The Arc Enemy

Our previous post was on good guys making a silly, funny and inconsequential mistake. This post is not.

Question B1 of Exam 2 for 2018 Northern Hemisphere Specialist Mathematics begins innocently enough. In part (a), students are required to graph the function \boldsymbol{f(x) = 10\arccos(2-2x)} over its maximal domain. Then, things begin to get stupid.

In part (b), the graph of f is rotated around the y-axis, to model a vase. Students are required to find the volume of this stupid vase, by setting up the integral and then pushing the stupid buttons on their stupid calculators. So, a reasonable integration question lost in ridiculous pseudomodelling and brainless button-pushing. Whatever. Just standard VCE crap. Then, things stay stupid.

Part (c) is a related rates question. In principle a good problem, though it’s hard to imagine anyone ever requiring dh/dt when the water depth is exactly \boldsymbol{5\pi} cm. Whatever. Standard VCE crap. Then, things get really, really stupid.

Part (d) of the problem has a bee climbing from the bottom of the vase to the top. Students are required to find the minimum distance the bee needs to travel.

Where to begin with this idiotic, 1-mark question. Let’s begin with the bee.

Why is it a bee? Why frame a shortest walk question in terms of a bug with wings? Sure, the question states that the bug is climbing, and the slight chance of confusion is overshadowed by other, much greater issues with the question. But still, why would one choose a flying bug to crawl up a vase? It’s not importantly stupid, but it is gratuitously, hilariously stupid.

Anyway, we’re stuck with our stupid bee climbing up our stupid vase. What distance does our stupid bee travel? Well, obviously our stupid, non-flying bee should climb as “up” as possible, without veering left or right, correct?

No and yes.

It is true that a bottom-to-top shortest path (geodesic) on a surface of revolution is a meridian. The proof of this, however, is very far from obvious; good luck explaining it to your students. But of course this is only Specialist Mathematics, so it’s not like we should expect the students to be inquisitive or critical or questioning assumptions or anything like that.

Anyway, our stupid non-flying bee climbs “up” our stupid vase. The distance our stupid bee travels is then the arc length of the graph of the original function f, and the required distance is given by the integral

    \[\boldsymbol{{\Huge \int\limits_{\frac12}^{\frac32}}\sqrt{1+\left[\tfrac{20}{1 - (2-2x)^2}\right]^2}}\ {\bf d}\boldsymbol{x}\]

The integral is ugly. More importantly, the integral is (doubly) improper and thus has no required meaning for Specialist students. Pretty damn stupid, and a stupidity we’ve seen not too long ago. It gets stupider.

Recall that this is a 1-mark question, and it is clearly expected to have the stupid calculator do the work. Great, sort of. The calculator computes integrals that the students are not required to understand but, apart from being utterly meaningless crap, everything is fine. Except, the calculators are really stupid.

Two brands of CAS calculators appear to be standard in VCE. Brand A will readily compute the integral above. Unfortunately, Brand A calculators will also compute improper integrals that don’t exist. Which is stupid. Brand B calculators, on the other hand, will not directly compute improper integrals such as the one above; instead, one first has to de-improper the integral by changing the limits to something like 0.50001 and 1.49999. Which is ugly and stupid. It also requires students to recognise the improperness in the integral, which they are supposedly not required to understand. Which is really stupid. (The lesser known Brand C appears to be less stupid with improper integrals.)

There is a stupid way around this stupidity. The arc length can also be calculated in terms of the inverse function of f, which avoid the improperness and then all is good. All is good, that is, except for the thousands of students who happen to have a Brand B calculator and who naively failed to consider that a crappy, 1-mark button-pushing question might require them to hunt for a Specialist-valid and B-compatible approach.

The idiocy of VCE exams is truly unlimited.

16 Replies to “The Arc Enemy”

  1. The amount of times you use the word ‘stupid’ was unbearable to read. You are a fucking tool – dishonest and bitter going on a ridiculous tirade against an institution you have some deep, emotional problems with.

    Good job in never constructing a salient point in any of your posts.

  2. One of the beauties of Mathematics (and there are many) is that we can disagree on assumptions (or axioms) but once we have agreed on the premises we cannot disagree on the conclusions.

    So I will therefore restrict myself to making the following counter-arguments:

    1. Most of Marty’s posts are not related to one institution, but more the environment(s) which have somehow evolved to allow mathematical crap to become so mainstream that it is either completely missed or assumed to be truth.

    2. Dishonest is a bit of a stretch considering Marty backs up his claims with references. Unlike some other published material.

    Debate is a wonderful thing, but debate the ideas, not the person; to do otherwise makes me think you have no actual ideas and must resort to insults.

  3. ad hominem … Please can the first respondent challenge the argument eg repetition of “stupid” but not the person lest you get mistaken for a trump twitter tirade?

    Steve R

  4. Thank you Number 8 and Steve R for your responses. I wasn’t sure whether to approve mathemagics’ comment, but figured that it’s only fair to permit strongly negative comments in response to my strongly worded posts. I didn’t bother replying myself, since there was evidently nothing of substance to address.

  5. *UPDATE* I finally managed to meet up with someone who is very familiar with Brand B calculators (or is it Brand C? Hardly matters I guess) and asked them about this situation. Without wanting to pass any judgement on VCAA, they made two observations:

    1. Brand A may be dominant in Australia, but internationally, Brand B is much more popular (80% at their estimate)

    2. There is a “vetter” (their words) for the VCAA exams for Brand A, but not for Brand B calculators.

    Make of this whatever you will. I know I did…

  6. So did I. That’s why I asked others. It seems no one can either confirm nor deny this rumour, but many admit to having heard it.

    These rumours have a strange habit of starting from truth in some form or another (although we both know truth seems to be a grey area for some in education circles).

    1. I’d need more than the the rumour being wide-spread to be convinced. Sure, I can understand the “starting from truth” aspect. Here, the underlying and undeniable truth is that the VCAA is arrogantly inept. So, in particular, the VCAA having an inept vetting process is believable, and indeed provable. But, to have a vetting process which doesn’t seek to check Brand B is a major step beyond simple ineptness.

      1. I agree. I also don’t expect anyone associated with VCAA to give the game away quite so easily. To be completely honest though I don’t think enough people would genuinely care enough to go Parisian on VCAA even if it were proven true beyond reasonable doubt. So I move on to more local politics…

  7. Fun fact for anybody doing this as a practice exam, Brand B calculators can easily give this arc length, in decimal, using the inbuilt arcLen function skipping the need for the student to think about the formula and integration entirely. I suspect that it is doing it numerically.

  8. What’s with all the CAS stuff? Did I Rip Van Winkle the last 20 years and the world changed?

    Problem b looks like a reasonable one (use washers or shells to solve) and familiar to me from 1980s AP Calculus. The antiderivative of the arccos is a required integral to know (or derive rapidly). Maybe a little on the hard side, given we also do the darned volume calc but definitely a standard topic. I wouldn’t imagine why a CAS is needed or expected. I think in the US nobody would expect a CAS to be needed. Even now! Maybe you get some radicals who say it should be used, but they haven’t yet won the fight.

    On d, yeah…they are trying to be fancy abut asking for arc length calculation. It concerns me that some knowledge of 3rd semester calculus or even calculus of cariations is needed to deside the path. They should just tell the kids “assume straight up, along the vase” or words to that effect to take away that distracting worry, so they know to reach for the arc length calculation. On the difficulty, been decades since I did math like this so I’ll take you on faith if you say it’s an overly hard integral for a standard test problem. Maybe just pick a simpler curve in that case.

    1. The CAS is huge in Victoria, and thus also infecting other states, primarily because of two powerful ideologues, and because mathematicians stopped paying attention to the perversion of school mathematics.

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