Elementary school students have difficulty analyzing and interpreting data. In one study, 80% of the first and second graders interviewed gave idiosyncratic or incomplete responses when they attempted to analyze data from a line plot and a bar graph.151 In another study, almost all the fourth and sixth graders could describe bar graphs, but fewer could interpret them, and many fewer still could use the graphs to predict.152

Learning About Chance

Although there has been substantial research on students’ probabilistic thinking over the past 50 years by both psychologists153 and mathematics educators,154 only recently has students’ learning about chance been examined with a view toward informing instruction. In this section, we examine what is known about students’ probabilistic thinking about five key concepts: sample space, probability of an event, probability comparisons, conditional probability, and independence.155

Sample Space

Students exhibit an understanding of sample space when they are able to identify the complete set of possible outcomes in a random experiment, an experiment in which the actual outcome cannot be determined ahead of time even though the set of possible outcomes can be determined. When two coins are flipped, for example, the possible outcomes may be represented as HH, HT, TH, and TT.

Several studies have addressed children’s thinking about sample space.156 Recent research has concluded that a substantial number of students in grades 1 through 3 are not able to list the outcomes of a one-dimensional experiment (such as rolling a single die) even after instruction.157 The students in these studies adopted a deterministic posture, maintaining that it was “always” possible to predict a particular outcome. The situation with respect to two-dimensional experiments (such as rolling two dice) is also problematic. Although some children as young as seven years can use efficient procedures for listing all outcomes,158 other children in grades 4 through 6 are reluctant or unable to list them all.159

Probability of an Event

Although probability tasks used in research with elementary and middle school students have typically involved equally likely outcomes, a number of



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