What even is a domain?
A return to Thorndike and Woodworth
On Twitter, Carl Hendrick posted a clip from a talk by Barak Rosenshine:
In the clip, Rosenshine describes a study that illustrates that knowledge is highly domain-specific. Apparently, expertise does not transfer between the medical specialisms of cardiology and endocrinology. Rather than possessing general purpose higher-order thinking skills, cardiology professors know a lot about cardiology. When asked to solve endocrinology problems, they stumble.
By the standards of education-related tweets, Hendrick’s has gone viral. I suspect this is because it involves a rare clip of Rosenshine, who has enjoyed a renaissance due to the exegesis of Tom Sherrington in the UK, and because it illustrates a counterintuitive realisation that has been slowly spreading through the teaching profession in recent years - that knowledge and skills are domain-specific.
Nevertheless, it is a worthwhile exercise to challenge this idea. Let’s deploy a little sophistry and begin by asking: What even is a domain? Do domains exist? If they do not, then knowledge cannot be domain-specific.
Such a question is, of course, an invitation to pseudo-profound chin-stroking. Consider, for example, asking what a ‘colour’ is and then following this up by suggesting that there is no clear boundary between orange and yellow in the visible spectrum. Where does orange end and yellow begin? See! There is no such thing as a colour!
In a sense, such a claim is trivially true. Colours are socially constructed concepts that vary across cultures and times. And yet, arbitrary as they are, colours still denote a difference - in this case in the range of frequencies of light. The claim against domains is similar. The existence of a continuum does not make it false to apply labels to different sections of that continuum. Nevertheless, this analogy does raise the more fruitful question of what it is that is varying along this continuum. What is the equivalent of frequency?
And when we consider Rosenshine’s example, the question becomes yet more salient. After all, Rosenshine is talking about two different specialisms within medicine. How come cardiology and endocrinology are the domains in this example and not simply medicine? Who moved the goalposts?
To answer this question, I think we need to return to an idea first pitched by Thorndike and Woodworth in 1901 as an answer to the problem of transfer of learning. Briefly, transfer of learning is the ability to apply something you have learnt in one domain to a problem set in another domain. Thorndyke and Woodworth proposed that this depends upon the existence of identical elements in the two domains. The extent to which they share identical elements dictates the extent of the transfer.
If we turn this argument inside out, we can sketch a rough understanding of the concept of a domain based upon the problems we wish to solve. By problem, I am referring to the widest possible sense of the word. A problem could be a closed mathematical problem, the problem of comprehending a text, the problem of evaluating a source or even the problem of creating a novel complex product that has value.
Drawing on Thorndike and Woodworth, we can propose that problems are set in the same domain if they draw on a large number of identical elements. In contrast, they are set in different domains if they draw on few, if any, identical elements. Yes, the boundary between these two conditions will ultimately be fuzzy and arbitrary but, like the boundary between orange and yellow, the distinction remains meaningful. Domains are ultimately defined by the problems you want to solve. Or, perhaps, domains reduce to problems and problems reduce to elements.
What is an element? In my mind, the elements of Thorndike and Woodworth are the same as the elements of cognitive load theory. This means that with increasing expertise, entire schemas held in long-term memory may be treated as elements. For instance, the problem of changing the subject of a linear equation consists of many elements for novices but few for experts. Elements therefore always interact with the expertise of the problem-solver. This is where we gain our intellectual superpowers from as a species. We can integrate large numbers of related elements into a single schema which, when called for in a different context, we can apply to that context.
This does not necessarily solve the problem of transfer. For some reason, we often possess an inverted view of reality when learning complex concepts. The few elements that are shared between different domains are often easily grasped and require little integration to form schemas. And yet we tend to think these elements are worth more.
For example, the law of conservation of energy was a profound discovery but, in my experience at least, is easily grasped. You could even get students to parrot it, if you wished. However, applying it to find the enthalpy of a reaction in chemistry or the speed of a trolley in physics is the hard part. And it is this hard part that is specific to chemistry or physics. Nevertheless, we have a tendency to dismiss the hard part as merely a procedure, whereas knowing the law of conservation of energy is characterised as conceptual understanding - a finer, nobler thing.
It is this phenomenon that is responsible for the rise of popular history and popular science books. I am pretty sure that if I sat down for a coffee with any reasonably well-educated adult, I could explain the basic principles that underpin the field of quantum physics - the uncertainty principle, wave-particle duality and so on. However, this would not make my companion a quantum physicist.
I suspect there is a rough principle that the more widely shared an element is across domains, the more it tends to be a discrete statement or proposition that does not depend on others, the easier it is to grasp and the less value it possesses. Technical and obscure sets of highly interacting elements that fit closely to a small class of problems are transformational in solving those particular problems, but are more difficult to grasp.
There are exceptions. Reading and writing are intensely technical skills that underpin pretty much all academic domains. Mathematics also punches above its weight. However, it is hard to think of many other examples. Critically, in these cases, the identical elements shared across domains are transparent, well-defined and do not require something in addition to Thorndyke and Woodworth’s model.
Of course, much of what I have written is mere speculation and Thorndike and Woodworth have had their critics over the years. To many, it is intuitive that there are general purpose thinking skills or that we can sharpen the mind in some general sense through cognitive work.
However, it is telling that in the 120 years since Thorndike and Woodworth’s paper, and with no lack of effort, we have yet to find, describe and demonstrate the effectiveness of training in general purpose cognitive skills.