Measuring the Ordination→Cardination Sequence

Abstract

During the short interval since the ordination+cardination sequence was reported, it has prompted a large amount of follow-up research. Interest in the sequence stems from the fact that, first, it is inconsistent with the prevailing view among educators that children's numerical concepts are basically cardinal (1, Ch. XI) and, second, it is inconsistent with predictions from Piaget's theory (1, ch. VI). Investigators who have conducted follow-up studies have devised some ingenious new procedures for measuring the ordination+cardination sequence (2, 3). Most of these new procedures seem valid. But, unfortunately, there is a problem with the method suggested by Williams (4, 5). The problem lies in Williams' definition of ordination. He claims that the key attribute of ordination is that it involves determination of number by Accually, counting is quite irrelevant to the logical definition of ordinal number (e.g., 1, Ch. 111) and, hence, it is irrelevant to ordination (which is the behavioral counterpart of ordinal number). Although counting may be used to generate instances of ordination, there are many other such instances that do not involve counting. It is important for researchers interested in number development to realize that counting is both an ordinal and a cardinal operation. In its most basic sense, counting is simply a method of correlating number names with objects. If Ss know that a certain number name, e.g., four, has a certain positional meaning (fourth), we have an example of ordination. If Ss know that a certain number name, e.g., four, has a certain numerousness meaning (quartet), we have an example of cardination. Finally, if Ss usually know positional meanings of number names before they know numerousness meanings, this might be viewed as an example of the underlying ordination+cardination sequence. In contrast, Williams examined Ss' tendency to conserve cardinal equivalence when they are allowed co count after transformation. Correct counting without subsequent conservation was viewed as an example of ordination+cardination. But, in line with wl~at I said about counting being a cardinal as well as an ordinal operation, note that the meanings of the number names generated in this situation are entirely cardinal, i.e., they always refer to numerousness and never to position. Therefore, it seems likely that the sequence Williams observed is not an example of ordination+cardination but, rather, is a novel sequence between earlier and later versions of cardination. This, at least, would be my interpretation.