WitCH 70: Troubled Relations

This is not an old one. It is from the 2019 Specialist Exam 2 and comes courtesy of student PURJ, who previously contributed to the discussion here. PURJ noted one, glaring, issue with the question below and its grading, but we think there are other issues as well. As with our previous WitCH, we’ll semi-update with excerpts from the examination report once people have had a chance to ponder and to comment.

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WitCH 69: Sines of the Times

Yet another old one, from the 2010 Specialist Exam 2 (CAS). We’ll look to semi-update with excerpts from the examination report once people have had a chance to comment. (Readers are of course free to peek at the examination report, anytime they so wish.)

Continue reading “WitCH 69: Sines of the Times”

SOCK-IT-TO-YA

We had planned to not write about this one, dearly hoping it would quietly fade away. That was never going to happen, however, and, having made The Washington Post and the like, there’s now no purpose in not writing about it.

A few days ago, an American schoolteacher went through what was apparently her standard trigonometry routine, donning a fake Indian headdress and chanting SOHCAHTOA as a war dance, or whatever. Yes, the schoolteacher is from another era and, quite possibly, another planet. This year, however, a Native American kid in the class filmed her and then gave permission for the video to be posted on Instagram, where it appeared complete with a campaign speech, and school name and contacts. The snowball then did what snowballs do, the teacher has been suspended and so forth.

It is interesting times when a member of the Hitlerjugend is almost universally portrayed as the victim.

WitCH 49: Trigged Again

The question below is from the second 2020 Specialist exam (not online), and was flagged by commenter John Friend in the discussion here. John has spelled out the problems, but the question is bad enough to warrant its own post, and there’s arguably a little more to be said.

MitPY 9: Team Games

This MitPY is from commenter HollyBolly, who asked on the previous MitPY for some advice on diplomacy.*

Can you guys after all the serious business give me some advice for this situation: on a middle school Pythagoras and trig test, for a not very strong group of students. Questions are to be different from routine ones provided with the textbook subscription. I try “Verify that the triangle with sides (here: some triple, different from 3 4 5) is right, then find all its angles”. After reviewing, the question comes back: “Verify by drawing that a triangle with sides…”

How do you respond if that review has come from:

A. The HoD;

B. A teacher with more years at the school than me but equal in responsibilities in the maths department;

C. A teacher fresh from uni, in their20s.

Regards.

*) Yeah, yeah. We’ll stay right out of the discussion on this one.

WitCH 17: Compounding Our Problems

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.

Update (25/08/19)

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 yxy + 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.

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?