Read the following text, paying particular attention to the highlighted words.
The Concept of Number
The long struggle forward in children's thinking comes out very clearly in the development of their number ideas - the part of Piaget's work which is now best known in this country. It offers a striking illustration both of the nature of his discoveries and of the basic pattern of mental growth. We can watch how the child starts from a level of utter confusion, without a notion of what number really means, even though he may be able to count up to ten or twenty; a level where number is completely mixed up with size, shape, and arrangement, or constantly shifts according to the way it is subdivided or added up. And we can see how, on an average two years later, children declare of their own accord that a number must stay the same, whatever you do with it, so long as you do not actually add to it or take away from it; or that whatever you have done with it, you can always reverse this and get back to where you started from; or that you can always show it to be the same by counting; and so on.
The following are a few examples of the ways in which Piaget's experiments bring out this pattern of growth:
1. Each child was presented with two vessels of equivalent shape and size containing equal quantities of coloured liquid. Then the contents of one of them was poured into (a) two similar but smaller vessels, (b) several such, (c) a tall but narrow vessel, (d) a broad but shallow one. In each case the child was asked whether the quantity of liquid was still the same as in the untouched vessel.
Piaget found that at a first stage, around 4-5 years, children took it for granted that the quantity of liquid was now different - either more because the level was higher, or more because there were more glasses, or less because the new vessel was narrower, or less because the levels in the two or more glasses were lower. In other words, there was no idea of a constant quantity, independent of its changing forms; if its appearance changed, the quantity changed and could become either more or less according to what aspect of the new appearance caught the child's eye. At a second stage, at about 5½-6, children had reached a transitional phase, in which they wavered uncertainly between the visual appearance and the dawning idea of conservation in their minds. Thus the quantity of liquid might be regarded as still the same when it was poured into two smaller glasses, but as greater when it was poured into three. Or as remaining the same if the difference in level or cross-section in the new vessel was small, but not if it was larger. Or the child might try to allow for the relation between cross-section and level, and experiment uncertainly without reaching any clear conclusion. In the third stage, 6½-8, children give the correct answers right away, either by reference to the height-width relation, or by pointing out that the quantity has not been changed: 'It's only been poured out.'
2. As a check on these results, Piaget carried out a similar set of experiments, with beads instead of liquids. In this way something closer to counting could be introduced (e.g. the child putting beads into a container one by one as the experimenter did the same into another vessel). Also he could be asked to imagine that the beads inside each vessel were arranged into the familiar shape of a necklace. The outcome was entirely the same. At the first stage, the children thought that the quantity of beads would be either more or less, according as the level looked higher, or the width greater, or there were more vessels, and this happened even when a child had put one bead into his vessel for each one that the experimenter placed in his. At stage 2 there is a struggle in the child's mind as before. This may show itself for example by his first going wrong when comparing levels between a wider and a taller vessel; then correcting himself if asked to think in terms of the necklaces ; but when the beads are spread over two or more containers, still thinking that the necklace will be longer. At stage 3 once more the children reply correctly and cannot be shaken, however the questions or the experiments may be varied.
(The Growth of Understanding in the Young Child, by Nathan Isaacs.)
Now try the exercises. Exercise a
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