WitCH 29: Bad Roots

This one is double-barrelled. A strange multiple choice question appeared in the 2019 NHT Mathematical Methods Exam 2 (CAS). We had thought to let it pass, but a similar question appeared in last week’s Methods exam (no link yet, but the Study Design is here). So, here we go. First, the NHT question: The examination report indicates the correct answer, C, and provides a suggested solution:

\Large\color{blue} \boldsymbol{ g(x)=f^{-1}(x)=\frac{x^{\frac15}-b}{a},\ g'(x) = \frac{x^{-\frac45}}{5a},\ g'(1) = \frac1{5a}}

And, here’s last week’s question (with no examination report yet available):

Update (19/06/20)

As commenters have noted, it is very difficult to understand any purpose to these questions. They obviously suggest the inverse function theorem, testing the knowledge of and application of the formula g'(d) = 1/f'(c), where f(c) =d. The trouble is, the inverse function theorem is not part of the curriculum, appearing only implicitly as a dodgy version of the chain rule, and is typically only applied in Leibniz form.

As indicated by the solution in the first examination report, the intent seems to have been for students to have explicitly computed the inverses, although probably with their idiot machines. (The second examination report has now appeared, but is silent on the intended method.) Moreover, as JF noted below, the algebra in the first question makes the IFT approach somewhat fiddly. But, what is the point of pushing a method that is generally cumbersome, and often impossible, to apply?

To add to the nonsense, below is a sample solution for the first question, provided by VCAA to students undertaking the Mathematica version of Methods. So, the VCAA has suggested two approaches, one which is generally ridiculous and another which is outside the curriculum. That makes it all as clear as dumb mud.

13 Replies to “WitCH 29: Bad Roots”

  1. So I guess whoever wrote the questions is not the same person who wrote the answer, because that solution doesn’t use the technique suggested by the question.

  2. Surely the only reason to ask about the derivative of an inverse is as an example of the chain rule: that dx/dy = 1/(dy/dx). There is no need to compute the functional inverse of f(x) and indeed it simply adds a level of algebraic confusion. Since f(0) = 1 it follows that g'(1) = 1/f'(0) which is all that should be required. There’s nothing wrong with the question per se, unless students are expected to solve it by the laborious method given by the examiners. But since it’s a multiple choice question, that becomes moot anyway. How I hate mathematical MCQs.

    1. With a CAS machine the method given by the examiners is not too laborious. In fact, it’s probably the most efficient method:

      The alternative method (either what the writer had in mind or a method the writer wanted available to students), gets you to 1/f'(0). Personally I don’t see a Methods student getting to g'(1) = 1/f'(0) (since it’s not on the course and will not have been taught) but even if they did, there are a few more steps they have to execute:

      You have to differentiate f, substitute x = 0 to get f'(0) = 5ab^4, use f(0) = 1 to get b = 1, and then get the final answer. All simple things to do with the CAS, but would a student do all this?

      I don’t know what the writer really wanted to test, but it would have been a ‘better’ question (it’s all relative) without the f(0) =1 and its implications.

      On a side-note, there is an on-going fascination by VCAA with using inverse functions in stupid ways, as well as butterflys. The exams are clearly being written by unimaginative lepidopterologists, not mathematicians.

  3. And let’s not forget Section A Q20 from the 2018 NHT Methods Exam 2 …. That one can only be done using the method explained by amca01. It’s unfortunate, however, that this method is not within the scope of the Maths Methods course (it is within the scope of Specialist Maths – another advantage for those students*).

    The 2018 NHT Methods question is totally inappropriate. And unfortunately ” … the laborious method given by the examiners …” is the only appropriate method (within the scope of the Maths Methods course) for the two 2019 questions. In my view those latter two questions are totally pointless as a consequence.

    There was a question a couple of years ago of the type “If int f(x) ……, then find ….. “, except that part of the integrand of what had to be found involved an f(x/2) or something similar. Tough within the scope of Methods, but simple (using a substitution) for the Specialist students.

    Going off-topic: I hate the MCQ’s too, they seem designed to test use of technology rather than mathematics (a classic case of this is this year’s Specialist Maths MCQ 4 – trivial with a CAS, thought provoking without technology. But I wonder how many students decided to think ….?). And to simply save money.

    1. Multiple choice tests are everywhere, and, quite often, the questions are not well framed.

      The only educational justification for multiple choice questions that I can see is for testing whether or not a candidate can recall a particular fact fairly quickly.

      You might enjoy Peter Hilton, The tyranny of tests, American Mathematical Monthly, 100(4) (1993), 365-369.

      1. VCAA don’t understand even that (clutching at straws) justification – you will find far too many bloated MC questions requiring the application of multiple skills on every VCAA exam. In an examination context, the simple fact is that there is no educational justification.

        Yes, unfortunately. MC tests are ubiquitous. They are a cheap and nasty tool intended solely to save money (and so of course have been particularly embraced by most universities).

      2. Thanks, Terry. It’s an interesting article, though I disagree with aspects. It’s now on my (long, long) list of things to blog about.

        It is worth noting, however, that the end of year “tests” are the least tyrannical aspect of VCE. The real tyranny comes from the SAC system, which means a student can’t take a shit without someone scoring it.

  4. Thanks to you all for the comments. (JF, if you can locate the f(x/2) question, I’m curious.) The other exam question to which JF refers is discussed here, and is hilarious.

    Of course the issue with these exam questions is whether they are meant to be solved by first finding the inverse, or by using the inverse function theorem (i.e. the chain rule plus a little sleight of hand). The criticism of requiring the first method is, as amca01 points out, it’s mathematically ridiculous. The criticism of requiring the second method is, as JF points, it’s not part of the curriculum.

      1. Thanks, JF. In principle it’s a good question, since it tests a sense of what integrals are and how they scale. Of course, the fact that only a quarter of students got the question correct says a lot.

        As to the broader question, of why the Methods curriculum doesn’t include integration by substitution, that’s an easy one: the people who wrote the Australian Curriculum, and the people who implemented that curriculum in Victoria, are as dumb as rocks and wouldn’t recognise a coherent maths curriculum if their faces were shoved into it.

        1. I agree that it’s a good question in principle. My beef is that there’s a simple way of doing it (substitution) available only to Specialist Maths students, whereas a ‘pure’ Methods student is essentially restricted to a scaling method that they will probably struggle with (and indeed did struggle with, for probable reasons that can be discussed another day). So there’s an inherent unfairness to the question.

          I had the same type of beef for quite a few years with Specialist Maths questions that gave an inherent advantage to students studying uni extension maths – the classic example being the question of finding a unit vector perpendicular to two given vectors. Even nowadays, I would argue that commonly appearing questions of the type
          “Find the value of a so that the vectors …. are linearly dependent”
          on Exam 1 (including this year’s exam) give an unfair advantage of efficiency to uni extension students.

          By the way, I think you’re insulting the intelligence of a rock.

          1. Thanks, JF. I included the “in principle” qualification for exactly the reason you suggest. What in isolation might be a good Methods question can be deeply unfair for a Methods-but-no-Specialist student, and the exam question you flagged is indeed deeply unfair. Of course that is a systemic problem, that students learning Specialist and/or who have a smart Methods teacher can avoid the maddening case-by-case substitution rules that the Methods curriculum pushes.

            More generally, there are plenty of topics in which Methods-but-no-Specialist students are screwed over by competing with Specialist students. It is the inevitable consequence of not having properly defined streams in VCE. It also may be that some of the batshit craziness of Methods comes from a futile and damaging attempt to lessen the subjects’ overlap.

            I think having to compete with extension (AP) students is more defensible and less of an issue. There will always be more advanced students that gain such an advantage. But, as always, the half-baked nature of VCE (all subjects) exacerbates the problem.

            1. Indeed. Most of these problems did not exist back in the 70’s because you could do General Maths as a single subject (the analogue of doing Methods nowadays), or Pure and Applied Maths as a double-subject package-deal (the analogue of doing Methods and Specialist nowadays).

              (Seriously, how hard would it be to go back to this sort of system ….?)

              And there was a clearly defined syllabus book that was 70% content and 30% administrative, as opposed to today’s piece of shit which is 10% content and 90% administrative. Exam questions should not be a clarifying supplement to the Study Design when it comes to content ….!!

              Re: The “futile and damaging attempt to lessen the subjects’ overlap”. Even this might not be so bad in the context of the exams if an attempt was made to lessen that overlap in exam questions.

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