Cognitive Development in School Context • Statistical Inference and Sampling

We assume that mature statistical understanding evolves from everyday intuitions -- intuitions that are strongly affected by the context in which problems and examples are situated. We believe that common sense understanding is often best characterized as pockets of disconnected intuition with piecemeal application. It is only after considerable effort and the support of external representations that these pockets develop the consistency of theoretical knowledge. This assumption about how to characterize everyday understanding is different from the assumption that understanding should be described as a set of coherent rules or principles that characterize one’s theory of some overall domain. diSessa’s (1983) construct of “knowledge in pieces” in the domain of physics provides an excellent example of the position we wish to emphasize. He argues that people’s understanding of the physical world comes in discrete pieces of intuitive understanding whose elicitation is contingent upon the problem context. “Scientific explanation begins with common sense observation, a principal characteristic of which is its appearance as disparate and isolated special cases.”  Although experts may have well-developed, coherent sets of principles, novices do not. Under this model, conceptual growth does not begin with first principles, such as the laws of thermodynamics, that are subsequently mapped into specific cases. Rather, the growth of understanding is characterized as a process of sifting through and reconciling the cases, “finding successively the more and more general and fundamental ones which serve as principles, explaining the more special cases." In the context of children’s notions of sampling and statistical inference from samples, we suggest that learning and understanding in statistical domains should also be characterized as the integration of pieces of contextual knowledge into fuller understanding. Our research has examined and supported 5th- and 6th-grade children's evolving notions of sampling and statistical inference. Our primary finding has been that the context, as well as the setting, of a statistical problem exerts a profound influence on children’s assumptions about the purpose and validity of a sample. A random sample in the context of drawing marbles, for example, is considered acceptable because chance is the only possible factor, whereas a random sample in the context of an opinion survey is not, because people’s traits cause their opinions. Children have difficulty working with randomness when situations offer causal explanations, and they cannot reconcile how random samples can reveal causal relations. In our design of instructional and assessment materials, we have tried to acknowledge and take advantage of the role that context plays in statistical understanding, while providing representational structures that extend beyond specific contexts.