Conservation

According to Piaget, notions of conserved quantity come fairly late.
Piaget's classical example is asking a child whether a tall glass or a
short glass has more liquid in it just after the liquid has been
poured from one to the other. Piaget's classical result is that
children younger than about seven pick the tall glass, citing the
height.^{15} Siegler in his textbook [Sie98]
asserts that conservation arises more gradually with different
conservation laws being learned at different times.

Suppose something appears and disappears. There are two kinds of mental models a person or robot can have of the phenomenon--flow models and conserved quantity models. Flow models are more generally applicable and apparently are psychologically more primitive. Thus water flows from the tap onto the hands, and water flows down the drain. This model does not require a notion of quantity of water. The same is true of a child's early experience with money. A parent gives you some and you buy something with it. When there is no way of quantifying the substance, as with the water flowing from the tap, the notion of conservation of water is of no help in understanding the phenomenon.

Siegler considers various conserved quantities--physical objects,
numbers and liquids. Conservation of physical objects comes first.
An object that has disappeared is regarded as being somewhere, and if
the object is wanted, it is worthwhile to look for it. Conservation
of number is not apparent to first graders, and they give silly
answers to questions like .^{16}

Let's take the designer stance. It would be good if the notion of conservation law were innate, and experience taught which domains it applied to. Alas, we aren't built that well.

The notion of conserved quantity is more abstract than other early
notions. The actor has to believe in there being a quantity of the
entity in question, e.g. of water. [Sie98] suggests, p. 42,
op. cit., that the child learns conservation of water via taking into
account the cross section of a glass as well as its height. My
opinion, which a suitable experiment might test, is that the abstract
notion of conserved quantity is learned, and talking about the width
of the glass is only window dressing, because not even Archimedes
could do the geometry needed to confirm the
conservation.^{17}

A mathematical description of a conservation law may be interesting. Here's a situation calculus axiom saying that the amount of a quantity is normally conserved.

Let denote the amount of quantity in reservoir in situation . We wish to say that if the occurence of the event in situation is not abnormal, then the amount of in all the reservoirs together remains constant, i.e. is conserved.

2008-09-18