I like gimmicks in my teaching. I’m not sure that it is always good that I like gimmicks, and I certainly don’t claim that you gotta get a gimmick. I hero worship the brilliant lecturers of yesteryear such as Mike Deakin and E. R. Love, as well as schoolteacher equivalents (about whom I must write), who were entirely free of gimmicks. But I am not them and I like the gimmicks. For me, mostly, I think they add value.
Some of these gimmicks are better considered eccentricities, and at least did not begin with the intention of being gimmicks. Sure, starting off a class or lecture by smashing a calculator with a hammer is a premeditated gimmick (and a very successful one, which I have employed sparingly), but most are not like that, or at least did not begin like that. I decline to wear shoes in my classes not so I will be known as “the weird teacher who doesn’t wear shoes”; I do not wear shoes simply because I hate shoes. Nonetheless, I am seemingly known as “the weird teacher who doesn’t wear shoes”, it has become a mark of difference, a gimmick. Similarly, as of 2016, I begin every (grown-up) lecture by downing a shot of vodka (at 9:15). That began as a joke, about Absolut Truth vodka, and as a toast to the wonderful Simon Pryor. But then it felt dishonourable to Simon to stop and so has become established practice. I have absolutely no idea what people think of this gimmick (and don’t much care).
Behavioural eccentricities aside, gimmicks within teaching have usually been concocted in the heat of teaching battle, to try to cut through the noise and to make a clear point. A good gimmick is discordant with predictable reality and so, like a good joke, it encourages the observer to step back a minute and to think outer-worldly about what is going on. One such gimmick, about which I have written, is USBB. The acronym stands for “Use Some Bloody Brackets”, and is my answer to the mindless fussing over BODMAS, particularly in upper primary and lower secondary school. Yes, kids must learn to interpret naturally ambiguous expressions, particularly when idiotic authorities are hellbent on manufacturing such ambiguities, but kids should also learn that BODMAS is primarily a pointless and idiotic game, and that the sensibly mathematical approach is to always attempt to write expressions in an unambiguous manner; the abruptness and contemptuousness of USBB is valuable in delivering this message. The byproduct of USBB and its directive is more use of brackets and thus more practice in understanding and manipulating brackets. Another acronym, which I have found even more useful than USBB, and which is the subject of this post, is “BISS”.
Explaining the role of definitions is a fundamentally tricky aspect of school mathematics. Kids naturally think of mathematics as stuff that is just out there, ready to be discovered (or, more accurately, handed to them). The idea that mathematicians may just be making it all up is a very foreign notion. This trickiness is made so much harder, of course, when textbook writers and curriculum authorities balls it up.
What to do. The first thing to do is a thing to not do: do not be a purist and go fussing on definitions when there is no proper purpose. If the kids are comfortable with the mathematical concepts in a topic at a functional level it is madness to then undermine this functional understanding with some masturbatory discussion on What This Really Means. One discusses a definition – in some careful manner, suitable for the kids at hand – when the lack of clear definition will likely be a source of confusion. I’ll consider three examples.
One topic where a lack of definition causes obvious, if not fatal, confusion for students is complex numbers. Complex algebra is very elegant and fundamentally easy, and students handle it well enough. That the students handle complex algebra, however, does not mean that they are at ease with it, that they are not unnerved by the other worldliness of square roots of negatives and so forth. This is not helped, of course, when the exam clowns or the textbook clowns screw it up, which of course they both do way too often, typically by the second sentence. These clowns must at least learn enough to stop writing 
.
That complex numbers are unnerving, however, is insufficient argument for providing a formal definition. Mathematicians fumbled productively with complex numbers for hundreds of years without really knowing what the things were, so kids can similarly get by for at least a year. Once the geometry kicks in, once kids have a picture of complex numbers in terms of the complex plane, they get it well enough, even if they still harbour doubts. It is a pretty quick step to turn that picture of complex numbers into a definition, as ordered pairs with appropriately defined operations, but that does not mean that the step should be taken. Formalism at high school almost always comes at a cost.*
The obvious topic where some broaching of definition is essential, and why I hammer it whenever possible, is the index “laws”. Here, the lack of definition screws up the kids no end. They understand, for example, what 
means, 3 multiplied by itself five times,** and then they naturally try to extend that understanding to 
: “three multiplied by itself –5 times”. The kids are doomed, of course, and the inevitable idiocy of the textbook ensures that they stay doomed, for years if not forever.
What to do. It is not as simple as declaring “give them the definition”, even if in the end that’s what it has to amount to. As I noted above, kids think of mathematics as stuff already out there, and at Year 8-ish they are not intellectually well prepared for a more nuanced approach. Particularly not when they are trying to get on with, for them, the main game of applying the index laws correctly. My approach is to first talk about prime numbers.
This plays out various ways depending upon what kind of (usually extension) class I have, and what is the focus of the class at that time. Typically, I will ask the class to start listing the prime numbers. They will begin, naturally enough, with 2, the more sure kids outnumbering the few who may tentatively suggest 1 is prime. At which point I will ask why not also 1. Are they sure 1 is not prime? I recite the standard line: “1’s only factors are 1 and itself”. So, why is 1 not prime? The surer kids will respond with the usual legalism, that “a prime number must have two different factors”, but usually they deliver this response hesitantly, with the good sense that this is a legal dodge and does not really explain why 1 is not prime.
I may at this stage go into the history, but usually I will first give the real reason why 1 is not prime: BISS. Which, after the puzzlement and/or laughter, I then spell out on the board:
Because I Said So
BISS is my abrupt and humorous way of stating 1 is non-prime by definition: I am telling you that 1 is not prime and, just like Hutton’s hams, there is nothing to argue about!
One could imagine this discussion being dismissed as the meanderings of a shoeless eccentric but there are two reasons why BISS seems to work very well here. The first reason is that it doesn’t matter much to the kids whether 1 is prime or not; they factorise happily or unhappily, for instance, without even thinking about 1. Since they are not that invested in what is or is not prime, they are more open to weird discussion about it. The second reason that it works very well is that 1 used to be prime.
That 1 used to be prime catches the kids off guard, of course, fundamentally conflicting with their idea of mathematics just being out there. As to why 1 used to be (defined to be) prime but is no more, there turns out to be a very simple and powerful analogy that the kids get: for exactly the same reason that Pluto used to be a planet but is no more.
Kids understand that Pluto is planet-ish but that we/scientists eventually decided Pluto wasn’t planet-ish enough and so we kicked it out of the club. Kids can then take their intuition that 1 is prime-ish, and so it’s not a great surprise if 1 was once considered a prime. Then, kids can take their intuition that 1, if prime-ish, is still kind of different, and so we/mathematicians can decide to kick it out of the club. 1 is now not prime, exactly as Pluto is now not a planet, because of BISS: because we now say so.
Whether 1 is prime is a very natural context in which to ponder the concept of definition with intuition and purpose, and BISS seems to be a clear and playful way of explaining how definition works: one cannot argue why a definition is true; a definition just is. Of course BISS cannot explain why we define something a certain way, which is important, of course, and which can be subtle and difficult to explain. Even for 1 and its disputed prime status, it is not quite as straight-forward as most people imagine. But the why a definition is chosen is secondary to the fact of a definition, and the former may be more or less appropriate for a given class.
With the idea of BISS more less understood, one can then approach the weird indices in a properly honest manner. Why is 
equal to 1? BISS. Why is equal to 
? BISS. Why is 
equal to 
? BISS. There is nothing to argue about!
The “why” underlying these definitions, the reasons the powers are defined as they are, is of course the index laws. We define the powers so that the index laws remain true beyond positive integer powers, and then can be proven to be true rather than accepted as commandments from God. That underlying “why” is much harder for kids to grasp and I’m happy enough if they come away with a decent inkling. But the not-why of the definitions, the BISS, can definitely be grasped and it is critically important that it be grasped, so that kids stop trying to make sense of powers in a manner that is fundamentally non-sensical and undermining of their functional understanding.
Once BISS has been established as a thing it becomes a very handy beacon. Although kids wrongly try to understand general powers in a manner that cannot work, the much more common problem is the opposite, of kids accepting something to be true without any proper thought. If I ask in a later lesson why 
equals 
, and even though I have previously explained why, there will always be a kid immediately suggesting “BISS”; they like things to be BISS, so they don’t have to think. This provides me the perfect opportunity to yell “No!”*** It is important that kids also realise that most mathematics is not BISS, that they can and should think about why something is true rather than simply accepting it on faith.
BISS also becomes a source of simple fun, an unmoored gimmick. The silliness of the word appeals to kids and there is plenty of entirely random calling out “BISS”, sounding exactly like the background Python knights calling out “Ni”. It is impossible not to laugh.
I really like BISS. It is my second favourite gimmick.
*) Fitzpatrick and Galbraith’s Pure Mathematics text from the 70s does the formal stuff as an optional, asterisked section. This seems the right way to do it.
**) Yeah, I am aware that is 3 multiplied by itself four times. Go away.
***) I really do yell.
The ad at the start reminded me of the one from Ulysses, which could be adjusted like this:
What is home without Plumtree’s Potted Meat? Incomplete. With it an abode of BISS.
A few weeks ago, I was teaching my Year 7 students about parallel lines and transversals. We arrived at alternate angles, corresponding angles, and co-interior angles. We saw that, if we know that vertically opposite angles are equal, then from the fact that co-interior angles are supplementary, it follows that alternate angles are equal to each other, and corresponding angles are also equal to each other. Near the end of the lesson, one student asked “But why are co-interior angles supplementary?” or words to that effect. I devoted the next lesson to presenting the answer. The reason is BESS. (E stands for Euclid – and his 5th postulate.)
Very nice.
This is exactly the sort of stuff that should be part of a Maths Method in teacher training.
I have a theory about teacher training. If I wanted to become an accountant, I would go to university and get a degree in accounting. If I wanted to become an engineer, I would go to university and get a degree in engineering. If I wanted to become a nurse, I would go to university and get a degree in nursing. Similarly, if I wanted to become a teacher I would go to university and get a degree in teaching.
If you wanted to become a carpenter would you go to a university and get a degree in carpenting?
Yes – if there were a degree in carpentry. I am not sure that there is such a degree but I expect that there would be qualifications in carpentry at TAFE.
My point above is that I reckon that this is how school students think. The thought that to be a mathematics teacher one would take a degree in mathematics first does not cross their minds.
If you’re dealing in hypotheticals proposed by a fictitious, ignorant student maybe signpost it? Plus, a degree in mathematics is of dubious value to a would-be teacher.
Do you really think that taking upper undergraduate math classes in college like “real analysis”, “algebra”, “analytical geometry”, “probability theory” has dubious value to a would-be math teacher? I assume that a middle/high school teacher should have ideas about what college level mathematics is, should be prepared (with Internet and textbook assistance) to answer non trivial student questions about mathematics. In addition to the “upper undergraduate math classes” there should be teaching specific math classes that discuss motivation behind sacrifices/shortcuts/simplifications common for school math and resulting intricacies.
P.S.: For the sake of argument there are college level “carpentry” programs: Pittstate has interesting “wood product manufacturing” major. CSU Long Beach has BFA and MA/MFA “Wood program”.
Thanks, Sergey. There’s something fundamentally odd about “college level” carpentry. On your main point, of course I recognise there is value in such undergraduate mathematics subjects, and I was being a bit of a Devil’s advocate. But I am not sure that such subjects are always or even typically as valuable as is knee-jerk assumed.
To begin, I think a teacher being able to answer “non trivial student questions”, while of course valuable, is largely off the point. The main point is that undergraduate mathematics courses should be, and to some extent are, valuable to the regular teaching of regular students on the regular material. To this end, for the good teaching of almost any mathematical topic at any level it is essential to have a sense of where that topic is headed, what the topic leads into: either the applications of the topic, or the next natural topic, or connected parallel topics, or a deeper and more mathsy coverage of the same topic. To this extent, pretty much all of undergraduate mathematics, where one sees the school topics applied and extended and connected and deepened, is worthwhile. But it is all evidently not as worthwhile as we would wish or might imagine. I shall consider the obvious example (and could similarly consider ten more at the drop of a hat).
How is it that every Victorian textbook screws up the index laws? Did the authors not do degrees in mathematics? Or were their subjects poor or unhelpful? Or were they bad students? All of the above?
I have no idea whether the authors were too ignorant to understand the nature of index definitions and the index laws, or whether they were too ignorant to understand that telling porkies about this really matters. Whichever, they screwed up. And I am not aware that any of the authors of any of the books are remotely concerned that they screwed up.
There is no reason to assume the textbook authors would be the strongest teachers in the state but, by Victorian standards, they are also presumably no slouches. But everything points to the fact that they just don’t get it, at least not nearly enough. Whatever benefit we might have hoped that the teacher-authors received from their undergraduate education, it was clearly not sufficient. Why not?
The obvious problem, it seems to me, is that too much of undergraduate mathematics is still cookbooky, and thus of limited worth in gaining an appreciation of mathematical thought. As universities become more mercenary and less principled, this problem is worsening. It is now at the stage where even “pure” subjects, such as real analysis and abstract algebra, are way too often way too much paint by numbers.
Even thirty years ago, undergraduate mathematics would not have been the best preparation for a to-be teacher. The goals overlapped a lot but still the goals were sufficiently different that specifically teacher-designed subjects would have been (and were) of greater benefit. But, the shallower undergraduate mathematics has become, the less there is overlap in the goals, and thus the less worth in undergraduate mathematics to a to-be teacher. To be sure, it is more valuable than the madness they receive in their education degrees, but that is to damn with no praise.
Took a while reading through this article then got to the reveal on what the acronym meant, and it made a very good point. I think there’s something to the fact that people who do well studying maths are the ones that can suspend their disbelief and accept the rules, without asking why too much. Asking why is usually a good thing, but with maths being so abstract, and with so much to learn, it can bog you down so much you never get anything done.
You’re making a slightly different point, but a reasonable point. I would say asking “why”, or not, at the right time, rather than too much or too little. Mathematicians get good at this but it’s pretty much impossible for a school student to have that intuition, and not at all easy for most teachers.
I well remember the index laws and how intuitive it was multiplying bases and adding the indexes, but got really confused when they said x to the power of -3. Yes, sometimes (very often) you have to accept the abstraction of mathematics, just accept the rule and move on.
Pretty much.
A teacher can and should give the (proper or approximate) motivation for the definition in some gentle form, but not much will normally stick. What a teacher cannot do, what those idiots at Cambridge cannot do, is pretend to *prove* the meaning of
and so forth.
Departments of Mathematics offer tailor-made courses on mathematics for engineers, or mathematics for computer scientists, or mathematics for biologists. Similarly they could offer courses tailor-made for prospective teachers. “Enrichment Mathematics for Secondary Students” would be a useful starting point.
I think there already are such offerings at some universities, and there are “Maths for Poets” subjects around, such as Burkard’s, which have a different focus but would definitely be of value for a to-be teacher and are taken by a few such students. I think it all amounts to window dressing.
In fact I wanted to go a bit further here and say the ability to simply accept abstract rules and apply them without necessarily understanding them is pretty much the core skill in learning mathematics. The rules of advanced mathematics (that you start hitting year 11 and beyond) are so abstract that you would drive yourself crazy trying to understand them there and then.
I know the point you were making, or hinting at, but I think you are making it much too strongly. It cannot just be an abstract game. At any stage, a student must have *some* sense of the “why”, even if the “why” simply amounts to BISS/definition.
Ok lets say the ability to suspend disbelief and get on with learning the rule is helpful when learning mathematics, ie having fractional or negative exponents.
Well, suspending disbelief is better than making shit up, which is what Cambridge does. But you don’t have to *just* suspend disbelief, and you shouldn’t.
“According to the formalist point of view, mathematics should be regarded as a formal game played with marks on paper.” (Paul Cohen (2006). “Set theory and the continuum hypothesis”, Dover, p. 3.)
I often use this idea when my students ask “Can we please play a game?”
RTQ, and his more extreme friend RTFQ were shared with me when I was a student teacher by a legendary (and edgy) mathematics teacher.
Kids love them and they sit on their shoulder, making a clear difference.
Now being put to good use by a generation after me.
I can guess the “F”, but what is RTQ?
Read The Question.
The F is employed for serial offenders.
Excellent.
The variation is similar to USBB versus USFB.