## On the threshold of what?

A contact, who I’ll not identify beyond saying that he’s another HE teacher junior even to me, is currently enduring an education course as part of his probation. Such courses often seem to reflect the research interests of the course leaders rather than the needs of the probationers; the course leader in this case is a big fan of “threshold concepts” (TCs) and has constructed the course around them. Thanks to my contact, I’ve found myself for the first time acutely rather than vaguely aware of the idea and forced to give some thought to what can or can’t be got out of it.

I’m not going to describe the idea in detail here, because Mick Flanagan at UCL has done so very thoroughly on his Threshold Concepts webpage, and has provided lots of links to other articles. (I’d not previously realised how widely the term has caught on: as so often in education, theory-friendly Australia seems to have embraced it with more enthusiasm than the cynical British Isles, and it has remained marginal so far in the mathematical education literature; but this may change.) To quote Flanagan quoting the original gurus, Jan (Erik) Meyer and Ray Land,

‘Threshold Concepts’ may be considered to be “akin to passing through a portal” or “conceptual gateway” that opens up “previously inaccessible way[s] of thinking about something”.

TCs apparently have roughly eight “characteristics”, but the defining one is summarised by Flanagan as follows.

Examples of the threshold concept must be transformative and involve a traverse through a liminal space. They are likely to be characterised by many of, but not necessarily all of, the other features listed above.

Rough translation: TCs change the learner’s way of looking at things, and there’s a messy transition period when the learner is confused about what they understand. (It’s interesting to note that my contact was given as a starting point a paper by Meyer & Land (2003) in which this defining criterion is lacking, allowing the notion to burgeon significantly more than Flanagan’s summary would allow. I’ll return to this below.)

The metaphor of a threshold is natural enough. We are aware as teachers that certain topics cost our students more to master than others, and both as teachers and learners that when we look back at some topics they have done far more than others to open new territory to us — so the rather Narnian analogy of stepping through a narrow gateway into a new landscape can often seem appropriate. Neither do the characteristics outlined by Flanagan seem unfamiliar: crossing these thresholds may be “transformative”, “troublesome”, “irreversible” and so on. At this stage, though, in encountering a new theory, I find myself craving specific examples, and this is where my doubts begin.

Let’s start with a fairly convincing example. Easdown (2009) [Int J. Math. Ed. Sci. Tech. 40(7), 941-949] describes mathematical proof as a threshold concept. This paper emphasises the tension between “semantic” and “syntactic” reasoning, and illustrates the vulnerability of syntactic reasoning when a semantic understanding is lacking. This semantic/syntactic tension echoes the tension between “concept images” and “concept definitions” described in Alcock & Simpson’s Ideas from Mathematical Education, and is indeed discussed in their chapter 3; a shorter discussion can be found in chapter 2 of Cox (2011). It is easy to fall into the trap of seeing concept images or semantic reasoning as non-expert forms of thought, and concept definitions or syntactic reasoning as expert forms (for an example see Wormley’s abstract on p. 111 of this conference booklet). However, even expert mathematical reasoning involves a complex play between these formal and less formal modes of thought, as Alcock and Simpson make clear; Davis & Hersh’s Mathematical Experience is a famous exposition of the point.

This need to hold different modes of thought together is what makes proof so hard, especially for students accustomed to think of maths as entirely procedural manipulation or of reasoning as driven by plausibility rather than proof. When definitions become detached from images (or syntax from semantics), the reasoner becomes vulnerable to careless “formulaic” errors of the kind discussed by Easdown — such as his wonderful one-line “proof” of Cayley–Hamilton. When images become detached from definitions, the reasoner is easily bamboozled by irrelevant associations or other deficiencies in his/her images. (This is one reason why Halmos’s advice to stockpile examples for any concept is so sensible.) Learning to construct and manipulate proofs often does take the learner through a “liminal space”, i.e. a period when they are somewhat confused about what they’re doing, and during which their apparent ability may even decrease for a while. James Atherton calls this “learning as loss”, and he notes its overlap with TCs; there’s a whole topic here related to “self-theories” and vulnerability to failure that I hope to return to in a later post.

Unfortunately, Meyer & Land (2003) offer two examples of “threshold concepts” in mathematics that seem less convincing than Easdown’s. The first is the concept of complex numbers; the second, of the limit of a function. These are evidently core concepts, in the sense that they are foundational to large chunks of subsequent mathematics; they are also, speaking from experience, concepts that quite a few students run into problems with. (It must be remembered, though, that for many students very few concepts in mathematics are unproblematic.)

I’ve taught introductory complex numbers to many hundreds of students over the years, mostly on engineering rather than maths courses, and I’ve yet to see convincing evidence of a substantial liminal space. Most students grasp the arithmetic and algebra of complex numbers readily, and also adapt quite easily to the interpretation of complex numbers as points in a 2D plane. (This is naturally presented as an extension of the “number line” familiar from primary school, which is a useful demystifier.) Students do often struggle with the process of calculating arguments, but less from lack of understanding than because an entirely different problem — the non-equivalence of “$\tan(\theta) = a$” and “$\theta = \tan^{-1}(a)$” — rears its head. (My evidence for this is that students have relatively little difficulty sketching numbers anywhere in the complex plane or calculating arguments in the range $-\pi/2 < \mathrm{arg}(z) < \pi/2$; it is in the other quadrants that problems occur.) Many students also struggle with the exponential form, but my impression is that this reflects simply their lack of fluency with exponentials in general — a topic that seems to receive too little attention in school.

Complex numbers, in fact, may be a good example of a concept that was liminal in the historical development of mathematics — hence the mystifying terminology such as “imaginary” and “complex” itself — but that need not be liminal for students. (As a side point, Meyer and Land’s attempt to explain real numbers as “those we all deal with in the ‘real’ world; numbers we can for example count on our fingers” suggests that they too may have been confused by the terminology.) Those students with a philosophical bent are liable to find complex numbers problematic if it is the first time they have been forced to ask what a “number” is — and there were several shifts in this concept over the millennia to incorporate first irrational numbers, then negative numbers, before the arrival of imaginaries. Nevertheless, mathematics and the philosophy of mathematics are not coterminous, and few sane people these days would suggest teaching maths in a manner intrinsically front-loaded with philosophy any more than they’d suggest teaching it starting from foundational set theory. (What such “New Math” approaches failed to recognise was the phenomenon we now call “didactical inversion”: it’s often easier to learn even a logically structured subject in messy and indirect ways and to rebuild the formal structure once one has a rough map of the territory.)

The limit of a function is also an interesting example, though not (I think) of a threshold concept. Meyer & Land (2003) state that:

The limit as x tends to zero of the function f(x) = (sine x)/x is in fact one (1), which is counter intuitive. In the simple (say, geometric), imagining of this limit is the ratio of two entities (the sine of x, and x) both of which independently tend to zero as x tends to zero…

I feel that the authors’ evident nervousness about the notation, as well as their problems with grammar, are revealing. The basic concept image of a limit is not generally hard to grasp, although as Artigue, quoted by Meyer and Land, indicates, the everyday associations of the word may cause problems (as they do with “imaginary” numbers). This verbal pollution of concept images is a widespread problem in mathematics, as in other fields (e.g. quantum physics or philosophy) where ordinary language is being repurposed, and a teacher must be alert to such semantic leakage. But there’s no need to restrict ourselves to verbal approaches to concept images. Plot the graph of a function; trace the graph with your finger; where your finger goes as $x \to 0$ is the limit of the function as $x \to 0$. What was notoriously troublesome, again historically, was making this intuitive definition rigorous and mopping up the awkward cases, and the story of mathematics from Newton to Cauchy is somewhat bloodstained as a result. (Berkeley’s Discourse Addressed to an Infidel Mathematician is the classic contribution.) But the difficulty here is once more that of learning to work with a formal concept definition which at first seems disjoint from the intuitive concept image, and later forces the image to be refined. Formal epsilon-delta arguments are infamously problematic for undergraduates, but for many these arguments are the first they have been forced to conduct rigorously except as highly circumscribed ritual exercises.

A paper by Szydlik (2000) [J. Res. Math. Ed. 31(3): 258-276] is relevant here. Szydlik argues convincingly that the idea of a limit involves several concepts for which students may have inadequate or misleading concept images: real numbers, functions and the (potentially) infinite. Students’ ability to make use of the formal concept definitions then depends both on their existing concept images and on their epistemology — whether they see mathematics as an arbitrary construction imposed by authority figures or as a system with its own internal consistency. The difficulty crossing the apparent threshold of “limit” thus seems to be merely a manifestation of the more pervasive problem of reconciling formal and informal modes of thought.

A paper by Scheja & Pettersson (2010) [Higher Ed. 59(2): 221-241], which looks in detail at students’ understanding of the notion of a definite integral as a limit, appears at first to support Meyer and Land’s identification of “limit” as a threshold concept. In fact, they focus on a particular instance, which is the limit appearing in the definition of the definite integral. The definite integral emerges from this paper as a much better candidate for a TC, because — as Scheja and Pettersson note — through the Fundamental Theorem of Calculus students are forced to connect integration as an already familiar algorithmic procedure (anti-differentiation) with the previously separate geometrical notion of calculating areas. This is genuinely heady conceptual stuff, which is why the Fundamental Theorem is called a Fundamental Theorem…

What Meyer and Land’s mathematical examples suggest is that, with a little misunderstanding and possibly an approach more informed by the history or philosophy of maths than by the subject itself, numerous troublesome concepts can be presented as threshold concepts when in fact they merely reflect troubles that are rooted elsewhere. Such an over-enthusiasm to identify TCs is also encouraged when, as in Meyer & Land (2003), no minimal definition is given. But does this really do any harm?

In one sense, no it doesn’t: Meyer and Land have an idea to sell as widely as possible, and if they want to think of $\sqrt{-1}$ as a threshold concept, it doesn’t hurt me any more than if Lacan wants to think of it as equivalent to his willy. Similarly, by considering whether a threshold is a useful way to look at $\sqrt{-1}$, I may be able to sharpen up my own thinking on how to teach it and where the problems really lie. (Lacan’s approach is probably not helpful here.) As a proponent of the use of multiple mythologies to inform teaching practice I’m reluctant to turn down another contribution to the myth-kitty. Before accepting the idea, though, I’m going to look at it in terms of that rough-and-ready distinction: can threshold concepts usefully be considered as a metaphor, a myth or a state religion?

First, let’s examine the metaphor of the threshold a little more deeply. Like so many metaphors in academic and everyday communication, there’s a strong spatial element to it. A threshold is at best an extended line; more typically, a much more localised feature like the narrow portal that illustrates TCs on Flanagan’s site. To think in terms of such a threshold, then, is to think of a concept that is localised and specific rather than pervasive: it primes us to see the “concept” in question as being, say, “limit” rather than “tension between image and definition”. In some disciplines this may be helpful; in mathematics I fear it can become misleading. From different starting points both Szydlik (2000) and Scheja and Pettersson (2010) seem to reach similar conclusions: students’ problems with limits have a great deal to do with their existing beliefs and epistemologies, or how they place limits in the wider context of the discipline. The “liminal” issues in mathematics, I suspect, are more often broad and pervasive than localised, and applying the localised threshold metaphor may make it harder for us to see the connections between individual difficulties.

The other problem with talking of “threshold concepts” is that it tends to suggest a problem with a fixed location, tied to particular features in the mathematical landscape. This contrasts with a view that sees difficulties arising from conflicts between the students’ existing habits or images and the new ideas they are trying to assimilate. Complex numbers, for example, may be liminal for students who have already developed a rigid concept of “number”, but much less so for those who have previously defined a number simply as something with which one does sums — an intuitive definition that is strangely consistent with the more sophisticated idea of a number as any object that satisfies the axioms for numbers. The temptation inherent in the threshold metaphor is to ignore differences between students — always a temptation for any teacher or academic administrator, and always dangerous.

At least the status of TCs as a metaphor is fairly hard to dispute. Moving to the next level of my personal classification system, I find it harder to see how they could be regarded as a myth; that is, as a story that points imaginatively beyond itself. It seems telling that the papers I’ve seen that have sought to identify TCs in mathematics have appeared rather unsure what to do with them once they’ve found them. Easdown (2009) mentions TCs in his first paragraph but never again, and the reference serves only as a theoretically “respectable” way of saying that proof is important and troublesome in mathematics education — very true, but not novel. Scheja and Pettersson (2010) grapple to draw out some consequences of identifying limits and integrals as a TC, but seem to get no further than they did by identifying these concepts as troublesome and as offering the opportunity to transform students’ understanding. It’s unclear what has been gained by introducing the word “threshold” at all.

As ever, the dangers really develop when a myth starts to become a state religion. As far as I can tell, TCs haven’t yet reached this status — although this is no thanks to the course organisers who are inflicting them on my contact, and no thanks to comments like “they constitute an obvious, and perhaps neglected, focus for evaluating teaching strategies and learning outcomes” (Meyer & Land 2003). (In justice to Meyer and Land, they do note the danger that TCs could become merely a tool of power, though it’s not clear that they realise that the entire idea, not just specific instances, could function in this way.) Supposing, though, that TCs do become part of the orthodox discourse of education, endorsed and enforced by university management: what ills might result?

The first consequence I can see is that far too many thresholds will be identified, because this identification becomes necessary for any component of the curriculum to be taken seriously. (It may also occur because probationers are under an obligation to find threshold concepts in their discipline or fail their probation.) The effect on education may be akin to a process of medicalisation, whereby sources of difficulty for students are eagerly diagnosed as TCs and — in the absence of any genuine insight — this diagnosis becomes an excuse for the situation rather than the means to a cure. As, by some cynical accounts, the multiplication of diagnosable psychiatric conditions has mostly served to enrich psychiatrists, the multiplication of thresholds is likely to serve mostly to enrich those who set themselves up as experts on TCs. And, because the phrase “threshold concept” couples a strong visual and spatial metaphor with an almost universally applicable abstract noun, it has a rhetorical power that could make it very appealing to charlatans.

The other danger I can see from the official endorsement of TCs is that once a threshold is identified, and treated as such, crossing it becomes a kind of initiation ritual. The idea, as I understand it, is that this should happen naturally:

On mastering a threshold concept the learner begins to think as does a professional in that discipline and not simply as a student of that discipline.(Flanagan)

Under official sanction, though, a newly defined threshold could readily become a locked gate to that space — rather like an end-of-level baddie from an old-fashioned computer game. Watching from some thousands of kilometres away the North American disputes over calculus reform, I get the impression that this is what has happened there. Rather than being a collection of core concepts, at least some of which are fairly intuitive, calculus seems to have become established as the threshold separating “elementary” from “advanced” mathematics. Thus it has been rendered a great deal more intimidating to students, who are not slow to pick up on either obstacles or formal progress criteria; and thus it has attracted the politics that initiation rituals always seem to attract, with fervent battles over the right to define the threshold and the right to determine by what means it may be crossed. (I’m reminded, with great sadness, of the energy that so many Christian churches put into deciding who is properly baptised, who may be admitted to communion, who is properly a church member and so on.)

So where does this leave us? Some threshold concepts can perhaps helpfully be identified, even in maths, and I’m prepared to believe that searching for them, using the criterion of “liminality”, could be a useful exercise for a teacher. Such a search, though, has to be very conscious of the other peculiarities of mathematical learning: in particular, that mathematical “concepts” tend to have a dual existence as definitions and images; that the historical or even the formal development of mathematics is not always a good model for its didactic development; and that the difficulties encountered by students often originate from mundane issues, such as lack of practice manipulating trig functions, rather than from deep ontological problems. Unless these peculiarities are acknowledged, mapping out threshold concepts may just end up building a Balkanised map of mathematics, divided by innumerable and heavily policed frontiers that both students and teachers are apprehensive of crossing.

The metaphor of the threshold is not a vacuous one, and treated as a metaphor, “threshold concepts” may be useful. Nevertheless they carry the danger of directing attention towards local causes and local problems rather than towards pervasive problems such as difficulty with formal proof. Treated as a myth, they don’t seem to point to much beyond themselves — certainly not to much that isn’t already covered by existing notions such as “learning as loss”. Treated as a state religion, they could become a breeding ground for charlatans as well as displaying the unappealing features of a sect with a passionate desire to police the border between belonging and not-belonging.

Unless we are careful deploying it, a handy minor tool for educators risks becoming, through the zeal and eloquence of its inventors, something approaching a dangerous fad. It would be a shame if a notion that places so much stress on the “integrative” were to become instead a way of fracturing mathematics education — leaving behind it a divided landscape and damaged students, victims of the rampage of a rogue metaphor.

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