For Piaget, 'the key tool of the constructivist is the group' (yes, the mathematical notion of a group!) and a child is understood as part of a feedback loop embedded in their environment consistent with much of the other cybernetic discourse, Norbert Weiner in particular.
Nothing in these books targets the actual teaching of mathematics.
Read and enjoy:
To begin his work on Genetic Epistemology, Piaget attempts to problematize the contemporary visions for scientific knowledge. Piaget believes that:
"Scientific thought, then, is not momentary; it is not a static instance; it is a process. More specifically it is a process of continual construction and reorganization." - GE, 2.
He offers two examples to support this. First, is Louis de Broglie who shifted from adhering to Niels Bohr's view of indeterminism. de Broglie later changed his mind, hence his knowledge is constantly in the process of being constructed.
Second example is Bourbaki. The mother structures weaknesses have been exposed, and Bourbaki's architectures were giving way to Eilenberg and MacLane's use of categories.
"As a result, today part of the Bourbaki group is no longer orthodox but is taking into account the more recent notion of categories."- GE, 3.
So, knowledge is not static, it is actively in construction.
Then, logical developments in children are subdivided into figurative and operative aspects. Figurative refers to imitative behavior whereas operative represents a transformation from one state to another. These operative modes are what matters if knowledge is in fact an active construction, and the actions on things are what matter.
"it includes actions themselves, which transform objects or states and it also includes the intellectual operations, which are essentially systems of transformation...to know is to assimilate reality into systems of transformations...knowing an object does not mean copying it-it means acting upon it. It means constructing systems of transformations that ccan be carried out on or with this object...knowledge, then, is a system of transformations that become progressively adequate." - GE, 15.
Piaget believes then that he has argued "the roots of logical and mathematical structures are to be found in the coordination of actions, even before the development of language..."(GE, 21)
Operating on something for Piaget means that:
An operation is an action that can be internalized
An operation is reversible
An operation always supposes some conservation
No operation exists alone
Thus, for Piaget, "every operation is related to a system of operations, or to a total structure as we call it."- GE 22.
The Bourbaki's had three 'mother structures'--algebraic, order, and topological. An example of an algebraic structure would be a group, ring, or field.
The group is Piaget's favorite tool. In mathematics, a group is a structure that consists of a collection of things and some action(s) on these things. Similarly, a group has its four axioms of closure, associativity, identity, and existence of inverses. These map closely to Piaget's description of operative knowledge, and it was deliberate.
Piaget described an interaction with the Bourbaki member Dieudonne at a meeting on 'Mathematical Structures and Mental Structures' to emphasize the connection:
Dieudonne gave a talk in which he described the three mother structures. Then I gave a talk in which I described the structures that I had found in children's thinking, and to the great astonishment of us both we was that there was a very direct relationship between these three mathematical structures and the three structures of children's operational thinking. We were, of course, impressed with each other, and Dieudonne went so far as to say to me: "This is the first time that I have taken psychology seriously. It may also be the last, but at any rate it's the first."-GE, 26
For Piaget, algebraic structures are readily observable through the logic of classes, i.e. classification. If you give children some different shapes, according to Piaget, around age 7 or 8 they will be able to classify them operationally as described above. Otherwise, Piaget noticed a range of pre-operational thinking in younger children.
For example, the lowest level was figurative behavior where children would make shapes of similar shapes, i.e. a triangle out of triangles, circle of circles. Later, children dismiss the figurative aspect and piles the shapes. Piaget claims that this child does not yet understand class inclusion however.
he cannot deduce, for instance, that the total class must necessarily be as big as, or bigger than , one of its constituent subclasses. A child of this age will
agree that all ducks are birds and that not all birds are ducks. But then, if he is asked whether out in the woods there are more birds or more ducks, he will say, "I don't know; I've never counted them.""- GE, 27.
So, the child is still not evincing all the characteristics of operational thinking. This relationship of class inclusion is what gives rise to THE OPERATIONAL STRUCTURE OF CLASSIFICATION. This structure is something like an algebraic structure, though not exactly as distributivity is not a feature. (birds + birds = birds, hence (bird + bird) - bird = 0 but bird + (bird - bird) = 0)
Note the similarity to how he is describing the changing characteristic of a child's knowledge based on group like behavior. Only when the child engages with classification tasks in such a way as to demonstrate all four axioms, they are demonstrating 'pre-operational' thinking, and these different forms are what he bases his stages of development on.
The other structures are there too. Piaget goes on to describe the example of an ordering structure through the problem of seriation, and that of topological through children's use of either topological, euclidean, or projective frames in geometric contexts. Piaget believed children think like topologist's first, and you can find more on this in his geometry book, The Child's Conception of Geometry.
I think you get a sense for how he uses structures as models for cognition here though, and if you're interested I suggest reading Genetic Epistemology and Structuralism. They're not long and very interesting to see how involved Piaget was with using mathematics and physics to build his system.
