By Edwards, Thomas G.; Hensien, Sarah M.
Teaching Children Mathematics , Vol. 6, No. 8
Mathematics instructional programs in elementary school should include opportunities for students to understand and apply basic concepts related to chance and probability. The discussion draft of Principles and Standards for School Mathematics suggests that "students in grades 3-5 begin to actively consider the likelihood of events. At these levels, informal explorations involving probability are appropriate" (NCTM 1998, 181).
These informal investigations might take the form of probability experiments. Students have a natural curiosity about such experiments. Many are quick to speculate about the likely outcomes of the experiments, and most have hunches about them. Students' hunches may enable them to correctly identify the more likely of two outcomes (e.g., a coin is more likely to land on heads than a number cube is to land on a 5). The goal, however, should be to help students begin to quantify these notions, that is, to state the chance of a coin's landing on heads as "one out of two" and the chance of a number cube's landing on a 5 as "one out of six."
The probability experiments and discussions described in this article are possible avenues for students to pursue their mathematical hunches about chance. When students examine their hunches, the mathematical concepts embedded in the experiments will emerge naturally. Students should come away from these experiences with a sense that numbers can play a useful role in describing the chance that an event will occur, as well as an understanding of the nature of events that are equally likely to happen.
In the vignette that follows, the students are fifth graders who have had little or no previous experience studying probability. The teacher, Ms. Johnson, has organized the class in six groups of four, and the groups are working on three different probability experiments. Johnson hopes that these experiments will help her students--
* construct a concept of equally likely events,
* assign a theoretical probability to events, and
* relate the theoretical probability of an event to the observed relative frequency of that event during the experiment.
Figure 1 shows the directions for each of the experiments on which the students are working. All three of these experiments involve events that have equally likely outcomes: flipping a single coin; spinning an evenly divided spinner; and tossing a single number cube, or die.
Johnson has chosen to have the six groups repeat two of the experiments 25 times and one, 30 times to yield a data pool of 150 or 180 repetitions of each experiment. In each instance, the total number of repetitions is divisible by the number of possible outcomes for the experiment. The results will facilitate students' discussions about how many occurrences of each event should be expected.
Johnson has decided to use three experiments because she does not want her students to develop misconceptions, such as that "equally likely events must involve just two (or any other specific number of) possibilities." She also expects that by doing three different experiments, her students will have three different opportunities to compare the theoretical probability of an event with its observed relative frequency. In discussing the issue of equal likelihood, Johnson believes that the idea of quantification of chance will emerge naturally. That is, students will begin to think of and express the likelihood of these events in numerical terms as a result of their direct experience with the experiments.
The following vignette was compiled from our analysis of several similar lessons. The description of the whole-class discussions represents our best attempt to capture the spirit of the actual discourse that occurred in those classrooms.
The groups have collected and recorded data at each of the stations and have finished pooling their results for each experiment to obtain data for the whole class. …