How the Other Maths Lives

A few days ago, the Sydney Morning Herald had an article on “the hardest question” from this year’s NSW Extension 2 Exam. The question is worth 5 marks, which equates to 9 minutes of a 3 hour exam (accessible here). The question, and another question (6-ish minutes), which apparently came along for the ride, are posted below. Readers’ homework exercise is to Compare and Contrast.

59 Replies to “How the Other Maths Lives”

  1. Some early thoughts (not having yet worked fully through either of the NSW questions):

    1. The NSW questions are clear in both their instruction and the requirement of the answer.

    2. Part (c) is quite a clever question which may require some “playing” before a solution is found.

    That said, I’m not sure how NSW uses numbering for their questions – the “different parts” often seem totally unrelated.

    Question (without notice): what types of calculator were permitted for this NSW exam paper? It makes quite a difference…

      1. I know a few of those brands. 8 to 10 digit display, scientific constants programmed in as well as order of operations etc.

        Makes a big difference.

      1. John, you’ve linked to itute’s suggested solutions. Was that intended? Also I don’t understand. What is the difference between the “graph” and “the shaded region”?

        1. Aha, wrong file. Not to worry, NSW made the same mistake as itute. Attached is the NSW report.

          Re: What is the difference between the “graph” and “the shaded region”?

          The graph is what you draw for Re(z) = Arg(z). It is the curve that is drawn (in both solutions). The shaded region is the answer and is correct. But the curve that is drawn is wrong. The curve does not exist for \displaystyle x \in \left( -\frac{\pi}{2}, 0\right).

            1. Forget about the shading and look at what’s left. A curve. A curve with equation y = x tan(x). The part of the curve corresponding to \displaystyle x \in \left( -\frac{\pi}{2}, 0\right) should not be there.

              Edit: My mistake. Itute got it right, NSW Report got it wrong.

              1. Ah, ok, I see now. I wouldn’t say the Report got it wrong. I don’t think they’re meaning to imply the second quadrant part of x tan x is being employed. But I agree that it’s potentially confusing, and it would have been preferable to not have included that part of the graph.

  2. Kind of cute, part B. I think they glue unrelated problems because each extended problem has a separate answer booklet, but it seems really awkward. Good luck to vic students on problems like this. After years of superficialness they would drop dead seeing these.

  3. I think, for me personally, question (b) shows the appeal of mathematics better than comparable VCE questions. Okay, it’s difficult and fiddly. But it’s nice getting lost in one’s own algebraic play, and remembering tricks to apply. The VCE model requires students to constantly touch base with the directions of the examiner, because everything is so bitsy.

    I’m probably biased because I’m from NSW but I remember enjoying questions like these and wanting to do more maths because of that. I guess different people will enjoy VCE maths and it will influence different people to pursue it? Maybe that’s fine though!

    1. Yeah, wst, you’re obviously just biased. It’s clearly implausible that any one state would have way, way, way, way better senior mathematics than another. That’s an absurd idea.

      1. Correct me if I’m wrong, but NSW has a totally different way of marking exams (seems to be more criteria-based than “you must show this line to get the mark(s)”.

        I reckon (and again, willing to be argued against) this rewards good mathematical working, rather than “memorise the answer from when it last appeared X years ago”

        1. Jesus. All these outlandish claims, that NSW tests mathematics, and that they award marks for mathematical thought. I just don’t understand where people get such notions.

          1. Problem is, I think NSW is in the majority here internationally. Look at papers from the various UK examination boards, look at the IB exams (which are hugely popular in Europe and North America) and you will see they are a lot closer to NSW than Victoria.

            So, when and where (and why) did Victoria get it all so wrong?

            Based on the 1966 Mathematics exams I’ve seen from Victoria, some time post 1966.

            May or may not coincide with the invention of VCE.

            1. Getting rid of the VUSEB was a major blow. But things still went OK under VISE and then VCAB – the HSC era.
              I think that the rot really set in with the establishment of the VBOS, then BOS and then, the crowning jewel, VCAA – the era of the VCE.
              Appointment of well-connected but under-skilled teachers to write and vet exams during this era hasn’t helped. However …

              Graphing Calculators came along in the early 90’s and this where the real trouble began: the TI-82 (, followed by the TI-83 etc. all the way up to the TI-Nsipid. The introduction of the CAS-calculator and the associated gravy train has been the single most destructive factor in mathematics education in Victoria.

              1. I think there have been about ten single most destructive factors. It is difficult to imagine a more poisonously stupid anti-mathematical swamp.

                1. Is it incompetence though or fingers in pies?

                  I will accept a “do not wish to answer that”.

                  I remember the 81, 83, 84, 84-Plus calculators well – they were allowed in IB exams as well…

                  Which makes me wonder if calculators are as big a part of the problem as I first thought…

                  1. Incompetence? I dunno. If you asked a duck to play Liszt and it failed, would you call it incompetent? It seems to me we got lots of ducks.

                    1. Dunno about ducks, but at least one rabbit can play Liszt. 🙂

                      On a more serious note, I completed HSC in Victoria in 1985, studying English (of course), Chemistry, Physics, Pure Mathematics and Applied Mathematics. My recent foray into teaching (also in Victoria) really opened my eyes with regards to how comprehensively things have degenerated. I’m still stunned. WTF happened?

                    2. Everything happened. Everything that might make mathematics education worse happened. And there is not a snowflake’s chance in hell that it will be fixed. There is not one single positive sign from institutional bodies.

                      Which means that anybody who compliantly and silently and actively participates with these institutions, some of whom read this blog and are definitely not idiots, should think hard about what the hell they are doing. They can fight the good fight, or they can spend their remaining years marking down students for leaving out “dx” in their integrals.

                    3. These dumb ducks can’t even play chop sticks.

                      Marty, I’m afraid I must agree. Unless there’s a complete reset (fat chance), it’s just re-arranging the deck chairs on the Titanic. Together with some fool and their thought bubble swaggering along every so often to make the iceberg bigger.

                      What’s needed is a different ship, different deck chairs and an icecube (ideally inside a drink).

      2. I just mean I feel nostalgic looking at this, and it adds to the appeal. Maybe VCE students will feel nostalgic about CAS calculators and stories about cars.

          1. Yeah, a car with infinite initial acceleration. Point it towards the stars and you’ve got a great idea for a new TV series: Car Trek.

            VCAA Examinations: the final frontier. These are the voyages of the stunted car Feckless. Its continuing mission: to explore inane new mathematics. To seek out new ways of screwing up. To stupidly go where no mathematician has gone (or will go) before!

            1. That feels like the opening line from a Methods SAC… be careful, I might borrow it… until my manager tells me to change the whole thing. Again.

              1. RF, feel free to steal. It sounds like you’ve already got someone you can model Captain Jerk on.
                And Captain Jerk graduated with honours from the Virgo Constellation Astronomy Academy. Which is sponsoring the voyages of the Feckless.
                At some point in the story I can hear Scotty saying … “But Captain, I canna’ change the laws of mathematics!”
                And Bonehead McCoy (also a VCAA graduate): “It’s mathematics, Jim. But not as we know it.”
                And don’t forget mathematics that causes the Vulcan Mind Mould.

                1. There has to be a Beverly Crusher joke in there somewhere…

                  Or something about boldly splitting infinitives (perhaps a hybrid function?)

                  1. Not to mention the Prime Directive …

                    (The Prime Directive of the Virgo Constellation Astronomy Academy is to screw up the mathematics and ensure inane contexts for questions).

    2. The two questions are both really nice. And very different to each other.

      (b): I agree, wst. It’s straightforward but a lot of algebra. And you’ve got to know what you’re doing. It tests a lot of skills. It’s an ‘obvious’ question to ask, and I thought I’d seen it somewhere in an old textbook, but I can’t find it. If original, it’s even nicer.

      (c): Not straight forward and very little algebra. You have to pause and think carefully. Probably the simplest approach is to use z = x + iy and consider each of the four quadrants separately. I really like it: simple, creative and unusual (in a good way). Again, it tests a lot of skills. I haven’t seen a question like it before. I might use the idea on a trial exam – very easy to modify. (And if the vettors complain, I can say that what’s good enough for NSW is good enough for Victoria). At 3 marks, it’ll be interesting to see the marking scheme. I’d consider making it worth 4 marks if I steal it.

      And not an apted stunted car to be seen!!

      These two questions are not a consequence of a better syllabus relative to Victoria. Both questions are well within the scope of Specialist Maths (nevertheless, a typical Specialist student would be eaten alive by these questions and the exam in general). They’re a consequence of having better exam writers and a much better examination structure. Part (b) would be trivialised if it appeared on a Specialist Maths Exam 2. It’d make a great Exam 1 question. But if all the questions that would be great for Exam 1 were put on Exam 1, Exam 1 would be 3 hours long and there’d be no time for Exam 2 …

      1. Another question for those who know more that I do (a rather large list…) in NSW, do students who take the Extension 2 paper also have to sit the Extension 1 paper? I recall reading this somewhere but wasn’t sure if it was still the case.

        Imagine if all Specialist students had to take the Methods paper as well in the same year… I know a lot do, but a growing number of Specialist students take Methods as Year 11s which has the occasional unintended consequence (such as when they test Methods skills on the Specialist exam…)

        1. Back in the day, there were two subjects called Pure and Applied Mathematics. They were a package deal. And there was none of this bogus acceleration crap of doing Form 6 (Yr 12) subjects in Form 5 (Year 11). I’d love to know the origins of that particular snake-oil – I’m sure it must have started as a marketing gimmick that quickly spread like the infection it is (the school marketing arms race). And all condoned by VCAA so that today we have the ludicrous situation that roughly 80% of students accelerate in at least one subject (back in the mid-2000’s there was a private school that moronically mandated that EVERY student had to accelerate in TWO subjects, regardless of his/her intellectual capacity. All in the name of gaining a marketing advantage. The stupidity only lasted a couple of years)

          NSW Question (b) smells like a question from an old Applied Mathematics exam.

          Re: Unintended consequences. Yes, if Methods and Specialist were a package deal, the murky pre-requisite issue would be a lot clearer. Including discrete probability from Methods in the NHT Specialist exams was despicable. But I will qualify this by saying that I don’t know if every NHT Specialist Maths student was also doing Maths Methods. That would make it slightly less despicable. Irrespective, the message it sends to the Southern Hemisphere, together with the mealy-mouthed statement by VCAA that everything in Methods is examinable in Specialist, is unconscionable and detestable.

      2. Re: (b). A comment left at the SMH article included:

        “Its in textbook Basic Physics 1 by Martin & Connor, Chapter 4: Particle Dynamics: 4.13 Shape of a Projectile Path – published in the 60’s …
        Whilst the book doesnt [sic] give the answer to that question “Fig 4.4 (b)”, it does show the calculation of the example “Fig 4.4 (a)” preceding it …
        Yes I actually have the book.”

        I thought it was too cool to be completely original. I plan to follow up this reference and check when I have the time – the text book might have other cool things in it.

        1. OK, I acquired the book. The SMH comment I quoted from the Age is misleading to the point of being total crap. The so-called
          “calculation of the example “Fig 4.4 (a)” preceding it”
          is nothing more than the routine calculation of the position vector of the path followed by the particle, that is, the result given in the question (“Do NOT prove this.”). There’s nothing interesting in this section of the book, not even in the Exercises. The SMH comment was a waste of space.

  4. hi,

    for those harping for “the good old pre CAS ” days here is an example of the S level paper
    set by London Board in 1978 (at A level Mathematics)

    non programmable green screen calculators had been introduced in 1976 to replace log books and other tables

    Steve R

    F 208 1978 A Level Maths iv

      1. Hi,

        I like the projectile question 6 for its clarity and the fact you get full marks by answering 7 questions well from 10 possibilities ( so if you don’t enjoy statistics or probability you still have 8 other questions)

        Steve R

        1. I agree, Steve. There was a time when many secondary school textbooks had pages and pages of questions just like Q6 (and part (b) of the NSW exam).

          All the questions have clarity. The absence of ‘real life’ context white noise and static is so refreshing. A simple focus on mathematical skills.

    1. And not a stunted car in sight! Thanks, Steve. I enjoyed those questions. NB: 1978 Victoria had decent maths exams too.
      @Marty: Would lives be at stake if the 1978 Victorian HSC Pure and Applied exams were posted?

      VCAA’s idiotic fixation to have ‘real life’ contexts – and VCAA’s stunted car with an an infinite initial acceleration is a wonderful example of this idiocy – together with the imposition of CAS technology is what makes the VCAA exams so terrible. Substandard vetting doesn’t help either. Nor does a light-weight syllabus that’s more froth than beer.

      And the ‘real life’ contexts are not only idiotic, they also enable (have enabled) greater opportunities for errors.

        1. Yeah, that’s what I thought. But I had to ask. No doubt VCAA would scream breach of copyright – a smoke-screen, of course. Because the reality is that VCAA wouldn’t want anyone seeing how badly standards have fallen. It would be a total embarrassment.

          It’s a genuine shame (but understandable – see above) that VCAA doesn’t have a publicly accessible archive of old exams going back to the VUSEB. The rich tradition of secondary school mathematics in Victoria has been insidiously eroded by VCAA and replaced with fairy floss. Now you have to search for it – in fewer and fewer libraries (until it’s not found anywhere because it’s seen as irrelevant and thrown out). Out with the old and in with the spew – the disposable society we live in. The MAV should be doing everything possible to preserve this tradition and enable all the old exams, reports etc. dating back to the VUSEB to be accessible to all teachers …

          1. It’s not obvious to me that VCAA would object to publishing such an old exam, even if it were me. But it’s not worth my while enquiring or just trying it.

          2. John,
            Despite being not very feasible, I will make every possible effort to save these exams and archive them, whether it be digital or hard copy… On my shelf, I have collected the hard copies of all VCE maths exams from 2000.

            These old gems tend to be thrown away. People often do not cherish these invaluable educational assets (the VBOS and previous eras). Not only exams, but also decent copies of the old textbooks (i.e: Fitzpatrick et al).

            I plan to dig the Unimelb libraries further early next year. I suspect the libraries at Clayton campus have already thrown many copies of those old maths textbooks…

            Occasionally I see eBay sellers posting some valuable resources…

    1. A lot has changed in 105 years.

      How come most disciplines have moved forward but Mathematics has gone backwards???

      Yes, I ask this a lot. I really want to know!

      1. Thanks very much, Lancelot. I wasn’t aware of that VCE archive. I’m not aware of a similar archive of older Victorian exams.

  5. The Tasmanian exams aren’t too shabby, either:

    Mathematics Specialised

    Scroll to the bottom, click on Supporting Documents etc. – More information.

    I notice that the TASC 2021 exams are already published too. The delay in publishing the VCAA exams is inexplicable and unacceptable.

    1. Tas, SA, Qld, England, IBO all seem to have a style of question that I have not seen in Victoria since the inception of the VCE. Something along the lines of the cubic polynomial p(x)=x^{3}-2x^{2}+5x-7 has roots \alpha,\beta and \gamma, q(x)=x^{3}+bx^{2}+cx+d has roots \frac{1}{\alpha},\frac{1}{\beta} and \frac{1}{\gamma}. Find the values of b,c and d

      Is this something that was once done in Victoria and, perhaps of more interest to me, is this a common question type elsewhere in the world?

      1. Hi RF. I I had a vague recollection that Vieta’s formulas were part of the Yr 12 (Form 6) maths curriculum about 50-60 years ago. But I can’t find any evidence that they’ve ever been part of a Victorian maths curriculum. I teach them (for cubics) in Specialist Maths because they’re useful for solving cubic equations under certain circumstances.

        1. This stuff appears a little bit in Fitzpatrick and Galbraith, but I have no memory of it being on the exams of that time (the 70s).

            1. Excellent – I knew they had a name, just couldn’t remember it…

              Went down the rabbit hole looking for the name and found Descartes Rule of Signs…

              And now I’m wondering if such an idea is SAC worthy…

              1. Hi RF,

                Definitely SAC worthy. In fact, if you look at 2001 VCE Spesh Exam 2, there was a complex number question (ERQ4) based on Vieta’s theorem (with quadratic equation).

                Also, 2021 NEAP Specialist Trial Exam 1 – Q10, the author covered as many approaches as possible, one of which uses Vieta’s theorem I believe, which is quite smart.

                Take a look at some IB HL maths exam questions around 2014-2018, as you mentioned, we usually see some good cubic equations related to the Vieta’s theorem.

                Wikipedia has a detailed page:
                The product formula can be related to some induction patterns, I guess (even though proof by induction is *not yet* part of VCE study design)

                I guess in VCE teachers teach it less often because time is too tight and usually in classroom we have to focus on ideas like “remainder/rational root theorem” or “equating the coefficients”, etc. Often we get bogged down with students struggling with the basic algebraic skills and wasted a lot of time during class…And we have to even cover necessary CAS operations within each limited lesson time!
                I believe the main problem is not that we lack decent and creative mathematical ideas…The crux is we have been suffering a decreasing trend of our education standards, while we witness the polarisation of students performance to the two extremes…

                1. For those who want to see the VCAA question – attached.

                  Re: NEAP question and solution. It was, MAYBE (see * below), reasonable to include a solution based on Vietta’s formulas. BUT … it would have been much better to include some background. A simple statement like “Using the Vietta formulas for a cubic polynomial …” prefacing the solution would have given the solution much more \displaystyle teaching value. As it is, these formulae (particularly * see below) are ‘rabbits out of the hat’. I think many teachers and students will look at this alternate solution with glazed eyes and ignore it.

                  I always teach the Vietta formulas for the product and the sum of roots of a quadratic and a cubic polynomial. They are simple to memorise.

                  * Anything beyond this (like what the NEAP solution does with “sum of the product of the roots taken two at a time”) is very unwise IMHO because it’s not easy to remember and not all that useful – students cannot look them up in Exam 1.

                  However, I do encourage the interested student to research further.

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