# Tribute to Invariance

## Piaget and Post-Piaget Experiments

A famous series of experiments by Jean Piaget (1896-1980) established the notion of conservation of number and demonstrated that children mostly lack it up to the age of 7. The idea has had a formative influence on the instruction of mathematics [McK]. Place two rows of different objects in front of a six year old. Ask the child whether there are more circles, more squares, or the same number of each kind. The expected answer is "The same." Rearrange one row as shown and ask the same question again. This time around a child would say "More squares."

One of the achievements of Piaget's research is the universal acceptance of the fact that children do not think like miniature adults, they think differently and in different categories. It is immensely gratifying that precisely on this basis later researchers began questioning Piaget's number conservation experiments. S.Dehaene [Deh] describes a similar experiment where row rearrangement had been carried out fortuitously by a teddy bear while the experimenter was ostensibly looking the other way. Turning to the arrangement, the experimenter would blame the bear for mixing things up and ask the same question. The same children who failed in the former experiment consistently gave the correct answer in the latter: "The same."

Dehaene further observes that, for three-and-a-half-year-olds, the order in which one recites the numerals is crucial, while the order in which one points toward objects is irrelevant. They find it perfectly acceptable to start counting at the middle of a row of objects, or to count every other object first, as long as one eventually counts all items once and only once. Some post-Piaget experiments are really startling. By measuring attention span of 5-month-old children, the results of experiments performed by K.Wynn (1992) indicate that infants know that 1+1=2 and 2-1=1 and appear surprised by attempts to be tricked into thinking that 1+1=1. (Children were shown 1 or 2 objects that subsequently have been hidden behind a screen. With screen lifted, children were gazing longer when presented with an unexpected number of objects but not when the size or shape of the objects has changed.)

Based on the later experiments, it will not be too far-fetched to speculate that, at least in some sense, children do conserve the number very early in life contrary to the Piaget's theory of stages.