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 7: North by Southwest

Our new WitCH, below, comes courtesy of Charlie the Enforcer. Once again, this WitCH is from the 2018 SCSA Mathematical Methods Exam (here and here): it’s the gift that keeps on giving. (And a reminder, WitCH 2 and WitCH 3 still require attention are still unresolved.)

Question 11 and the solution in SCSA’s marking key are below. Happy hunting.

Update

John has pretty much caught it all. The killer issue is the use of the term “deceleration” in part (c) which, the solution implies, refers to the drone speeding up in the southerly direction. This is arguably permissible, since deceleration can be (though is far from universally) defined as a negative acceleration, and since way back in part (a) it was implied that North coincides with the positive x direction.

Permissible acts, however, can nonetheless be idiotic: voting Liberal or Republican, for example. And, to use “deceleration” on a high stakes exam to refer implicitly to increasing speed is idiotic. Moreover, to use “deceleration” in this manner immediately after explicitly indicating the “due south” direction of motion is truly ruly idiotic. Still not as idiotic as voting Liberal or Republican, but genuinely special-effort idiotic.

That’s enough to condemn the question, even by SCSA standards. But, the question is also awful in many other ways:

  • The question is boring and butt ugly.
  • No indication is given whether exact or numerical solutions are permitted or required.
  • Having a drone an arbitrary 5m up in the sky for a 1D problem is asking for trouble. For example:
  • The “displacement” of x(0) = 0 for a drone 5m up is pretty stupid.
  • “Where is the drone in relation to the [mysterious] pilot?” Um, kind of uppish?
  • “How far has the drone travelled …” is needlessly wordy and ambiguous. If you want a distance, for God’s sake say “distance”.
  • Given the position function x(t) is at hand, part (c) can easily and naturally be solved by hand. But of course why think about things when you can do mindless calculator crap?

WitCH 6: Parallel Reality

In this WitCH we will again pick on the Cambridge text Specialist Mathematics VCE Units 3 & 4 (2019): see the extract below. (We’d welcome any email or comment with suggestions of other generally WitCHful texts and/or specific WitCHes.) And, a reminder that there is still plenty left to discover in WitCH 2 , WitCH 3 and Tweel’s Mathematical Puzzle.

Have fun.

Update

Below, we go through the passage line by line, but that fails to capture the passage’s intrinsic awfulness. The passage is, as John put it pithily below, a total fatberg. The passage is worse than wrong; it is clumsy, pompous, circuitous, barely comprehensible and utterly pointless.

Why do this? Why write like this? Sure, ideas, particularly mathematical ideas, can be tricky and difficult to convey; dependence/independence isn’t particularly easy to explain. And sure, we all write less clearly than we might wish on occasion. But, if you write/proofread/edit something that the intended “readers” will obviously struggle to understand, then all you’re doing is either showing off or engaging in a meaningless ritual.

An underlying problem is that the entire VCE topic is pointless. Yes, this is the fault of the idiotic VCAA, not the text, but it has to be said, if only as a partial defence of the text. No purpose is served by including in the curriculum a subtle definition that is not then reinforced in some meaningful manner. Consequently, it’s close to impossible to cover this aspect of the curriculum in an efficient, clear and motivated manner. The text could have been one hell of a lot better, but it probably never could have been good.

OK, to the details. Grab a whisky and let’s go.

  • First, a clarification. The definition of “parallel vectors” appears in a slightly earlier part of the text. We included it because it is clearly relevant to the main excerpt. We didn’t intend, however, to suggest that the discussion of dependence began with the “parallel” definition.
  • For the given definition of “parallel vectors” it is redundant and distracting to specify that the scalar k not be 0.
  • As discussed by Number 8, the definition of “parallel vectors” should not exclude the zero vector. The exclusion may be natural in the context of geometric proofs, but here it is a needless source of fussiness, distraction and error.  As an example of a blatant error, immediately following the above passage the text begins a proposition with “Let a and b be two linearly independent (i.e. not parallel) vectors.” A second and entirely predictable error occurs when the text later goes on to “resolve” an arbitrary vector a into components “parallel” and “perpendicular” to a second vector b.
  • The definition of “linear combination” involves a clumsy and needless use of subscripts. Thankfully, though weirdly, subscripts aren’t used in the subsequent discussion. The letters used for the vector variables are changed, however, which is the kind of minor but needless, own-goal distraction that shouldn’t occur.
  • No concrete example of linear combination is provided. (The more abstract the ideas, the more critical it is that they be anchored immediately with very specific illustration.)
  • It is a bad choice to begin with “linear combination”. That idea is difficult enough, but it also leads to a poor and difficult definition of linear dependence, an unswallowable mouthful: “… at least one of its members [elements? vectors?] can be expressed as a linear combination of [the] other vectors [members? elements?] …” Ugh! What really kills this sentence is the “at least one”which stems from the asymmetry hiccup in the definition. (The hiccup is illustrated, for example, by the three vectors a = 3 + 2j + k, b = 9i + 6j + 3k, c = 2i + 4j + 3k. These vectors are dependent, since b = 3a + 0c is a combination of a and c. Note, however, that c cannot be written as a combination of a and b.)
  • As was appropriately done for “linear combination”, the definition of linear dependence should be framed in terms of two or three vectors staring at the reader, not for “a set of vectors”. 
  • The language of sets is obscure and unnecessary.
  • No concrete example of linear dependence is provided. There is not even the specialisation to the case of two and/or three vectors (which, again, is how they should have begun).
  • If you’re going to begin with “linear combination” then don’t. But, if you are, then the definition of linear independence should precede linear dependence, since linear independence doesn’t have the asymmetry hiccup: no vector can be written as a combination of the other vectors. Then, “dependent” is defined as not independent.
  • No concrete example of linear independence is provided. 
  • The properly symmetric “examples” are the much preferred definition(s) of dependence. 
  • The “For example” is weird. It is more accurate to label what follows as special cases. They are not just special cases, however, since they also incorporate non-obvious reworking of the definition of dependence.
  • No proof or discussion is provided that the “example[s]”  are equivalent to the definition. 
  • No genuine example is provided to illustrate the “example[s]”.
  • The simple identification of two vectors being parallel/non-parallel if and only if they are dependent/independent is destroyed by the exclusion of the zero vector.
  • There is no indication why any set of vectors including the zero vector must be dependent. 
  • The expression “two-dimensional vector” is lazy and wrong: spaces have dimension, not vectors. (Ditto “three-dimensional vectors”.)
  • No proof or discussion is provided that any set of three “two dimensional vectors” is dependent. (Ditto “four three-dimensional vectors”.)
  • The “method” for checking the dependence of three vectors is close to unreadable. They could have begun “Let a and b be linearly independent vectors”. (Or, with the correct definition, “Let a and b be non-parallel vectors”.)
  • There is no indication of or clarification of or illustration of the subtle distinction between the original “definition” of linear dependence and the subsequent “method”.

What a TARDIS of bullshit. 

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 4

Well, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle are still there to ponder. A reminder, it’s up to you, Dear Readers, to identify the crap. There’s so much crap, however, and so little time. So, it’s onwards and downwards we go.

Our new WitCH, courtesy of New Century Mathematics, Year 10 (2014), is inspired by the Evil Mathologer‘s latest video. The video and the accompanying articles took the Evil Mathologer (and his evil sidekickhundreds of hours to complete. By comparison, one can ponder how many minutes were spent on the following diagram:

OK, Dear Readers, time to get to work. Grab yourself a coffee and see if you can itemise all that is wrong with the above.

Update

Well done, craphunters. Here’s a summary, with a couple craps not raised in the comments below:

  • In the ratio a/b, the nature of a and b is left unspecified.
  • The disconnected bubbles within the diagram misleadingly suggest the existence of other, unspecified real numbers.
  • The rational bubbles overlap, since any integer can also be represented as a terminating decimal and as a recurring decimal. For example, 1 = 1.0 = 0.999… (See here and here and here for semi-standard definitions.) Similarly, any terminating decimal can also be represented as a recurring decimal.
  • A percentage need not be terminating, or even rational. For example, π% is a perfectly fine percentage.
  • Whatever “surd” means, the listed examples suggest way too restrictive a definition. Even if surd is intended to refer to “all rooty things”, this will not include all algebraic numbers, which is what is required here.
  • The expression “have no pattern and are non-recurring” is largely meaningless. To the extent it is meaningful it should be attached to all irrational numbers, not just transcendentals.
  • The decimal examples of transcendentals are meaningless.

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.

What is this Crap Here?

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:

  • If you believe you can pinpoint the awfulness in the excerpt then feel free to spell it out in the comments, in small or great detail. You could also offer suggestions on how the ideas could have been presented correctly and coherently. You are also free to ponder how this nonsense came to be, what a teacher or student should do if they have to deal with this nonsense, whether we can stop such nonsense,* and so on.
  • If you don’t know or, worse, don’t believe the excerpt below is awful then you should quickly find someone to explain to you why it is.

Here it is. Enjoy. (Updated below.)

* We can’t.

Update

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.

  • The major crime is that the initial proof is ass-backwards. 91/2 = √9 by definition, and that’s it. It is then a consequence of such definitions that the index laws continue to hold for fractional indices.
  • Beginning with 91/2 is pedagogically weird, since it simplifies to 3, clouding the issue.
  • The phrasing “∛5 is irrational and [sic] cannot be expressed as a fraction” is off-key.
  • The expression “with no repeated pattern” is vague and confusing.
  • The term “surd” is common but is close to meaningless.
  • Exploring irrationality with a calculator is non-sensical and derails meaningful exploration.
  • Overall, the page is long, cluttered and clumsy (and wrong). It is a pretty safe bet that few teachers and fewer students ever attempt to read it.