The Frequency with which a Group of Unselected College Students Experience Colored Dreaming and Colored Hearing

The principal aim of this report has been to clear the air with regard to number development theory and research. We began with a brief overview of the status of the number concept in pure mathematics and psychology. It was argued that although there are clear methological differences between pure mathematical and psychological approaches to the study of number, the two approaches are substantively complementary. Both are attempts to found the number concept on underlying psychological processes.

Three influential mathematical theories of number and three metamathematical criteria by which such theories are judged (consistency, completeness, categoricalness) then were summarized. The first theory, originally proposed by Peano (30) and von Neumann (49), postulates that the number concept may be reduced to a wholistic property of the natural numbers: viz., their inherent ordering. It was found that Peano-von Neumann's ordinal theory allows one to construct natural numbers without contradiction (consistency) and to construct all the natural numbers (completeness). However, the ordinal theory is not only a theory of number; it is a theory of all ordered progressions. The second theory, originally proposed by Russell (39, 40, 41, 42, 51, 52) and Frege (15), postulates that the number concept may be reduced to an atomistic property of the natural numbers: viz., the fact that each natural number is a class which includes all classes containing a particular number of elements. It was found that Russell-Frege's class theory does not allow one either to construct natural numbers without contradiction (inconsistency) or to construct all the natural numbers (incompleteness). The third theory, originally proposed by Piaget (31, 32), is a combination of the Peano-von Neumann and Russell-Frege theories; i.e., it founds number simultaneously on the wholistic property of order and the atomistic property of class. Piaget's theory was found to be contradictory (inconsistent) and to contain a superfluous undefined concept. A comparative analysis of the three theories revealed that although none of them is universally accepted as an ultimate theory of number, Peano-von Neumann's ordinal theory is by far the most satisfactory of the three by metamathematical criteria.

Each of the mathematical theories of number was translated into a cognitive developmental theory by substituting “developmental priority” of “mathematical priority.” The first theory postulated that children's number concepts derive from a prior understanding of the quantification of ordinal relations (ordination). The second theory postulated that children's number concepts derive from a prior understanding of the quantification of classes (cardination). The third theory postulated that children's number concepts derive from a prior understanding of both ordination and cardination. Published research which pertains to the postulates of each of these theories was reviewed (2, 8, 9, 10, 23, 33, 45, 46). Because of incomplete data analyses and poor operational definitions, it was concluded that the existing evidence does not clearly support any of the three theories.

Behavioral methods of assessing ordination, cardination, and natural number competence in children's thinking then were constructed. Care was taken to insure that each behavioral test was the most obvious counterpart of the mathematical definition of the notion that the test was designed to measure. It was argued in this regard that scrupulously precise operational definitions—as opposed to the loose operational definitions that characterize Piaget's (33) original studies of number development—are essential if we hope to decide which of the competing theories of number development is the most tenable. Two developmental studies were reported. In the first study, there were three principal findings: (a) Ordination was found to emerge in three stages (no ordering, spatial ordering, ordination). (b) Cardination was found to emerge in three stages (no correspondence, one-to-many and many-to-one correspondence, one-to-one correspondence). (c) Ordination was found to emerge long before cardination. In the second study, there were three principal findings: (a) The major findings of the first study were replicated. (b) Ordination was found to emerge prior to natural number competence. (c) Cardination was not found to emerge prior to natural number competence. It was concluded that the theory of number development which corresponds to the most satisfactory of the three mathematical theories—namely, the ordinal theory—is consistently supported by the developmental findings.

Possible explanations of the cognitive-developmental sequences reported in the two studies were considered. It was suggested that Flavell's (14) notion of item structure probably accounts for all the sequences. Flavell's notion of item structure specifies that a developmental sequence of the form X₁ → X₂ obtains between two cognitive items when the use of X₂ logically presupposes the use of X₁. It was argued in this regard that the sequences reported in the two studies each reflect logical relationships mentioned in the earlier review of mathematical number theories. Finally, the epistemological question of why such a close relationship exists between logical priorities and cognitive-developmental sequences was considered. It was concluded that there must be at least some structural isomorphism between the domains of cognition and pure mathematics.