## Beyond the pail

Earlier this week, a colleague tells me, a student walked out of one of his lectures in protest.

This wasn’t exactly the spirit of ’68 revived. The lecture was part of an introductory course in classical mechanics, and the student’s protest was against the use of vectors. More specifically, my colleague had been setting up a problem involving a particle moving under gravity, and had introduced the gravitational acceleration g as a vector. At this point, the student announced loudly that he knew from Advanced Higher Physics that this was nonsense and that there was no need to do mechanics using vectors, and he walked out.

So, what the hell was going on here? First, a look at Education Scotland’s material for Advanced Higher Physics makes it fairly clear where the student had got his ideas from. It’s a curriculum apparently designed to be broad but shallow, and to involve the student in as little maths as possible. Although vectors haunt the mechanics section, for example when forces or accelerations are resolved into components in perpendicular directions, the philosophy seems to be to conceal them wherever possible. When they are occasionally mentioned, the mention is not necessarily helpful, as when a torque is described as a vector “pointing out of the page” — meaningful, perhaps, for 2D problems, but liable to present vectors as a mystifying rather than a unifying concept.

Second, it is possible to teach basic mechanics without explicit reference to vectors (as far as I recall, Feynman gets as far as he can without them); it’s just that it makes things a great deal harder in the long run. Although kinematic quantities can be encountered in one dimension, this involves some important specialisations: in particular, students brought up on 1D kinematics often struggle to understand how in higher dimensions a particle can accelerate without its speed changing (as, for example, a body moving in a circle under a central force). Some aspects of momentum conservation are similarly overspecialised: in 1D, the velocities of two bodies after a collision are completely specified if it’s known to be perfectly elastic, but in 2D they aren’t (or the skill involved in snooker would be substantially reduced). Conservative forces are hard to teach without the concept of the gradient operator, and as noted above, central forces become almost incomprehensible. Later, to take just one more example, the dynamics of rigid bodies can be treated only in a fairly childish manner if vectors, and their big brothers tensors, aren’t available.

When we designed the mechanics class in question (yes, that was a plural “we”), we discussed the stage at which vectors should be introduced. We decided to bring them in as early as possible, before students new to mechanics had the chance to become hypnotised by 1D problems, and so that students brought up on the Higher and Advanced Higher version could get to grips with them while still in the comfort zone of their physical understanding. Evidently, this didn’t quite work. What seems to have happened is that the student in question has recognised the conflict between the “new” vectorial approach to mechanics and the one he “knows” from school, and has dealt with this conflict by angrily rejecting the new approach.

Possibly the right way to view this is through the lens of cognitive dissonance. It seems to fit the bill: for this student, evidently, accommodating vectors into his mental scheme of mechanics is very hard, and generates a strong emotional resistance — strong enough to drive him to challenge the social norms of the lecture room, in which passive rather than active resistance is very much the order of the day. There’s a particular aspect of dissonance that seems worth pursuing:

…counter-intuitively, perhaps — if learning something has been difficult, uncomfortable, or even humiliating enough, people are less likely to concede that the content of what has been learned is useless, pointless or valueless. To do so would be to admit that one has been “had”, or “conned”.

[Atherton J S (2011) Learning and Teaching; Cognitive Dissonance and learning [On-line: UK] retrieved 5 February 2013 from http://www.learningandteaching.info/learning/dissonance.htm]

The question this bears on is exactly why this student is so keen to defend his school-level understanding against the encroachments of university. (I don’t, by the way, believe that he is alone, merely that he presents the disease in an almost textbook manner. I’ve had students complain before that we must be teaching X wrong because it’s not what they were taught at school, and I’m aware of at least one academic appeal, at quite a high level, that was lodged on the basis that a subject “wasn’t the same as it was at school”.)

Threat and humiliation might well be part of it. I don’t know this student’s background, but I can imagine that some of our students found Advanced Highers “difficult, uncomfortable or even humiliating”, and thus would experience a strong revulsion to the idea that their effort hadn’t been worthwhile. I can also believe that some students find the demands of university education actively threatening, and kick rather blindly against it whenever an implicit slight is detected. I also suspect, though, that part of the problem is that, as so very often, the student’s existing understanding has been presented to him as a closed rather than as an open system.

By presenting something as an open system of understanding, I mean making it explicit that, for those who desire to go deeper, further modifications or extensions to their understanding will be necessary. For example, when the concept of a derivative is first introduced as the idea of “zooming in” to a curve, it costs the teacher little effort to say that for those who take maths next year, they’ll see a way to talk about “zooming in” more precisely. Next year, students meet the concept of a limit, and see the definition of a derivative in these terms. Again, the teacher can tell them that if they suspect this is still not quite watertight (Bishop Berkeley, where are you when we need you?) they should consider taking a calculus class at university. In their first-year undergraduate calculus class, they’ll meet epsilon-delta proofs, which rest on the properties of real numbers, and the lecturer might like to add that those who now want to ask what exactly a real number is would be well advised to take real-variable analysis next year… The idea, very much in the spirit of modern science, is not to present each level of understanding as a complete system that must be accepted or overthrown, but as a tentative and gappy theory which may need to be stretched and patched and generally accommodated, but which is an approximation one stage closer to the best there is.

The trouble is that — and perhaps especially for mathematics — the deck is stacked against any attempt to present a subject in open terms, particularly in school. For one thing, we have the tyranny of the crammed syllabus and procedure-heavy assessment regime, which make it so much easier to ignore digressions and uncertainties and to teach everything as a closed and complete system which will equip the students to perform their most important duty: satisfying the examiners. For another, we have the psychology of many students who opt to study maths. Frequently, to judge by my conversations with them, they like certainty. They like the fact that there are right answers (by which, typically, they mean “right” procedures to follow rather than rigorous proofs or close accordance with reality). They do not like to be told to worry about basic concepts or to refine an approach (for example, to revision) that they believe has served them well so far. And what applies to maths students, I fear, applies to at least some of their teachers, who have either had their enthusiasm for loose ends and open questions battered out of them by the demands of the chalkface, or never possessed much of this enthusiasm at all.

Where does that leave us? My colleague and I will try to identify the student who walked out; try to discover if we’ve reconstructed his motives accurately; try to persuade him that if he never learns to work in vectors then mechanics will be indescribably more painful than if he does. We’ll try to explain more generally to the class, yet again, why we’re teaching them the way we are; we’ll show yet more examples and set yet more exercises that demonstrate the power of vectors to unify and simplify the subject. But at the end of the day, I fear, we’ll always be up against that unarguable authority, second only to My Mate In The Year Above Says: the Olympian voice of But My Teacher Told Me To Do It This Way. Many teachers, I suspect, sometimes wish for the power to shape our students’ minds irrevocably: it’s as well to remember that this wish invokes the curse of the monkey’s paw.

Advocates of “real education” often quote an aphorism which is usually attributed to W. B. Yeats but more probably a paraphrase from Plutarch’s De Auditu: “Education is not the filling of a bucket but the lighting of a fire.” For university teachers, all too often, education feels like trying to light a fire in a bucket that’s already been filled to the brim with water.