Understanding Conditional Probability

THE fundamental aim of education is the development of understanding: the mastery of concepts and procedures that permit the intelligent solution of problems in new and novel situations. In mathematics this goal is particularly challenging, for students are often adept at memorising the rules and formulas necessary to solve welldefined test and textbook problems, without ever coming to terms with the meaning or logic of the arguments they employ. As research shows, outside of the classroom even the most familiar operations can be ignored in favour of intuitive judgements (Gardner, 1991; Piattelli-Palmarini, 1994). But while commonsense may provide serviceable solutions to many problems, experience also reveals that our natural theories can lead us into serious errors. To improve their intellectual skills, students must learn to recognise the limits of their spontaneous problemsolving strategies and develop secondary intuitions for the more precise and powerful reasoning tools developed by mathematicians (Fischbein, 1987).