What changes in children's drawing procedures? Relational complexity as a constraint on representational redescription

Children's ability to modify their drawing procedures changes in their rst decade. Young children make size/shape changes and end-of-sequence insertions/deletions of drawing elements. Older children also make middle-of-sequence insertions/deletions and position/orientation changes in drawing elements. Why do modi cations occur in this order? We argue that older children's modi cations require processing ternary relations, which according to a relational complexity theory, is beyond the working memory capacity of young children. Introduction: Redescription in children's drawings Karmilo -Smith (1986, 1992) hypothesized that, more than just behavioral mastery, cognitive development includes a process of reorganization of learned, internal representations so as to facilitate more exible, creative behavior. In what she calls the representational redescription hypothesis, two (broad) levels of development are postulated. At the implicit level, knowledge in the system is usable in a very narrow context. At the explicit level, such knowledge is reformatted so as to be accessible by other cognitive processes for use in other domains. One source of support for representational redescription was found in a study on children's drawing procedures. Karmilo -Smith (1990) studied two groups of children on their ability to modify their drawing procedures. The younger (4-6 years) and older (8-9 years) groups were asked to draw three types of objects: a man, a house, and an animal. Both groups were then asked to draw a man, a house and an animal that does not exist (e.g., a man with two heads). The second task probes the sorts of changes children can make to their \normal" drawing procedures in producing \nonexistent" objects. Karmilo -Smith observed four types of changes: (1) Size/shape change (4-6 years): Younger children modi ed their drawings by changing the size or shape of one of its elements (e.g., a man with a square head). (2) End-of-sequence deletion/insertion (4-6 years): Elements were deleted/inserted elements at the end of a drawing after which no more elements were drawn (e.g., a house with no windows, where windows are the last elements of a normal drawing procedure). (3)Middle-of-sequence deletion/insertion (8-9 years): Older children inserted elements between other elements in their drawing procedure (e.g., man with two heads). (4) Position/orientation change (8-9 years): Elements were placed in di erent positions or orientations (e.g., rotated head). Although a notion of redescription is intuitively appealing, providing a mechanism has proven di cult. Progress has been hampered by a lack of details about the nature of the process. In particular, if redescription is to be a theory of cognitive development it must explain why the observed changes to drawing procedures occur in the order they do (Boden, 1994). For example, why do children have the capacity to make end-of-sequence insertions and deletions before they have the capacity to make middle-of-sequence insertions and deletions? Karmilo -Smith (1990) explains the di erence in terms of serial constraints and interrupts. The execution of young children's procedures is constrained to operate in a xed, serial order. Middle-of-sequence insertions/deletions require interrupting that procedure to insert a new subprocedure. But this explanation almost begs the question by appealing to terms so closely related to descriptions of the data. Cognitive development is replete with observations of task orderings. We have explained a number of these observations in terms of relational complexity: the maximum number of interacting dimensions of information that must be processed in a single decision (Halford, 1993; Halford, Wilson, & Phillips, in press). Brie y, the capacity to process higher arity relations increases with age: unary relations (at median age one year); binary (two years); ternary ( ve years); and quaternary (11 years). Therefore, tasks requiring higher arity relations appear later. Relations are ubiquitous in psychological models, and have a formal de nition in computer science (see Appendix A). In this paper, we argue that relational complexity also explains the order of drawing modi cations: middle-of-sequence deletions/insertions and position/orientation changes performed by older children require ternary relational information, which exceeds the binary relational information limit of young children's working memory capacity (Halford, 1993; Halford et al., in press). Relations, associations and explicit/implicit dimensions of variation Informally, a relation is an abstraction of the world that identi es \connections" between its entities and the roles the entities play within these connections.