How Does Piaget View the Learning of Mathematics?

Natural numbers are regarded as a gift from God, from which humankind has created the entire world of mathematics.

As someone outside the field of mathematics, my most immediate impression of this subject is that it is extremely practical: through mathematical models we can describe and calculate a wide range of physical phenomena and economic behaviors, and apply them across disciplines. Mathematics seems to possess a capacity for abstracting the essence that lies behind the surface appearances of things, which gives one a sense of certainty. Piaget, from an epistemological perspective, analyzed three essential characteristics of mathematics, thereby responding to this phenomenon.

1. Although it rests on very few and relatively impoverished concepts or axioms, it is remarkably fruitful.

Euclidean geometry, for instance, derives an entire vast edifice of geometry from a handful of simple axioms, and can even develop new branches. The essence of mathematics' continual extension into new knowledge is a form of "operations upon operations." However complex a mathematical system may be, it is fundamentally built up from the most basic notion of the "set," through the continual manipulation, combination, and abstraction of sets; mathematics is a hierarchy of sets. The most central feature of mathematical thinking is that mathematicians no longer concern themselves with "what things are," but with "what the relations between things are."

2. It possesses necessity.

Once a mathematical result has been derived, it carries absolute necessity, and no one can refute it. Mathematics is neither "discovery" nor "invention," but a construction of necessity: once constructed, it becomes necessary. It is like building with Lego: one is not discovering a castle that already exists, nor can one build arbitrarily; one must build in accordance with the interlocking rules of the bricks themselves. Of course, the structures built by mathematics can themselves be divided into "strongly structured classes" and "weakly structured classes," where "strongly structured" means that the rules are rigorous, such that the emergence of a contradiction is fatal.

3. Although it is entirely deductive, it accords with experience and physical reality.

Mathematics is derived wholly within the human mind through logical deduction, without the need for experiments. Yet physicists have found that the real world conforms remarkably well to these formulas derived by pure formal operation; that is, mathematics and reality are isomorphic (identical in structure). Moreover, mathematics frequently anticipates reality, actively preparing a framework for it in advance. A case in point is the century-long encounter between non-Euclidean geometry and general relativity: the real universe turns out to operate according to a "logical game" that mathematicians had worked out in their minds.

Piaget then sought to analyze, from the perspective of children's cognitive development, why human beings are able to construct the world of mathematics, that is, the psychological and biological roots of mathematical construction.

Piaget held that mathematics is the inevitable outcome of human cognitive development. In the early cognitive activity of primitive peoples or young children, we can observe a primitive, intuitive form of mathematical thinking, manifested as "coordination of actions," which later develops into, for example, pure formal operations. That is to say, our biological endowment grants us the cognitive capacity for mathematical thinking, namely form thinking. The level this capacity ultimately attains depends on whether one's environment demands continued use of the cognitive function of form thinking, for instance, whether one's life requires the application of mathematics to solve problems.

How should those of us working in early childhood education understand the process by which children's mathematical cognition develops? Of course, not everyone needs to become a mathematics teacher, at least not at the stage of early education. Moreover, cultivating the capacity to learn mathematics is not the same thing as learning mathematical knowledge.To answer this question, we need to understand how children accumulate "logico-mathematical experience" in daily life.

Through observation, we find that children's initial mathematical activity appears to be physical activity rather than formal deduction, for example, manipulating an abacus by combining or separating its beads, or classifying and ordering objects. Piaget, by distinguishing logico-mathematical experience from "physical experience," explained the essence of the former:

‍ ‍When children accumulate physical experience, they are perceiving "the objects themselves." For example, the weight of a stone has nothing to do with the action of "weighing" it; it is an intrinsic property of the stone.

‍ ‍When accumulating logico-mathematical experience, by contrast, children are perceiving "their own actions." For example, when water is poured from a "short and wide" glass into a "tall and thin" one, although the water level rises, the child knows that the quantity of water has not changed. This "unchanged" is something the child constructs in the mind through the reversible action of "being able to pour it back"; it is this reversibility that endows the child with the mathematical logic of "conservation of volume." Whatever the object may be, so long as the structure of the action remains the same, the experience the child obtains remains the same. Such experience is just as the child himself puts it: "When you know once, you know always." As experience accumulates, children gradually become able to internalize "actions" into mental operations (formal operations) without actually having to carry them out by hand; this is deduction.

Mathematical deduction therefore possesses necessity because what it studies is not "what the world is," but "the internal form of coordination of the actions by which we act upon the world." Accordingly, in understanding how children develop the capacity for learning mathematics from early mathematical experience, we should attend to and analyze what kind of "coordination of actions" children achieve in their activities, and when they achieve the "internalization" of "actions," thereby truly forming a certain kind of mathematical thinking.