This PoSWW (as is the accompanying WitCH) is from Cambridge’s Mathematical Methods Units 1 and 2. and is courtesy of the Evil Mathologer. (A reminder, we continue to post on Cambridge not because their texts are worse than others, but simply because their badness is what we get to see. We welcome all emails with any suggestions for PoSWWs or WitCHes.)
We will update this PoSWW, below, after people have had a chance to comment.
Similar to Witch 6, the above proof is self-indulgent crap, and obviously so. It is obviously not intended to be read by anyone.
One can argue how much detail should be given in such a proof, particularly in a subject and for a curriculum that systemically trashes the concept of proof. But it is difficult to see why the diagram below, coupled with the obvious equations and an easy word, wouldn’t suffice.
With John the Impatient’s permission, I’ve removed John’s comments for now, to create a clean slate. It’s up for other readers to do the work here, and (the royal) we are prepared to wait (as is the continuing case for WitCh 2 and Witch 3).
This WitCH is probably difficult for a Specialist teacher (and much more so for other teachers). But it is also important: the instruction and the example, and the subsequent exercises, are deeply flawed. (If anybody can confirm that exercise 6G 17(f) exists in a current electronic or hard copy version, please indicate so in the comments.)
It is obviously long, long past time to sort out this godawful mess. We apologise to all those industrious commenters, who nailed the essential wrongness, and whose hard work was left hanging. This will also be a very long update; you should pour yourself a stiff drink, grab the bottle and get comfy. For the benefit of the Twitter addicts, however, who now find it difficult to concentrate for more than two paragraphs, here’s the punchline:
Cambridge‘s notion of “non-vertical asymptote” is so vague, falling so short of a proper definition, it is close to meaningless and it is pointless. This leads to the claim in Example 31 being flawed in three distinct ways. In particular, it follows from Cambridge that the “curve” is a “non-vertical asymptote” to the function .
The source of Cambridge‘s confusion is also easy to state:
More complicated asymptotes cannot be defined, interpreted or computed in the manner possible for simpler asymptotes.
Unfortunately, the discussion in Cambridge is so sparse and so far from coherent that directly critiquing the excerpt above would probably be incomprehensible. So, we’ll first try to make very careful sense of “non-vertical asymptotes”, taking some minor whacks at Cambridge along the way. Then, with that sense as foundation, it will be easy work to hammer the excerpt above. To simplify the discussion, we’ll only consider the asymptotes of a function as . Obviously, the case where can be handled in like manner, and vertical asymptotes are pretty straightforward.
OK, take a swig and let’s go.
1. Horizontal Asymptotes
We’ll begin with the familiar and demon-free case, for asymptotes as horizontal lines. Consider, for example, the function
We can write , noting the second term is tiny when huge. So, we declare that has the horizontal asymptote .
Formally, the asymptote in this example is captured with limits. The underlying functional behaviour is or, more officially, . The limit formalism is of no benefit here, however, and is merely likely to confuse. The informal manner of thinking about and writing limits, as is standard in schools, suffices for the understanding of and computation of horizontal asymptotes.
Note also that horizontal asymptotes can easily be spotted with essentially no calculation. Consider the function
which is Example 29 in Section 6G of Cambridge. Here, it is only the highest powers, the terms in the numerator and denominator, that matter; the s cancel, and so the function has horizontal asymptote . This simplification also applies to suitable rooty (algebraic) functions. The function
for example, also has horizontal asymptote , for the same reason.
2. Linear Asymptotes
We’ll now consider general linear asymptotes, which are still reasonably straight-forward (and which include horizontal asymptotes as a special case). There are, however, two demons lurking, one conceptual and one computational.
Cambridge begins with the example
The function can be rewritten as , and it seems simple enough: a similar “when is huge, the leftovers are tiny” intuition leads us to declare that has the linear asymptote . The trouble is that the example, and the discussion that follows, are too simple, so that the demons remain hidden.
3. A Conceptual Demon
How do we formally (or semi-formally) express that a function has a linear asymptote? For the example above, Cambridge writes “f(x) will approach the line y = 8x – 3”. This is natural, intuitive and sufficiently clear. It is considering the asymptote to be a geometric object, as the line, which is fine at the school level.
What, however, if we want to think of the asymptote as the function ? The problem is that both and are zooming off to infinity, which means that writing , or anything similar, is essentially meaningless. Those arrows are shorthand for limits and, fundamentally, the limit of a function must be a number, not another function. (For vertical asymptotes we consider infinity to be an honorary number, which may seem dodgy, but which can be justified.)
The obvious way around this problem is simply to ignore it. As long as we stick to asymptotes of reasonably simple functions – rational functions and carefully chosen others – Cambridge’s intuitive approach is fine. But in the, um, unlikely event that our situation is not so simple, then we have to carefully consider the limiting behaviour of functions. So, we’ll continue.
Suppose that the function appears to have the linear function as asymptote. To capture this idea, the simple trick is to consider the difference of the two functions. If is huge then this difference should be tiny, and so being the asymptote to is captured by the limit statement
Again, none of this is really necessary for capturing linear asymptotes of simple functions. But it is necessary, at least as underlying guidance, if we wish to consider less simple functions and/or more general asymptotes.
WARNING: It may seem as if our boxed definition of linear asymptote would work just as well for non-linear, but there is a trap. There is another, third demon to deal with. Before that, however, our second demon.
4. A Computational Demon
It is very import to understand that the simple, “highest powers” trick we indicated above for spotting horizontal asymptotes does not work for general linear asymptotes. Cambridge‘s introductory example illustrates the issue, albeit poorly and with no subsequent examples. For a better illustration, we have Exercise 12 in Section 6G, which presents us with the function
Here, it would be invalid to divide the by the and then declare the linear asymptote to be . The problem is the in the denominator has an effect, and we are forced to perform long division or something similar to determine that effect. So, , and then it is clear that the linear asymptote is .
All that is fine, as far as it goes. Cambridge routinely performs long division to correctly determine linear asymptotes (including horizontal asymptotes, for which the simpler highest powers trick would have sufficed). What, however, if the function is not rational? Consider, for example, Example 31 above. Can we safely ignore the in the denominator of the function, as Cambridge has done? And, if so, why?
In order to stay in the linear world for now, we’ll leave Example 31 and instead consider two other examples:
So, are we permitted to ignore the and the in the denominators of and ? It turns out that the answers are “Yes” and “No”: the function has linear asymptote , but has linear asymptote .
The behaviour of and is, of course, anything but obvious. In particular, the “highest powers” and long division tricks are of no assistance here. Moreover, similar difficulties arise with the analysis of non-horizontal asymptotes of pretty much all rooty functions. It is essentially impossible for a Victorian school text to cover non-horizontal asymptotes of non-rational functions. The text must either work very hard, or it must cheat very hard; Cambridge takes the latter approach.
5. A Nonlinear Demon
Almost there. We have non-linear “asymptotes” left to consider, which are fundamentally demonic. This is demonstrated by Cambridge‘s very first example, Example 26, which considers the function
Dividing through by gives , after which Cambridge baldly declares “The non-vertical asymptote has equation “. But why, exactly? All we have to go on is Cambridge‘s intuitive approach to above. Intuitively, the function approaches the parabola . That’s fine, and we can formalise it by setting . Then guided by our boxed definition above, we can write , and we seem to have our asymptote.
The problem is that there are a zillion functions that will fit into that blue box. It is also true, for example, that . This means that itself just as good an asymptote as . So, yes, one can reasonably declare that is asymptotic to , but we cannot declare to be THE asymptote to .
Of course we want asymptotes to be unique, so what do we do? There are two ways out of this mess, the first solution being to restrict the type of function that we’ll permit to be an asymptote. That’s intrinsically what we did when considering horizontal and linear asymptotes, and it is exactly why the multiple asymptote problem didn’t arise in those contexts: linear functions only had to compete with other linear functions. Now, for general rational functions, we must broaden the notion of asymptote, but it’s critical to not overdo it: we now permit polynomial asymptotes, but nothing more general. Then, every rational function will have a unique polynomial (possibly linear, possibly horizontal) asymptote.
But now, what do we do for non-rational functions? These functions may naturally have rooty asymptotes, as illustrated by Example 31 above. Such functions can also be dealt with, by suitably and carefully generalising “polynomial asymptote”, but it’s all starting to get fussy and annoying. It is making a second solution much more attractive.
That second way out of this mess is to just give up on unique nonlinear “asymptotes”. Then, our blue box becomes a definition of two functions being “asymptotically equivalent”. We don’t get unique asymptotes, but we do get a clear way to think about the asymptotic behaviour of functions. (You can’t always get what you want, But if you try sometimes, you just might find…)
One final comment. The computational demon we mentioned above is of course still around to bedevil non-linear asymptotes. This is well illustrated by Exercise 17, which considers the function
As discussed by commenters below, this exercise used to have a part (f), asking for the vertical asymptote. That part was deleted in later editions, and it is easy to guess why; the in the denominator of turns out to matter, so that the “highest powers” technique that Cambridge invalidly employs on such examples gives the wrong answer.
6. The Devil in Cambridge‘s detail
Finally. We can now deal with the Cambridge excerpt above. There is no need to write at length here, since everything follows from the discussion above, as indicated. We begin with a clarification of the punchline:
The claim in Example 31 is triply flawed: it is meaningless (conceptual demon); the calculation is invalid (computational demon); and the conclusion is wrong (nonlinear demon).
To be fair, it turns out that the -1 in the denominator of the function in Example 31 can be ignored, but not because of the highest powers trick that Cambridge appears to employ. This also means y = √x will be the non-linear asymptote to y = (x + 1)/√(x-1) if we suitably generalise the notion of “polynomial asymptote”.
That’s plenty wrong, of course. But to round off, here are the other problems with the excerpt:
The “graph of y = f(x)” doesn’t approach anything. It is the function that approaches the asymptote.
Underlying the punchline, the definition of “non-vertical asymptote” is hopelessly vague and does not remotely mean what Cambridge thinks it means.
We do not “require √(x – 1) > 0; we require x – 1 > 0.
Our new WitCH, below, comes courtesy of Charlie the Enforcer. Once again, this WitCH is from the 2018 SCSA Mathematical Methods Exam (here and here): it’s the gift that keeps on giving. (And a reminder, WitCH 2 and WitCH 3 still require attention are still unresolved.)
John has pretty much caught it all. The killer issue is the use of the term “deceleration” in part (c) which, the solution implies, refers to the drone speeding up in the southerly direction. This is arguably permissible, since deceleration can be (though is far from universally) defined as a negative acceleration, and since way back in part (a) it was implied that North coincides with the positive x direction.
Permissible acts, however, can nonetheless be idiotic: voting Liberal or Republican, for example. And, to use “deceleration” on a high stakes exam to refer implicitly to increasing speed is idiotic. Moreover, to use “deceleration” in this manner immediately after explicitly indicating the “due south” direction of motion is truly ruly idiotic. Still not as idiotic as voting Liberal or Republican, but genuinely special-effort idiotic.
That’s enough to condemn the question, even by SCSA standards. But, the question is also awful in many other ways:
The question is boring and butt ugly.
No indication is given whether exact or numerical solutions are permitted or required.
Having a drone an arbitrary 5m up in the sky for a 1D problem is asking for trouble. For example:
The “displacement” of x(0) = 0 for a drone 5m up is pretty stupid.
“Where is the drone in relation to the [mysterious] pilot?” Um, kind of uppish?
“How far has the drone travelled …” is needlessly wordy and ambiguous. If you want a distance, for God’s sake say “distance”.
Given the position function x(t) is at hand, part (c) can easily and naturally be solved by hand. But of course why think about things when you can do mindless calculator crap?
Tanya Plibersek, Australian Labor’s Shadow Minister for Education, has just been reaching out to the media. Plibersek has objected to the low ATAR sufficient for school leavers to gain entry to a teaching degree, and she has threatened that if universities don’t raise the entry standards then Labor may impose a cap on student numbers:
We [should] choose our teaching students from amongst the top 30 per cent …
This raises the obvious question: why the top 30 per cent of students? Why not the top 10 per cent? Or the top 1 per cent? If you’re going to dream an impossible dream, you may as well make it a really good one.
Plibersek is angry at the universities, claiming they are over-enrolling and dumbing down their teaching degrees, and of course she is correct. Universities don’t give a damn whether their students learn anything or whether the students have any hope of getting a job at the end, because for decades the Australian government has paid universities to not give a damn. The universities would enrol carrots if they could figure out a way for the carrots to fill in the paperwork.
The corruption of university teaching enrolment, however, has almost nothing to do with the poor quality of school teachers and school teaching. The true culprits are the neoliberal thugs and the left wing loons who, over decades, have destroyed the very notion of education and thus have reduced teaching to a meaningless, hateful and hated profession, so that with rare exceptions the only people who become teachers are those with either little choice or little sense or a masochistically high devotion to civic duty.
If Plibersek wants “teaching to be as well-respected as medicine” then perhaps Labor could stick their neck out and fight for a decent increase in teachers’ wages. Labor could fight for the proper academic control of educational disciplines so that there might be a coherent and deep Australian curriculum for teachers to teach. Labor could fight against teachers’ Sisyphean reporting requirements and against the swamping over-administration of public schools. Labor could promise to stop, entirely, the insane funding of poisonously wealthy private schools. Labor could admit that for decades they have been led by soulless beancounters and clueless education hacks, so as much as anyone they have lost sight of what education is and how a government can demand it.
But no. Plibersek and Labor choose an easy battle, and a stupid, pointless battle.
None of this is to imply that Labor’s opponents are better. Nothing could be worse for education, or anything, than the sadistic, truth-killing Liberal-National psychopaths currently in power.
But we expect better from Labor. Well, no we don’t. But once upon a time we did.
Tanya Plibersek has announced a new Labor policy, to offer $40,000 grants for “the best and the brightest” to do teaching degrees, and to go on to teach in public schools. Of course Plibersek’s suggestion that this will attract school duxes and university medal winners into teaching is pure fantasy, but it’s a nanostep in the right direction.
Below, we go through the passage line by line, but that fails to capture the passage’s intrinsic awfulness. The passage is, as John put it pithily below, a total fatberg. The passage is worse than wrong; it is clumsy, pompous, circuitous, barely comprehensible and utterly pointless.
Why do this? Why write like this? Sure, ideas, particularly mathematical ideas, can be tricky and difficult to convey; dependence/independence isn’t particularly easy to explain. And sure, we all write less clearly than we might wish on occasion. But, if you write/proofread/edit something that the intended “readers” will obviously struggle to understand, then all you’re doing is either showing off or engaging in a meaningless ritual.
An underlying problem is that the entire VCE topic is pointless. Yes, this is the fault of the idiotic VCAA, not the text, but it has to be said, if only as a partial defence of the text. No purpose is served by including in the curriculum a subtle definition that is not then reinforced in some meaningful manner. Consequently, it’s close to impossible to cover this aspect of the curriculum in an efficient, clear and motivated manner. The text could have been one hell of a lot better, but it probably never could have been good.
OK, to the details. Grab a whisky and let’s go.
First, a clarification. The definition of “parallel vectors” appears in a slightly earlier part of the text. We included it because it is clearly relevant to the main excerpt. We didn’t intend, however, to suggest that the discussion of dependence began with the “parallel” definition.
For the given definition of “parallel vectors” it is redundant and distracting to specify that the scalar k not be 0.
As discussed by Number 8, the definition of “parallel vectors” should not exclude the zero vector. The exclusion may be natural in the context of geometric proofs, but here it is a needless source of fussiness, distraction and error. As an example of a blatant error, immediately following the above passage the text begins a proposition with “Let a and b be two linearly independent (i.e. not parallel) vectors.” A second and entirely predictable error occurs when the text later goes on to “resolve” an arbitrary vector a into components “parallel” and “perpendicular” to a second vector b.
The definition of “linear combination” involves a clumsy and needless use of subscripts. Thankfully, though weirdly, subscripts aren’t used in the subsequent discussion. The letters used for the vector variables are changed, however, which is the kind of minor but needless, own-goal distraction that shouldn’t occur.
No concrete example of linear combination is provided. (The more abstract the ideas, the more critical it is that they be anchored immediately with very specific illustration.)
It is a bad choice to begin with “linear combination”. That idea is difficult enough, but it also leads to a poor and difficult definition of linear dependence, an unswallowable mouthful: “… at least one of its members [elements? vectors?] can be expressed as a linear combination of [the] other vectors [members? elements?] …” Ugh! What really kills this sentence is the “at least one”, which stems from the asymmetry hiccup in the definition. (The hiccup is illustrated, for example, by the three vectors a = 3 i + 2j + k, b = 9i + 6j + 3k, c = 2i + 4j + 3k. These vectors are dependent, since b = 3a + 0c is a combination of a and c. Note, however, that c cannot be written as a combination of a and b.)
As was appropriately done for “linear combination”, the definition of linear dependence should be framed in terms of two or three vectors staring at the reader, not for “a set of vectors”.
The language of sets is obscure and unnecessary.
No concrete example of linear dependence is provided. There is not even the specialisation to the case of two and/or three vectors (which, again, is how they should have begun).
If you’re going to begin with “linear combination” then don’t. But, if you are, then the definition of linear independence should precede linear dependence, since linear independence doesn’t have the asymmetry hiccup: no vector can be written as a combination of the other vectors. Then, “dependent” is defined as not independent.
No concrete example of linear independence is provided.
The properly symmetric “examples” are the much preferred definition(s) of dependence.
The “For example” is weird. It is more accurate to label what follows as special cases. They are not just special cases, however, since they also incorporate non-obvious reworking of the definition of dependence.
No proof or discussion is provided that the “example[s]” are equivalent to the definition.
No genuine example is provided to illustrate the “example[s]”.
The simple identification of two vectors being parallel/non-parallel if and only if they are dependent/independent is destroyed by the exclusion of the zero vector.
There is no indication why any set of vectors including the zero vector must be dependent.
The expression “two-dimensional vector” is lazy and wrong: spaces have dimension, not vectors. (Ditto “three-dimensional vectors”.)
No proof or discussion is provided that any set of three “two dimensional vectors” is dependent. (Ditto “for three-dimensional vectors”.)
The “method” for checking the dependence of three vectors is close to unreadable. They could have begun “Let a and b be linearly independent vectors”. (Or, with the correct definition, “Let a and b be non-parallel vectors”.)
There is no indication of or clarification of or illustration of the subtle distinction between the original “definition” of linear dependence and the subsequent “method”.
The Evil Mathologer is out of town and the Evil Teacher is behind on sending us our summer homework. So, we have time for some thumping and we’ll begin with the Crap Australian Curriculum Competition. (Readers are free to decide whether it’s the curriculum or the competition that is crap.) The competition is simple:
Find the single worst line in the Australian Mathematics Curriculum.
You can choose from either the K-10 Curriculum or the Senior Curriculum, and your line can be from the elaborations or the “general capabilities” or the “cross-curriculum priorities” or the glossary, anywhere. You can also refer to other parts of the Curriculum to indicate the awfulness of your chosen line, as long as the awfulness is specific. (“Worst line” does not equate to “worst aspect”, and of course the many sins of omission cannot be easily addressed.)
The (obviously subjective) “winner” will receive a signed copy of the Dingo book, pictured above. Prizes of the Evil Mathologer’s QED will also be awarded as the judges see fit.
This one’s shooting a smelly fish in a barrel, almost a POSWW. Sometimes, however, it’s easier for a tired blogger to let the readers do the shooting. (For those interested in more substantial fish, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle still require attention.)
Our latest WitCH comes courtesy of two nameless (but maybe not unknown) Western troublemakers. Earlier this year we got stuck into Western Australia’s 2017 Mathematics Applications exam. This year, it’s the SCSA‘s Mathematical Methods exam (not online. Update: now online here and here.) that wins the idiocy prize. The whole exam is predictably awful, but Question 15 is the real winner:
The population of mosquitos, P (in thousands), in an artificial lake in a housing estate is measured at the beginning of the year. The population after t months is given by the function,.
The rate of growth of the population is initially increasing. It then slows to be momentarily stationary in mid-winter (at t = 6), then continues to increase again in the last half of the year.
Determine the values of a and b.
Go to it.
As Number 8 and Steve R hinted at and as Damo nailed, the central idiocy concerns the expression “the rate of population growth”, which means P'(t) and which then makes the problem unsolvable as written. Specifically:
In the second paragraph, “it” has a stationary point of inflection when t = 6, which is impossible if “it” refers to the quadratic P'(t).
On the other hand, if “it” refers to P(t) then solving gives a < 0. That implies P”(0) = 2a < 0, which means “the rate of population growth” (i.e. P’) is initially decreasing, contradicting the first claim of the second paragraph.
The most generous interpretation is that the examiners intended for the population P, not the rate P’, to be initially increasing. Other interpretations are less generous.
No matter the intent, the question is inexcusable. It is also worth noting that even if corrected the question is awful, a trivial inflection problem dressed up with idiotic modelling:
Modelling population growth with a cubic is hilarious.
Months is a pretty stupid unit of time.
The rate of population growth initially increasing is irrelevant.
Why is the lake artificial? Who gives a shit?
Why is the lake in a housing estate? Who gives a shit?
Finally, it’s “latter half” or “second half”, not “last half”. Yes, with all else awful here, it hardly matters. But it’s wrong.
The marking schemes for the exam are now up, here and here. As was predicted, “the rate of growth of the population” was intended to mean “population”. As is predictable, the grading scheme gives no indication that the question is garbled garbage.
The gutless contempt with which certain educational authorities repeatedly treat students and teachers is a wonder to behold.
The VCE maths exams are over for another year. They were mostly uneventful, the familiar concoction of triviality, nonsense and weirdness, with the notable exception of the surprisingly good Methods Exam 1. At least two Specialist questions, however, deserve a specific slap and some discussion. (There may be other questions worth whacking: we never have the stomach to give VCE exams a close read.)
Question 6 on Specialist Exam 1 concerns a particle acted on by a force, and students are asked to
Find the change in momentum in kg ms-2 …
The problem of course is that the suggested units are for force rather than momentum. This is a straight-out error and there’s not much to be said (though see below).
Then there’s Question 3 on part 2 of Specialist Exam 2. This question is concerned with a fountain, with water flowing in from a jet and flowing out at the bottom. The fountaining is distractingly irrelevant, reminiscent of a non-flying bee, but we have larger concerns.
In part (c)(i) of the question students are required to show that the height h of the water in the fountain is governed by the differential equation
The problem is with the final part (f) of the question, where students are asked
How far from the top of the fountain does the water level ultimately stabilise?
The question is typical in its clumsy and opaque wording. One could have asked more simply for the depth h of the water, which would at least have cleared the way for students to consider the true weirdness of the question: what is meant by “ultimately stabilise”?
The examiners are presumably expecting students to set dh/dt = 0, to obtain the constant, equilibrium solution (and then to subtract the equilibrium value from the height of the fountain because why not give students the opportunity to blow half their marks by misreading a convoluted question?) The first problem with that is, as we have pointed out before, equilibria of differential equations appear nowhere in the Specialist curriculum. The second problem is, as we have pointed out before, not all equilibria are stable.
It would be smart and good if the VCAA decided to include equilibrium solutions in the Specialist curriculum, along with some reasonable analysis and application. Until they do, however, questions such as the above are unfair and absurd, made all the more unfair and absurd by the inevitably awful wording.
Now, what to make of these two questions? How much should VCAA be hammered?
We’re not so concerned about the momentum error. It is unfortunate, it would have confused many students and it shouldn’t have happened, but a typo is a typo, without deeper meaning.
It appears that Specialist teachers have been less forgiving, and fair enough: the VCAA examiners are notoriously nitpicky, sanctimonious and unapologetic, so they can hardly complain if the same, with greater justification, is done to them. (We also heard of some second-guessing, some suggestions that the units of “change in momentum” could be or are the same as the units of force. This has to be Stockholm syndrome.)
The fountain question is of much greater concern because it exemplifies systemic issues with the curriculum and the manner in which it is examined. Above all, assessment must be fair and reasonable, which means students and teachers must be clearly told what is examinable and how it may be examined. As it stands, that is simply not the case, for either Specialist or Methods.
Notably, however, we have heard of essentially no complaints from Specialist teachers regarding the fountain question; just one teacher pointed out the issue to us. Undoubtedly there were other teachers bothered by the question, but the relative silence in comparison to the vocal complaints on the momentum typo is stark. And unfortunate.
There is undoubted satisfaction in nitpicking the VCAA in a sauce for the goose manner. But a typo is a typo, and teachers shouldn’t engage in small-time point-scoring any more than VCAA examiners.
The real issue is that the current curriculum is shallow, aimless, clunky, calculator-poisoned, effectively undefined and effectively unexaminable. All of that matters infinitely more than one careless mistake.
The exam Reports are now out, here and here. There’s no stupidity so large or so small that the VCAA won’t remain silent.