A Choice of Difficults

You have a choice. The two questions below come from different exams:

Each question was (arguably) last year’s most difficult exam question on the most difficult mathematics subject in that state. Each question was effectively allocated just under 20 minutes to complete (11/100 x 180 and 13/80 x 120).

Now, you must choose: which question is better, in any sense of the word “better”?


NSW  (Formula Marking guide and sample solution are here.)


VIC  (Briefly discussed here, marking guide and sample solution are in your dreams.)

40 Replies to “A Choice of Difficults”

  1. Purely cause I’m in NSW, the extension 2 questions is better. Though, having a questions that is succinct is preferable over something that is verbose and clunky.

      1. Yes, particularly the first part. In the same 2020 Extension 2 exam question there is a pulley questions that sets up the context in far fewer lines.

  2. First I should declare a vested loathing of VCAA. If I have to choose between VCAA and Brand X I will always choose Brand X. So … The better question is from the NSW Mathematics Extension 2 Exam. For reasons too numerous to all be listed.

    Here are five of my many reasons:

    1) Clarity. The VCAA question is a dogs breakfast. I pity the ESL student that has to wade through that shit-heap.
    I should point out that the VCAA question is a shit-heap mainly because VCAA totally ruined the mechanics topic. Look at how beautiful mechanics was back in the late 70’s as part of Applied Mathematics, partner in crime to Pure Mathematics. Specialist Maths and Maths Methods are not fit to lick the boots of that long lamented dynamic duo.

    2) Simplicity and beauty of statement.

    3) Imitation is the most sincere form of flattery. I guarantee if integration by parts becomes part of the Specialist Maths curriculum, this NSW question will get plagiarised. (In fairness, the NSW question has been plagiarised from countless University exams). The VCAA question will never get plagiarised.

    4) It’s not a VCAA question.

    5) It’s not a VCAA question.

  3. Ooof. That extension 2 question *is* a classic though. The VIC question is, I don’t know what. Confusing.

    1. In its own way, the VIC question is a classic, too.

      I deliberately tried to be fair and “same” in my choices, but I don’t think it would have mattered. Pick the dumbest question on NSW and the best question on VIC, and I’m sure the comparison would still be stark. Maybe the next post …

    2. And therein lies the key point (to my mind):

      Any *good* student who had prepared well for the HSC exam would have seen *similar* problems to this induction question. The goal of the question is clear and it is actually *good mathematics*.

      The VCE question is excessively wordy, so immensely unfair to a lot of students (unless the aim of the exam is to test their reading comprehension as well…), far too much button-pushing and not at all clear what, exactly the objective is meant to be.

      I would choose any exam that rewarded mathematical thinking over button-pushing at the best of times, but (like J(N)F) I have no love for VCAA. Sadly though, my feelings towards Specialist Mathematics these days are probably best described as “apathy”.

  4. The two questions deal with quite different topics. The NSW question is a question about pure mathematics. The Victorian question is a question about applied mathematics. Questions about applied mathematics often require more explanation in order to set the context.

    JF mentions the difficulty that a students whose first language is not English might face. In my experience, even some students whose first language is English have difficulty in reading examination questions. We are often reminded that we live in a multicultural society and that the students in our classes may have different language backgrounds. To what extent is this considered in VCE mathematics examinations? Probably a PhD in this.

    1. OK, a third and fourth post: pure v pure; and applied v applied. It won’t matter. The VIC question isn’t wordy because it’s applied, it’s wordy because the writers don’t know how to write. Which is exactly your second point: the VIC question is painful for *anyone* to read. It is shit.

        1. I put the first paragraph of the Victorian question through a Gunning Fog index calculator and obtained the following result. “Your text has an average grade level of about 12. It should be easily understood by 17 to 18 year olds.”

          1. VIC is not an English comprehension question. A year 12 maths question does not need to be in Year 12 level language, laid out like an essay. I know nothing of Gunning-Fog, but the VIC question above is shit. What you done is test Gunning-Fog, not the VIC question.

    2. TM, VCAA claims that its exams are vetted for ESL suitability. The claim is possibly true. But, just like the other vettors, the ESL vettor is obviously asleep at the wheel. No PhD required – on paper VCAA tick the ESL box, but in reality the tick is worth as much as a school’s policy on bullying.

      Again, it’s easy to blame the writers (and rightly so). But it’s the vettors that should be publicly outed and pilloried. Then sacked.

    3. Sorry Marty, I agree with Terry.
      The NSW questions are posed as “prove” or “show”. Underneath these simple instructions there are a host of assumptions, but the candidate is expected to know the deal. With the pulley question there is no simple word to convey the situation. Even the wordy version that we are given is insufficient. A complete wording would specify that the platform is horizontal, and that the string is parallel to the platform – but perhaps the examiners thought the diagram was enough. And a “smooth pulley” is a vague beast. To act as a pulley the string needs to rotate the wheel – but then the moment of inertia of the pulley needs to be negligible. Possibly it is acting as a smooth peg? The wording could benefit from some commas. For example, “the motion of magnitude F_1 newtons” had me confused. I give the examiner 7/10. That’s for the wording; the question is shit.

      Are we being too protective in requiring exam questions to use only the simplest English? At VU we were trying to make students employable. To this end one of our electives was an extra English subject which those weakest in English were required to “elect”. It was an automatic pass for a student who attended and showed any effort. But many Vietnamese students hated it – “We are doing Maths and Computing because it does not need English”

      1. I should add, I have a lot of sympathy with your second point, that questions should not be vetoed simply because they are wordy, and that questions should be phrased in the natural language of the subject rather idiot-proof language. In this regard, I have little sympathy for ESL students: the natural language of a subject is part of the subject. It’s the one, small aspect of “numeracy” that has some merit.

        But, again, the VIC question is the worst possible example for your argument. There is no way on God’s Earth it could be described as “simplest” English. It’s crabbed, pedantic, cover-our-asses English. And it’s bad English.

        It is fine to include non-simple English if it is a natural part of what you choose to examine. But the English shouldn’t be gratuitously non-simple and, in particular, it shouldn’t be non-simple just because the exam writer is illiterate or pig-lazy.

        1. The real problem is that the mechanics topic has been ravaged to the point where nothing non-trivial or interesting can be asked without creating convoluted contexts (rather than just rehabilitating the topic). And unfortunately it takes a steaming pile of shit to define these convoluted contexts.

          No doubt VCAA will interpret criticism of this question as validating the removal of mechanics. (Rather than validating the removal of the idiots who are calling the shots).

          1. This is a very good point. The more contrived and empty your subject, the harder it is to ask a question, particularly a longer question, that is natural and sensible. And of course contrived questions for a contrived subject will more difficult to word in a clear manner.

            So, whose fault is that?

      2. No need to apologise, tom; I’m quite happy for you to be wrong along with Terry.

        Actually, you’re right, Terry has a point and I was too quick to dismiss it. The, VIC question is more naturally wordy. But you’re also wrong. The VIC question, as well as being directionless and clunky and trivial, is poorly thought out, poorly laid out, poorly diagrammed and poorly worded. It is a shit question, presented in a lazy, shitty manner.

        I already agreed that there were other ways to try to compare like-with-like. I went for hardest-with-hardest, but I’ll try to get to applied-with-applied and pure-with-pure and dumbest-with-best. I have absolutely no doubt who will win in each case, but these will be interesting tests.

    4. No need to apologise, tom; I’m quite happy for you to be wrong along with Terry.

      Actually, you’re right, Terry has a point and I was too quick to dismiss it. The, VIC question is more naturally wordy. But you’re also wrong. The VIC question, as well as being directionless and clunky and trivial, is poorly thought out, poorly laid out, poorly diagrammed and poorly worded. It is a shit question, presented in a lazy, shitty manner.

      I already agreed that there were other ways to try to compare like-with-like. I went for hardest-with-hardest, but I’ll try to get to applied-with-applied and pure-with-pure and dumbest-with-best. I have absolutely no doubt who will win in each case, but these will be interesting tests.

  5. Umm … Marty
    Did you need to rewrite the pulley question or is it a direct cut and past? There seem to be things missing.

    1. Tom, I *think* it’s all there, but you can check the exams at the links. If something is missing, please let me know and I’ll fix.

  6. Not answering the question (tangential as usual) let me discuss the units in the pulley question. I think the convention is to use the singular form. So “newton” not “newtons” and “kilogram” not “kilograms”.
    The use of ms^{-2} is definitely non-standard, unless the acceleration has the impossible unit of millisecond to the power -2. The convention is to use m s^-2 with either a gap or a high dot between the m and the s. See https://physics.nist.gov/cuu/Units/checklist.html. Of similar concern is the velocity unit in part d.

    1. Nice observations, tom. Units are one of the many touchy points with VCAA – that gang of idiots still can’t acknowledge that the unit for change in momentum is kg m/s NOT kg m/s^2. (And VCAA’s goons have wilfully attempted to cover this up).

  7. So… I’ve now gone to look over the NSW Ext 2 exams since 2016 (parent-teacher night, needed something to do between saying “homework” and “read the question” every 3 minutes…)

    They are in general:
    1. MUCH less wordy
    2. Less “structured” as in there is a lot less breaking down the questions into 1-mark steps.
    3. More open to a variety of methods of solution.
    4. Not bound by a story or theme, they are just mathematical questions.

    Not sure if any of this makes them better or worse than VCAA questions, but the difference in style, content and structure is remarkable.

    1. Yes, you are sure. It makes them better. I will try to post some more this-versus-that comparisons in the next few days.

      1. OK, to be *totally* fair, I still don’t think the NSW HSC exams are as good as the IB exams, but are a darn sight closer. My reasons:

        1. In the HSC exams, there seems to be a bit of an odd way of setting out the sub-parts of a question; 1a and 1b need not even be related on some HSC papers. In the IB exams, they at least start a new page with a new idea.

        2. In the IB world, “command terms” are very clearly defined and very clearly used.

        3. The balance between short-answer and extended-response is really nice and (in my opinion) provides a very fair challenge.

        4. The marking schemes put out (albeit for sale – although $3USD is not a huge price to pay) by the IBO are really, really good and can be used as an excellent learning tool.

        5. Because there are Northern Hemisphere (in two time zones for Standard Level) and Southern Hemisphere (OK, so they are known as May and November, not North and South…) papers that are all equally good, students get more than enough exam practice, making it a fair assessment.

        And yet I still really like the way the NSW HSC papers examine analytical geometry in a way rarely seen elsewhere in Australia… and the HSC papers are free.

        1. Thanks, RF. In reply:

          0. I had thought of also including a “comparable” IB question, but couldn’t see an official public copy to use. I’ma big fan of IB, but I actually know little of HSC. At this stage, I don’t know which I would prefer, although I’d guess IB. The main point is, they are so way, way preferable to the current VCE garbage.

          1. Yeah, I know, it’s weird. VIC Pure and Applied also used to be like that. But, although it’s weird, I don’t think it matters: they’re just separate questions.

          2. A “command term” meaning “show” or “verify” or whatever? Is there a reason to think HSC is not so good in this regard?

          3. I haven’t thought about it.

          4. Yes, they’re excellent. But the HSC marking guide looked very good to me, and obviously much better than VCAA’s scribbles.

          5. Good point. I think HSC has been pretty stable for a long time, so it may not be that bad.

          6. The HSC analytic geometry looks very good, although I’m not a fan of analytic geometry.

          1. Yeah, there are no “public copies” of IB exams unfortunately.

            I don’t have an issue with the way HSC uses what the IB calls “command terms” but the sheer fact that the IB goes to the trouble to clearly define them I think avoids a few of VCAAs problems in this regard.

            As for geometry, I get that examining it can be a bit problematic, I just really like it as a branch of mathematics for its own sake. Personal preference I guess.

              1. Now take a look at any IB Further Mathematics HL exam if you want to truly appreciate the galactic wide chasm between top level VCE mathematics and top level IB mathematics … VCE mathematics is not fit to lick the boots of the IB.

                1. I’ve seen some commercial VCE trial Specialist mathematics exams rip off IB exam questions. Pity VCAA don’t believe that imitation is the best form of flattery…

                  1. That wouldn’t surprise me at all. But now I’m curious – which trial exams?

                    I expect VCAA will rip off many IB (and NESA HSC) maths questions from 2023 onwards if proof by induction, integration by parts etc. become part of the Specialist 3/4 course. And the commercial trial exams will follow suit in droves.

                    1. I noticed one recently, on circular functions, from TSFX 2010. The question was from an IB exam in 2007.

                      I feel like I’ve seen plenty of others, but I’d have to dig to find the specific years.

                      VCAA bashing aside, assuming that the proposed changes to the SM study design are actualised, then as you note, past IB exams will be a useful resource for teachers / students for the new topics.

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