FORMAL AND CONCRETE THOUGHT PROCESSES

This project was an empirical investigation of predictions derived from a theory “by Piaget concerning the development of intelligence. In particular, three content areas were to be studied, making use of group testing procedures and special statistical techniques for analysis. Piaget defines a mental operation as an internalized act, i.e., the external object is represented symbolically in the mind of the subject who instigates behavior with respect to the symbol. Concrete operations are operations on the symbols of objects that can be manipulated through the senses. Formal operations are operations on concrete operations. His theory implies a steplike development of intelligence and the possiblility of demonstrating that there are children who can carry out concrete operations but not formal and that all children who can carry out formal operations can also carry out concrete operations. An empirical finding reported by Piaget is that children in general change from the concrete to the formal operations between their twelfth and fifteenth birthdays. Of course, some children make the change before this period and some afterwards, and others again probably never do. Conditions which facilitate or inhibit this change are not explicitly stated by the theory. In the area of arithmetic, for example, Piaget's theory of intelligence is interpreted as representing the development of intelligence as a process of building up a set of elements and an operation with respect to this set, such that the conditions for a semigroup are satisfied. In this way the stage of concrete operations is established. Later in the developments of the individual the condition of an inverse is added and this with the unit changes the semigroup into a group. This is the stage of formal operations. The theory was examined with respect to three content areas, arithmetic, probability and inequalities (also referred to as “comparisons”). In the area of arithmetic, pairs of problems were prepared, one of which required reversibility of operations in the sense that an operation combined with its inverse leaves the original number unchanged, whereas the other could be solved by normal computational operations. In the area of probability theory formal operations were examined through the notion of independence. The children in effect were given in one problem that a certain number of object A and a certain number of object B were placed in a box and were asked which object, A or B, was most likely to come out by random drawing. In another problem it is given that one more of either object is in the box but that it has been removed by chance drawing before they have to decide which is most likely to come out. Formal operations in this case lead to the notion of independence which is an important condition for operations on probabilities. In the case of the comparisons items the items differed solely on the basis of generality. Such items do not examine the theory as stated, but were included for other reasons related to Piaget's work and to evidence found in the literature. The effect of generality was examined also in the other content areas. Data were obtained by administering 74 items, some of which had several parts, to children from Grades IV, VI, VII, VIII, and X. Approximately two hundred from each of the first two grades and three hundred from each of the upper three grades were tested. These data were punched onto IBM cards and analyzed in terms of the relationship between matched pairs of items. The results of this analysis indicated that six of the seven arithmetic item pairs were related in a way suggested by the theory. The seventh item pair required operations with fractions; further study of such items is suggested. For the probability items effects which could be attributed to lack of appreciation of the notion of independence were obtained with Grade IV and Grade VI children, Children in Grades VII, VIII, and X behaved somewhat differently, but this could have been due in part to the effect of the influence of a variable not anticipated in the planning of these items. Further study of this type of item is suggested. No effects were obtained with the inequalities items. Specific suggestions are made as to how items, in this area could be constructed that require reversibility of operations. In general, it was concluded that the group testing situation could be used experimentally with systematic variation of the content of items. Piaget's theory was used as the basis of this variation and certain predicted results were obtained. Some specific suggestions for further studies arose from the results of this study.