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 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.
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:
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 . 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.
July 20th was the 50th anniversary of Neil Armstrong’s walking on the moon. Well, maybe.
I still have vivid-grainy memories of watching Armstrong’s first steps. A random few students from each class in Macleod State School were selected to go to the library to watch the event on the school’s one TV. I was not one of the lucky few. But Mr. Macrae, our wonderful Grade 4 teacher, just declared “Bugger it!”, determined which student in our class lived closest to the school, and sent out a posse to haul back the kid’s 2-ton TV. We then all watched the moon landing, enthralled and eternally grateful to Mr. Macrae.
But did it really happen?
There have been plenty of questions and questioners, suggesting that the moon landings were faked. How, for example, is the flag in the above photo flapping, given there is no atmosphere to flap it? Then, there is the fake photo of astronauts playing golf on the moon. And the lack of stars in moon photos. And the killer radiation that didn’t kill. And the strange links to Stanley Kubrick. And on and on.
Can all this evidence of doctoring be discounted? Did Man really walk on the moon?
The answers, of course, are Yes and Yes.
The idea that the moon landing was faked is completely ridiculous, and it takes a wilful stupidity to believe it. Which includes about 5% of Americans, and 10% of millenials. (The funniest take is that Kubrick did indeed stage the moon landings, but he was such a perfectionist that he went to the moon to do it.)
There is more to this story, however, than a bunch of conspiracy clowns and gullible slobs. As the brilliant Matt Taibbi writes, there is, indeed, plenty of sleight of hand going on.
The important question is why so many people are willing to believe something so patently false? The answer must be some combination of an inability to discern truth with a lack of concern for truth. And why might that be? Well, just perhaps one factor is an extended history of media and government authorities willing to misdirect and to obfuscate and to flat out lie about everything else. Just perhaps people don’t trust authorities because authorities have abused people’s trust for too long. As Taibbi writes:
“… the flowering of conspiracy theories has an obvious correlation, to a collapse of trust in institutions like the news media and the presidency. … It’s simple math. You can only ask the public to swallow so many fictions before they start to invent their own. The moon story is a great illustration.”
Which is a huge problem. It doesn’t matter a damn if people believe moon landing conspiracy crap. But if they believe that crap then they’ll also believe, more easily, climate change conspiracy crap. And then, an authority that has lost authority is powerless to convince them otherwise. And then, we’re doomed.
But at least we can laugh at the dumb slobs while the Earth goes down in flames.
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.
The ![\[\mbox{\bf Solve } \boldsymbol{\frac{1}{k-0} \int\limits_0^k \left(\frac1{2x+1}\right)dx = \frac16\log_e(7) \mbox{ \bf for $\boldsymbol k$}.\ k = 3.}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-70d480a648919be6645199a724b963cd_l3.svg "Rendered by QuickLaTeX.com")
. 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.
