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 i + 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 “for 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.
Two issues in the first two lines…
Yes, that’s where the rot begins.
And then all of a sudden, later on, zero vectors are allowed…
This is less difficult than other WITCH, perhaps?
I don’t think so. Be detailed and precise: what exactly is the issue(s) above with zero vectors?
Moreover, though zero vectors are the source of the problem, there is also plenty more to critique. The question isn’t what is the falsity here; the question is what is the crap here. Of course being false is one way to be crap but, alas, there are plenty of other ways.
Read the excerpt. It is awful, painful to read. It is obviously crap. But why?
The existence of the crappiness is clear, but sorting out the reasons for and the nature of that crappiness takes some thought.
OK (still wanting to allow other crap-hunters their share of the fun) – you can divide the crap onto this page into two very broad categories, although there is a lot of overlap, most of the crap is dominant in one of the categories.
Category A – ideas which are just wrong. Either because they leave out key parts or are totally misleading.
Category B – ideas which might be correct but are totally useless to anyone (especially the supposed audience – students and perhaps teachers)
The whole thing is just a big over-bloated mess – a total fatberg. Imagine a student or inexperienced teacher trying to make sense of it.
Most experienced teachers cannot make sense of it.
Aren’t vectors that are are opposite in direction and different in magnitude called anti-parallel?
I don’t think you’ll find this distinction in most textbooks. An alternative definition is that two vectors are parallel if their cross product (not on the Specialist Maths course) is equal to zero.
Potii, I think John’s right. The term anti-parallel exists, but I haven’t seen it much used. More to the point, John’s suggested “alternative definition” (which can be rephrased with no reference to cross product) is quite standard, though not universal. And, very much to the point, Cambridge’s definition and John’s are not equivalent.
Two vectors are parallel if u=k*v where k is any real scalar. If u is the zero vector it is parallel to all other vectors, which can be seen by setting k=0. If k=0 is not allowed, then the zero vector is not parallel to anything.
This then means that any set of vectors which contains the zero vector will be linearly dependent. The justification is trivial, just set the coefficients of the non zero vectors to zero and the coefficient of the zero vector to anything other than zero.
This point is made in the excerpt, but by making the error about the zero earlier on, the page contradicts itself.
Thanks, Number 8. Yours is the preferred (though not universal) definition of parallel vectors in this context. I don’t see where the text has contradicted itself. Rather, the text has painted itself into a corner.
I’m not an expert, but I think many non-experts are interested in these posts as well.
I can’t find any mathematical inconsistency in the excerpt. If i’m not mistaken,then, the crappiness is “merely” pedagogical.
1. The definitions for “parallelism” and for “linear dependence” are given independently.
Parallelism is simply Undefined for the case of a zero vector.
The two definitions are later related in the first “example”.
2. The so called examples do not apply the definitions per se, but present an alternative definition of
dependence without proof of equivalence.
3. The final method for checking dependence for three vectors is…. well :
Redundant for one thing.
And, the condition c=ma+nb (which is sufficient for dependence from the previous definition) is not proven
to be necessary. Thus the student may take this as a “third” definition that applies for the case where a and b are
non-zero AND non-parallel (i.e linearly independent)
So, without a single numerical example, the student is given FOUR different “definitions” and a big: WHY?
What did I miss?
Thanks for your comment, Zouheir. I care most about the “non-experts”. My primary aim in this blog is to give people, and teachers and students in particular, the license and the courage to treat any and all maths ed with scepticism and, when appropriate, with contempt. If you struggle or fail to make sense of something in maths ed then I’d suggest the odds are high that it’s not you, it’s them.
You are correct that there is no inconsistency (of which I am aware) in the above excerpt. It is, however, undeniable crap. Very few teachers could make heads or tails of it, and no reasonable mathematician could read it without throwing up.
As for your substantive comments, you’ve missed nothing that I can see. Stemming from an ill-considered and irrelevant definition of “parallel”, combined with a disregard for language and style, the text presents a jungle of “examples”, amounting to semi-equivalent definitions, none of which is explained or justified.
While walking the dogs after reading the above, a car, owned by a business, went by, and the ad on the side of the car told me that the business was an “independant consultant”. Sigh.
Anyway, back to Marty’s original post. Yes, the section quoted is hard to understand. I note that in his classic work on vector spaces, Halmos does not mention parallel vectors as far as I can see.
The post, and the discussion, made me wonder why this is in the syllabus at all. I would prefer to see these introduced in a university subject on linear algebra, and then discussed fully in this context.
Psychology students are likely to meet these concepts when they come to multivariate analysis as they might do in the second year of their degree. How they deal with this with very little mathematical background (certainly not Specialist Mathematics) is beyond me.
Thanks, Terry. Your “hard to understand” is a massive understatement, but you’ve always been (too) gently spoken. It’s notable but not surprising that Halmos doesn’t mention parallel vectors. I checked a number of standard cookbook LA texts, and few use the word “parallel” except in a vague and undefined manner. And of course you’re right, the independence stuff in Specialist serves absolutely no purpose whatsoever.
In his essay “Of education”, John Milton wrote in the 17th century: “Hence appear the many mistakes which have made Learning generally so unpleasing and so unsuccessful; first, we do amiss to spend seven or eight years meerly in scraping together so much miserable Latine and Greek, as might be learnt otherwise easily and delightfully in one year. “
Great quote!
“No purpose is served by including in the curriculum a subtle definition that is not then reinforced in some meaningful manner.” – Why teach math at all? It cannot serve any meaningful purpose.
Real numbers are not objects of nature, therefore they must be false. Real numbers are points on a straight line, but there is no straight line in the universe, because everything in the universe is continuously moving. Since the foundations of mathematics are false, entire mathematics must be false too. Such a false system has created major problems in our society. Our engineering is unreliable, crashes, and pollutes the environment, all because of false math, false science, and false money (note – money is a real number also). Money is increasing poverty in the world. Everything started with math.
Dear idpnsd, if you want to engage with the ideas then do so. But please stay on topic.
There is plenty to debate about the nature of mathematical ideas, the utility of mathematical models and the utility of mathematics education. It is all interesting, and I have some sympathy with your apparent point of view. But none of that can justify teaching mathematics poorly, which is what this post, and this blog generally, is about.
“But none of that can justify teaching mathematics poorly,…” The idea is that if a subject is false, like mathematics, then that subject cannot be taught correctly.
id, how about you give an example, relevant to the theme of this thread (that is, linear independence), that demonstrates your assertion.
idpnsd, if mathematics is “false” in the way you describe then so is any attempt to make sense of the world and our thoughts. You seem to want to reduce us to cavemen staring at rocks.
I know I’m rusty on my math, but something about this text just bugs me stylistically. It’s not that they are right/wrong, but more that they don’t do a good job explaining things. I pick up other old texts and am drawn back into the topic. But this stuff reads strange. Like they are so worried about making a mistake that they give an awkward introduction instead, with too much pseudoproofy derivation. Fatberg…indeed.
I worry that kids using this text aren’t being moved towards familiarity and mastery. I guess my best hope is they just use the texts for the homework problems.
“something” bugs you about the text??? How about everything?
This is more minor, but I don’t like the excessive fonts (italics, bold, and script, and that double-lined R) and colors (4 font colors, plus the light green shading with darker green shade underbar) for emphasis. There are a huge amount of them. It’s distracting. Some emphasis is helpful (e.g. numbering equations and spacing them out of the text…and maybe bold). But it’s an orgy of colors/fonts, especially with the shading. Just hard to read.
Me thing, but: I also am not a fan of designating vectors with bold. I prefer the “arrow on top” as the most easy to see. And especially, easy to replicate on paper (or blackboard).
Again very minor, but…also, am not a fan of the fancy font R (for reals). It’s just the sort of fancy font, prissy pure math grad student bullshit. Just say real numbers. That font is not replicable by students (easily) in writing. At least the fookers didn’t jam in some Fraktur (why I say my comment is minor). Or use the Greek letters “squiggle” or “other squiggle”.