WitCH 38: A Deep Hole

This one is due to commenter P.N., who raised it on another post, and the glaring issue has been discussed there. Still, for the record it should be WitCHed, and we’ve also decided to expand the WitCHiness slightly (and could have expanded it further).

The following questions appeared on 2019 Specialist Mathematics NHT, Exam 2 (CAS). The questions are followed by sample Mathematica solutions (screenshot corrected, to include final comment) provided by VCAA (presumably in the main for VCE students doing the Mathematica version of Methods). The examination report provides answers, identical to those in the Mathematica solutions, but indicates nothing further.

UPDATE (05/07/20)

The obvious problem here, of course, is that the answer for Part (b), in both the examination report and VCAA’s Mathematica solutions, is flat out wrong: the function fk will also fail to have a stationary point if k = -2 or k = 0. Nearly as bad, and plenty bad, the method in VCAA’s Mathematica solutions to Part (c) is fundamentally incomplete: for a (twice-differentiable) function f to have an inflection point at some a, it is necessary but not sufficient to have f’’(a) = 0.

That’s all pretty awful, but we believe there is worse here. The question is, how did the VCAA get it wrong? Errors can always occur, but why specifically did the error in Part (b) occur, and why, for a year and counting, wasn’t it caught? Why was a half-method suggested for Part (c), and why was this half-method presumably considered reasonable strategy for the exam? Partly, the explanation can go down to this being a question from NHT, about which, as far as we can tell, no one really gives a stuff. This VCAA screw-up, however, points to a deeper, systemic and much more important issue.

The first thing to note is that Mathematica got it wrong: the Solve function did not return the solution to the equation fk‘ = 0. What does that imply for using Mathematica and other CAS software? It implies the user should be aware that the machine is not necessarily doing what the user might reasonably think it is doing. Which is a very, very stupid property of a black box: if Solve doesn’t mean “solve”, then what the hell does it mean? Now, as it happens, Mathematica’s/VCAA’s screw-up could have been avoided by using the function Reduce instead of Solve.* That would have saved VCAA’s solutions from being wrong, but not from being garbage.

Ask yourself, what is missing from VCAA’s solutions? Yes, yes, correct answers, but what else? This is it: there are no functions. There are no equations. There is nothing, nothing at all but an unreliable black box. Here we have a question about the derivatives of a function, but nowhere are those derivatives computed, displayed or contemplated in even the smallest sense.

For the NHT problem above, the massive elephant not in the room is an expression for the derivative function:

    \[\color{red} \boldsymbol{f'_k(x) = -\frac{x^2 + 2(k+1)x +1}{(x^2-1)^2}}\]

What do you see? Yep, if your algebraic sense hasn’t been totally destroyed by CAS, you see immediately that the values k = 0 and k = -2 are special, and that special behaviour is likely to occur. You’re aware of the function, alert to its properties, and you’re led back to the simplification of fk for these special values. Then, either way or both, you are much, much less likely to screw up in the way the VCAA did.

And that always happens. A mathematician always gets a sense of solutions not just from the solution values, but also from the structure of the equations being solved. And all of this is invisible, is impossible, all of it is obliterated by VCAA’s nuclear weapon approach.

And that is insane. To expect, to effectively demand that students “solve” equations without ever seeing those equations, without an iota of concern for what the equations look like, what the equations might tell us, is mathematical and pedagogical insanity.


*) Thanks to our ex-student and friend and colleague Sai for explaining some of Mathematica’s subtleties. Readers will be learning more about Sai in the very near future.

84 Replies to “WitCH 38: A Deep Hole”

  1. Thanks Marty for your accurate and condensed summary. Greatly appreciated.

    The title is very sophisticated – a “deep hole”…
    I wonder, Is there any method for the commenters to post images?Or I need to send you the images in future?

    Back to the topic, my deep concern is that the sample solution only assumes the function having no cancellable factors, which exhibit general “tick shape”.

    So, interestingly, when I approached this question last year, I somehow spotted that if k=0, then the numerator becomes x^2 + x = x(x+1), which gets rid of the same (x+1) at the bottom. In this case, f(x) = x/(x+1) =1 – 1/(x+1), and that is a hyperbola where no stationary point exists. Similarly, once k = -2, the stationary point does not occur as well.

    I think teachers and students will encounter similar problems when they deal with the quotient functions involving two polynomials, where students find struggling to decide when the function will have a “hole”, and when it will have vertical asymptote(s).
    I might not be 100% in defining concepts, but normally what I tell students is:
    1) when you have cancellable factor(s) for both numerator and denominator, the root(s) will generate “hole” (point of discontinuity)
    2) when the quotient function is in the simplest form, such as partial fractions or proper fractions, it is normally safe to decide where the vertical asymptotes are.

    Sadly, such detailed discussions are totally the onus of teachers……Neither did our study design give us a clear indication or definition to these cases, nor any textbooks…

    1. Hi P.N. I think what you’re telling your students is very good. Indeed, there’s no discussion in either textbooks or the Study Design about the type of discontinuity (‘hole’, or more technically, a removable discontinuity) that arises here.

      I wonder who at VCAA wrote those Mathematica solutions (otherwise known as the smoking gun)? Maybe the work experience kid, maybe some DuLLard. Who knows ….? Obviously someone with just enough mathematical competence to dig this ahem …. hole for themselves. The question is only worth two marks so I doubt the exam writer understood the correct answer either.

      As I’ve said elsewhere but I’ll say again:

      The troubling aspect of this error is that VCAA has published detailed Mathematica solutions to this exam and gets the same wrong answer for Q2(b) as the report. So the error is clearly not a typo in the report. So in all likelihood the marking scheme has the same wrong answer (and therefore incomplete solution) and therefore every single student who sat that exam has been incorrectly marked.

      One could say meh it’s just another mathematical error among a laundry list of errors. True. But this latest example of mathematical incompetence by VCAA is special because this time there’s actual empirical evidence showing the thinking that led to the error. There’s a smoking gun.

      If only we knew who wrote those Mathematica solutions we might get better insight into why there are so many errors year after year in the VCAA exams and Reports. But given the availability of these Mathematica solutions, I’m sure the author will become known sooner or later.

      The implications of this error are extremely serious, and nothing short of a clear and transparent explanation by VCAA as to how the NHT 2019 Q2(b) was marked and, in particular, whether students who included k = 0 and k = -2 in their answers were penalised, is acceptable.

      1. I think the approach provided by the DuLLard here could have been saved if instead of asking the question “where do I have no solutions to f_{k}' (x) =0 ?” they asked “when do I have no solutions to f_{k}' (x) = 0 within the the domain of the function?” Then I think (I don’t have Mathematica) they could have excluded x =  \pm 1 in their Solve blah-blah-blah command and wouldn’t really need to think too hard about what they function looks like and how cancelling factors affect things.

        I really wonder though, what is the point of asking this question? What are they trying to encourage students to learn and do by asking it?

      2. I am sure there was no intention to have the question include such special cases. It was meant to be just another idiotic exercise in pushing buttons. So, as for “saving” the question, there was nothing worthwhile saving.

    2. Thanks, P.N. I assume some other SM teachers may have noticed the error, but it’s remarkable that it has remained uncorrected on the VCAA site.

      I looked at allowing commenters to attach images et al (although I’m not sure it’s a good idea), but it didn’t seem to work. I’ll investigate further.

      1. In this particular case the error is in the so-called Report, not the exam, and so it will go unnoticed by teachers for even longer because:

        1) It’s not a report, it’s just a list of answers so there’s no useful insight and so no real point in giving it more than glance (just to check if it’s suffered a case of mistaken identity and is actually a report). Unless a teacher has an incentive to carefully check these answers, wrong answers on an NHT report will not get noticed by teachers.

        2) The report comes out so late that it becomes mostly irrelevant anyway.

        3) Availability of NHT exam solutions is limited. None are produced by itute, the MAV don’t bother (probably because there’s no money to be made). So there’s nothing to re-ignite interest in these questions. Who’s going to bother writing solutions and finding mistakes in the reports?

        I assume VCAA rely on teachers to pick up mistakes in reports (although my experience is that VCAA does its very best to obstruct any discussions on this), so if no teacher notices a mistake in a NHT report (quite likely for the above reasons) then VCAA will be oblivious and so no correction will get made.

        My bet is that this is only the tip of the error-berg.

        1. I have heard that there was only 8 NHT Specialist Maths Students in 2017. I can’t imagine that number rose significantly in 2018 and 2019 – the NHT VCE runs mainly in China, and according to a reliable source most of those partnership schools don’t provide Specialist Maths as an option. Only a few of them have Specialist, one of which is in partnership with a well-known private school (with ambitions of national and global domination).

          So it’s VERY likely that only one person marks those exams, which makes it very UNlikely that errors in the marking scheme will be detected (particularly if that one person is ahem …., well, you know ….). On the other hand, there are dozens of markers of the November Specialist Exams, greatly increasing the chance of errors in the marking scheme being detected (many more Swiss cheese-slices – see below).

          I don’t think VCAA knows or even cares about the Swiss cheese model of accident causation (and whatever cheese-slices VCAA does use has numerous gigantic holes):

    3. So, for k=0,-2 this rational function is reduced to a simple hyperbola, when k=0 it has the rule x/(x-1)
      when k=-2 : (x+2)/(x-1), so just one va , x=1.
      You are correct in that these two hyperbolae (bar a point , to be strict) do not have s.p.

      Similar situation appears in y11 and y12 texts of MM: “find the values of parameter k for which
      (k-1)x^2 …. has exactly 1 solution.
      Students (as well as – frequently- us teachers) work out the discriminant…
      What about k=1 case? The quadratic in question reduces to a linear eq which (typically) results in a sole solution.
      They come in a variety, asking about values of k where 2 solutions exist, some interval results from considering delta and k is inside of it.. I do not think you can say texts do not prepare for such cases , regretably theanswers/solutio s do not always reflect this aspect.
      I can not remember exams testing this, the problem raised seems to be the first of this kind.
      Out role is to emphasise such an issue, similarly for the problem.

        1. Jac 1st edition mmy11.
          Ex 2G q5 has 9 , 4 have k related qudr coefficient, every one of them partially flawed due to the reduction to linear equation
          Essential 1/2 5th ed. Ex3J q4 omission of p=1/2 which reduces quadratic to a linear

          Some of the qns would get away with these had they been phrased specifically solution to these quadratics. But only some.
          Y12 texts usually don’t ask such questions

          1. B. S.:

            Actually, your remarks are very important on the reduced case to linear.

            2018 Q8 part b is directly related to this.

            In practice both teachers and students seem very used to the discriminant in terms of finding number of solutions…However, such type of questions must also consider the reduced linear case for completeness.

  2. Re: Part (c). A necessary but not sufficient condition for a point of inflection is that f” = 0. It’s disappointing that a check for change in concavity at x = 0 when k = 1 was not explicitly done.

    1. J.F.:

      I agree. f”(x) = 0 and change of concavity near this point are compulsory for a point of inflection to exist.
      2017 SM Ex2 ER. Q1… 2 marks including rejection of (0,0) quite worried me.

      Given that the last part is only 1 mark, I think button pressing is the quickest hedging method… I do think one mark for the sign testing + correct values of k is too tight, under exam conditions. On the other hand, I do see many questions set in such a style – one marking carry two or more info, such as many questions you wrote for Spesh trial exams, or some other commercial trial paper starting with ‘K’ written by a TI expert every year.
      Personally, I am ok with 1 mark awarded to students who stated k = 1, which must be shown from correct justification as you mentioned.

      You probably have noticed the even weighting of each question. To save marks for subsequent questions, such a sacrifice must be made, especially when writing up and balancing a full exam questions.
      Again I have to use my ‘opportunity-cost’ theory, but this time, it is interpreted from the exam-setting panel perspective. To minimise the level of uncertainty and balance of the weighting in each area of study, it makes them overlooking minor, even ‘trivial’ details, in their intended questions which were used to assess students.

      I knew the total number of Spesh Students doing NHT exams should be less than 20 (in 2019, and in 2017 there were just 8 kids doing the NHT Spesh Exams), if I have not mistaken. Looks like for this special cohort NH kids every year, comparative marking was applied, just as if it is a small VCE subjects with low enrolment numbers. So, SADLY, the whole paper SEEMS being used to examine the students holistically, whereas in Sth HT each November, the number of students is much larger than NHT and thus more seriousness SHOULD be taken to guarantee consistency and accuracy……


      It raised my concerns that just ‘SOME’ SM teachers noticed the matter. How about the others?
      Many first-year-out teachers may not have very strong and in-depth maths background as you or John have, and it would tend to be an inheritable flaw to be carried on in future others may not even notice for a long time.

      1. Hi P.N.

        Re: I knew the total number of Spesh Students doing NHT exams should be less than 20 (in 2019, and in 2017 there were just 8 kids doing the NHT Spesh Exams)

        Where are you getting this data from? Is there data available for NHT Methods?

      2. Of course, as a 1-mark question there was no intent that students test for concavity. That just means that it was a screwed 1-mark question. As for the Mathematica solution, it’s difficult to know what to make of it. I’ve corrected the screenshot, to include the final comment, indicating that f”(0) = 0 is “necessary”. It’s not clear that sufficiency was also meant to be implied, but it’s also not clear what the message is supposed to be. Whatever the message was supposed to be, the message received is “Don’t think, just push the button”.

        I won’t necessarily be critical of SM teachers, since I don’t know how they use the past NHT exams. But I’m ready to be. The question is, to what extent does anybody work through such problems in a proper mathematical way: if one does, one is much, much more likely to spot the special cases.

        1. Marty, I disagree. The question(s) is: Who are the cretins that are meant to be vetting these exams and their marking schemes? Who is the fool that signs off on those things with his elephant stamp?

          That’s where the blowtorch should be aimed.

          1. Well, yes, that is Question Number 1, and hence the WitCH. But I think the provided solutions raise another question: the general thoughtless push-button method to answer such questions. That thoughtlessness would be there even without the error, and that thoughtlessness is endemic. It is expected. It is demanded.

            1. Indeed. It’s in 10 m high neon lights for all to see in those Mathematica solutions. Who needs to think when you can push buttons.

      3. Yes, sometimes questions worth 1 mark should be worth 2 marks. However, the solutions should always be complete and thorough and show the full two marks worth of solution so that teachers don’t get misled.

        For the present question, a check for change in concavity should have been included – as it is many teachers might get misled by those Mathematica solutions into thinking that f” = 0 is sufficient.

  3. Apologies if this has been mentioned elsewhere, but as a daily user of the software in question, I cannot see why any competent user would define a function f_{k} when working through this problem as it is a complete waste of time…

    More comments specific to the program coming, just need to get some screen-shots ready.

    1. Hi, RF. I’ll look to see if I can rejig the site to allow commenters to post images. Of course the wisdom or otherwise of defining f_k is not my concern here, but there have been suggestions that VCAA’s Mathematica solutions are often less than, um, optimal. I’ll be interested in your comments.

      1. Marty, the word you’re looking for (and it’s a charitable word at that) is sub-optimal (but I guess you knew that). A better word is Quatsch.

        VCAA have absolutely no clue how relatively uber-powerful Mathematica is compared to a CAS calculator in the Exam 2 setting. And doesn’t care. It simply wants to bulldoze through the whole Mathematica CBE shambles and turn the subjects into Computational Specialist Mathematics and Computational Mathematical Methods where the focus is on coding rather than mathematics.

  4. I’m not familiar with Mathematica, but what’s with the arrows where a “=” would normally be written? (ie. k -> –1 rather than k = –1).

    1. Yeah, that is Mathematica output form. For those used to the CAS calculators Mathematica actually has five different commands that on the CAS mean solve:

      Solve, NSolve, Reduce, FindRoot and FindInstance. All can be done over restricted domains and all are useful in their own ways, but one has to know what one is trying to do first. Which, given the limited amount of time available to teach the course, let alone the button-pushing makes things problematic.

      In the hands of a skilled user (of which there are few in high schools, teachers or students) the software gives a seriously unfair advantage, especially in multiple choice, but of course, one has to spend a lot of time learning how to game the system first!

      Marty, I’ll try to email some screen-shots soon (using your definition of soon…)

          1. I had to change the comment format to permit images etc to be attached. I decided to moderate a little more closely. But if you put in your name, your should appear, and if you’ve commented previously (maybe since I adjusted things), your comment shoudl be approved automatically.

      1. It should also be notable that Reduce will give more familiar notation, using ==, && and || in place of several things on top of Solve.

        RF, you’re too right with how effective Mathematica is with trivializing technology in the right hands, however, I’ve seen very few students come out of the CBE trial wielding it to its full capability. For instance, you can predefine keyboard shortcuts that will replace letter combinations with code. In fact, let me tease this idea by the following function I designed below. It calculates the important information of a function on top of Manipulate, also factoring in the edge cases where k=0 and k=-2. (Side comment: What would an assessor think of this, if they saw that k=-2 had no stationary points? Would they have to concede their point??)

        Food for thought, if anyone would like to think about it. How would you use Wolfram Mathematica, or any CAS/programming language to find vertical asymptotes and oblique asymptotes of a function? The question itself actually asks: “If you could trivialize this concept and its calculation, how would you?” which is quite a hilarious but not quite practical problem to answer.

        1. You can try ==ComplexInfinity with Solve if you want to have some fun… and go nowhere in a hurry.

          (Mathematica users only)

          When graphing asymptotes, they tend to get drawn as solid lines. I wonder if students get penalised for this or if they are taught to use the ExclusionStyle-> sub-command. I doubt it some how.

          1. RF, once again you bring up some intriguing points that I’ve voiced in a previous post. Do students get penalized for how they draw their graph in the CBE exam, or rather what code they use? It’s the million-dollar question that undeniably goes unanswered with the DuLL Mathematica “marking schemes” or whatever you could feasibly call them. And as for the Exclusion Styles, I can tell you that most students wouldn’t be aware of half the plotting options.

            As Marty alluded early, there will be fun for all in due course. And it’s quite assuring to hear that there are more pedantic people who’ve decided to snoop around for details in the CBE exam, and you’ve arrived at a similar answer to I have, with regards to scaling, an entirely different can of worms. You also mention a training session for CBE, I’m curious as to what they went over and what other nonsense they piled up.

            Steve R, this link here (http://scriptedmath.com/home.html) may be an intriguing read for you, as it contains an example of a library of functions that automate certain tasks for the TI Nspire. It makes you wonder, how many kids would have used something like this in the exams?

            1. MK,

              Thanks for the link .

              I have found the interface for standard TI functions for the CX with LUA scripting
              to be rather clunky even if the keyboard emulator is used.

              Nevertheless I know of a couple of able candidates from previous years who added customized functions similar to the Mathematica examples above to the standard build .

              Steve R

            2. Hi MK,

              As a 2019 leaver, I can tell you that the number of fellow classmates whom I saw utilising existing files with pre-defined functions in SACs and exam 2 is quite numerous – after all, most schools do not have a requirement to clear memory.

              I too was one of these students, and had all sorts of cool things like calculating mean, median, variance and sd of pdfs, vector resolutes etc. Of course, the power lies in being familiar with the inbuilt quirks of the TI e.g. when approximate solutions sometimes fail to generate if one uses > instead of >= etc.. as well as simple PC-like keyboard shortcuts; otoh customising for one’s own style and not defining too many custom functions that could seldom be used or were too rigid was similarly important e.g. I always manually found the DE for tank problems and manually found SPs and intercepts instead of using poi() given in the above link as grabbing the result from that poi() was always too cumbersome imo.

              1. Thanks, IC. Do you (anyone here) have a sense of the relative ease and relative power of TI versus Casio? Is there an advantage of one over the other?

    2. It denotes a “rule” which can then be applied to any expression involving k in order to perform the substitution. i.e., k + 1 /. { k -> -1 } evaluates to 0.

  5. Here is one option for approaching the problem with Mathematica. One problem of course, is what counts as working… but if you just need an answer, this works for these types of question:

    1. A nice duplicate of this diagram
      (with a couple of graphs with varying k values) will be very good to attract M1.

  6. Which TI users could not do (or if they did draw multiple graphs they would run out of time and/or room).

    I still feel that the scales are being tipped here…

      1. The problem is I think they do care. I get the distinct feeling that someone (or more than one) at VCAA really want Mathematica to succeed (I’ve heard rumours that other software such as Maple was considered also at one point) and I’ve seen nothing but evidence in support of this belief.

        MK – nice one. Such a function could quite easily be shared with students as they are permitted a USB drive of files in an exam. It doesn’t matter if the students are brilliant users, a competent enough teacher with a USB of pre-loaded files could be a major point of difference; the public/private school debate also comes to mind a lot here…

        1. I meant that VCAA doesn’t care that Mathematica has such an advantage over a CAS-calculator. But I retract that – I think VCAA does care inasmuch as it will gradually force more and more schools to choose to use Mathematica, making VCAA’s inevitable imposition of Mathematica and CBE much ‘cleaner’ ….

          When CAS-calculators were first introduced, there were separate CAS and non-CAS Exam 2’s in Methods. The points of difference in each exam were not large, but they existed and each cohort was scaled separately. I’m not sure that’s happening with Mathematica.

          A very strong case can be argued that the Mathematica cohort should be scaled separately in both Methods and Specialist, and that in fact this cohort should be given different exams in both Specialist and Methods. But VCAA won’t do this, mainly for the reason given above, but also because I think the clowns calling the shots are just too dumb and arrogant to understand the lack of parity.

          By the way, I hear a ReplyAll email (capturing dozens of teachers) has been sent to DLL about this error on the NHT exam. But the ‘Report’ remains unchanged and VCAA remains silent. DLL really does not care.

          1. Random thought… do scaling reports ever get released for NHT? Because that would make for an interesting comparison, if one was very clear about the sample sizes (which should be freely available but do not seem to be anywhere I’ve looked).

              1. Sure, but they still need to calculate a scaled study score somehow for the purposes of calculating the ATAR.

                The VCE is full of “small studies” which require special calculations for their study scores. The statistical data for these is still published annually.

                So, I’m genuinely curious how they calculate study scores for these NHT examinations (and if their SACs are ever audited, but that is another story altogether!)

            1. Indeed, RF. Should be freely available ….. But isn’t (because of embarrassment?) From my June 24 comment:

              I have heard that there was only 8 NHT Specialist Maths Students in 2017. I can’t imagine that number rose significantly in 2018 and 2019 – the NHT VCE runs mainly in China, and according to a reliable source most of those partnership schools don’t provide Specialist Maths as an option. Only a few of them have Specialist, one of which is in partnership with a well-known private school (with ambitions of national and global domination).

              How to scale study scores with a cohort of 8 is a great question. And there are lots of great questions about the SACs of shit.

              One could do a dodgy extrapolation for the number of NHT Methods students and get a couple of hundred.

              You have to be a member of the VCAA inner sanctum of teachers to get this data. So information such as marking schemes, numbers of enrolled students (in NHT subjects and Maths Methods CBE) is all confidential.

                1. OK, it’s a conjecture based on the fact that this data cannot be found anywhere, including the VCAA website. A conjecture that I doubt will be proved wrong.

              1. So, they do get study scores…? Or do they do these VCE exams as well as their own local program with the sole intention of getting enough units to receive an Australian qualification?

                Any inkling of a clue? I know a few people who went to the school you are implying (with its current size it is difficult not to know someone who went there, worked there or has children who go there…)

                This is all I can find from VCAA:

                “The external assessment results of the NHT cohort will be subject to equating to ensure comparability with the results of the larger Victorian cohort assessed in the previous calendar year.”

                equating… to WHAT?

                1. R.F.:

                  It doesn‘t matter this year…
                  Eventually these NHT kids will sit the same exam in November.

                  I am deeply concerned about the stage 3 lock-down from today on… Would VCE external assessment successfully operate?More importantly would it be safe then?For all the good wellbeing of our students and teacher colleagues…

                  1. Perhaps. The premier seems keen to not make any further changes to VCE assessment, but schools will no doubt have to guess their way through the VCAA guidelines, much as they do through the curriculum.

                    SACs are the bigger issue for the moment, as I don’t see how they can possibly be “verified” for “integrity and consistency” when so many schools were doing them off-site, or under very, very different conditions.

                    I would have much preferred just dumping SACs altogether for 2020, but that was never going to happen!

        2. RF ,

          Clearly both you and MK may have used Mathematica more often than the average respondent. As I interact mainly with non CAS students and undergrads I wondered if you could elaborate as to what is allowable in terms of pre-loaded functions ,programs,libraries etc on a TI Nspire CX say going into a Specialist VCE exam.

          I believe the current version of the TI may use LUA scripting rather than Mathematica.

          Clearly any one with a pre-loaded customized graphing function such as MF’s would be at a significant advantage for many questions

          steve r

          1. I quote from the cover page of the 2019 Specialist Maths Exam 2:

            Calculator memory DOES NOT need to be cleared.

          2. Steve, it has been a while, but I was invited to attend a training session for teachers using Mathematica for Computer Based Examinations (CBE) before it was a genuine option for schools.
            When I later spoke to teachers at one of the four schools “invited” to pilot CBE (two in regional Victoria, one in inner Melbourne and one in outer Melbourne) they told me that students were permitted to bring a USB pre-loaded with Mathematica notebook files into the CBE exams.
            Unlike a bound-reference, one can cut-and-paste from these files into an electronic exam paper…
            And JF, from what I gathered, there is no separate scaling.

  7. Has anyone raised the sheer craziness of the final statement that f''(x)=0 is a necessary or sufficient condition for a point of inflection?

    I’m thinking they need to see graphs of y=x^{4} at (0,0) or y=x^{\frac{1}{3}} at (0,0) as examples of why this is crap.

    1. JF raised the point about the lack of sufficiency. In this context I’m ok with the report’s suggestion of necessity, taking the assumption of differentiability as implicitly understood. But, you are correct that this is an assumption, and it probably shouldn’t be implicit in an examination report.

    2. Hi RF. More broadly, it’s regrettable that the Study Design doesn’t at least mention that points of inflection can occur at points where a function is continuous but not differentiable and give some examples. Your example of y = x^{1/3} is a classic and salient example.

      What would VCAA make of y = x|x| ….? The double derivative doesn’t exist at x = 0 but it has a point of inflection at (0, 0) (and the function and its first derivative are continuous over all real numbers, so the graph looks ‘nice’).
      Answer: VCAA wouldn’t (because its too dumb) and couldn’t (because it’s not within the scope of the impoverished Study Design) make anything of this function.

      What would VCAA make of \displaystyle y = |x^3 + x^2| with a point of inflection at (-1, 0) (its ‘corner’) (as well as a PI where x = -1/3) ….? Nothing, its too stupid – I say this because of MCQ12 on the 2017 Specialist Exam 2 and the subsequent Examination Report. From the given options, VCAA does not understand that something like

      y = |f(x)| where \displaystyle f(x) = -\frac{1}{20} x^5 + \frac{1}{4} x^4 - \frac{1}{2} x^3 + \frac{1}{2} x^2 - \frac{9}{20} x - \frac{3}{4}

      (so all the convoluted conditions imposed by the question on f(x) are satisfied) has a point of inflection at its ‘corner’ (https://mathematicalcrap.com/2019/10/18/witch-22-inflecting-the-facts/).

      VCAA obviously does not understand the following:

      A necessary and sufficient condition for a function f to have a point of inflection at x = c is that g”(c – e) and g”(c + e) have opposite signs in the neighbourhood of x = c (that is, there is a change in concavity). The function is NOT required to be differentiable at x = c. In the case where the second derivative does exist, a necessary but NOT sufficient condition is that g”(c) = 0.

      VCAA can include something technical like strictly increasing/decreasing in the Methods Study Design (but then don’t know how to meaningfully use it anywhere), but can’t include the above technical definition of a point of inflection in the Specialist Study Design. Figure that one out!

      (Actually I can partly figure it out: Some idiot wanted to include points of inflection but only as an application of the double derivative. But I can’t figure why strictly increasing/decreasing was put on the Methods Study Design … VCAA’s lame-brained “Find the intersection of points of the function and its inverse function” question fetish gave the perfect opportunity to justify its inclusion. But they didn’t).

      And to top things off, VCAA obviously doesn’t understand the difference between necessary and sufficient otherwise it would explicitly emphasis this in the resources it provides.

      1. Those modulus composed-functions are really great case-studies, but VCAA examiners, even in Specialist, seem to have gone off them of late.

        The modulus in the answer to one of the parts in 2019 Paper 2 Q2 caught me by surprise when I first saw it (I will admit not considering it when I first did the paper), so they do still know of its existence.

        I think what all this comes down to (all the excellent points raised above) is that teachers of these studies have to make quite a number of guesses and if they guess wrong, it is the students who ultimately suffer. Maybe not more than a point or two in their study score, but in VCAA land where everything is calculated to six decimal places (according to the briefing I went to on scaling) such things are significant.

  8. Marty, just quickly, I don’t believe that Mathematica is wrong here. The solutions ARE, that is beyond reasonable doubt, but from the evidence I’m seeing, Mathematica is doing exactly what the operator has asked for in both cases. When asked to solve for a variable, the software gave the solutions (in terms of the parameter).

    When asked to then solve an inequality, again it did so.

    What the software didn’t do is think.

    The operator (and author of these solutions) failed to correctly interpret the output. This is a failure of the operator.

    But your much larger point perhaps encompasses all of this; expecting a CAS tool to answer the question about a derivative when you don’t bother to look at what the derivative actually is… this is not even bad Mathematics, it is an absence of Mathematics.

    (In my unqualified opinion, as I don’t entirely know what Mathematics actually is; every time I thought I knew something came along to prove me wrong!)

    1. RF, your former conclusion on Mathematica being thoughtless is mostly fine. However, if you use Reduce, there is an additional condition imposed on the equation, that is for the given x coordinate and k value, the equation 1+x+kx cannot be equal to 0, which in turn will throw out the invalid solution if k=-2 or k=0. Now, this thing is actually existent in Solve (consider the ConditionalExpression output on Solve) which can impose a restriction on variables if you come across problematic solutions that either become undefined or become a complex number. This is evident with logarithms and exponentials (and trigonometric functions with generalized solutions). So the question is, why doesn’t Mathematica apply the same following of logic to realize that a solution is invalid?

      Another gem that Mathematica screws up is the function f(x)=tan(x)+sec(x). The input Solve[Tan[x] + Sec[x] == 0 && -[Pi] <= x <= [Pi], x] returns x=-Pi/2, which by inspection is a hole. Now, Solve will fail to return a general solution in this case, but this goes to show behavior with holes is unpredictable to say the least (Reduce for some reason returns the general solution). I’d encourage readers to try this with f(x)=(x-1)/(x^2-4) and a standard logarithm function to find the holes of their reciprocal function. So, not all holes are equal.

      While all of this could be written off as semantics regarding the functioning of the Wolfram language, it’s a matter of time till VCAA stumbles into something extremely problematic. Additionally, what happens when a TI nspire or ClassPad is asked to solve the same thing?

      1. What it boils down to are the subtle differences in meaning between the Mathematica function Solve and the the mathematical definition solve. They might look like the same thing 90% of the time but that doesn’t mean they’re the same thing all the time. The trouble arises in assuming they still mean the same thing the other 10% of the time. This is exacerbated when getting no solution with no restrictions but getting some solutions with restrictions.

        Look at how many different ‘solve’ functions Mathematica has …. Obviously they can’t all have the same meaning as the mathematical definition of solve.

      2. MK,

        I plugged f(x)=tan(x)+sec(x) into an recently inherited TI Nspire CX with updated OS to check the reliability of solve() in this case which is I believe is a LUA script.

        solve(f(x)= 0,x) returned false which is encouraging however when graphing f(x) and finding the intersections with g(x) =0 on the graphing page with no domain restriction(menu 813 is the keyboard short cut for those who dislike typing) it returned x=-Pi/2

        As far as “pushing the envelope” with this calculator I have found the many turkvids by Sean Turkington to be most useful but as he says in the first link below “having CAS is not a substitute for having brains”


        I have found it difficult to track down the actual source code to solve() from TI or github. Can you ,RK or anyone help with this request?

        Steve R

        1. Perhaps Mysterious Knucklehead can help with the source code, although I imagine it is secret. As for CAS not being a substitute for brains, yes, sure. But that’s not an excuse for CAS systems delivering incorrect answers.

        2. Steve, most of the commercial CAS companies are closed source and do not allow for source code viewing (this includes Wolfram, Casio and Texas Instruments). On a tangential note, I have found a company that produces a CAS calculator that is open source, which may be a first step as to what’s going with the ridiculous behavior of this function here:https://www.numworks.com/

          And that MVT example is pretty hilarious, but also quite confusing, once again which is a brainless function of a CAS. If provided an interval, it’s the job of the CAS to solve over that interval, without missing a single solution. It’s also important that Mathematica’s FindRoot will do a similar thing, but it was prefaced. To add to this, it shouldn’t be a hard task to just plug the value found back into the function to see if it does return the intended value (e.g tan+sec) which would evidently set off a huge red flag, since directly plugging -Pi/2 will alarm Mathematica that an indeterminate form has been encountered.

    2. RF, it’s a tautology that a program will do what it’s programmed to do. The problem is, the program doesn’t do what the name suggests it does, and what most people think it does: Solve doesn’t “solve”. Despite what you claim, the software did not give the solution to the NHT problem.

      Your own comment below gets to it best: if a program is “more reliable” or “less reliable” then it is unreliable. And a black box that is unreliable is just a fucking brick.

        1. It’s the name, and people’s understanding of what the name implies. Whether or not Solve is useful, in general or in school, is an important question. But, even if the program is useful, people at least have to be aware that Solve doesn’t solve.

          1. Sure. I was just thinking of the (insert name of any Year 11 Methods text here) exercise, usually in Chapter 1 that asks students to “solve” (for example) x=\frac{a-1}{a+1} for a.

            Of course transpose would be a better command term, but I’ve always considered this a valid use of “solve”.

            Willing to admit I’m wrong though.

            1. RF, I think that is a reasonable point, and I’ll agree that “solve for” is in practice confusing, and confuses the natural meaning of “solve”. But what would you expect a student, or a machine, to do with “solving” x(x+a) = 3x for a? What if the student/machine were to “solve” the same equation for x?

              1. I would expect a “solution” of the form a=3-x in the first instance and x=0 or x=3-a in the second.

                Neither are truly satisfying as the equation itself is as contrived as a VCAA paper 2 extended response question or a typical methods “modelling” question where the answer doesn’t need to have meaning because the question didn’t.

                When you have an “equation” and it isn’t clear what are variables and what are parameters, then “solve” is a difficult concept to grasp. When you have an equation with only one variable and no parameters, the meaning should be clear enough, or have I (again) missed the point?

                1. OK, RF, I’m uncomfortable with your suggested “solving” of x(x+a)=3x for a, but I’ll leave it.

                  I’m assuming you agree with MK, that “solving” tan x + sec x = 0 to give (e.g.) x = -pi/2 is not “solving” in any meaningful sense of the word.

                  1. Yeah, I forgot to add if x is non zero in my solving for a which is a pretty significant omission!

                    And yes, finding an x-intercept and solving an equation are quite different things.

                    Some textbook authors may disagree.

                    1. Rf, the “if x is not zero” is exactly the kind of omission Mathematica makes. That’s the point.

                      In regard to tan x + sec x = 0, I think you’re missing the point (pun intended).

                    1. And there’s a big difference between solving and finding solutions that don’t exist.

  9. Do an internet search for “Wolfram RealOnly Add-on”.

    Basically, there is a package (which is essentially a nb file, so can be used under the CBE rules) which tells the program that you only want real output to be considered…

    I’ve talked with many teachers who are or were in CBE schools. None of them seemed to know that such packages exist. Is this a failure of VCAA or the teacher themselves? I don’t blame the teacher, for what it is worth.

    As for your issue with solve, whenever solving trig equations, try FindRoot instead. It is much more reliable in this instance.

  10. Thanks for the update, Marty. An excellent summation of the ‘state of the union’. I think your conjecture that

    “this [is] a question from NHT, about which, as far as we can tell, no one really gives a stuff.”

    is true, at least for NHT Specialist Maths (where probably no more than a dozen students at most sat the exam in 2019). I say this because the error in both the solutions and the Examination Report was reported to VCAA over 3 weeks ago. But despite VCAA explicitly acknowledging the error, the Examination Report currently remains unchanged.

    1. Update: I’ve just noticed that the Examination Report has been amended. The following explanatory comment has been added to Q2(b):

      “Note that the endpoints are included as the resulting forms of 𝑓𝑘 (𝑥) do not have a stationary point when k takes those values.”

      It’s a useful and welcome amendment. But unfortunately, the amendment has been bungled: The answer did not get corrected! Can you believe it? (They must have got the work experience kid to do it). Hopefully this might change in the coming days. And in fairness to VCAA, the amendment was made in good faith.

      1. Update: I’ve just noticed that the Examination Report has been amended once more. This time VCAA gets it right. This time the answer has been fixed. Hallelujah!

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