The following WitCH is pretty old, but it came up in a tutorial yesterday, so what the Hell. (It’s also a good warm-up for another WitCH, to appear in the next day or so.) It comes from the 2011 Mathematical Methods Exam 1:
For part (a), the Examination Report indicates that f(g)(x) =√([x+2][x+8]), leading to c = 2 and d = 8, or vice versa. The Report indicates that three quarters of students scored 2/2, “However, many [students] did not state a value for c and d”.
For Part (b), the Report indicates that 84% of students scored 0/2. After indicating the intended answer, (-∞,-8) U (-2,∞) (-∞,-8] U [-2,∞) or R(-8,-2), the Report goes on to comment:
“This question was very poorly done. Common incorrect responses included [-3,3] (the domain of f(x); x ≥ -2 (as the ‘intersection’ of x ≥ -8 with x ≥ -2); or x ≥ -8 (as the ‘union’ of x ≥ -8 with x ≥ -2). Those who attempted to use the properties of composite functions tended to get confused. Students needed to look for a domain that would make the square root function work.”
The Report does not indicate how students got “confused”, although the composition of functions is briefly discussed in the Study Design (page 72).
Our second (and last for now) NHT WitCH is due to the ever-vigilant John the Merciless (who shall, to begin, hold his fire …). It comes from the 2019 Exam 1 of Specialist Mathematics (calculator-free):
The examination “report” gives the answers as: (a) (51,65); (b) 0.02, 0.03 accepted.
We’ve finally found some time to take a look at VCAA’s 2019 NHT exams. They’re generally bad in the predictable ways, and they include some specific and seemingly now standard weirdness that we’ll try to address soon in a more systematic manner. WitCHwise, we were tempted by a number of questions, but we’ve decided to keep it to two or three.
Our first NHT WitCH is from the final question on Exam 2 (CAS) of Mathematical Methods:
As usual, the NHT “Report” indicates nothing of how students went, and little of what was expected. In regard to part f, the Report writes,
p(x) = q(x) = x, p'(x) = q'(x) = 1, k = 1/e
For part g, all that the Report provides is the answer, k = 1.
The VCAA also provides sample Mathematica solutions to schools trialling Methods CBE. For the questions above, these solutions are as follows:
It seems it might be worthwhile itemising the outstanding WitCHes, and inviting a general discussion about the WitCHES, and perhaps the blog in general. So, first to the outstanding WitCHes:
WitCH 8 is a jungle, that will presumably not be further unjungled. It’s still open for discussion, but I’ll update soon.
WitCH 10 has turned out to be very interesting. It is done, except for one (in the opinion of at least some mathematicians) major issue. There is now included near the end of the comments an (admittedly cryptic) clue.
WitCH 12 is not a deep one, though there are aspects that really annoy me. The absence of comments suggests others are less bothered (or more resigned). I’ll update soon.
WitCH 18 is a semi-WitCH, and commenters have pretty much highlighted the absurdity of it all. I’d suggest the analysis could be a bit more mathsy, but it’s no big deal, and I’ll update soon.
WitCH 19 has just been posted. It’s not deep, but we’ll see what commenters make of it.
Now, as to the WitCHes in general, what do people think of them? Are they interesting? Are they just nitpicking? Is there any value in them? Which WitCHes are Column A and which are Column B?
Of course I have my own reasons for posting the WitCHes, and for writing this blog generally in the manner I do. But I’m genuinely curious what people think. What is (arguably) interesting here is an (ex)-mathematician’s blunt criticism crashing into teachers’ and students’ reality, notably and unexpectedly highlighted by WitCH 10. But do commenters, and teachers and students in particular, regard this as interesting and/or entertaining and/or helpful, or merely demoralising and/or confusing and/or irritating?
To be clear, I am inviting criticism. It doesn’t mean I’ll agree (or pretend to agree) with such criticism. It doesn’t mean I’ll switch gears. But to the extent that people think this blog gets it wrong, I am willing and keen to hear, and will treat all such criticism with due respect. (I presume and know that this blog actively irritates many people. It seems, however, that these people do not wish to lower themselves to comment here. Fair enough.)
The following WitCH is from VCE Mathematical Methods Exam 2, 2009. (Yeah, it’s a bit old, but the question was raised recently in a tutorial, so it’s obviously not too old.) It is a multiple choice question: The Examiners’ Report indicates that just over half of the students gave the correct answer of B. The Report also gives a brief indication of how the problem was to be approached:
Though undeniably weird and clunky, this question clearly annoys commenters less than me. And, it’s true that I am probably more annoyed by what the question symbolises than the question itself. In any case, the discussion below, and John’s final comment/question in particular, clarified things for me somewhat. So, as a rounding off of the post, here is an extended answer to John’s question.
Underlying my concern with the exam question is the use of “solve” to describe guessing/buttoning the solution to the (transcendental) equation . John then questions whether I would similarly object to the “solving” of a quintic equation that happens to have nice roots. It is a very good question.
First of all, to strengthen John’s point, the same argument can also be made for the school “solving” of cubic and quartic equations. Yes, there are formulae for these (as the Evil Mathologer covered in his latest video), but school students never use these formulae and typically don’t know they exist. So, the existence of these formulae is irrelevant for the issue at hand.
I’m not a fan of polynomial guessing games, but I accept that such games are standard and that“solve” is used to describe such games. Underlying these games, however, are the integer/rational root theorems (which the EM has also covered), which promise that an integer/rational coefficient polynomial has only finitely many candidate roots, and that these roots are easily enumerated. (Yes, these theorems may be a less or more explicit part of the game, but they are there and they affect the game, if only semi-consciously.) By contrast, there is typically no expectation that a transcendental equation will have somehow simple solutions, nor is there typically any method of determining candidate solutions.
I find something generally unnerving about the exam question and, in particular, the Report. It exemplifies a dilution of language which is at least confusing, and I’d suggest is actively destructive. At its weakest, “solve” means “find the solutions to”, and anything is fair game. This usage, however, loses any connotation of “solve” meaning to somehow figure out the way the equation works, to determine why the solutions are what they are. This is a huge loss.
True, the investigation of equations can continue independent of the cheapening of a particular word, but the reality is that it does not. Of course, in this manner the Solve button on CAS is the nuclear bomb that wipes out all intelligent life. The end result is a double-barrelled destruction of the way students are taught to approach an equation. First, students are taught that all that matters about an equation are the solutions.They are trained to give the barest lip service to analysing an equation, to investigating if the equation can be attacked in a meaningful mathematical manner. Secondly, the students are taught that that there is no distinction between a precise solution and an approximation, a bunch of meaningless decimals spat out by a machine.
So, yes, the exam question above can be considered just another poorly constructed question. But the weird and “What the Hell” incorporation of a transcendental equation with an exact solution that students were supposedly meant to “solve” is emblematic of a an impoverishment of language and of mathematics that the CAS-infatuated VCAA has turned into an art form.
We have lots of catching up to do, WitCHes to burn and whatnot. However, we’ll first try to get in a few quick topical posts (give or take a couple weeks …). This first one is half-post, half-WitCH. We had planned it as a post, but it then seemed worth letting readers have a first whack at it; as always, readers are welcome and encouraged to comment below.
Serena Williams was back at Wimbledon this year, for the ninety-fifth time, almost grabbing her eighth title. This phenomenal athlete was also the subject of some media fluff, of the type that always accompanies these events. It was reported, prettymucheverywhere, that
“One in eight men think that they could score a point off Serena Williams”.
The Serena fluff came courtesy of British polling firm YouGov, popular with those comforted by the illusion that someone cares what they think. Specifically, YouGov asked:
“Do you think if you were playing your very best tennis, you could win a point off Serena Williams?”
YouGov announced the result of the poll on Twitter, with catchy headline and accompanying graph:
“One in eight men (12%) say they could win a point in a game of tennis against 23 time grand slam winner Serena Williams”
Note that 3% of women also answered that they could win a point; we could see nothing in the media reports questioning, much less ridiculing, this percentage. (The missing percentages correspond to people who answered “don’t know”.)
On the YouGov website, the poll is also broken down by age and so on, but there is little information on the nature of the polling. All we are told is:
“1732 [Great Britain] adults were questioned on 13 Jul 2019. Results are weighted to be representative of the GB population.”
OK, so now the WitCH aspect. What is wrong with the poll? What is wrong with the reaction to it and the reporting of it? As always, feel free to respond in the comments. (You might try to keep your answers brief, but it won’t be easy.)
Finally, to state explicitly what should be obvious, we are not in any way having a go at Serena Williams. She is a great athlete, and throughout her career she’s had to put up with all manner of sexist and racist garbage. We just don’t believe the YouGov poll is such an example, or at least so clearly so.
Similar to our parallel WitCH, it is difficult to know whether to focus on specific clunkiness or intrinsic absurdity, but we’ll first get the clunkiness out of the way:
John comments that using x and y for angles within the unit circle is irksome. It is more accurately described as idiotic.
The 2π*k is unnecessary and distracting, since the only possible values of k are 0 and -1. Moreover, by symmetry it is sufficientto prove the identity for x > y, and so one can simply assume that x = y + α.
The spacing for the arguments of cos and sin are very strange, making the vector equations difficult to read.
The angle θ is confusing, and is not incorporated in the proof in any meaningful manner.
Having two cases is ugly and confusing and was easily avoidable by an(other) appeal to trig symmetry.
In summary, the proof could have been much more elegant and readable if the writers had bothered to make the effort, and in particular by making the initial assumption that y ≤ x ≤ y + π, relegating other cases to trig symmetry.
Now, to the general absurdity.
It is difficult for a textbook writer (or a teacher) to know what to do about mathematical proofs. Given that the VCAA doesn’t give a shit about proof, the natural temptation is to pay lip service or less to mathematical rigour. Why include a proof that almost no one will read? Commenters on this blog are better placed to answer that question, but our opinion is that there is still a place for such proofs in school texts, even if only for the very few students who will appreciate them.
The marginalisation of proof, however, means that a writer (or teacher) must have a compelling reason for including a proof, and for the manner in which that proof is presented. (This is also true in universities where, all too often, slovenly lecturers presentincomprehensible crap as if it is deep truth.) Which brings us to the above proof. Specialist 34 students should have already seen a proof of the compound angle formulas in Specialist 12, and there are much nicer proofs than that above (see below). So, what is the purpose of the above proof?
As RF notes, the writers are evidently trying to demonstrate the power of the students’ new toy, the dot product. It is a poor choice, however, and the writers in any case have made a mess of the demonstration. Whatever elegance the dot product might have offered has been obliterated by the ham-fisted approach. Cambridge’s proof can do nothing but convince students that “proof” is an incomprehensible and pointless ritual. As such, the inclusion of the proof is worse than having included no proof at all.
This is doubly shameful, since there is no shortage of very nice proofs of the compound angle formulas. Indeed, the proof in Cambridge’s Specialist 12 text, though not that pretty, is standard and is to be preferred. But the Wikipedia proof is much more elegant. And here’s a lovely proof of the formula for sin(A + B) from Roger Nelson’s Proof Without Words:
To make the proof work, just note that
x cos(A) = z = y cos(B)
Now write the area of the big triangle in two different ways, and you’re done. A truly memorable proof. That is, a proof with a purpose.
I guess if you’re gonna suggest a painful, ass-backwards method to solve a problem, you may as well fake the solution:
Checking directly that P(1 – i√2) = 0 involves expanding a cubic, and more, which the text does in one single magic line.
The painful multiplication of the products for part b is much more naturally and easily done as a difference of two squares: (z – 1 – i√2)(z – 1 + i√2) = (z – 1)2 + 2, etc.
After all that the third factor, z – 1, is determined “by inspection”? Inspection of what?
AS RF notes, it is much easier to spot that z = 1 solves the cubic. Then some easy factoring (without long division …) gives P = (z – 1)(z2 – 2z + 3). Completing the square then leads to the linear factors, answering both parts of the question in the reverse, and natural, order.
Alternatively, as John notes, the difference of two squares calculation shows that if z – 1 + i√2 is a factor of P then so is the quadratic z^2 – 2z + 3. That this is so can then be checked (without long division …), giving P = (z – 1)(z^2 – 2z + 3), and so on, as before.