I first encountered the notion of threshold concepts in David Didau's marvelous book. He provides links there to this article, by Jan Meyer and Ray Land—who first wrote about threshold concepts—and this one, by Glynis Cousin, which provides a briefer introduction to the idea. Meyer and Land describe threshold concepts this way:
Reading this, I recall my freshman year in college, when I took a lot of 101 courses: Anthropology 101, Biology 101 (I think it was 150 actually), even Theology 101 (though I'm quite sure it wasn't called that). What I enjoyed about these courses—though even then I was certain that I was not going to be a biologist, anthropologist, or theologian—was that they delivered these "previously inaccessible way[s] of thinking about something."
Sure, I had some ideas about the world that fell within the purview of each of these fields—some ideas about how human societies function, how the human body functions, and some ideas about how we collectively think gods get their work done. But it was clearly not a goal in these courses to stretch my previous or intuitive understandings into something more mature and rigorous. What was intended was a welcoming into an academic community—a community that looked at the world in a specific set of ways that had served it well over the decades or centuries; one that had developed useful schemas and language for investigating the specific slice of the universe that interested it; one that had essentially constructed, at its boundaries and deeper within, threshold concepts which initiates had to come to terms with in order to navigate and contribute to the community; and one which—and this seems like a characteristic of academic communities that is often overlooked by K–12 education—continuously reinforces and overturns these concepts through free and transparent criticism and debate.
I highly recommend the more detailed exposition given at the links above.
The Cheese Stands Alone
When I think about the characteristics of these concepts, as outlined by Meyer and Land—they are transformative, often irreversible, integrative, bounded, and likely to be troublesome—it seems to become clear that threshold concepts are also, in notable ways, isolated. The authors use complex numbers as one example:
Notice the phrase "beyond . . . intellectual grasp." The authors themselves can't help but talk about a threshold concept as a node that lies above something cognitively "below" them, such that one can reach up from "below" to grasp them intellectually (or not). But if we accept the authors' characteristics of threshold concepts, it seems we should reject this implicit picture. Rather, threshold concepts stand more or less alone and disconnected (from below, at least). A threshold concept cannot be transformative and irreversibly alter our perspective while also simply being the last domino in a chain of reasoning. We can and do make sense of threshold concepts as continuations or extensions of our thinking, but we do so by employing a few tricks and biases.
We Can Make Up Stories About Anything
In his book You Are Now Less Dumb, featuring captivating descriptions of 17 human fallacies and biases (and ways to try to overcome them), David McRaney alludes to one way in which we are practically hard-wired to misunderstand stand-alone concepts—the narrative bias:
Thus, we need for threshold concepts like complex numbers to be the middle part of some storyline. So we simply invent an educational universe in which we believe it is always possible to "motivate" this middle by writing some interesting beginning. In reality, this might be at best intellectually dishonest and at worst delusional. Given their characteristics, threshold concepts may only truly make sense as the beginnings of stories. Yet our very own narrative biases can actually cause these concepts to be troublesome, because we search for ways in which they follow from what we know when those ways don't in fact exist. To master a threshold concept, one may need to take a leap across a ravine, not a stroll over a bridge. Louis C.K.'s mom said it well:
Another blockade that exists and which prevents us from embracing the very idea of threshold concepts is the implicit assumption that learning is continuous and linear—and that it always moves forward. You can see that this is in some way related to the narrative bias discussed above. I'll simply quote Didau on this, as he deals with it at length in his book:
Dealing Honestly with Threshold Concepts in Mathematics Education
In education, I think we could stand to be more comfortable recognizing concepts that simply have more outgoing arrows of influence than incoming arrows. These kinds of concepts do more powerful work in the world shedding light on other ideas and problems. Thus, it may be a far better use of our time to treat them as the first acts of our stories—and a waste of time to treat them as objects of investigation on their own. One would think that lighthouses make a lot more sense when you look at what their light is shining on rather than directly at their light.
It can seem disquieting, to say the least, to think that it might be better to approach certain concepts by simply "living inside them" for a while—seeing out from the framework they provide rather than trying to "understand" them directly. But the point I'd like to press is that this discomfort may be a result of our bias to see 'understanding' from just one perspective—as a causal chain or story which has as one of its endings the concept we are interested in. Some understanding may not work that way. More importantly, there is no reason—other than bias or ideological blindness—to believe that understanding has to work that way.
Another reason threshold concepts may be so troublesome is that perhaps we misunderstand historical "progress" within a field of study in precisely the same way we misunderstand "progress" for students: as a linear, continuous, always-forward movement to higher planes. How, you might ask, can I expect students to simply come to terms with ideas when humanity certainly did not do this? But your confidence in how humanity "progressed" in any regard is likely informed by a narrative bias writ large. You simply wouldn't know if chance had a large role to play, because the historians involved in collating the story, along with all the major characters in said story, are biased against seeing a large role for chance. Marinating in ideas over time, making chaotic and insightful leaps here and there—such a description may be closer to the truth about human discovery than the tidy tales we are used to hearing.