Triadic education and learning

Triangulation of Incommensurable Concepts for Global Configuration (Part #6)

Triadic education: David Clarke, Sandra Frid and Carne Barneti (Triadic Systems in Education: categorical, cultural or coincidence. 1993) argue that recent research in education. and mathematics education. in particular, has led to the identification of independent categorizing systems intended to mirror the .structures found in such diverse fields as teacher professional development (Barnett. 1992); student writing in mathematics (Clarke. Stephens. and Waywood. in press); and student acquisition of calculus knowledge (Frid. 1992). There are particular characteristics of .. these categorizing systems which display a tantalising similarity:

Contextual similarity: the common location of all three studies within educational environments;
Structural similarity: the "three-valued" (triadic) structure of all three categorizing systems;
Conceptual similarity: categories in each system resemble each other in the nature of their conceptual distinctions.

Threefold learning cycle: Following from his insights as the developer of the Bell helicopter, and from its control in flight, these were articulated in terms of a threefold and fourfold learning cycles by Arthur Young (The Geometry of Meaning, 1984):

In discussing the (fourfold) learning cycle, I implied that fourfold analysis is insufficient as a general description of process....the cycle includes not only actions but other factors as well. This is the threefold cycle of stimulus, response, and result. The three aspects in this case are again different categories, for which it is quite difficult to find sufficiently general names. ( p.25)
So the generalized description of the threefold cycle is...relationship, act, state. In returning to a state, which it must do to be a cycle, the process does not necessarily return to the same state that initiated it; it only returns to a state of some kind....The threefold is not limited to this cycle. In fact, the cycle, as an analytic concept, does not fully describe the threefold. In the more general sense, this is a. way by which wholes divide into three inter related factors, often, but not always, in the form of two elements plus that which is between them....Since it is basically nonconceptual, it cannot be defined... (p.26-27)
The threefold operator, represented analytically as equidistant points on a circle, is actually a three-dimensional activity, whose measure gives only its analytic aspect. The analytic aspect, which is in two dimensions, does not convey the full meaning of the cube root; it is like the shadow of a solid figure. The threefold nature of the cube root is nonanalytic. It involves categories which differ from one another more profoundly than those of the fourfold. ( p.55)
We may think of the threefold as an endless line (curving though space in an arbitrary fashion). Between any two points on this line, we can say which is before, but we cannot look at a piece of it without getting off the line....It is a reality which is always with us, yet perpetually eludes analytic description ....If we try to analyze what it is that the threefold describes, we are in a bind, for it is just that element of participation in life that analysis cannot, and does not even pretend to, cope with. (p.57]
The threefold is an entirely different way of cutting the cake. It is much more fundamental (than the fourfold), and it cannot be analyzed....In fact, the threefold is the natural way we move in life. We see something, buy it, and enjoy it; food, eating, satisfaction; stimulus, response, result. There are three categories of terms: relations, acts, and states. The reader will recall the difficulty of finding sufficiently general words. ( p.99)
But when the stimulus causes wrong action and the result is not achieved, the (fourfold) learning cycle becomes necessary. Thus the learning cycle occurs only when there is an obstacle in the larger, threefold cycle. (p.24)
In sum, we have shown that the threefold, despite the seamless quality which makes it difficult to get "hold of", is still in the measure formulae of science. These expressions, which constitute a basic vocabulary of science, fall into three groups of four: actions, states, relations." ( p.102).

Some earlier tentative adaptations of Young's insights are presented in:

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Triadic education and learning