Here’s a suggestion that probably sounds more cynical than it actually is: the pressing question for educational theories should not be “how do students learn?” but “how do students avoid learning?”

Let me try to motivate this a little. Given any reasonably sane teaching method and any reasonably sane educational theory, it is possible to produce a decent defence of the method on the basis of the theory. As I discovered while I was doing my teaching certificate, this game can be played equally well with methods that are already widely used and with methods that aren’t, although the former is considered to be cheating. The only real difference between existing and untried methods, though, is that the former have already been shown to be flawed, while the latter have not yet been shown to be flawed. Despite this, in practice educational theories only tend to be used to justify innovations rather than tried-and-tested methods. This may go some way towards explaining why they are so often resisted by practitioners.

The other trouble with theories is that they rarely help us to analyse what’s gone wrong with our practice. (During the “peer assessment” exercises in my certificate, all discussion of what had gone wrong with a class was explicitly forbidden, possibly for this reason.) This isn’t the theories’ fault: they have generally been developed to answer the question “how do [or how should] students learn?”, while what we want to answer is “how did students fail to learn in this setting?” — or, if you want to sound less judgemental, “what did they learn instead of what I intended them to learn?”; or “what understanding did they construct instead of the one I was hoping for?”; or etc. However you phrase it, the tool is simply wrong for the job.

As all teachers know, our students find a remarkable variety of ways in which to fail to learn what we intend. We carefully provide multiple representations of concepts; we supply peer learning and feedback opportunities; we revisit key ideas in a diligent spiral; we design situated problems and integrate across the curriculum; we nurture reflection, metacognitive skills and self-directed learning; and somehow a disturbing proportion of our students negotiate this as if it were an unwelcome obstacle course, developing all sorts of ways to carry out the tasks in front of them without taking the intended benefit from them. If we’re to do better, surely we have to understand the strategies that they develop — however innocently — to circumvent learning.

I’m not thinking here of direct avoidance or disengagement, though of course these are a problem. (It’s a lot harder to help a student who isn’t there and doesn’t do the work than to help one who is and does.) What’s more interesting is engagement that is diligent and apparently sincere, but largely fruitless; neither the student’s nor the teacher’s “fault”, but nonetheless intensely dispiriting for both. What’s going on here?

One context in which this does seem to have been studied is that of “word problems”. More than two decades of work (see e.g. Ruesser 1988; Schoenfeld 2012) have investigated a pretty universal observation: when mathematical tasks are presented in a “problem-solving” context, students often manage to “solve” them in ways that are unrelated to the logic of the task in front of them. This can be demonstrated by setting students problems that don’t in fact have a solution, and observing how many students nonetheless “solve” them. Here’s an example from Ruesser (1988), as described by Schoenfeld:

Reusser taped [primary-school] students working on the following problem:

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?A typical “solution” was

, this is too big… and , this is still too big… while . That works. I think the shepherd is 25 years old.

According to Schoenfeld, some studies have found that students become more likely to force solutions this way, rather than less, as their mathematical education proceeds.

Ruesser (1988) suggests — and subsequent work seems to agree — that students see the context “this is a maths problem” as setting up an implicit contract, the articles of which include: the problem will have a solution; this solution will be unique; all the facts stated in the question will be used and will be relevant; the mathematical operations will not be longer or more complicated than the context suggests; the answer will come out nicely as whole numbers or simple fractions. This contract is developed, presumably, by exposure to lots of problems for which these articles do apply, and the trouble is that by assuming that it applies students can often find ways to reach a correct answer without following a correct line of reasoning.

Having developed this theory of where the trouble lies, we can in principle do something about it: we can set problems that — repeatedly and in various ways — violate the implicit contract; we can try to find ways of assessing the students’ attempts that make it clear that the final answer is less important than the working. (“In principle”, at least for me, because by the time students reach university they’ve had a dozen years of reinforcement of the contract, and while I try hard I simply don’t have the time to overturn all their misconceptions.)

Note that a theory of non-learning needn’t be divorced from theories of learning: presumably bad ideas and habits can be formed and sustained through much the same mechanisms as good ones, so understanding these mechanisms ought to help us in both cases. The emphasis might not be the same: I suspect, for example, that a fully developed theory of non-learning would draw more from behaviourism than is currently fashionable for theories of learning. The big difference is that we would have to look much more at what is implicit and unintentional in our practices, with a very sharp eye for the law of unintended consequences. For example, we don’t like setting students problems with messy numbers, because we’re worried that then they will be derailed by arithmetical manipulations and unable to engage with the real point of the exercise. This is laudable, except that for students the exercise then ends up providing further evidence that the “expect nice numbers” heuristic works — not the real point we were trying to make.

Often, I fear, trying to develop our theories of non-learning, or to act upon them, will leave us feeling that we’re violating our students’ trust, trying to trick them or catch them out. We’re trying, after all, to find out what they actually do and why, not what they think they’re doing or what they think will please us if they do it. We may have to be sneaky — as a very simple example, never setting multiple-choice questions (e.g. indefinite integrals) in which it’s easier to work backwards from possible answers than to work forward from the question. We may have to be a lot more sceptical about giving a student credit for being “almost right”, and will certainly need to turn down our telepathy when we mark poorly-written work. We will probably, in short, be perceived by our students as total bastards. And somehow, throughout this, we will have to retain enough of their trust that they will continue to engage with the tasks we set them, in the hope that sense will be made in the end.

None of this, I suspect, is peculiar to maths: indeed, Ruesser provides a wonderful example in which education students were given an deliberately nonsensical passage presented as “the latest educational theory” and asked to discuss it — which they did, conscientiously and without challenging the exercise. It may be worse in maths than elsewhere, since so many students choose to study it at university for exactly the “wrong” reasons: they like everything to have a definite answer and to be neatly self-contained and unambiguous. But perhaps, for that reason, maths education is where it should start.

Any volunteers?

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