Cognitive Development in School Context • Proportional Reasoning

Mathematics, diagrams, and other explicit representations help scientists discover and organize complex empirical relations, and this may also be true for the developing child. The proposal that external representations and organizing activities contribute to learning and development comes from Vygotsky’s foundational insight that cultural forms mediate the relation between stimulus and response. We have been conducting studies in the area of proportional reasoning to determine specific properties of cultural forms that support cognitive development. In particular, the studies examine the significance of mathematics, a cultural form, in helping children develop complex physical knowledge about balance. In these studies, mathematics does not help children learn primitive perceptual categories of causality, such as weight, torque, or balance. Instead, mathematics helps children structure complex causal relations; for example, when children need to coordinate multiple parameters of distance and weight to determine balance. Results show that when children receive easy to count stimuli that lend themselves to arithmetic operations, or when they are encouraged to “invent” math they exhibit more advanced development using classic developmental tasks.

An issue for views of development that emphasize culturally mediated change is the generality of a given cognitive change. If specific situations and external supports give rise to learning and development, then it is not clear that the resulting knowledge  will spontaneously generalize beyond those specific situations. So, rather than developing a new cognitive structure that is spontaneously applied across situations, children may develop a particular, culturally mediated organization of knowledge (e.g., a proportion) that happens to have general application. It is like learning the concept of variability. The concept is narrow, but its range of application is broad, and it changes the way people can reason about very many situations. By this account, we should not anticipate children who successfully learn to reason proportionally about one situation to exhibit spontaneous proportional reasoning in all amenable situations. Instead, we should expect the continued development of proportional reasoning to be characterized by imperfect and poorly generalized applications of the newly acquired cultural representation. By this account, development depends on the use and transfer of cultural representations.