I’m not sure I really feel like writing this post, particularly since I’ve already written it once: readers could simply read this PoSWW (and this WitCH would also not hurt), and I could go back to watching the Australians cheat at cricket. After the recent “right angle” screw-up on SM1, however, this post might be useful and the cricket will have to wait.
Definitions are one of the ricketiest aspects of school mathematics. This is not all writers’/authorities’ fault. Yes, when Cambridge nitwits totally stuff up the index laws I’m gonna yell at them, but the role of definitions is fundamentally tricky. To begin, one has to tread very lightly and much more often definitions will be implied rather than explicitly stated; the definition-theorem-proof formality that mathematicians love is simply not appropriate for the vast majority of school mathematics. Secondly, as students progress, inevitably some implied or stated definitions will change. It would probably not be a great idea, for example, to introduce multiplication to prep kids as a commutative and associative binary operation in a complete ordered field. Even if “repeated addition” can cause problems down the track, that very intuitive idea is where one must begin.
I have (at least) two posts about definitions on my to-do list. So, I will try to leave the general sermonising for the future and stick close to the matter at hand. Which is the notion of vectors, in this case velocity vectors, being at “right angles”.
Vector mathematics is the archetypal example of where the definitioning is tricky, and very year-dependent. As “vectors” change from physical “magnitude and direction”, to geometric “arrows”, to algebraic “coordinates” (and, finally, to elements of a vector space), their use and the language with which to use them changes as well. The clash that concerns us here is the concept of “right angles” and the zero vector.
In geometry, the concept of a right angle is clear enough: a quarter way round a circle, or half a line, or whatever. This is then fine for (geometric or coordinate) vectors as well, as long as we steer clear of the zero vector. But, for the zero vector we’re obviously screwed. It simply makes no sense to talk about the angle – a fundamentally geometric notion – between some vector and the zero vector. As such, the SM1 exam question is badly in error. The more interesting question is whether this issue with the zero vector can be fixed and, if so, how.
The most obvious “fix” is to simply try to duck the issue, at least on exams. There will inevitably be wonky definitions and half-truths at the edges of school mathematics, and it is simply sensible practice for exam writers to steer well away from these edges. In the classroom, it is possible and reasonable to note and briefly consider the edge issues but one is certainly not obligated to worry them or resolve them.
The angle 
between non-zero vectors 
and 
is of course beautifully captured by the dot product:
A geometric concept has been captured by an algebraic computation, which means we can, if we wish, broaden our definition. We can, if we wish, define two vectors and , including possibly the zero vector, to be perpendicular if 
.
So, with this definition non-zero vectors would be perpendicular exactly when they are at right-angles and, as well, the zero vector would be perpendicular to everything. Should we so define perpendicularity of vectors? I have two things to say in response:
1. Make up your bloody mind.
2. Yes.
Formality is not generally a good solution to edge issues in school mathematics but sometimes the authorities/writers simply have to state clearly what they mean and this simply does not occur with the concept of perpendicular vectors. At the bottom of the post are what amount to definitions from the three major SM textbooks (as of 2023), and they are an absolute mess. Moreover, to the extent that a definition can be discerned, the zero vector is precluded from being perpendicular to anything, which directly contradicts the “right angle” vectors in the SM1 exam question. VCAA, of course, is utterly silent on the whole damn thing.
Point 1 is self-evident but Point 2 takes an argument. Obviously “perpendicular”, if avoiding “angle”, still has the smell of geometry about it, and it is reasonable enough in the context of vectors to take “perpendicular” and “right angles” to be synonyms. One should not be distracted by word games, however. The mathematical question is whether the way the dot product works is an argument for switching to a new, broader definition, whatever we might call it.
Mathematicians have decided yes, and their term for the newly captured concept is orthogonal. That is, two vectors and are orthogonal if . This is now an entirely algebraic definition, the zero vector is orthogonal to everything, and no one has geometric grounds for objection. The new term “orthogonal” does not come pre-loaded with any denotations or connotations (unless one is Greek).
Why did mathematicians decide to admit the zero vector into the right-angle/perpendicular/orthogonal club? Because, in the end, and very quickly, the definition works better that way. This strays into a very general point about mathematical definitions but I will keep it short and localised, with the obvious illustration from Specialist Mathematics.
One of the standard topics of SM is vector projections/resolutes. So, given a vector 
and a non-zero vector 
, we can split up into two bits, one parallel to and one perpendicular to . So for example, with 
and 
, the parallel bit is 
and the perpendicular bit is … 
.
There is the point. For what we naturally do with vectors we inevitably wind up in situations where it makes sense to think of the zero vector as perpendicular. Does that clinch the case for Point 2? Absolutely not. There is an intuitive cost to including the zero vector, particularly if one insists with also sticking to the geometric language of “perpendicular”. And, at least at the school level, when one runs into this edge issue one can simply duck the question, as VCAA and all the textbooks do. I think it is better to bite the bullet and, in SM, go with the ultimately natural algebraic definition, but it’s a judgment call.
If Point 2 is a judgment call, however, Point 1 definitely is not: make up your bloody mind.
***************************************************************************
CAMBRIDGE
NELSON
JACARANDA
I was introducing my Year 7 students this term to the ideas about angles. They could illustrate the meaning of acute angles, obtuse angles, and reflex angles on the board. But when I asked them to describe – maybe with a diagram – the meaning of an angle of zero degrees, the class erupted into a debate. I love debates in mathematics lessons!
Nicely done.
The question of definitions appears to be woefully underdone. I started wondering what the definition of multiplication was a while ago and on googling it, remembered vaguely Pure Maths at uni and what you quoted above. But many appropriately qualified teachers wouldn’t have done that much Pure Maths at uni. You’re right – at the end of the day, it is grey and a matter of pedagogy and judgement calls on what is rigorous enough at what level.
I think of myself as teaching students to think like mathematicians, but I’m not actually one. But reading on this site and elsewhere has reinforced to me that my gut response is pretty good mathematically.
This is a bit rambling, but it’s an interesting topic that doesn’t really get raised elsewhere.
I agree that it is not helpful to over emphasise definitions in school mathematics. But it is helpful if the teacher is aware of definitions. I have worked in other fields which are relatively devoid of formal definitions – and education is one such field. What are the meanings of words such as “numeracy”, “patriotism”, “good”, “evidence based”. Even in mathematics the meaning of some words are elusive; e.g. “probability”, “surd”. On the other hand, graph theory, which students meet in Year 10 or 11, has too many definitions; these spoil an introduction to the subject. On the other hand, it is useful for students to know about the idea of a definition. You should know what you are talking about!
Thanks, Terry. Your second sentence is key. One should not be overemphasising the logical structure of the mathematics but the teacher needs at least some decent awareness of it. And, yes, because “proof” is relegated to a topic, within that topic the logical framing tends to be fetishistic rather than natural or illuminating.
Should we present a proof of Pythagoras’s theorem to all students? If so, when?
The reason that I ask is that it is taught at all levels of secondary school.
All that students remember is
. When I give them examples of numbers 
where this is not true they are confused. But that is what we were taught last year!
BTW, I now write the identity as
to remind students that 
is the length of the hypotenuse. Also in handwriting it is difficult to distinguish between 
and 
.
Yes. Primary school.
Just having teachers and students aware there is more to “it”, than stating “it” as a god-story would be a win.
In a previous life I was asked to do a little text book writing. It was a short lived experience; the publishers had no time for anything but god-stories.
Yes, “god-story” really captures it. David, of textbook fame, rails against the phrase “index laws” because these “laws” are inevitably presented and treated as if they were commandments from God.
In particular I don’t like the numbering of these index laws. For the life of me, I can’t remember what is meant by “index law 3”. I note that the Cambridge texts do not use this numbering.
We use the word “law(s)” in various ways. Laws of nature, Newton’s laws of motion, natural law, laws of chess, law of the excluded middle, law of war, Gestalt laws of perception. I interpret the meaning of the word by its context.
Do you also note that Cambridge does not have a damn clue why the index laws are true?
yes
Perhaps “rules for indices” would be better than “index laws”.
In teaching students about logarithms, I always provide the students with the following definition. “The logarithm of a number is the power to which the base is raised to give the number.” I like it because it tells you how to calculate a logarithm.
I recall an episode when I was coaching a young student. The conversation went like this.
Student: Today we learned about logarithms.
Me: Let me tell you what a logarithm is.
S: Oh, our teacher told us that today.
M: What did your teacher say?
S: The logarithm of a number is the power to which the base is raised to give the number.
Me: Wow!
Out of curiosity, since you say ‘Wow!’ — what other definitions are used?
I said “Wow” because she remembered the definition; it was not a comment on the definition. I would guess that most students could not define a logarithm even after it had been taught to them.
This captures the question of depth rather nicely – understanding is not binary – it’s more like layers of an onion and as teachers, the question is how far further to the centre of the onion should we go. Re: textbooks – their goal is simply to make money – https://dylanwiliam137385.substack.com/p/onions
Yeah, I guess …
Not fully refined but I’ve been doing a bit of pondering and here’s why I think it’s relevant. If we unpack the assumptions that underlie your comments, it would appear to be that both VCAA and academic publishers should have absolute rigour in their works.
If we take the metaphor that understanding is a gradual process like an onion, then we set far higher standards for some activities than others. In my guess that means that both works of VCAA and academic publishers should be able to fault-free when checked even by the most expert mathematician.
We accept that there are going to be faults in ordinary tests in classrooms because the effort required to get no faults is pretty significant and the people writing them don’t necessarily have the sufficiently high understanding that the top mathematician would (nor do they need to – you don’t have to be a top mathematician to be able to move people closer to the centre of the onion and improve their understanding. And of course as teachers we say ‘this definition is OK at this level but it’s not actually the real one’ – like your multiplication definition. We acknowledge that at this point we are using an approximation and that’s what the textbooks should acknowledge, rather than pretending there are numbered laws etc. I’m not sure if either VCAA or academic publishers would necessarily hold assumptions you do. And that’s part of the problem.
When I teach kids, my aim is to have them think like a mathematician. There’s a certain pattern of thinking that goes with any trade. One of mathematicians’ assumptions is that mathematics always works. Hence the considerable effort devoted to refining the concept and definition of limits. And the textbooks are not reflecting this. I suspect this assumption is lacking in both VCAA and academic publishers.
An interesting question would be, what principles guide the preparation of exams. To what standards? A simple letter to the Minister with this one question could be informative. I’m not sure what Bennett said on this.
It often helps to surface underlying principles. Another interesting letter to the minister would be simply to ask on what grounds is a CAS Calculator used. There’d be no other question in the letter, so it would be pretty clear whether or not it was answered and Minister’s letters always get some form of response.
Just a few thoughts ….
Thanks, JJ. I’ll think about it. I thought the Dylan thing was true but pretty trite, my general feeling about all those guys. But I think your querying about the standards of VCAA and the publishers is interesting and worth thought. I’ll reply properly later.
Thanks again, JJ.
I don’t think the “like an onion” thing works any better here than it did for Shrek. And, the idea that questions to VCAA and the Minister, no matter how clear and respectful, would receive anything other than boilerplate replies is pure optimism; feel free to give it a shot and let us know how you go. But the issues of the very low quality/accuracy of the textbooks and the VCE exams are there, and you raise interesting points.
I can see four general reasons why a textbook/examSAC/etc may not be up to scratch:
a) The curriculum and VCAA’s mandates handcuff the person(s) involved
b) The person(s) involved do not have sufficient time
c) The person(s) involved do not know enough
d) The person(s) involved do not care enough
i think all of these reasons can be involved at all levels, and the reasons bleed into each other. But, in brief, it seems to me:
*) For teachers, the weight is primarily on a) and b), with a bit of c).
Almost all SACs and school tests are bad, or worse. Yes, I really wish VCE (and all) teachers knew more mathematics, and if they did then they would assess the kids much better. But, the primary causes are that teachers have no time to to give the mountain of assessment proper attention, and VCAA’s garbage curriculum and SAC poison makes it virtually impossible anyway.
*) For publishers, I think it is primarily (c) and (d), with a decent amount of (a) and a little (b).
Clearly, the majority of textbook writers do not know the mathematics sufficiently well, and very clearly the publishers don’t give a stuff. Yes, VCAA makes it hard for publishers just as it is hard for teachers, but it is not *that* hard; there is no excuse for the crap they publish. And, yes, publishing is a busy job and there will always be rushed bits and errors and so forth. But if the publishers cared then they would at least fix the glaring errors, and they very seldom do.
*) VCAA, it is a huge amount of (a) and (b).
VCAA’s maths clowns do not know enough, VCAA’s inner sanctum of MAV and MathsEd twats do not know enough, and none of them give a flying fuck that they do not know enough. Awful, awful people.
I think the upshot of this is that I only partially agree with you. Yes, the lack of understanding is a big part of it all. And, there’s a way it is more than it appears: if they understood more then they would less vulnerable to the other reasons for failing. But, there are multiple reasons.
Finally, I think you have to be careful with terms such as “absolute rigour”. School mathematics should not and cannot be absolutely rigorous; there are always grey areas and judgment calls. And, a textbook is a large undertaking; it cannot be perfect. Even the legendary Fitzpatrick and Galbraith texts have clunks and errors. That’s life. But, what you must have is authors with sufficient mathematical nouse to be aware when they’re in the grey areas and to be able navigate those grey areas in a safe and sensible manner.
Finally finally, I am not sure the proper goal is to have school kids should think like mathematicians, even if this goal is infinitely more sensible than VCAA”s goal of turning them all into button-pushing automatons. I’ll have to think about this.
Of course publishers must make money. That’s true of all of us. However, for publishers, like other businesses, making money is not their sole aim. Cambridge University Press (CUP) has been around for a long time – about 500 years. I imagine that, initially, their purpose was to publish works by academics at the University – this was the case for most university presses. Now they have a broader purpose, but their core business is still publishing. One only has to look at text books over time to see how publishing text books has changed. To publish a book, you have to convince the publisher that the book will sell enough copies. Russell and Whitehead published Principia Mathematica with CUP – but Russell had to pay for the typesetting!
Making money is not the sole aim of publishers? How are the publishers’ actions any different from if making money *were* their sole aim?
If making money were their sole aim, they would choose a more profitable business such as mining.
The prosecution rests.
Sorry Terry – Chatgpt says “The academic publishing industry is significantly more profitable than the mining industry, with top academic publishers consistently achieving profit margins of 30-40%, which is substantially higher than the average for major mining companies.”
JJ, Thanks for the input; no need to be sorry. My point was that if the sole aim of a firm were to maximise profits, the firm would choose its industry accordingly.
And thus we forgive all firms in all but one industry.