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.
*) Yeah, yeah. We’ll stay right out of the discussion on this one.
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.
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?
So Plimpton 322, the inscrutable Babylonian superstar, has suddenly become scrutable. After a century of mathematics historians puzzling over 322’s strange list of Pythagorean triples, two UNSW mathematics have reportedly solved the mystery. Daniel Mansfield and Norman Wildberger have determined that this 3,800-ish year old clay tablet is most definitely a trigonometry table. Not only that, the media have reported that this amazing table is “more accurate than any today“, and “will make studying mathematics easier“.
Babylonian mathematics is truly astonishing, containing some great insights. It would be no surprise if (but it is by no means guaranteed that) Plimpton 322 contains.great mathematics. What is definitely not great is to have a university media team encourage lazy journalists to overhype what is probably interesting research to the point of meaninglessness.