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.
“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.
Eddie Woo is reportedly concerned about private tutoring. His warning comes courtesy of SMH‘s education editor, Jordan Baker, in an article entitled ‘Be very, very careful’: Experts raise warning on private tutoring. The article begins,
Maths teachers including high-profile mathematician Eddie Woo have sounded an alarm on private tutoring, warning that bad tutors could be “fatal” to students’ future in the subject.
Eddie said it, so it must be true. And, Baker quotes another expert, the chief executive of the Australian Tutoring Association, Mohan Dhall:
I am absolutely dismayed at the lack of creativity and lack of real-world applicability most tutors bring to maths …The main problem stems from this idea that they focus on the outcome – ‘this is what students need to know’, rather than ‘this is what kids need to learn to be interested and engage’.
Finally, Baker quotes expert Katherin Cartwright, a lecturer in mathematics education at The University of Sydney. Cartwright, according to Baker, is concerned that poor tutoring could lead to a lack of confidence:
If it becomes about skill and drill and speed, and it becomes an anxious, emotional issue for students, then they are not going to like it, and they will not want to take it further.
Yep, of course. The most important consideration when framing an education is to be sure to never make a student anxious or emotional. Poor, fragile little petals that they are.
Baker’s fear-mongering is nonsense. Almost every line of her article is contentious and a number contain flat out falsehoods. Beginning with the title. Woo and Dhall and Cartwright are “experts” on the issues of tutoring? According to whom? Based on what? Perhaps they are experts, but Baker provides no evidence.
OK, we could concede Baker’s point that Eddie is a mathematician. Except that he isn’t and we don’t. Not that it matters here, since most mathematicians are unlikely to know much about the role of tutoring in Australian education. But the false and pointless puffery exemplifies Baker’s unjustified appeals to authority.
What of the declared concerns of Baker’s “experts”? Cartwright is supposedly worried about “skill and drill and speed”. This in contrast to school, according to Baker:
Most schools no longer emphasise speed and rote learning when teaching maths, and now focus on students’ understanding of key concepts as part of a concerted effort to improve engagement in maths across the system.
This hilarious half-truth undercuts the whole thrust of Baker’s article. It is true that many schools, particularly primary schools, have drunk the educational Kool-Aid and have turned their maths lessons into constructivist swamplands. But that just means the main and massive job of competent Year 7 maths teachers is to undo the damage inflicted by snake-oilers, and to instil in their students, much too late, an appreciation of the importance of memory and skill and efficient technique. Such technique is critical for formal success in school mathematics and, which is sadly different, for the learning of mathematics. Baker seems entirely unaware, for example, that, for better or worse, Year 12 mathematics is first and foremost a speed test, a succession of sprints.
As for Dhall, does he really expects tutors to be more offering of “creativity” and “real-world applicability”? Dhall seems blissfully unaware that most “real-world” applications that students must suffer through are pedagogically worthless, and are either trivial or infinitely tedious. Dhall seems unaware that some subjects have warped “applicability” into a surrealist nightmare.
And Eddie? What worries Eddie? Not much, as it happens, but too much. Eddie’s quoted comments come from a NSW podcast, which appears to have been the genesis of Baker’s piece; stenographic fluffing is of course the standard for modern reportage, the cheap and easy alternative to proper investigation and considered reflection.
Eddie’s podcast is a happy public chat about teaching mathematics. Eddie is demonstrably a great teacher and he is very engaging. He says a number of smart things, the half-hour podcast only being offensive for its inoffensiveness; Eddie, or his interviewer, was seemingly too scared to venture into a deep public discussion of mathematics and the sense of it. The result is that, except for the occasional genuflection to “pattern”, Eddie may as well have been talking about turtle farming as teaching mathematics.
Eddie’s comments on tutoring are a very minor part of the podcast, a response in the final question time. This is Eddie’s response in full:
When I think about external tuition – again just like before this is a really complex question – there is tuition and then there is ‘tuition’. There is some which is enormously helpful to individual students to come in at a point of need and say “you have got gaps in your knowledge, I can identify that and then help you with those and then you can get back on the horse and off you go, fantastic”. There are other kinds of tuition which are frankly just pumping out an industrial model of education which parents who are very well intentioned and feel like they cannot do anything else, it is like “at least I can throw money at the problem and at least they are spending more time on maths hopefully that will help”. Maybe it does and maybe it is making your child hate maths because they are doing it until 9pm at night after a whole day? That to me is heartbreaking.
I think that students need to be very, very careful and parents need to be very, very careful about how they experience mathematics. Because yes the time is a worthwhile investment, it is a practical subject, but if you are just churning through, often tragically learning things which actually are just machine processes. I have students come to me and they say “I can differentiate, I am really good at that, I am only fifteen years old”. You don’t need to know what differentiation is, but they come to me with this ability to turn a handle on this algorithm this set of steps. Just like me; I don’t know how to bake, but I can follow a recipe. I have no idea what baking powder does or why 180 degrees Celsius is important but I can follow steps. That is okay for a cake because you can still eat it at the end, but that is fatal for mathematics because you don’t know why you are doing any of the things that you are doing. If that is what you are, you are not a mathematician, you are a machine and that is not what we want our children to become. We have to be careful.
Eddie says plenty right here, touching on various forms of and issues with tutoring, and school teaching. The issues do not get fleshed out, but that is the nature of Q & A.
Eddie also gets things smugly wrong. Sure, some tutoring might be characterised as “industrial”. But more so than schools? How can mass education not be industrial? This isn’t necessarily bad: mostly, it just is. Unless, of course, little Tarquin’s parents have the time and the money to arrange for individual or small-group lessons with an, um, tutor.
All the concerns Baker and her experts raise about tutoring apply as much or more so to school education and, as a matter of business necessity, are largely a reflection of school education. And, how do tutors and tutoring companies deal with this? Some well, some poorly. But mostly with industry, which is not a dirty word, and with good and honest intent.
Baker notes the underlying issue, seemingly without even realising it:
However, Australian students’ performance in maths has either stalled or declined on all major indicators over that period, and academics have raised concerns about students arriving at university without the maths skills they need.
Why do parents employ tutors? Having enjoyed and suffered forty years of tutoring, in pretty much all its forms, we can give the obvious answer: there’s a zillion different, individual reasons. Some, as Eddie suggests, are looking for a little damage control, the filling of gaps and a little polishing. Some, as Eddie suggests, think of mathematics, falsely, as a syntactic game, and are looking for lessons in playing that dangerously meaningless game. Some believe, correctly or otherwise, that their teacher/school is responsible for little Johnny’s struggling. Some are trying to get darling Diana into law school. Some are hothousing precious little Perry so he/she can get a scholarship into Polo Grammar or Mildred’s College for Christian Ladies.
But, underlying it all, there is one obvious, central reason why parents employ tutors: parents are unsatisfied with the education their child receives at school.
Why are parents unsatisfied? Are they right to be? Of course, it depends. But, whatever the individual analyses, the massive growth of the tutoring industry indicates a major disconnect, and either a major failing in schools’ performance or a major blindness in parents’ expectations, or both.
That would be a much more worthwhile issue for Baker, and everyone, to consider.
Following the lead of France and Ontario, the Victorian Labor government has decided to ban mobile phones in government classes. One stated reason is to combat cyberbullying, but they’re probably lying. The good and blatantly obvious reason is that smart phones destroy concentration.
Still, any change, no matter how compelling, will have its detractors. There is the idiotic argument that the ban is unenforceable; the claim is almost certainly false, but if true points to such a profound loss of authority that schools may as well just give up entirely. And, there is the argument – one in a stream of tendentious half-truths – that occasionally the internet is down, meaning a lesson can only continue aided by a mobile’s hotspot. The argument is based upon a falsehood but in any case is much worse than wrong; any teacher so addicted to the internet for their teaching may first wish to heal thyself. They may also wish to consider a new profession. Please.
And of course there is discussion of the suggested educational benefits of smart phones, proving only that there is no idea so idiotic that some educational hack cannot be found to support it.
Luckily, it would appear that the Labor government is holding firm, and students will be able to get back to the intended lessons. On their fucking iPads.
It would appear that the Ramsey Centre‘s Degree in Western Civilisation will now be a thing. This comes after the ANU rejected the idea out of concerns about Ramsey’s autocratic meddling. And, it comes after Sydney University shot itself in the foot by censoring its own academics. But, the University of Wollongong is hellbent on offering Ramsey’s Bachelor of Arts in Western Civilisation. This comes with the news that the University Council overruled Wollongong’s academic senate because, after all, what would those silly academics know about academic integrity?
Jillian Broadbent, UoW’s chancellor, claimed that the council had “full respect for the university’s academic process”. If only Broadbent had a modicum of respect for the meaning of English words.
Underlying all of this is the question of the meaning of “Western civilisation”. UoW advertises that in Ramsey’s degree a student will:
Learn how to think critically and creatively as you examine topics in ethics, aesthetics, epistemology, metaphysics, philosophy of religion and political philosophy.”
The irony is palpable. But, at least it makes clear what is meant by “Western civilisation”. It means the power of a business-bloated gang to use Orwellian language while ramming through the selling out of a public institution to rich bigots.
We intend these words, of course, with the fullest of respect.
Either you are horrified by the persecution of Julian Assange or, like Lisa Millar and Patricia Karvelas and Peter Greste and Michael “Gold Star” Rowland, you support the fascistic, war-mongering motherfuckers out to get him. There is no middle ground.
ABC’S Four Corners has just aired a 2-part program on Julian Assange. It is well-made, interesting and, in keeping with modern journalistic style, entirely without self-awareness and entirely off the fucking point.
The WitCHfest is coming to an end. Our final WitCH is, once again, from Cambridge’s Specialist Mathematics 3 & 4 (2019). The section establishes the compound angle formulas, the first proof of which is our WitCH.
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 sufficient to 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 present incomprehensible 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.
This WitCH comes from one of our favourites, the Complex Numbers chapter from Cambridge’s Specialist Mathematics 3 & 4 (2019). It is not as deep or as beWitCHing as other aspects of the chapter. But, it’s still an impressive WitCH.
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.
OK, playtime is over. This one, like the still unresolved WitCH 8, will take some work. It comes from Cambridge’s Mathematical Methods 3 & 4 (2019). It is the introduction to “When is a function differentiable?”, the final section of the chapter “Differentiation”.
We wrote about this nonsense seven long years ago, and we’ll presumably be writing about it seven years from now. Nonetheless, here we go.
The first thing to say is that the text is wrong. To the extent that there is a discernible method, that method is fundamentally invalid. Indeed, this is just about the first nonsense whacked out of first year uni students.
The second thing to say is that the text is worse than wrong. The discussion is clouded in gratuitous mystery, with the long-delayed discussion of “differentiability” presented as some deep concept, rather than simply as a grammatical form. If a function has a derivative then it is differentiable. That’s it.
Now to the details.
The text’s “first principles” definition of differentiability is correct and then, immediately, things go off the rails. Why is the function f(x) = |x| (which is written in idiotic Methods style) not differentiable at 0? The wording is muddy, but example 46 makes clear the argument: f’(x) = -1 for x < 0 and f’(x) = 1 for x > 0, and these derivatives don’t match. This argument is unjustified, fundamentally distinct from first principles, and it can easily lead to error. (Amusingly, the text’s earlier, “informal” discussion of f(x) = |x| is exactly what is required.)
The limit definition of the derivative f’(a) requires looking precisely at a, at the gradient [f(a+h) – f(a)]/h as h → 0. Instead, the text, with varying degrees of explicitness and correctness, considers the limit of f’(x) near a, as x → a. This second limit is fundamentally, conceptually different and it is not guaranteed to be equal.
The standard example to illustrate the issue is the function f(x) = x2sin(1/x) (for x≠ 0 and with f(0) = 0). It is easy to to check that f’(x) oscillates wildly near 0, and thus f’(x) has no limit as x → 0. Nonetheless, a first principles argument shows that f’(0) = 0.
It is true that if a function f is continuous at a, and if f’(x) has a limit L as x → a, then also f’(a) = L. With some work, this non-obvious truth (requiring the mean value theorem) can be used to clarify and to repair the text’s argument. But this does not negate the conceptual distinction between the required first principles limit and the text’s invalid replacement.
Now, to the examples.
Example 45 is just wrong, even on the text’s own ridiculous terms. If a function has a nice polynomial definition for x ≥ 0, it does not follow that one gets f’(0) for free. One cannot possibly know whether f’(x) exists without considering x on both sides of 0. As such, the “In particular” of example 46 is complete nonsense. Further, there is the sotto voce claim but no argument that (and no illustrative graph indicating) the function f is continuous; this is required for any argument along the text’s lines.
Example 46 is wrong in the fundamental wrong-limit manner described above. it is also unexplained why the magical method to obtain f’(0) in example 45 does not also work for example 46.
Example 47 has a “solution” that is wrong, once again for the wrong-limit reason, but an “explanation” that is correct. As discussed with Damo in the comments, this “vertical tangent” example would probably be better placed in a later section, but it is the best of a very bad lot.
And that’s it. We’ll be back in another seven years or so.
The problem, as commenters have indicated below, is that there is no parabola with the indicated turning point and intercepts. Normally, we’d write this off as a funny but meaningless error. But, coming at the very beginning of the introduction to the parabola, it most definitely qualifies as crap.