This WitCH is a companion to our previous, MitPY post, and is a little different from most of our WitCHes. Typically in a WitCH the sin is unarguable, and it is only the egregiousness of the sin that is up for debate. In this case, however, there is room for disagreement, along with some blatant sinning. It comes, predictably, from Cambridge’s Specialist Mathematics 3 & 4 (2020).
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:
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 (as is the accompanying PoSWW) is an exercise and solution from Cambridge’s Mathematical Methods Units 1 and 2, and is courtesy of the Evil Mathologer. (A reminder that WitCH 2, WitCH3, Witch 7 and WitCH 8 are still open for business.)
As Number 8 and Potii pointed out, notation of the form AB is amtriguous, referring in turn to the line through A and B, the segment from A to B and the distance from A to B. (This lazy lack of definition appears to be systemic in the textbook.) And, as Potii pointed out, there’s nothing stopping A being the same point as C.
And, the typesetting sucks.
And, “therefore” dots suck.
This PoSWW (as is the accompanying WitCH) is from Cambridge’s Mathematical Methods Units 1 and 2. and is courtesy of the Evil Mathologer. (A reminder, we continue to post on Cambridge not because their texts are worse than others, but simply because their badness is what we get to see. We welcome all emails with any suggestions for PoSWWs or WitCHes.)
We will update this PoSWW, below, after people have had a chance to comment.
Similar to Witch 6, the above proof is self-indulgent crap, and obviously so. It is obviously not intended to be read by anyone.
One can argue how much detail should be given in such a proof, particularly in a subject and for a curriculum that systemically trashes the concept of proof. But it is difficult to see why the diagram below, coupled with the obvious equations and an easy word, wouldn’t suffice.
OK, Dear Reader, you’ve got work to do.
So far on this blog we haven’t attacked textbooks much at all. That’s because Australian maths texts are, in the main, well-written and mathematically sound.
Yep, just kidding. Of course the texts are pretty much universally and uniformly awful. Choosing a random page from almost any text, one is pretty much guaranteed to find something ranging from annoying to excruciating. But, the very extent of the awfulness makes it difficult and time-consuming and tiring to grasp and to critique any one specific piece of the awful puzzle.
The Evil Mathologer, however, has come up with a very good idea: just post a screenshot of a particularly awful piece of text, and leave others to think and to write about it. So, here we go.
Our first WitCH sample, below, comes courtesy of the Evil Mathologer and is from Cambridge Essentials, Year 9 (2018). You, Dear Reader, are free to simply admire the awfulness. You may, however, go further, and what you might do depends upon who you are:
Here it is. Enjoy. (Updated below.)
* We can’t.
Following on from the comments, here is a summary of the issues with the page above. We also hope to post generally on index laws in the near future.