Learning from experience is one of the most basic and important cognitive problems, for both adults and, especially, for young children. Much of this learning takes the form of conditional probability judgments: judging what outcomes are likely given some evidence. This project explores the development of such conditional probability judgments. Specifically, how do children use examples they know about to make predictions about unfamiliar cases? The significance of some examples depend on 1) which population one is concerned about (e.g., the conditional one is assessing) and 2) the population sampled from (e.g., how the evidence was generated). Asking how and when children are sensitive to differences in conditionals and sampling addresses longstanding debates about the role of inferential and similarity-based processes in cognitive development. The results of this work will illustrate ways in which basic similarity models must be elaborated to account for children's conditional judgments. In addition, a developmental perspective is critical for understanding why adults do or do not adequately account for conditionals and sampling in their judgments. Finally, conditional probability judgments are just such a basic feature of cognition that understanding their development is central to a wide range of psychological phenomena and theories.
Understanding how children make conditional probability judgments is important for a number of social and educational concerns. For example, stereotyping often involves a failure or bias in conditional judgment. From evidence that students who commit school shootings are likely to have played violent video games, people often, erroneously, conclude that students who play violent video games are likely to commit school shootings. This project will allow us to better understand the conditions under which children form stereotypes, and provide guidance for efforts to reduce stereotyping. The reasoning abilities explored in this project are also critical in educational contexts, especially for scientific reasoning. The logic of experimental design and hypothesis testing is based on conditional probability, including ideas about sampling and implications of evidence. Conditional probability judgments are involved when interpreting statistical claims in the media and in public policy debates. By understanding how children approach conditional probability judgments we will be able to design more effective instruction to improve scientific literacy.
. Conditional probabilities are central for learning and prediction. For example, learning that hot stoves burn could involve learning that p(burn|touched hot stove) is high (the probability of a burn given that you've touched a hot stove). Conditional probability has a very specific, mathematical definition. Do children's representations of predictive/conditional relations work like conditional probabilities? The general answer from this project is that they do not. Knowing how they do not has significant implications for children's learning and predictive judgments. Much of the project focused on the idea of symmetry, or "the inverse fallacy". Research with adults has shown that people often assume that p(a|b) = p(b|a). Unfortunately this is rarely the case. Young children seem especially susceptible to this assumption. One consequence is that children often think they are learning "more" than they really are. For example, if I learn that most hot stoves burn, p(burn|stove)~1, I am not really learning anything about the frequencies of different kinds of burns. I should not expect that most burns are caused by stoves, p(stove|burn)~1. Children often act as if they are learning both these conditional probabilities from experience that is actually informative about only one. This kind of error has direct implications for stereotyping: Learning that most criminals are young men does not mean that most young men are criminals. This also helps us understand children's hypothesis testing and scientific reasoning. The evidence that one would want to decide whether most stoves burn is very different than the evidence that most burns are caused by stoves. Our project suggests that young children do not adjust their exploration or hypothesis testing to focus on different conditional probabilities. Finally, a sort of opposite effect is learning "too much" from counter-examples. Learning that most burns are not caused by stoves can convince children that most stoves do not burn. A second line of investigation addressed a different sort of expansion or unwarranted inference from conditional probability. Roughly, that half the a's in the world are <1, and half the b's in the world are <1, does not mean that p(b<1|a<1) is high. The ways a's and b's are distributed does not provide much information about p(a|b) or vice versa. Our work suggests that young children are particularly inclined to see consistency among different (but related) distributions. One consequence is "drift" in learning. For example, animals "naturally" divide such that whales seem like fish rather than cows. The tug to make conditional probabilities line up may lead someone who has already learned that whales are mammals shift their boundary and start calling whales fish. Our research has demonstrated just this kind of drift. One general conclusion is that children operate as if they were in a pretty friendly world in which various probabilities are all consistent. For example p(a|b) = p(b|a) is a special case (true when p(a) = p(b)), but children may assume this is the norm. It is actually not at all clear how "bad" these assumptions really are given people's everyday experience. Moreover, it is also unclear how much parents and teachers act to structure children's experience so that it is "learnable" (e.g., conforms to their expectations). Now that we know more about the assumptions children make about conditional probabilities we are in a much better position to design instruction. This project has already generated hypotheses about how to structure practice problems to best support learning of particular conditional probabilities. These results have been shared with faculty in teacher education at UW-Madison, and have been incorporated into courses for pre-service teachers and educational psychologists. This project has also provided the foundation for a funded research proposal to the Institute for Education Science to study the effects of different forms of practice with mathematical problem solving. We have also been working with the Madison Children’s Museum to test some implications for exhibit design.
