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# What is not a Threshold Concept?

It is is important to establish what does not count as a threshold concept.

A threshold concept is not merely an important (or key, or core) idea. That is not to say that a threshold concept might not also be important, key or core; but that is not what makes it into a threshold concept.

So, for example, the ratio of the circumference of a circle to its diameter is π. That is very important and key to substantial areas of mathematics. But it is not a threshold concept, because it does not change the way you look at maths. What might be a threshold concept is the idea that the ratio of the circumference of the circle to the diameter is always the same. And the discovery that π is an irrational number (one whose decimal expansion never ends or repeats, for our purposes) may well be a threshold concept, as indeed is the very notion that there are such things as irrational numbers. It is certainly “troublesome”.

A colleague suggests (arising from a discussion of a visit to the battlefield at Hastings) that important though it is, knowing the date of that battle and who won it is not a threshold concept. It may be core to an understanding of the pattern of English history, but it does not radically change the way one sees it. On the other hand, she points out, seeing the Bayeux tapestry and understanding that “history is written by the victors” is transformative. One can never seen history in the same way again. Dealing with that point provides almost the whole basis for historiography and historical research methods and hermeneutics… And it’s disconcerting and troublesome, too; can I really trust anything I’m supposed to “know” about the past? It leaves me in a liminal state, teetering on the edge of thinking like a proper historian—but it is quite likely I will continue to function on a day-to-day basis as if it were not true, for a while at least.

In basic book-keeping, the double-entry method is standard. [Hang on, it wasn’t Pacioli in 1494; it was Cotrugli 1458. No, there’s evidence of it in 1215… But history is written by the victors, remember?]Everyone learns to do it, and it is practised almost universally. But that knowledge, even when grasped, tends to be “ritual knowledge”, in Perkins’ terms. You don’t have to understand it, you just have to do it. So it is not a threshold concept. But understanding why the method has been adopted so universally since its invention by Pacioli in the fifteenth century is. As one commentator put it, “Double entry bookeeping is the beginning of the organization of western business structure. Nothing else has come remotely close to providing the level of organization necessary for large economic endeavors.”

## What is the opposite of a Threshold Concept?

By definition, the opposite of a threshold concept must be a dead-end, a cul-de-sac; an idea which, however initially attractive it may appear to be, is not going to get you anywhere, and may seriously mislead you. And it is interesting to note this “however initially attractive it may appear to be” quality, because the opposite of a threshold concept is unlikely to be troublesome; it is likely to appear common-sensical and intuitively correct. Much of the history of science is about progress made by challenging such superficially plausible ideas. Drop two stones—one heavy and one lighter—from the leaning tower of Pisa and of course the heavy one will hit the ground first!

See these video clips on science learning for illustrations;

On photosynthesis

On electrical circuits

And given that we cannot bear not to understand or have some angle on what is going on, it's not surprising that our heads are populated by all kinds of myths and misconceptions; tales learned at our mother’s knee (or father’s—women have no special claim on misconceptions), ideas from the playground, or mistaken “discoveries” of our own. Such ideas have to satisfy no test beyond meeting the needs of the immediate moment, and they may be very well established before they are challenged. Moreover, they can have consequences; prejudices can translate into action. They may indeed seem to be threshold concepts—“I see it all now! All the problems which beset us are the fault of (insert name of ethnic/religious/political group of your choice).” And perhaps in a sense they are.

But it is misconceptions and preconceptions which make troublesome much of the knowledge which characterises true threshold concepts, because they have to be challenged before one can learn the real stuff.