WitCH 10: Malfunction

It’s a long, long time since we’ve had a WitCH. They have been not-so-slowly accumulating, however. And now, since we’re temporarily free of the Evil Mathologer, it is the WitCHing hour. Due mostly to the hard work of Damo, all of the outstanding WitCHes have been resolved, with the exception of WitCH 8. That one will take time: it’s a jungle of half-maths. Our new WitCHes are not so tricky, although there is perhaps more to be said than indicated at first glance. The first of our new batch of WitCHes is from the VCE 2018 Specialist Exam 1:

The Examiners’ Report gives the answer as \int_0^{\frac34}\left(2-t^2\right)dt. The Report also indicates that the average score on this question was 1.3/5, with 98% of students scoring 3 or lower, and over a third of students scoring 0. Happy WitCHing.

Update (16/02/20)

This problem is ridiculous and, more importantly, it is wrong. First, the wrongness.

As indicated by the examination report, the examiners imagined that they were, in essence, asking for students to determine the speed function \boldsymbol{v(t)} of the particle. The distance is given by \boldsymbol{d=\int\limits_0^{3/4} v(t)\,{\rm d}t}, and a non-trivial calculation gives \boldsymbol{v(t) = 2 +0t - t^2}. Then, the coefficients \boldsymbol{a = -1, b = 0, c = 2} can be read off. 

That is not, however, the question the examiners asked. What did the examiners really ask? They asked for integers \boldsymbol{a,b,c} for which \boldsymbol{d=\int\limits_0^{3/4} at^2+bt+c\,{\rm d}t}. But

    \[\boldsymbol{d=\int\limits_0^{3/4} 2  - t^2\,{\rm d}t = \dfrac{87}{64}\,.}\]


    \[\boldsymbol{\int\limits_0^{3/4} at^2+bt+c\,{\rm d}t = \frac{9}{64}a+\frac{9}{32}b+\frac{3}{4}c \,.}\]

So, multiplying out the fractions and cancelling out a 3, what the examiners really asked for were integer solutions to the equation

    \[\boldsymbol{3a+ 6b+ 16c  = 29\,.}\]

This equation has infinitely many integer solutions, meaning the examination report is missing infinity minus one valid solutions.

This is a flat out, undeniable error (which the Trumpian VCAA will never concede), but is it a problem? As commenters here have noted, there is little chance of a VCE student being actively misled to chase the infinitely many solutions. In, particular, the method to find all solutions requires first finding the particular solution the examiners had in mind. We are not convinced such direct concerns should be so quickly dismissed, and we discuss this further below. Still, the extra solutions require thought to even contemplate, and significant work to compute, which is an important point.

Whatever the immediate practical concerns, however, mathematicians are aghast at this error. They are aghast because the exam question is simply not testing mathematics. Yes, the students went through the ritual and attempted to compute what was intended and were graded accordingly. And, yes, teachers can now coach current and future students on the required ritual. But none of that is mathematics and, indeed, it is worse: it is antimathematics. It is teaching students to ignore mathematical meaning, to see no value in mathematical precision, to respect only ritual.

OK, that is the awful wrongness of the exam question. Now, the sundry ridiculousnesses:

  • The question is badly and needlessly opaque. There is no a priori reason to imagine the distance as being given by the integral of a quadratic. Asking for (more accurately, attempting to ask for) the speed function in this overly cute manner adds no value, only confusion. The confusion is enhanced by the arbitrariness of the 3/4 limit and, especially, by the pointless specification that the coefficients of the quadratic be integers.
  • Independent of the opacity, the wording of the question is lazy and clumsy. The distanced travelled “in three-quarters of a second” is not the same as the distance travelled in the first three-quarters of a second and, indeed, is not anything. The phrases “moving along a curve” and “travels along a curve” are just verbiage. The units are pointless.
  • The question would be much more natural as an arc-length question, rather than a distance question.
  • The answer in the examination report is incorrect, even in the intended terms. The question asked for the values of the coefficients, not the integral. Yes, this is a nitpick, but it is exactly the kind of nitpick that the examiners routinely employ in their sanctimonious whacking of VCE students. So screw ’em. Sauce for the gander.
  • Last, and far from least, there is something very strange about the score distribution for the question. The average score was 1.3/5, which is depressing, although not surprising: computing the speed (without CAS) requires a level of care and facility beyond most CAS-drunk students, and the question contains a hidden absolute value to negotiate. What is strange is that, whereas 2% of students received the full 5/5 for the question, apparently 0% of students received 4/5. It is difficult to see how that could occur with any sensible grading scheme.

75 Replies to “WitCH 10: Malfunction”

  1. I feel like this doesn’t get to the heart of the matter, but I’m not convinced that the distance travelled by the particle in “three quarters of a second” is given by an integral with lower and upper limits of, respectively, 0 and 0.75. Couldn’t it be any integral with lower and upper limits of x and x + 0.75?

    1. I felt the same thing, but was willing to throw VCAA a bone and assume that out of all the possible 3/4 second time intervals, they were specifying the one from t = 0 to t = 3/4.

    2. Thanks, SRK. Of course you are correct. Yeah, it’s not the main point and the required answer makes clear the time interval. But the wording is simply wrong, and needlessly wrong.

      And, true, one might consider, as John does, to throw the VCAA a bone. But think about it. This is lazy “Who gives a shit?” wording from the same nasty obsessives who self-righteously nitpick students to death. So screw it. Throw the bone at them.

      1. Indeed. It would have been no skin off their chin to have added two simple words and said “… in *the first* three-quarters of a second….”

        1. Exactly. I can completely understand a harried teacher writing such a sloppy expression. And at times there is a tricky balancing act to be made between accuracy and clarity. But not here. Here, it is just laziness. Or idiocy. Or both.

  2. Was thinking a bit more about this one, because this topic is coming up shortly. Again, I feel like I am not seeing the main problem Marty sees with this one, but:

    I think it is unnecessary and, for reasons given below, somewhat harsh (perhaps even malicious) to write this as as an arc-length question. Given what was required to achieve full marks for this question, I think it would have been better to ask students to find an expression for the speed of the particle. Reasons:

    (1) students weren’t expected to evaluate the definite integral. (Indeed, they were required to write their final answer as the definite integral). Maybe there’s a point here about assessing whether students know the arc-length formula, but it’s given on the formula sheet so that seems pointless.

    (2) A savvy student would realise that writing each line of working with a definite integral is asking for trouble – very easy to forget to include a terminal or a “dt”. So they would just work out the expression for the speed, and in the final line write down the answer as a definite integral in the required form. I wonder if this is what VCAA had in mind…

    Edit: actually, re-reading the question, I realised the definite integral is completely pointless, because students were only required to find the values of a, b, c.

    1. Thanks, SRK. It’s been so long since I could get back to this, I kind of forgot myself! You don’t quite raise the point I have in mind, but I agree entirely: asking for the speed would have been a much more natural question. Although, as John indicates below, the whole framing in terms of kinematics is not natural for the given function.

  3. I also don’t feel like I’ve seen the main problem that Marty has with this question. As such, I’ve gone looking and I think that I may have something, but maybe I’m overthinking it…

    The question is phrased as a Kinematics question, which means that we can find the velocity and acceleration vectors. These are:

    v(t)= and a(t)=

    Both of these are undefined when t=1, which is within the stated domain of the function. In particular, the magnitude of the acceleration goes to neg infinity as t goes to 1. Which, when you consider the force that would need to be applied for this humble particle to follow this particular path, well it’s pretty heroic.

    1. Yes, using a kinematics context for this question is pretty dumb, because then things have to make physical sense. Which they don’t.

      I think the velocity is OK: v = t^2 i + 2Sqrt[1 – t^2] j and there’s no trouble here as t –> 1.

      It’s the acceleration where things go South. It makes no physical sense (to me, anyway) that at t = 1 the object is at r = i/3 + pi/2 j, |v| = 1 m/s and yet |a| –> oo m/s^2.

      To use a kinematics context here simply to, I assume, also test whether students understood that distance travelled is length along a curve is just stupid and clumsy. The curve should just have been defined parametrically and the length between two points asked for: “A curve is defined by ….. Show that the length of the curve between …. is given by ….” The fact that the square root of a perfect square had to be simplified by noting the domain is a nice touch, but ruined by everything else.

      I find the constant desire by VCAA to contextualise their questions with irrelevant, contrived, artificial and unphysical scenarios simply to make the mathematics look ‘relevant’ (I suppose) hilarious – better to do nothing and be thought a bonehead than do something and remove all doubt.

      1. Yep, the kinematics context just makes the whole question a little absurd. The other thing that contributes to the absurdity is the manner in which they have defined the domain to be 0<=t<=1, but then asked for the length of the curve from t=0 to t=3/4. I am assuming that they were seeking to avoid an improper integral, but it adds an unnecessary and confusing element to the question.

        A quick edit of my above post – the magnitude of the acceleration goes to infinity, the j component of the acceleration goes to neg infinity. Also, I don't know what happened to my equations for velocity and acceleration, I'm mostly confident that I typed them in.

        1. The integral for length isn’t improper when calculated over the entire domain, but given past experience I don’t think checking this possibility would even occur to anyone at VCAA.

          Since the resulting integral is readily calculated, I’m surprised a value wasn’t required.

          Wouldn’t it be great if there was a ‘Meet The Exam Writing Panel’ each year ….

          1. A genuine question here: because you substitute the derivative into the arc length integral, and because the derivative is undefined when t=1, does this not make the integral improper? Even if it can be simplified, through cancelling, to a form that could be integrated?

        2. Hi Damo, quickly on your missing equations. I don’t know whether this is the issue, but I noticed that wordpress doesn’t seem to like any html code in comments: it will reject the whole comment. However, once the comment is up, I can edit the comment and insert the html code. That’s what I do to get superscripts and links etc in comments. You and everyone also now have the ability to edit your own comments (for thirty minutes after posting).

  4. Damo, I overlooked the subtlety that dy/dt is indeterminant at t = 1. So if integrating to t = 1, technically a limit *would* have to be set up first, so I guess the integral to t = 1 *is* improper. You’re right. I haven’t come across that before. Very clever. So I doubt that’s what stopped VCAA from asking.

    1. Yes, I doubt it as well, particularly given past history. But if not that, why? I second the idea of a Meet the Exam Writing Panel…

  5. Thanks to Damo and John, and to SRK to whom I replied above, for the extended discussion, and sorry to be so slow to get back to comment.

    All your criticisms are (eventually) spot on, and the question in total is half-baked weirdness in the ways you all indicate. But, interestingly, no one has picked up on the issue that first caught my eye. To me, the issue is glaring and the exam question is simply and ridiculously wrong. Perhaps teachers, however, are used to this type of thing, and would regard the issue as a nitpick, and would simply ignore it. (Versions of this error/nitpick appear in other exam questions but, to me, the issue is particularly glaring in the question above.)

    Quickly on the improperness. Yes, at t = 1 the derivatives of arcsin(t) and √(1-t2) are undefined, which implies the velocity is undefined, which implies the speed is undefined. This necessarily makes the distance integral improper at t = 1. The improperness is “removable”, in the sense that the speed function can be extended to be continuous at t = 1. (A simpler example exhibiting the same behaviour would be integrating the function x/|x| on [0,1].) But, the improperness is there, and that is undoubtedly why the examiners wisely, and with astonishing ineptness, avoided integrating over the whole interval.

      1. Hi, SRK. I haven’t had a close look at the NHT exams. I don’t see what you’re seeing on the NHT question (and am curious), but it’s not what I’m thinking of for the question above.

        1. As mentioned above, was really just fishing. I was wondering if it had something to do with either (i) saying “distances are measured in metres” (as opposed to displacements), or (ii) not explicitly mentioning that i, j are perpendicular unit vectors.

  6. I am now at the guessing stage: is it that the distance that the particle travels is actually given by the arc length integral, which simplifies to their integral?

  7. This is a nitpick , but I don’t think it’s what Marty has in mind: The use of brackets around the integrand irks me. Not so long ago VCAA said that brackets around integrands were NOT required, that the integral symbol and the ‘dx’ (or whatever) acted as defacto brackets. And yet here’s VCAA using brackets.

    1. Thanks, JF. Yes, that’s not what I had in mind, but I agree that the redundant brackets are annoying.

    1. Thanks, JF. It is odd phrasing. Perhaps an artifact left over from an arc length version of the question.

  8. Just a quick follow-up. Given that commenters are not seeing what I see as a central flaw in the exam question, I asked five people – four mathematicians and one (very smart) non-mathematician – to take a *one-minute* look at the exam question, and to tell me what they saw. All four mathematicians noted the flaw (plus some of the other nonsense), and the non-mathematician did not.

    This is turning out to be a very interesting WitCH.

    1. OK, so I’m going to think outside the teacher-box with all its implied assumptions and get technical:

      For the position of a particle to be specified on a curve, the particle is either point-like or the position vector specifies the location of the centre of mass, say, of the non-point-like particle. None of which is said in the question.

      I really hope this or something like this is *NOT* the perceived flaw (because there are so many ‘better’ flaws already mentioned), but I’m assuming there’s some technicality that us non-mathematicians have automatically overlooked as a matter of implied assumption. Given how much ink mathematicians have used to prove 1+1=2, I’m guessing it might be something along these lines (I just assume 1+1 = 2 and hope for the best).

      1. Hi JF, I understand the frustration non-mathematicians have with logical formalism and pedantry, but it’s definitely not that kind of flaw/nitpick.

        1. I don’y have a problem with logical formalism and pedantry – I totally understand their importance. It’s just not my cup of tea unless I’m forced to drink it (I got through 3 years of real analysis and have worked through Rudin but my teeth were gritted all the way).

  9. Is it that dy/dx = 2*sqrt(1-t^2)/t^2, which means that it is undefined (unbounded?) when t=0, which means that the integral is improper (despite what appears to be an attempt to avoid that very thing)?

    1. Nice try, Damo. But dy/dx is not needed for calculating the distance, only dx/dt and dy/dt (which are both defined over the interval of integration). dy/dx simply gives information about the gradient of the path and says that the gradient is undefined at (0, 0). This does not constitute a technical infelicity requiring methods beyond the scope of the course.

      (My money’s still on the boot studs being 1mm longer than the rules allow).

      1. Well, commenters here can eventually decide whether they think the flaw is analogous to boot stud length. But I can assure you that mathematicians think it more akin to Ted Whitten using a flick pass.

  10. Probably my final suggestion:

    Since distances are given as measured in metres, specifying that d is in metres is redundant. In fact, the introduction of d itself is redundant. Taking this and many earlier comments into account, the question could be better worded as:

    “The position vector ….. ” The distance travelled by the particle in the first three-quarters of a second is given by int….. where a, b, c e Z. Find the values of a, b and c.

    Or even better:

    A curve is defined parametrically by x = … and y = … where 0 <= t <= 3/4. Find the length of the curve.

    Mathematicians value elegance, rigour, beauty …. So I can well imagine mathematicians immediately considering this question ugly. Teachers on the other hand see this ugliness year after year after year. It's viewed with such jaded eyes that the ugliness no longer even consciously registers. Just like living next to a train line, there comes a time when you just don't hear the passing trains anymore …..

    So here's my final suggestion:

    The main flaw is the that the question is coyote ugly (and, unfortunately, you can't chew your arm off to get away from VCAA )

    1. JF, you’re right again: the “metres” is redundant. And, you’re right, that, overall, the question is appallingly ugly (and clunky). Do teachers become blind to such ugliness? I don’t know. Commenters on this blog seem to pick up the ugliness quickly.

      1. Speaking for myself, I’m not so much blind as just weary and resigned …. I might have a fleeting thought about it and then shrug my shoulders, let it pass as par for the course and forget about it. When you have muppets who refuse point-blank to acknowledge the existence of serious mathematical errors such as functions given as probability density functions that are no such thing (eg. Maths Methods 2016 Q3(h)), what hope do you have when it’s just simple ugliness because they’re a bunch of tryhards ….? It just goes without saying.

        My theory is that most commenters on your blog generally pick up ugliness quickly because they’re explicitly being invited to comment on witCH. But in this case, I think explicitly acknowledging ugliness is over-shadowed by the desire to find mathematical errors (a desire no doubt motivated by the ugliness). I know that’s the case with myself.

        1. Yeah, I guess. I was tutoring a Methods student today, going through a practice SAC. The SAC contained a question that was so poorly worded that it was effectively impossible (and the teacher’s suggested solution was wrong), and another question that was blatantly wrong. But beyond that, the whole fucking SAC was meaningless, klutzy, pointless, ugly as sin pseudo-modelling. Just complete and utter trash from beginning to end. So, which was worse: the two isolated errors, or the endemic awfulness?

          And I’d suggest the same with the exam question above. The question is clumsy, and clumsily worded, to the point of awfulness. But, as well, the question contains a (for mathematicians) blatant error. Which is worse? Which should teachers be more alert to? Tough call.

          1. That’s what happens when the instructions you’re following mandate klutzy, pointless, ugly as sin pseudo-modelling. And you have to write a different three every year under those ridiculous guidelines.

            Errors are worse than ugliness. Then again, if it’s ugly enough it will be unintelligible, in which case it doesn’t matter whether there are errors or not.

            And the problem is that SACs rarely get vetted competently, if at all. There’s simply no time, everyone assumes that the other guy(s) vetted it …. The teacher is simply the (time)poor sap who had to write it.

  11. Does the error you refer to exist in a re-worded version?:

    A curve is defined parametrically by x = … and y = … where 0 <= t <= 3/4. Find the length of the curve.

  12. OK, I haven’t responded to this post for a while because I feel that others in the discussion are far superior in their Mathematical knowledge, but I will offer just one suggestion: is there a problem with how “distance” or “length” is interpreted? A simple definition of distance would be the Euclidean distance from the start and finish points, done using a simple right-angled triangle.

    If not… any hints?

    1. Hi RF, no that’s not it. There is an interesting question there, when you generalise from straight-line distance to path-length, but that’s not an issue here.

      As for a hint, maybe later this week. I’m flat out preparing a talk. After that, if no one has gotten it by then.

  13. So, using the arc length formula from the VCAA formula sheet, the positive root is implied. But what then happens when (2-t^2)^2 and (t^2-2)^2 both have the same square? Is it possible that some students, factorised 4-4t^2+t^4 as (t^2-2)^2 and then when taking the square root wrote a=1, b=0, c=-2?

    Yes t^2-2 is negative for the domain given, but students have been caught by VCAA on this before.

    Back to the wine…

    1. Thanks, RF. There’s no choice over the square root, since we’re after a distance. But it’s natural to factorise as you indicate, forget the domain of t, and miss the answer by a minus sign. Intended or otherwise, it’s a trick in the question, and I imagine it’s why so few students scored more than 3/5.

    2. Well, Specialist Maths students should certainly know that Sqrt[a^2] = |a|, particularly since the modulus function is on the course (for more reasons I assume than just to solve integrals of the form int[1/u, u]).

      Sure, it’s a ‘trick’ in the question, but only if the falsehood Sqrt[a^2] = a has been ‘encouraged’ throughout the year.

      1. JF, by labelling it a trick I wasn’t meaning to suggest it was unfair, just that many students would have been tricked.

        1. Unfortunately, I think the reason many of them would have been tricked is that they might not have met Sqrt[a^2] = |a|. This is something that has to be explicitly taught and then reinforced (within broader examples) throughout the year and therefore requires a certain level of experience on the part of the teacher. And as we know, there is a steady exodus of experience ….

          1. Thanks, JF. I agree that √(a2) = |a| must be taught, and I’m sure it is not taught sufficiently. It is of course natural to think of squaring and roots as inverses, and it takes training to treat that semi-truth with proper respect. But also in the exam question above, the issue is disguised by the complicated square.

            I think the question in that respect is fair and good, but I wouldn’t bash a student (or teacher) for making the error. On the other hand, since 0% of students (approximately, and maybe exactly) scored 4/5 on the question, what does that tell you about the grading?

  14. OK, here’s a small hint, for those who still care to look. At one point SRK writes something that is seemingly obvious but is false.

    1. This might be asking for a bit too much hand-holding, but of all the false things I have said in this thread, I think the most “seemingly obvious”-looking is the assertion that any definite integral from x to x + 0.75 gives the distance travelled in three quarters of a second. So I’m going to take a punt that digging into why this is false is going to reveal the flaw with the question….

  15. OK, random stab (haven’t checked whether this is true or not…) the distance travelled by the particle is different for different 0.75 second time intervals…(?)

    Or have I, yet again, missed the point?

    1. Hi RF, it’s not the missing error. But you are correct, that the distance travelled in 0.75 seconds depends upon the time interval, which means that the wording of the exam question is stuffed. SRK pointed this out, correctly, by suggesting what the exam question was literally asking for.

  16. My other guess was going to be to do with displacement (change in position) as opposed to distance (total path length), but as with my previous comment, I haven’t had a chance to investigate further as yet.

  17. When you try to solve this problem by evaluating the square root of the sum of squares of the derivatives for x(t) and y(t), you actually end up with two possibilities for the integrand: 2 – t^2 and t^2 – 2, To determine which one is true you need to do a bit more work. Also, there’s the curious a, b, c, in Z statement at the end, which would seem to imply that there are multiple solutions, of which students are only required to produce those which are integer valued. It’s certainly a stupid question, and stating it in a pseudo-physical setting makes it absurd. No wonder students are turned off studying maths. Of course arc length questions tend to be a bit silly, as there are only relatively few simple curves for which the arc length can be given as a closed form of elementary functions. For your enjoyment, check out this fine piece of …. whatever it is: http://www.ijsrp.org/research-paper-0813/ijsrp-p20103.pdf

    1. Re: The “fine piece” ….. What a load of Farooquen BS. Two minutes of my life I won’t get back.

      Re: The curious a,b,c. Your observation boils down to saying that the actual value of the arclength could be given by the integral of any quadratic with appropriate coefficients, and the question requires the specific quadratic with integer coefficients. I don’t have a problem with this, because the integer coefficients emerge naturally from the calculation.

      But I don’t see the point in asking only for the integral when they went to all the trouble of defining a curve whose arclength integral is tractable (‘by-hand’). They could have used any curve, regardless of whether or not the arc length can be given as a closed form of elementary function within the scope of the course. So it’s a bit of a waste. Unless ….

      As for the two possibilities for the integrand, I have no problem with this. It’s a reasonable way of sneaking in the modulus function and will have the effect of forcing sqrt[a^2] = |a| into the classroom next year. But I wonder if explicit recognition of this was valued, or whether students could simply conjure the correct choice out of thin air …. Perhaps only the integral was required because then students wouldn’t calculate an answer, discover it was negative, and then switch to the *other* possibility as a result ….

      Clearly we all agree with “… stating it in a pseudo-physical setting makes it absurd.” I just don’t understand this stupid push to dress up everything within some dopey ill-fitting context. It’s like forcing everyone to wear a suit and tie at a Bar-B-Q. What is so wrong about having maths questions that stand alone as maths questions!? At least there wasn’t a bee walking along the curve ….

      1. “But I wonder if explicit recognition of this was valued, or whether students could simply conjure the correct choice out of thin air …. ”

        John, at the “Meet the Assessors” run by the MAV earlier this year, I recall the presenters saying that most students who proceeded to a correct answer did not include the line sqrt((t^2 – 2)^2) = |t^2 –2|, but rather went directly from the square root to 2 – t^2.

        I wonder if this is what lies behind the 0% of students who received 4 out of 5 marks… Perhaps students who recognised how the domain mattered didn’t bother using the modulus function; students who didn’t recognise how the domain mattered didn’t include a step which could have gained them an additional mark.

  18. That’s a very interesting comment, SRK. I’m sure you’re right.

    It’s not saying much that

    “… most students who proceeded to a correct answer did not include the line sqrt((t^2 – 2)^2) = |t^2 –2|, but rather went directly from the square root to 2 – t^2”

    because only 2% of the state got 5/5 (2% of ~4500 = 90 ….), and they were probably among the top hundred or so in the state …. So 90 students in the whole state understood that the integrand depended on the domain, and probably only a couple of dozen of them explicitly stated this ….

  19. I’ll write more later, but in brief the final issue is what amca01 noted, that there are potentially (and actually) multiple solutions to the problem. That is, the values of a, b and c given in the Examiners’ Report are not the only valid solution.

    So, the question is, why are mathematicians so alert to and bothered by this? Or, if you like, why are non-mathematicians so unbothered?

    1. So a = –1, b = 3/2 and c = 23/16 gives a definite integral with the same value. I’m not yet seeing why I should be bothered by this (at least in the context of this question); I’m more bothered by the fact that I didn’t notice this possibility sooner.

  20. I’m unbothered that there may be no unique solution because the solution VCAA is looking for emerges naturally from the algebra. I don’t see how another solution could be reasonably found …. I haven’t checked, but are there other solutions with integer values of a, b, c that give the correct numerical answer? And how would you come across or justify them in a natural way within the context of the obvious algebra arising from the arclength formula?

    My only concern would be if a student did find – somehow – integer values different to those that emerge naturally from the calculation that give the correct value and was not given full marks.

    Not wantnig to sound churlish, but there is an obvious method leading to an obvious answer. Which is what teachers and students would be focused on. Other possible answers are not on the radar in this context. How would you possibly justify them – you couldn’t until you found the obvious answer, did the integration and then demonstrated that the other answers gave the same value …. In which case, why would you even bother doing this in an exam?

  21. Thanks, SRK and JF. I think the teacher-mathematician disconnect on this aspect of the question is fascinating, and worth a separate and general post. But let me ask it this way.

    The question is undeniably wrong, in the sense that the full answer is much more complicated than indicated in the Report. (Yes, there are other integer solutions.) So, first, just in and of itself, does that not bother you? Secondly and more specifically, what is the source of that wrongness? Then, even if the wrongness per se was insufficient to bother you, does the origin of this specific wrongness not bother you?

    1. Here is an attempt to describe the wrongness in the question. The question instructs students to “find a, b, c, where a, b, c are integers”. The question does not – as far as I can tell – state that these values are unique. But perhaps more seriously it does not state that students are only required to find *one* set of integer values for a, b, c that gives the arc length. So perhaps the answer to the question (as it is written) requires students to find all possible integer values for a, b, c. In which case, the full answer is definitely much more complicated than the examiners report indicates.

      1. The problem is all gummed up in requiring the integrand only for the specific time value [0, 3/4]. Why they chose 3/4 is a mystery; they could have asked for an integral expression for the time interval [0, k] with k <= 1, which would in many ways have been more natural. And the answer (with k instead of 3/4) would be correct. But they are asking for a, b and c for a particular value of d, in fact d = 87/64. If you expand the integral expression for this d, you get the equation 48c + 18b + 9a = 87, which of course has infinite solutions. The problem thus only works if you expect students to blindly follow a set of mathematical rules without worrying too much – or at all – what they mean. In this sense it's barely a step up from rote learning, and in many ways this problem neatly encapsulates what's wrong with school mathematics.

        P.S. Does this site support LaTeX in its comments?

        1. Thanks, amca01. I’ll let others comment. Hadn’t thought of latex in the comments, but I believe it should work, just by chucking code between dollar signs. Hmmm, but no. I’ll investigate.

    2. Well, it bothers me much less than a lot of other things (like improper integrals, incorrect units, functions claimed to be pdf’s that don’t integrate to 1 etc.) I’m much less bothered mainly because if it’s approached using the arclength formula, there is one inevitable answer arising from the one inexorable integrand of 2 – t^2. Yes, it’s true that

      “The problem thus only works if you expect students to blindly follow a set of mathematical rules without worrying too much – or at all – what they mean. In this sense it’s barely a step up from rote learning”.

      However, the question does test a variety other skills such as differentiation, algebraic manipulation and simplification etc. so I’d argue it’s more than just barely a step above rote learning.

      Would amca01 have students derive the arclength formula before using it? If not, then pretty much every arclength question is defective by his/her reasoning. The Arclength formula IS on the course (the argument of whether it should be on the course is a completely different debate). Given this fact of life, it cannot be ignored and must be tested. The question therefore works in the only way that it can work. To worry about the existence of other integer solutions, whilst mathematically worthwhile, is pragmatically moot.

      OK, there are an infinite number integer answers (one such example is a = 3, b = 22, c = -7). Maybe some students thought that finding the infinite number of possible of answers was required and attempted to find them. I strongly doubt it. Why would they? *Why would anyone in this context?* (Would a mathematican, *in this context*).

      And note: 48c + 18b + 9a = 87 (which establishes the existence of an infinite number integer answers) only arises as a consequence of rote use of the arclength formula to get 87/64 in the first place. But once you’ve used it and got the integral that gives this value, why on earth would you keep going, get the value and then look for other possible answers? No offence, but that’s crazy. Every student is taught (or should be taught) never to over-engage with an exam question.

      This error is way down the scale of what I would consider a major error. No offence, but I’d call it a hair-splitting error. In fact, at the end of the day, the only error (in this context) is one of the following:

      1. VCAA asking for the form of the integrand rather than a numerical value of the arclength, or

      2. VCAA not specifying a, b and c in such a way that their values are uniquely a = -1, b = 0 and c = 2. (Mathematicans might go down the infinite rabbit hole, but students and teachers live in a more pragmatic world – a world where one obvious answer is assumed to be what is required. I’ll bet dollars to doughnuts that VCAA never intended “Find [all possible values of] a, b, c, where a,b,c eZ”).

      I’ve said it before, I’ll say it again: VCAA should have asked for the value. Trying to be cute usually has unintended and ill-considered bad consequences. This is a good example.

      So in short, I’m not bothered by this error. Why?

      1. It doesn’t require techniques beyond the scope of the course (like perpetrating improper integrals).

      2. It’s not plain wrong (that is, it’s not in the same ball park as wrong units, or pdf’s that don’t integrate to 1).

      3. The error is trivial (particularly trivial compared to the multitude of other errors the question has) and does not lead to incorrect mathematics.

      4. The other possible answers can only be found once the obvious answer is found. And since the obvious answer answers the question, why would you keep going when the clock is ticking?

      5. I doubt any student will have spent any time trying to find other answers, so it does not impact on student performance.

      Two last things:

      “It’s raining in the window. You’d better do something.”

      I wonder how a mathematican would have answered not just this question, but the whole exam under timed examination conditions. Because the answer to this question has to be taken in the broader context to answering the whole exam. I wonder whether the error would still be considered so serious.

      1. JF, I’ll try to stay out of the argument, at least for now. I want know what people think. But your closing remark 2 is simply incorrect. The exam question is flat out wrong, in the manner that has been indicated. One might argue for the triviality of that wrongness, as you have. But the question is plain, unarguably, undeniably wrong.

      2. To put it more succintly:

        To get all the answers you first need to get the obvious answer, after which you’ve answered the question, so why look for more answers?

        Within the context of Specialist Maths it is more than reasonable to assume that only the obvious answer is required. In which case the only error is one of interpretation, I don’t see a mathematical error.

        So the difference between the mathematician and the teacher is that the mathematician is not looking at the question within the context of the Specialist Maths course, whereas the teacher is.

  22. JF, here is what I am starting to understand as the problem, and how it might arise without already knowing that d = 87/64.

    Integrating at^2 + bt + c between t = 0 and t = 3/4 gives 64d = 9a + 18b + 48c. If this equation has at least one set of integer solutions for a, b, c, d (and we are told by the question that it does), then it has infinitely many.

    As to the impact upon student performance: I share your view that, most likely, very few (if any) students proceeded this way or suspected that there may be multiple solutions. But that’s speculation, and in high stakes exams VCAA shouldn’t be betting on students not getting tripped up by a flaw in a question because they’re using a formula on auto-pilot.

    1. Hmmm … Looking at it from outside the scope of Specialist Maths, I agree. But is it within the scope of the course to know that

      “If [e = 9a + 18b + 48c] has at least one set of integer solutions for a, b, c, [e] …. then it has infinitely many.”

      (I think you mean e not d, where e = 87).

      And even if you do know this, the value of e is required before any headway can be made on answering the question and finding the integer values of a, b and c. In which case, the method you propose is a dead-end for actually answering the question.

      I see no way of avoiding explicit use of the arclength formula to get the obvious answer, after which you’ve answered the question, so why look for more answers? (I don’t think any reasonable person with experience with VCAA and the Specialist Maths course could possible think that more than the obvious answer is required. Not withstanding the ambiguous wording).

      1. JF, how does your “within the context of the course” defence of this question sit with (I assume) your contempt for the f = f-inverse questions that Methods loves?

        1. You’ve just hit the big red button, Marty. But I’ll be brief:

          The ‘f = f-inverse questions’ Methods loves to ask in Exam 1 is in an error league way above the ‘values of a,b,c’, mainly because the method of solution consistently given by VCAA is incomplete at best and defective at worst. Either way, students and teachers learn incorrect mathematics from VCAA as a consequence. The solution consistently given by VCAA is a total disgrace and should be condemned by everybody.

          What are the consequences of the ‘values of a,b,c’ error …? (I acknowledge a possible error exists, but it mainly centres around interpretation. As I’ve said, to get the infinite set of values you first have to get the obvious set of values, at which point the question is answered, depending on your interpretation [which in turn will be strongly influenced by ones experience]).

          So, making the comparison, Sinead OÇonnor puts it nicely: “Nothing compares to U ….”

          (And in answer to your earlier question, I’d argue that mathematicians saw an error where teachers didn’t simply because mathematicians are completely unencumbered by experiences with VCAA and the Study Design. Nothing fascinating here, simply an indictment of what teaching the VCE does to people).

          1. Thanks, JF. For me, and I presume for amca01, the equivalent awfulness is that the actual question being asked is not the question the VCA *thought* they asked, is not the question that anybody answered, and that this disconnect is either never noticed or never considered sufficiently important to be of concern. Moreover, in both cases, the this-question or that-question distinction is not technical, but stark and large and fundamental.

            In particular, in the distance question here, the VCAA is under the impression that they are asking for a specific function (i.e the coefficients of the quadratic that gives the speed). But that’s not what they asked: they actually asked for the solutions of a Diophantine equation. That’s a hell of a difference.

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