WitCH 26: Imminent Domain

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\backslash(-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).

WitCH 24: The Fix is In

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:

Make of it what you will.

WitCH 23: Speed Bump

Our second WitCH of the day also comes from the 2017 VCE Specialist Mathematics Exam 2. (Clearly an impressive exam, and we haven’t even gotten to the bit about using inverse trig functions to design a brooch.) It is courtesy of the mysterious SRK, who raised it in the discussion of an earlier WitCH.

Question 5 of Section B of the (CAS) exam concerns a boat and a jet ski. Though SRK was concerned with one particular aspect, the entire question is worth pondering:

The  Examiner’s Report indicates an average student score of 1.4 on part a, and comments,

Students plotted the initial positions correctly but significant numbers of students did not label the direction of motion or clearly identify the jet ski and the boat. Both requirements were explicitly stated in the question.

For part i, the Report indicates an average score of 1.3, and comments,

Most students found correct expressions for velocity vectors. The most common error was to equate these velocity vectors rather than equating speeds. 

For part ii, the Report gives the intended answer as (3,3). The Report indicates that slightly under half of students were awarded the mark, and comments,

Some answers were not given in coordinate form.

For part i, the Report suggests the answer {\sqrt{(\sin t - 2\cos t)^2 + (1 + \sin t + \cos t)^2}} (with the displayed answer adorned by a weird, extra root sign). The report indicates that a little over half of the students were awarded the mark, and comments,

A variety of correct forms was given by students; many of these were likely produced by CAS technology, including expressions involving double angles. Students should take care when transcribing expressions from technology output as errors frequently occur, particularly regarding the number and placement of brackets. Some incorrect answers retained vectors in the expression.

For Part ii, the Report indicates the intended answer of 0.33, and that 15% of students were awarded the mark for this question. The Report comments,

Many students found this question difficult. Incorrect answers involving other locally minimum values were frequent.

The Report indicates an average score of 1.3 on part d, and comments;

Most students correctly equated the vector components and solved for t . Many went on to give decimal approximations rather than supplying the exact forms. Students are reminded of the instruction saying that an exact answer is required unless otherwise specified.

Lots there. Get hunting.

WitCH 22: Inflecting the Facts

We’re back, at least sort of. Apologies for the long silence; we were off visiting The Capitalist Centre of the Universe. And yes, China was great fun, thanks. Things are still tight, but there will soon be plenty of time for writing, once we’re free of those little monsters we have to teach. (Hi, Guys!) In the meantime, we’ll try to catch up on the numerous posts and updates that are most demanding of attention.

We’ll begin with a couple new WitChes. This first one, courtesy of John the Merciless, is a multiple choice question from the 2017 VCE Specialist Mathematics Exam 2:

The Examiners’ Report indicates that 6% of students gave the intended answer of E, and a little under half the students answered C. The Report also comments that

f”(x) does not change sign at a.

Have fun.

WitCH 21: Just Following Orders

This WitCH come courtesy of a smart VCE student. It concerns the newly instituted VCE subject of Algorithmics, and comes from the 2017 exam:

The Examiners’ Report indicates that half of students gave the intended answer of A, and notes

It is important for students to understand that Big-O notation describes an upper bound, and so is only used for analysis of worst-case running times.

Have fun.

WitCH 20: Tattletail

This one is like complaining about the deck chairs on the Titanic, but what the Hell. The WitCH is courtesy of John the Merciless. It is from the 2018 Specialist Mathematics Exam 2:

The Examiners’ Report notes the intended answer:

H0: μ = 150,   H1: μ < 150

The Report indicates that 70% of students gave the intended answer, and the Report comments on students’ answers:

The question was answered well. Common errors included: poor notation such as  H0 = 150 or similar, and not understanding the nature of a one-tailed test, evidenced by answers such as H1: μ ≠ 150.

Have fun.

Which WitCH is a WitCH?

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.)

WitCH 19: A Powerful Solvent

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:

    \[\mbox{\bf Solve } \boldsymbol{\frac{1}{k-0} \int\limits_0^k \left(\frac1{2x+1}\right)dx = \frac16\log_e(7) \mbox{ \bf for $\boldsymbol k$}.\ k = 3.}\]

Have fun.

Update (02/09/19)

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 \mathbf {\frac1{2k}{\boldsymbol \log} (2k+1) = \frac16{\boldsymbol \log} 7}.  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.

WitCH 18: Making Serena Pointless

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, pretty much everywhere, that

“One in eight men think that they could score a point off Serena Williams”.

Oh, those silly, silly men.

Twitter, of course, lit up over these “delusional” men and the media gleefully reported the ridicule, and more often than not piled on. A rare few articles gave tepid consideration to the idea that the men weren’t delusional, and none more than that.

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.