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.

WitCH 16: The Root of Our Problem

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.

Update (11/08/19)

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.

WitCH 15: Principled Objection

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”.

Update (12/08/19)

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.

WitCH 14: Stretching the Truth

The easy WitCH below comes courtesy of the Evil Mathologer. It is a worked example from Cambridge’s Essential Mathematics Year 9 (2019), in a section introducing parabolic graphs.

Update

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.

WitCH 13: Here for the Ratio

The WitCH below is courtesy of a clever Year 11 student. It is a worked example from Jacaranda’s Maths Quest 11 Specialist Mathematics (2019):

Update (11/08/19)

It is ironic that a solution with an entire column of “Think” instructions exhibits so little thought. Who, for example, thinks to “redraw” a diagram by leaving out a critical line, and by making an angle x/2 appear larger than the original x? And it’s downhill from there.

The solution is painfully long, the consequence of an ill-chosen triangle, requiring the preliminary calculation of a non-obvious distance. As Damo indicates, the angle x is easily determined, as in the following diagram: we have tan(x/2) = 1/12, and we’re all but done.

(It is not completely obvious that the line through the circle centres makes an angle x/2 with the horizontal, though this follows easily enough from our diagram. The textbook solution, however, contains nothing explicit or implicit to indicate why the angle should be so.)

But there is something more seriously wrong here than the poor illustration of a poorly chosen solution. Consider, for example, Step 5 (!) where, finally, we have a suitable SOHCAHTOA triangle to calculate x/2, and thus x. This simple computation is written out in six tedious lines.

The whole painful six-step solution is written in this unreadable we-think-you’re-an-idiot style. Who does this? Who expects anybody to do this? Who thinks writing out a solution in such excruciating micro-detail helps anyone? Who ever reads it? There is probably no better way to make students hate (what they think is) mathematics than to present it as unforgiving, soulless bookkeeping.

And, finally, as Damo notes, there’s the gratuitous decimals. This poison is endemic in school mathematics, but here it has an extra special anti-charm. When teaching ratios don’t you “think”, maybe, it’s preferable to use ratios?