt to guess the likely intended solution (by VCAA) to part g. The meaning of “greatest possible minimum” should be inferred from usage of terminology elsewhere in the question. Then turn to the graphing calculator or exam laptop and look at the graph when a=1 (the lower bound of a) where the global minimum is -√2. Increase the value of a, and watch the graph change. Observe that the graph goes below -√2. Conclude that this is true for all a>1 and therefore that the answer is -√2 for 1 mark.

      VCAA would say that it is not too difficult.

      I assume that prior to the early eighties, the use of calculators and laptop computers in examinations was a little more rare compared to now.

      1. Another thing to note is that I have seen in too many places the suggestion that the answer to part g. is -1 as a becomes large, based on looking at a graph of an example of for instance a=10^4. This appears to be the case for a small section of the graph, but is clearly incorrect if one zooms out far enough. I must admit that I was guilty of this error during the examination and so lost the mark. This sort of thinking and exam strategy, just looking at some graphs with some values plugged in, is really encouraged and a direct result of the effect of CAS and the style of questions such as part g. that can be reasonably answered using only such “tricks”.

      2. Scientific calculators that could perform calculations using the normal distribution were in common usage in the early eighties. The Kaye and Laby Four Figure Mathematical Tables ( https://webmaths.wordpress.com/2014/03/31/a-short-history-of-mathematics-teaching-learning-pt-1/ ) – in common usage for many decades in the mathematics classroom – was also permitted. Students of yesteryear learnt to use these tables like students of today learn how to use a CAS. Pages from these tables pertinent to the normal distribution were reprinted and included on the Exam Formula Sheet until the late 90’s. Using Kaye and Laby required far greater mathematical acumen than using a CAS. This was all back in the days of yore when the mathematics syllabus was unambiguous and contained mathematics, and the exams actually examined mathematics.

        People will say that things are never as good as you remember them, but I guarantee the mathematics syllabi and textbooks back then were vastly superior to what is available today.

        1. Your last sentence is true without a doubt.

          I would also suggest that, even now, it is way, way, way more valuable to introduce, for example trig calculations to year 9 students, by printing off a table of S-C-T from 0 to 90 and having the students use that.

    2. Thanks, tom. The current tolerance for incompetence is astounding. On E. R. Love, I have a small post planned about him as soon as I escape these exams.

  2. I don’t think the essence of the crap has been captured at this blog. The true essence is what the hell does “the greatest possible minimum value” mean!!??

    As I’ve commented elsewhere – for a given value of a there is a set of local minima. Each of these sets has a value whose magnitude is smallest and a value whose magnitude is largest. You can take each of these values for each value of a and construct two new sets – the set with the largest value and the set with the smallest values.

    Is the question asking for the largest or the smallest value in each of these two sets.

    The only answer that makes sense once all this crap is processed is \displaystyle -\sqrt{2} corresponding to when a = 1. The limiting value of -2 is never attained and so the question must intend the value with minimum magnitude out of all possible minimum values for all possible values of a.

    Maybe I’ve made heavy weather trying to explain this, but to me this is the true crap:

    What does “the greatest possible minimum value” actually mean. Is it possible to explain what it means using only VCE language (which is what I’ve tried to do above).

    1. Agreed John – it is ambiguous. And it needed your full analysis to make that clear. If the question refers to just some fixed value of a then there are at least 2 meanings. If we allow the optimum over all possible a (not in the syllabus) then we get 2 more possibilities.

  3. Quick correction: in your update, you seem to have written “Part (a)” and “Part (b)” when you mean “Part (b)” and “Part (c)” respectively.

  4. A couple of points:

    1) Re: Part (g). I think it’s worth noting that the average mark for the state was 0.0. Yes, out of the 16,000 or so students who did Maths Methods, the number of students who got a correct answer was less than 800 or so. This is a damning indictment and should be a total embarrassment to VCAA, assuming it’s capable of feeling embarrassment (which I very much doubt).

    I got the 0.0 datum from the External Assessment Summary that every school receives. I got it on 23 February. Which begs the perennial question:
    Why, as of 17 April, are the mathematics Examinations Reports not available?
    Laziness? Collusion? Disrespect? Difficulty deciding how to address the many exam errors? Who knows, because VCAA refuse to give a reason(s) despite being asked many times.
    The data has clearly been available since at least 23 February. All other subject Examination Reports are available. The issue of late reports is unique to mathematics, despite grand promises by VCAA last year that things would be different … 5 MONTHS since the mathematics Exams and still no Examination Reports.

    2) Re: Update to part (b).
    Despite your justifiable derision of this question, only 60% of students got the 1 mark. I find this absolutely astounding, even allowing for the fact that some students might not have got to Question 5. 70% of students got the 1 mark for part (d) (equally astounding, by the way) so I’d guess that at least 70% of students attempted part (b).

    3) Re: Update to part (c).
    “If you mean the functions f(h-x) and f(x) are equal then damn well use the word “function”. And, why not define a new function, Fh or whatnot?”

    I have no problem with not using the word function. f(x) has been clearly defined and hence also f(h – x). As an example of what I mean, if we’re doing differentiation from first principles we don’t explicitly state that f(x + h) is a function or define a new function.

    However, I (obviously) strongly agree with your sentiment that VCAA have become too fond of wanting students to use ‘pictures’ to get answers. I have no problem with VCAA wanting students to make a conjecture based on a ‘picture’. BUT students must then be required to \displaystyle PROVE their conjecture. VCAA doesn’t require this, and possibly doesn’t even understand the importance of doing it.

    Of course, what we than get is a question where the conjecture is worth 1 mark and the proof is worth another mark. VCAA would probably argue that since the conjecture is correct a proof is not required and the ‘proof mark’ can be better allocated elsewhere. Which completely misses the point.

    Part (c) should be written “… and \displaystyle prove your answer.”

    4) Re: Update to part (e) (ii). Well, the sentence is certainly a lot worse than “It was a dark and stormy night …”, that’s for damn sure.

    I think your comment “The question itself could be a good test of knowledge of trig symmetry” is a bit misleading, and should be expanded upon and generalised:
    i) There is no guarantee that the integral over one period of a periodic function will be equal to zero in general. There are obviously plenty of periodic functions whose integral over a period is not zero.
    ii) The question only works because the integral of this particular periodic function over its period is zero.
    iii) The question works for any function whose integral over an interval is equal to zero.

    The question exploits (I assume, but we’ll have to wait for the Examination Report to know) the simple fact that if \displaystyle \int_{a}^{b} f(x)dx = 0 then the areas bounded by y = f(x) above and below the x-axis over the interval [a, b] must be equal. Requiring students to explicitly use this fact is what would have made the question more interesting.

    1. Thanks very much, John. There’s been so much idiocy since this idiocy, I had forgotten about this idiocy. In quick response to the points you make:

      1) Yes, obviously a question where the average mark is 0.0/1 is pointless, and obviously a screw-up. And of course VCAA is taking so long with the examination report because they want to make sure the wording of their acknowledgement and their apology is perfect.

      2) I cannot really respond to your comment on (b), but I assume the question is trivial with some stupid CAS trick.

      3) No, John, you are simply wrong. The fact that it is two functions being equated is critical, and this is not remotely clear from the wording.

      4) I think you’re misinterpreting my phrase “trig symmetry”, but no big deal. We’re making essentially the same point.

      1. Thanks, Marty.

        Re 1): Ha ha ha. Yes, of course. Why didn’t I think of that!
        But I would have thought they screw up so often and hence are soooo practiced in making apologies that it would take no time at all to make the apologies the 2021 Reports require.

        Re 2): Yes. The question is totally trivial. I can only assume that the students who got it wrong either misread their CAS or did not round correctly. Otherwise ….

        Re: 3). Marty, I’m genuinely curious – how would you have worded part (c)? Perhaps something like

        “Consider the function \displaystyle g: R \rightarrow R, g(x) = f(h - x). Find the smallest positive value of \displaystyle h such that \displaystyle f = g for \displaystyle x \in R.”
        ?

        Re: 4). Yes, I probably am. I wouldn’t have bothered including the word “trig”.

        1. Hi, John. On (3), I agree that it’s not obvious how to word a question such as (c), and I reply below, but don’t be distracted. What is obvious, and the point I was making, is that the don’t-give-a-damn wording used in the exam is idiotic and incompetent.

          Yes, we will all use f(x) ambiguously, sometimes to refer to the function f and sometimes to refer to the value of f at x, with x maybe less or more specified. It is helpful to have both interpretations available. But it takes a special anti-mathematical genius, in one quick sentence, to introduce an h and then have f(h-x) refer to the function of x, with h fixed, except that you’re looking for the one special h. Seriously, who does that?

          Yeah it’s just one mark and, yeah, you can probably figure out the one reasonable interpretation pretty quickly. But you shouldn’t have to. It shouldn’t be your job as an exam-taking student to make up for the indifference and incompetence of VCAA’s writers and vetters.

          Is it as bad as (e)(ii)? No. But it is a product of the same lazy incompetence. These people cannot write, they have no conception that they cannot write, and they have no conception that they are in any sense subject to judgment. Year after year after year they do this crap, and clearly no one at VCAA gives a damn. It is entirely unnecessary, and thus it is insane.

          Now, how might you word such a question? Yes, I’d go with something like what you suggested, probably introducing h as a real number first, and leaving out the x \in R at the end. But I think the critical points are: (i) you have to explicitly announce the presence of the new function, as you did; (ii) you have to slow down, and split the question into two sentences, as you did.

          There is always a cost to making questions longer, but if you have too much to say in one sentence then you must accept the fact and write more slowly. A simple human truth of writing, which is entirely unknown to VCAA nitwits.

      1. It is pathetic that VCAA pontificates that “-2 was a common incorrect answer” but does not have the guts to admit that students who gave this answer quite reasonably chose interpretation (2) (see 15/11/2021 update above). Students who gave -2 as the answer are victims of the stupid, ambiguous wording of the question and, more generally, VCAA’s incompetence and cowardice. One might even argue that students were led to this interpretation by part (f) …

        It should be noted that -2 is incorrect NOT because there is only one correct interpretation (according to VCAA). It is incorrect because -2 is the \displaystyle limiting value of \displaystyle M_A. \displaystyle M_A is never equal to -2. However, given that VCAA clearly wanted students to use a \displaystyle Magritte non-proof for getting its preferred answer of \displaystyle -\sqrt{2}, it’s hardly a sin for students to use the same style of non-proof in this context to get the non-preferred but equally reasonable answer of -2 …

        I’ll bet London to a brick the fool who wrote Question 5 has no clue how to \displaystyle PROVE the VCAA preferred answer nor how to \displaystyle PROVE that -2 is a limiting value …

        VCAA, you are a total disgrace.

        1. A solution to part (g) came my way a few days ago and I have permission to share it – attached. Readers of this blog might find it interesting. VCAA, particularly the writer and vettor(s) of Question 5, are recommended to read it and maybe even learn from it.

          2021mm-exam 2-Q5g-solution

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