The present paper is a report on my study of school children's ability of seriation of complicated arranging materials. The author attempted to clarify the developmental process of their conception of series on the “stage of concrete operations”, by analysis of errors made by the children in their answered papers. 241 school children from the 1st grade to the 6th grade were examined in the form of group test, class by class. Two types of task were given; numeral series and series of geometrical figures such as circles, triangles and squares. Each type of task demanded the subjects to complete the series in one-dimensional cyclic order.
1. In the case of the numeral series task, frequency of the errors is highest in 1st graders' a nswers, 2nd to 5th graders rank second, frequency of the errors is lowest in 6th graders' papers. The number of errors decrease in the order of→(1st graders)→(2nd, 3rd, 4th and 5th graders)→(6th graders). In the case of the geometrical figure series task, frequency of the errors is highest in the 1st grade Ss ; 2nd to 6th grade Ss rank second. The number of errors decrease in the order of: (1st graders)→(2nd, 3rd, 4th, 5th and 6th graders). From the proportion of the errors between the numeral series and the geometrical figure series; the 6th grade Ss show higher frequency of errors in the latter, but the Ss from the 1st to the 5th graders show no such remarkable difference.
Whereas from an investigation of the errortypes, the following points are revealed: Frequency of the errors of type A (unstable about phase operation) is higher in the numeral series papers of the 1st graders ; but on the contrary the subjects of 4th to 6th grade show higher frequency of errors in _geometrical figure series papers. And almost all the Ss's, errors of the type B and C (unstable about simple seriation) are more frequent in the numerals task.
This proves that the seriation of numerals are easier than that of geometrical. figures if school children, can undertake. phase operation, but that the relation is diametrically opposed so long as the series used appeal only to children's direct visual perception.
2. The total frequency distribution, as a whole, of each type of errors in each grade reveals that there are the first developmental crisis at the 2nd grade, and the second crisis at the 6th grade. This is shown as a diagram →(1st graders)→(2nd, 3rd, 4th and 5th graders)→(6th graders).
From the above findings, we may conclude (1) that almost all the children of the 1st grade still stay on the level of pre-operational seriation,(2) that almost all the children of the 6th grade are already on the level of formal reasoning, and (3) that almost all the children's operational structure from the 2nd to the 5th grade are yet unstable and unflexible. Therefore in the complicated situations, their seriation is likely to be suspended in spite of their capability of phase operation.
But we find that there are still two sub-stages with the developmental stage stated in (3); The first sub-stage covers the 2nd and 3rd grades, and the second covers the 4th and the 5th grades. And particularly the latter substage has the characteristics of a transitional state from the “stage of concrete operations” to the “stage of formal operations”.
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