But the end of Mr. Brooke’s pen was a thinking organ, evolving sentences, especially of a benevolent kind, before the rest of his mind could well overtake them.
(George Eliot, Middlemarch, chapter XXX)
Last year I was involved in an HEA workshop on “contemporary challenges” affecting university maths teaching. It was a good workshop, though not for any reason connected with my contribution. I was supposed to be speaking on the use of technology in lectures, but the more time I spent preparing my talk, the more sceptical I became that it was a useful topic to address. There are two reasons for this. One of them I’ll try to describe only briefly; the other I want to discuss in more detail, and with reference to a pickled fish.
The first reason why I’m sceptical that technology in lectures is a good topic is that I’m increasingly convinced that — pace conventional wisdom — lectures are not where the greatest problems in university mathematics lie. Yes, there are plenty of crap lectures and crap lecturers out there, and there is plenty of space for improvement. However, there is no evidence that this is the real bottleneck in the learning process. It’s been argued repeatedly, and I believe accurately [see e.g. Pritchard (2010)] that there is no point assessing the educational value of maths lectures in isolation. Lectures are only meant to be effective as part of a wider programme of activities, such as the classical lecture–homework–tutorial cycle, and if students complain that they aren’t learning from lectures this is as likely to reflect a misunderstanding of how learning occurs as it is to reflect an actual deficiency in the lectures. Indeed, there are a few straws in the wind that suggest the problem lies outside class: evidence that students who rely on lectures rather than on other resources perform better [Inglis et al. (2011)]; lack of evidence that the affective benefits of clickers (which definitely do spice up lectures) translate into improved performance [King & Robinson (2009)]; and above all the persistent evidence [e.g. Bekhradnia (2012)] that maths students only spend about as much time working outside class as they spend in class, a ratio that no maths lecturer I know regards as sufficient.
If the problem lies with students’ independent study rather than with what happens in class, then no amount of deploying audio-visual equipment, clicker systems and the like will improve their learning much — it’s like widening part of an A-road to dual carriageway while other stretches remain single-track with passing places. (Actually, that’s not a bad description of the old A9, but I digress.) There are plenty of technologies that on the face of it could do plenty to enrich students’ independent study: plotting and visualisation tools; computer algebra systems; even the maligned craft of programming. (The 2011 JMC report Digital Technologies and Mathematics Education gives a nice round-up of the possibilities.) I’ve asked around, though, and I know I’m not the only lecturer who is often bemused by his or her students’ resistance to using most of these technologies — there seems to be a mental barrier separating “technology” from “maths” for many students. It might, of course, be that all their lecturers are so technologically illiterate that we drag the subject back into the dark ages with the rest of our practice, but frankly I don’t think so — with the sole exception of Wikipedia in final-year projects (sigh), in my experience it’s almost always the lecturer urging the student to accept a technological tool, not vice versa.
We’re left with a quite unsettling apparent paradox. Mathematics underpins practically every piece of modern technology; this technology promises to assist the learning of maths in many ways; yet attempts to deploy this technology in maths education are often highly disappointing. Why does this happen? This is where the pickled fish comes in.
The pickled fish in question is to be found in Samuel H. Scudder’s classic essay, which describes the first days of his scientific apprenticeship under the geologist and zoologist Jean Louis Rodolphe Agassiz. On his first morning in the lab, Scudder was presented with the fish and told to look at it. Within ten minutes, he was bored. After a few hours, he had become very bored, and in desperation had resorted to drawing the fish. A little later, discovering more and more features of the fish that he had failed to notice before, Scudder was hooked. He spent three days in total looking at that one fish, a time he later described as “the best entomological lesson I ever had — a lesson, whose influence has extended to the details of every subsequent study; a legacy… of inestimable value”.
Scudder’s pickled fish is the epitome of a task, and of an educational philosophy, that can be found in many other disciplines: for example, in Prof. Jennifer Roberts’s requirement that her art history students spend three hours in close study of a single painting. Prof. Roberts describes her approach as “teaching strategic patience”; Prof. Daniel Willingham, in the blog post from which I rediscovered Scudder’s fish, argues that this is a skill that is underdeveloped in many contemporary students. Briefly, he argues that students who have grown up in an information-saturated environment, with immediate distractions always readily available, are likely to have learned how to make very rapid judgements about the interest value or otherwise of a resource or an activity, but not to have learned how to persist with a mildly boring task until it becomes interesting. At a purely anecdotal level, this fits many of my students well. They’re far happier than I remember being at making quick (and often accurate) decisions of the sort to be found in quizzes and multiple-choice questions, but they are also far readier to declare that something is “boring” or “unhelpful”. This attitude isn’t necessarily a form of intellectual passivity, though it easily slides into passivity: rather, it’s an expectation about the timescale over which the reward of an activity becomes apparent.
Here, I think, we can start to explain the mental barrier that, for many students, separates technology from mathematics. Consider the rhetoric that surrounds anything marketed as “technology”. (From this point onwards, I’ll capitalise the word Technology when I’m referring specifically to that which is presented or perceived as technological in this sense, so a smartphone is Technology whereas, say, an orbital gravity measuring system is merely technology.) A device such as a smartphone is supposed to be intuitive to use; it’s supposed to produce results rapidly; it’s supposed to facilitate multitasking or rapid switching between activities (Facebook, texting, music…) While such a device can be used in other ways, certain values are associated with it, and these values are orthogonal to those of fish-sketching and associated practices — into which category, I’d argue, genuine mathematics frequently falls.
I’d like to suggest the following. When equipped with non-Technological equipment, such as a pencil and paper, students may be able to accept that they are working within the world of mathematics, where values of persistence and concentration apply. (By no means all my students do accept this, to judge by their work, but enough of them seem to enjoy doing long calculations by hand that I suspect there’s something to it.) When equipped with a piece of Technology, in contrast, they’re pulled in two directions: the task may ostensibly demand attention, but the Technology creates quite a different expectation, in which speed and multitasking are valorised. (I’ll leave aside for now the evidence that multitasking is in any case an illusion [Sanbonmatsu et al. (2013)]: what matter for this explanation are students’ expectations rather than their abilities.) A useful way to express this might be to say that the “didactic contract” between teachers and students is at least partly formed through the expectations introduced by technology. Interestingly, although I have seen the argument [Pierce et al. (2010)] that this occurs, the assumption was that handheld technology was more likely to deepen learning than to impair it — unless, of course, undermined by those rotten ol’ technophobic teachers…
The hypothesis I’ve outlined above implies that the most popular applications of technology among maths students should be those that least confound the Technological expectations of easy comprehensibility and swift gratification. Thus, quizzes or tests consisting of short questions to be tackled quickly should be popular; material that is presented as easily-digested and downloadable — the educational equivalent of iTunes tracks — ought also to appeal, though perhaps to be praised more than it is actually used; powerful tools such as Maple or Matlab, which demand a period of apprenticeship to use effectively (and which involve “tasks completely absent in the mainstream computer use culture”, to quote the 2011 LMS policy statement on ICT) should be the least popular. Approximately, at least, this is what I seem to observe: students are reasonably enthusiastic about clickers; they demand reams of downloadable material but rarely seem to read it; and many of them give the impression that if trapped in a Maple worksheet they would gnaw off their own limbs to escape.
Does this really matter? Here I think we can identify two schools of thought, though intermediate positions can be formulated. One school, which tends to use phrases like “blended learning”, argues that it is always worth trying to provide new resources and to harness new technologies, allowing students to combine them in whatever way works best. This attitude seems, for example, to inform the 2010 NUS report on students’ perspectives on technology. The other and more cynical school, to which I belong, observes that providing extra resources and using them both carry a cost, even if this cost is solely the investment of time and effort required for students to decide which resources to use. This school notes that in practice students seem to settle exclusively on a small subset of the available resources [Inglis et al. (2011)]; it notes that since many of these students are much better at making snap judgements than considered decisions their choices are not likely to be optimal; and it concludes from this that a form of Gresham’s law operates: bad learning methods can drive out good ones.
None of this is remotely novel. Roughly two millennia ago, Plutarch wrote his famous piece of advice to a young man with an interest in philosophy, cautioning him against the accessible but undemanding performances of the sophists and commending to him instead a much more demanding course of action. A few centuries before, a philosopher had informed a king that there were no royal roads to learning. (Whether the philosopher was Euclid, addressing King Ptolemy, or Menaechmus, addressing Alexander the Great, apparently depends on whether one believes the Prologue to Proclus’s Commentary on the Elements or Stobaeus’s Eclogues, II.XXXI.115. I’m not taking sides.) The trouble with educational technologies that don’t challenge the expectations of rapid reward and minimal effort is that they suggest to students — and, more worryingly, to managers and policy-makers — that royal roads to learning have now been built. At eighteen years old, I’d have been daft enough to believe this: why shouldn’t my students be the same?
Where does this leave us? Similar anxieties in the computer-science community were what initiated the back-to-geek-basics Raspberry Pi project, and although we’ve yet to see the, er, fruits of this, the spirit seems to be commendable. Can we find ways to embed mathematical fish-sketching into Technology in a manner that doesn’t compromise the former (and so perhaps subverts the values of the latter)?
In principle, there is. One approach might be called in general “experimental mathematics”, in which a flexible computational tool is used to explore the behaviour of a mathematical system and thus to develop or refine conjectures which can then — perhaps — be proved or disproved. The tool acts to allow students to bypass some of the cognitive burden associated with lower-level processes and focus on the more interesting parts of the problem [vide Perkins & Unger (1994)]. Examples might include using a package like GeoGebra to investigate relationships between lines drawn in a circle and the angles between them, or using ODE solvers to simulate the behaviour of an elementary but complex mechanical system [e.g. Peter Lynch’s excellent pages on the spring pendulum], or using a package like Maple to generate thousands of terms in a numerical sequence and search for patterns in them.
The catch, in comparison with fish-sketching, is that while one can probably find at least one distinct and accessible fish per biology student per year, it is hard to find one distinct and accessible mathematical investigation per maths student per year. There’s thus an awful tendency for such investigative projects to become the equivalent of the “experiments” I did in early secondary school, in which the aim was to obtain the result on the next page of the workbook, and a smart but impatient adolescent readily concluded that turning over the page was an efficient alternative to doing the experiment. (In my defence, three quarters of these experiments seemed to involve titration, which is no fun when you’re colour-blind.) A gap thus opens between the ostensible purpose of the exercise, “determine what happens if…” and the actual purpose, “learn how to find out…”. (This gap is one explanation for some of the behaviour we class as plagiarism, but that’s another story…)
The other catch is that mathematical fish-sketching, whether technology-assisted or not, is not going to capture for maths the affective benefits (if benefits they are) of Technology. Plenty of people hope it will. Here’s the JMC Digital Technologies report again:
Unless we can develop mathematics education in a more stimulating way, which takes into account the modern world and students’ interests we are in danger of turning mathematics into an increasingly “dead language” and alienating groups of students whose mathematical potential will remain undeveloped.
Note the assumption: mathematics in itself can’t possibly be stimulating enough; it doesn’t have the values that students come in with; so maths is at risk of becoming a “dead language”. Because, as we all know, the point of university cannot be to draw students into a different and wider world than the one they knew beforehand; and nobody has ever found the study of a dead or an unfamiliar language to yield them any insights of that sort whatsoever. Rather, we should be telling students more of what they already know, letting them do more of what they already do, and warping every subject into a shape already familiar to the ignorant. Sorry, but no. If maths isn’t good enough for my students it isn’t good enough for them; I’m happy to do all I can to bring out the intrinsic interest and satisfaction that it possesses, but I’m buggered if I’m going to try to compete in the “stimulation” stakes with Sugar Rush Saga or whatever other distractions my students are used to using to get the endorphins sloshing about their brains. If it’s a choice between being stimulating and efficient and Technological and not doing maths, and being inefficient and antiquated and doing maths, I know which I’d prefer.
Which brings me to my final thought, which is that being inefficient — in the sense that the acolytes of Technology mean it — might not be such a bad thing after all, because I’m increasingly persuaded of the importance of what one might call “serendipitous inefficiency”.
An example of serendipitous inefficiency is that trip to the library in search of a piece of information that nobody had yet bothered to digitise, which leads to a journey through unexpected worlds that intersect with the expected in a single classmark on the library shelves. Julio Alves hymned this discovery of “unintentional knowledge” in an article in the Chronicle of Higher Education last year, and it’s something I’ve experienced myself: my best research paper — or at any rate my most cited — was the consequence of a dusty afternoon in the stacks of the Engineering library of the University of Bristol. Conversely, I’m increasingly conscious of the mental impoverishment that many of my students face precisely because they’ve managed to screen themselves from all sorts of unintentional knowledge.
Perhaps another example of serendipitous inefficiency can be found in the example that technoevangelists like to trot out against people like me. Didn’t your kind, they ask, make exactly the same sort of objections when aural learning was displaced by writing? Didn’t they decry the ill effects and mental ill-discipline that would result when people committed their knowledge to a parchment or a papyrus instead of to their brains? And don’t all these protests seem rather silly now? Well, yes and no. Of course we learned to use the new technology efficiently, and of course civilisation didn’t collapse overnight; but in most disciplines the ability to commit large chunks of material to memory and retrieve it intact remains hugely beneficial. Listen to a competent advocate, or a successful scientist, or anyone else who makes it their business to marshall and deploy ideas, and tell me that aural learning was completely superseded. Maybe at the time when writing started to displace memory there were those who argued that listening to and retaining information was now a “dead language”. We don’t remember those people, and perhaps there’s a reason for that.
My last example of serendipitous inefficiency is rather trendier. The recent study by Mueller & Oppenheimer (2014) which compared note-taking using laptops with note-taking in longhand has drawn the attention of periodicals including Scientific American and the New York Times. Their suggestion, which seems to resonate with other recent work both on handwriting and on classroom technology, is that the very difficulty of taking notes longhand forces students to summarise and to condense as they write: they record less information verbatim, and as a consequence they retain more of what they hear. I don’t know how this translates to the maths classroom, where verbatim copying of equations has traditionally had a more substantial role, but the general principle behind it seems very plausible: working in a resistive medium, one takes more time and more thought, and with practice the very resistance of the medium produces something more tightly focused and more memorable. Visual artists sometimes talk about the benefits of working in a medium such as oil paint that demands time and effort from the painter. I know from my own experiments in poetry that writing within a scheme of rhyme and meter is harder than writing without it, but also that a poem that results from such a process has a far better chance of being, in Don Paterson’s terms, an efficient “machine for remembering itself” (101 Sonnets, xiv).
In their opening paragraph Mueller & Oppenheimer also make a fascinating little observation, which is that although students tend to believe that using laptops in class will improve their learning, those students who do use them tend to be less satisfied with their education than those who don’t. In other words: after being seduced by Technology, omne animal triste. Those much-vaunted affective benefits, it seems, might not last long.
So, back to Scudder’s fish. I can’t give each of my students their own individual pickled fish to study, because I don’t have one. I can’t persuade them that studying a pickled fish is what they’ve always wanted to do, because it isn’t. I can’t tell them that studying a pickled fish will be fun, at least in any way recognised by the outside world as fun, because it won’t. I can tell them to sit down in front of it, to remove all the distractions they can, and to start working with those most resistive of tools — the pencil, the memory and the intelligence — to learn with them, and through them, and to get to know that pickled fish as it never knew itself. Once they’ve done that, they can use whatever technology or Technology they like to record their findings and to share them with anyone who might be interested. But first I want them to do things the hard way, and to learn that fish.