WitCH 11: Impartial

The following WitCH comes from the (CAS permitted) 2018 Specialist Mathematics Exam 2:

The Examiners’ Report indicates that about half of the students gave the intended answer of D, with about a third giving the incorrect answer B. The Report notes:

Option B did not account for common factors and its last term is not irreducible, so should not have Dx in the numerator.

WitCH 9: A Distant Hope

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.)

Update

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.

WitCH 8: Oblique Reasoning

A reminder, WitCH 2, WitCH 3 and WitCH 7 are also open for business. Our new WitCH comes courtesy of John the Merciless. Once again, it is from Cambridge’s text Specialist Mathematics VCE Units 3 & 4 (2019). The text provides a general definition and some instruction, followed by a number of examples, one of which we have included below. Have fun.

Update

With John the Impatient’s permission, I’ve removed John’s comments for now, to create a clean slate. It’s up for other readers to do the work here, and (the royal) we are prepared to wait (as is the continuing case for WitCh 2 and Witch 3).

This WitCH is probably difficult for a Specialist teacher (and much more so for other teachers). But it is also important: the instruction and the example, and the subsequent exercises, are deeply flawed. (If anybody can confirm that  exercise 6G 17(f) exists in a current electronic or hard copy version, please indicate so in the comments.)

WitCH 5: What a West

This one’s shooting a smelly fish in a barrel, almost a POSWW. Sometimes, however, it’s easier for a tired blogger to let the readers do the shooting. (For those interested in more substantial fish, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle still require attention.)

Our latest WitCH comes courtesy of two nameless (but maybe not unknown) Western troublemakers. Earlier this year we got stuck into Western Australia’s 2017 Mathematics Applications exam. This year, it’s the SCSA‘s Mathematical Methods exam (not online. Update: now online here and here.) that wins the idiocy prize. The whole exam is predictably awful, but Question 15 is the real winner:

The population of mosquitos, P (in thousands), in an artificial lake in a housing estate is measured at the beginning of the year. The population after t months is given by the function, \color{blue}\boldsymbol{P(t) = t^3 + at^2 + bt + 2, 0\leqslant t \leqslant 12}.

The rate of growth of the population is initially increasing. It then slows to be momentarily stationary in mid-winter (at t = 6), then continues to increase again in the last half of the year. 

Determine the values of a and b.

Go to it.

Update

As Number 8 and Steve R hinted at and as Damo nailed, the central idiocy concerns the expression “the rate of population growth”, which means P'(t) and which then makes the problem unsolvable as written. Specifically:

  • In the second paragraph, “it” has a stationary point of inflection when t = 6, which is impossible if “it” refers to the quadratic P'(t).
  • On the other hand, if “it” refers to P(t) then solving gives a < 0. That implies P”(0) = 2a < 0, which means “the rate of population growth” (i.e. P’) is initially decreasing, contradicting the first claim of the second paragraph.

The most generous interpretation is that the examiners intended for the population P, not the rate P’, to be initially increasing. Other interpretations are less generous.

No matter the intent, the question is inexcusable. It is also worth noting that even if corrected the question is awful, a trivial inflection problem dressed up with idiotic modelling:

  • Modelling population growth with a cubic is hilarious.
  • Months is a pretty stupid unit of time.
  • The rate of population growth initially increasing is irrelevant.
  • Why is the lake artificial? Who gives a shit?
  • Why is the lake in a housing estate? Who gives a shit?

Finally, it’s “latter half” or “second half”, not “last half”. Yes, with all else awful here, it hardly matters. But it’s wrong.

Further Update

The marking schemes for the exam are now up, here and here.  As was predicted, “the rate of growth of the population” was intended to mean “population”. As is predictable, the grading scheme gives no indication that the question is garbled garbage.

The gutless contempt with which certain educational authorities repeatedly treat students and teachers is a wonder to behold.

WitCH 3

First, a quick note about these WitCHes. Any reasonable mathematician looking at such text extracts would immediately see the mathematical flaw(s) and would wonder how such half-baked nonsense could be published. We are aware, however, that for teachers and students, or at least Australian teachers and students, it is not nearly so easy. Since school mathematics is completely immersed in semi-sense, it is difficult to know the rules of the game. It is also perhaps difficult to know how a tentative suggestion might be received on a snarky blog such as this. We’ll just say, though we have little time for don’t-know-as-much-as-they-think textbook writers, we’re very patient with teachers and students who are honestly trying to figure out what’s what.

Now onto WitCH 3, which follows on from WitCH 2, coming from the same chapter of Cambridge’s Specialist Mathematics VCE Units 3 & 4 (2018).* The extract is below, and please post your thoughts in the comments. Also a reminder, WitCH 1 and WitCH 2 are still there, awaiting proper resolution. Enjoy.

* Cambridge is a good target, since they are the most respected of standard Australian school texts. We will, however, be whacking other publishers, and we’re always open to suggestion. Just email if you have a good WitCH candidate, or crap of any kind you wish to be attacked.

Update (06/02/19)

The above excerpt is indicative of the text’s entire chapter on complex numbers. It is such remarkably poor exposition, the foundations so understated and the direction so aimless, it is almost impossible to find one’s way back to sensible discussion.

Here is a natural framework for a Year 12 topic on complex numbers:

  • First, one introduces a new number \boldsymbol i for which \boldsymbol i^2=-1.
  • One then defines complex numbers, and introduces the fundamental operations of addition and multiplication.
  • One then at least states, and hopefully proves, the familiar algebraic properties for complex numbers, i.e. the field laws, \boldsymbol {u(z + w) = uz + uw} and so forth. All these properties are obvious or straight-forward to prove, except for the existence of multiplicative inverses; one has to prove that given any non-zero complex \boldsymbol z there is another complex \boldsymbol w with \boldsymbol {zw = 1}.
  • That is the basic complex algebra sorted, and then one can tidy up. This includes the definition of division \boldsymbol {\frac{z}{w} = zw^{-1} = w^{-1}z}, noting the essential role played by commutativity of multiplication.
  • Then, comes the geometry of complex numbers, beginning with the definition and algebraic properties of the conjugate \boldsymbol {\overline{z}} and modulus \boldsymbol {|\boldsymbol z|},  the interpretation of these quantities in terms of the complex plane, and polar form.
  • Finally, the algebra and geometry of complex numbers are related: the parallelogram interpretation of addition, the trigonometric-polar interpretation of multiplication, roots of complex numbers and so forth.

Must complex numbers be taught in this manner and in this order? No and yes. One obvious variation is to include a formal definition of a complex number \boldsymbol {z = a + bi} as an ordered pair \boldsymbol {(a,b)}; as Damo remarks below, this is done as an asterisked section in Fitzpatrick and Galbraith. Though unnervingly abstract, the formal definition has the non-trivial advantage of reinforcing, almost demanding, the interpretation of complex numbers as points in the complex plane. More generally, one can emphasise more or less of the theoretical underpinnings and, to an extent, change the ordering.

But, one can only change the ordering and discard the theory so much, and no more. Complex numbers are new algebraic objects, and defining and clarifying the algebra is critical, and this fundamentally precedes the geometry.

What is the Cambridge order? The text starts off well, introducing \boldsymbol i with \boldsymbol {i^2  = -1},  and then immediately goes off the rails by declaring that \boldsymbol {i  = \sqrt{-1}}. Then, in brief, the text includes:

(a) an invalid treatment of the square roots of negative numbers;

(b) complex addition stated, presumably defined, with the inverse \boldsymbol {-z} introduced but not named;

(c) complex subtraction, followed by an almost invisible statement of the relevant field laws, none of which are proved or assigned as exercises;

(d) scalar multiplication;

(d) the complex plane and “the representation of the basic operations on complex numbers”;

(e) complex multiplication defined, with an almost invisible statement of field laws, none of which are proved or assigned as exercises, and with no mention of the question of multiplicative inverses;

(f) the geometry of multiplication by \boldsymbol i;

(g) the modulus of a complex number defined, with algebraic properties (including {\boldsymbol {|\frac{z}{w}| = \frac{|z|}{|w|}}) stated and assigned as exercises;

(h) the conjugate of a complex number defined, with algebraic properties stated and either proved or assigned as exercises.

(h) Finally, as excerpted above, it’s on to reciprocals of complex numbers, multiplicative inverses in terms of modulus and conjugate, and division.

(i) This is followed by sections on polar form, de Moivre’s theorem and so forth;

(j) CAS garbage is, of course, interspersed throughout. (Which is not all Cambridge’s fault, but the text is no less ugly for that.)

At no stage in the text’s exposition is there any visible concern for emphasising or clarifying foundations, or for following a natural mathematical progression. There is too seldom an indication of what is being defined or assumed or proved.

What is the point? Yes, one can easily be overly theoretical on this topic, but this is Year 12 Specialist Mathematics. It is supposed to be special. The students have already been introduced to complex numbers in Year 11 Specialist. Indeed, much of the complex material in the Year 11 Cambridge text is repeated verbatim in the Year 12 text. Why bother? The students have already been exposed to the nuts and bolts, so why not approach the subject with some mathematical integrity, rather than just cutting and pasting aimless, half-baked nonsense?

Now, finally and briefly, some specific comments on the specific nonsense excerpted above.

  • division of complex numbers has already appeared in the text, in the list of (unproved) properties of the modulus.
  • the algebraic manipulation of \boldsymbol {\frac1{a+bi}} is unfamiliar and unmotivated and, as is admitted way too late, is undefined. There is a place for such “let’s see” calculations – what mathematicians refer to as formal calculations –  but they have to be framed and be motivated much more carefully.
  • There is no need here for a “let’s see” calculation. The critical and simple observation is that \boldsymbol {(a + bi)(a-bi) = a^2 + b^2} is real. It is then a short step to realise and to prove that \boldsymbol {\frac{a}{a^2 + b^2} - \frac{bi}{a^2 + b^2}} acts as, and thus is, the multiplicative inverse of \boldsymbol {a + bi}.
  • Having finally admitted that \boldsymbol {\frac1{a + bi}} has not been defined, the text goes on to not define it again. The text states the multiplicative inverse of \boldsymbol z, but it is not clear whether this statement amounts to a definition or a conclusion.
  • Division of complex numbers is then defined with needless subscripts and, more importantly, with no mention of the fundamental role of commutativity of multiplication.
  • Throughout, the use of conjugate and modulus is muddying rather than clarifying.
  • At no stage is it made clear why \boldsymbol {\frac1{a + bi}} makes sense in contrast to, for example, the non-sense of \boldsymbol {\frac1{M}} for a matrix.

WitCH 2

Well, WitCH 1 is still not satisfactorily resolved, and Tweel’s puzzle is also still out there. But, we may as well get another ball rolling.

The second in our What is this Crap Here series comes from Cambridge’s textbook Specialist Mathematics VCE Units 3 & 4 (2018). Enjoy, and please get to pondering, and posting.

Update

Thanks to Damo for their hard work below.

The main problem with the above excerpt is that it should not exist. It is pointless to introduce complex numbers with more than a sentence on complex roots, and it is almost impossible to do so in a sensible manner. The nonsense of the text’s approach is encapsulated by the equation

\color{blue}\boldsymbol{ \sqrt{-4}= 2i\,.}

This equation is best thought as false and, in the context of the excerpt above, must be thought of as meaningless. As is, thus, the discussion leading up to this equation.

How did they get there? To begin, i is introduced as a number for which i2 = -1, which is fine and good at the school level. Then, they note that the equation x2 = -1 has the two solutions i and -i, which is significantly less fine; since general complex numbers, and -i in particular, have not yet been defined, the notation -i is thus far meaningless, as is the notion of squaring this number. Still, if the sentence were more carefully worded, it would be reasonable in an introductory paragraph. The cavalier attitude to definition and meaning, however, is the sign of much worse to come.

The text continues by “declaring” that √(-1) = i, and then heads on its merry calculating way. But the calculation is complete fantasy. The declaration amounts to a (bad) definition of a specific root which cannot, in and of itself, tell us what any other root means or how it might be manipulated. So, √(-4) is as yet undefined, and the manipulation of this quantity is unjustified, as yet unjustifiable, and is best thought of as wrong.

In the real context we use √x to distinguish the positive root but it is fundamental that complex roots are multiple-valued. And, for the polynomial focus of VCE mathematics, multiple values are perfectly fine and perfectly natural. The quadratic formula remains true without change and the purportedly troublesome identity

\color{blue}\boldsymbol{\sqrt{a}\times \sqrt{b} = \sqrt{ab}}

is always true (modulo the understanding that if x is a positive real then√x is now ambiguous). Moreover, with this natural interpretation, the text’s declaration that √(-1) = i is false, as is the equation √(-4) = 2i.

Admittedly, at some point it is valuable, and essential, to introduce principal values of roots, by which the text’s equation can be interpreted to be true. But principle roots are intrinsically awkward, must be introduced with great care and should only be introduced when there is a purpose. Which is not on page 1 of a school text, and arguably not ever in a school text.

Apart form the utter pointlessness and utter meaningless of the excerpt, we note:

  • The text conflates the introduction of imaginary numbers in the 16th century with the introduction of the symbol i in the 18th century.
  • The text implies 0 is an imaginary number, which is ok though a little peculiar.
  • The real numbers and imaginary numbers are not subsets of \Bbb C.
  • The characterisations {\rm Re}: \Bbb C \to \Bbb R and {\rm Im}: \Bbb C \to \Bbb R are grandiose and pointless.