One of the unexpected and rewarding aspects of having started this blog is being contacted out of the blue by students. This included an extended correspondence with one particular VCE student, whom we have never met and of whom we know very little, other than that this year they undertook UMEP mathematics (Melbourne University extension). The student emailed again recently, about the final question on this year’s (calculator-free) Specialist Mathematics Exam 1 (not online). Though perhaps not (but also perhaps yes) a WitCH, the exam question (below), and the student’s comments (belower), seemed worth sharing.
Have a peek at Question 10 of Specialist 2019 Exam 1 when you get a chance. It was a 5 mark question, only roughly 2 of which actually assessed relevant Specialist knowledge – the rest was mechanical manipulation of ugly fractions and surds. Whilst I happened to get the right answer, I know of talented others who didn’t.
I saw a comment you made on the blog regarding timing sometime recently, and I couldn’t agree more. I made more stupid mistakes than I would’ve liked on the Specialist exam 2, being under pressure to race against the clock. It seems honestly pathetic to me that VCAA can only seem to differentiate students by time. (Especially when giving 2 1/2 hours for science subjects, with no reason why they can’t do the same for Maths.) It truly seems a pathetic way to assess or distinguish between proper mathematical talent and button-pushing speed writing.
I definitely appreciate the UMEP exams. We have 3 hrs and no CAS! That, coupled with the assignments that expect justification and insight, certainly makes me appreciate maths significantly more than from VCE. My only regret on that note was that I couldn’t do two UMEP subjects 🙂
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.
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.
Reflecting on the comments below, it was a mistake to characterise this exercise as a PoSWW; the exercise had a point that we had missed. The point was to reinforce the Magrittesque lunacy inherent in Methods, and the exercise has done so admirably. The fact that the suggested tangents to the pictured graphs are not parallel adds a special Methodsy charm.
No one appears to have a bad word for Eddie Woo. And no, we’re not looking to thump Eddie here; the mathematics videos on Eddie’s WooTube channel are engaging and clear and correct, and his being honoured as Local Australian of the Year and as a Top Ten Teacher is really cool. We do, however, want to comment on Eddie’s celebrity status and what it means.
What do Eddie’s videos exhibit? Simply, Eddie is shown teaching. He is explaining mathematics on a plain old whiteboard, with no gizmos, no techno demos, no classroom flipping, rarely a calculator, none of the familiar crap. There’s nothing at all, except a class of engaged students learning from a knowledgeable and engaging teacher.
Sure, Eddie tapes his lessons, but Eddie’s charmingly clunky videos are not in any way “changing the face of mathematics teaching“. Eddie’s videos are not examples of teaching, they are evidence of teaching. For actual instruction there are many better videos out there. More importantly, no video will ever compare to having a real-live Eddie to teach you.
There are many real-live Eddies out there, many teachers who know their maths and who are teaching it. And, there would be many, many more real-live Eddies if trainee teachers spent more time learning mathematics properly and much less time in the clutches of Australia’s maths ed professors. That’s the real message of Eddie’s videos.