In grades K-4, the mathematics curriculum should include experiences with data analysis and probability so that students can--

• collect, organize, and describe data;
• construct, read, and interpret displays of data;
• formulate and solve problems that involve collecting and analyzing data;
• explore concepts of chance.

Focus

Collecting, organizing, describing, displaying, and interpreting data, as well as making decisions and predictions on the basis of that information, are skills that are increasingly important in a society based on technology and communication. These processes are particularly appropriate for young children because they can be used to solve problems that often are inherently interesting, represent significant, applications of mathematics to practical questions, and offer rich opportunities for mathematical inquiry. The study of statistics and probability highlights the importance of questioning, conjecturing, and searching for relationships when formulating and solving real-world problems.

A spirit of investigation and exploration should permeate statistics instruction. Children's questions about the physical world can often be answered by collecting and analyzing data. After generating questions, they decide what information is appropriate and how it can be collected, displayed, and interpreted to answer their questions. The analysis and evaluation that occur as children attempt to draw conclusions about the original problem often lead to new conjectures and productive investigations. This entire process broadens children's views of mathematics and its usefulness.

Statistics and probability are important links to other content areas, such as social studies and science. They also can reinforce communication skills as children discuss and write about their activities and their conclusions. Within mathematics, these topics regularly involve the uses of number, measurement, estimation, and problem solving.

Discussion

This standard recognizes the importance of having all students develop an awareness of the concepts and processes of statistics and probability. The curriculum must emphasize that statistics is more than reading and interpreting graphs: It is describing and interpreting the world around us with numbers, and it is a tool for solving problems. Children need to recognize that many kinds of data come in many forms and that collecting, organizing, displaying, and thinking about them can be done in many ways.

In the early grades, actual objects should be displayed so that their characteristics can be observed and discussed. In this work, each unit used on scales for graphs should represent 1. Later, pictorial and symbolic graphs should be constructed and discussed; by grades 3 and 4, children should be able to use scales representing other units, such as 2, 5, or 10. Computer graphing programs make a wide variety of explorations accessible to older children after they have learned to create graphs on their own. Exercises like the following encourage children to use pictures and symbols to characterize and group objects.

Pretend we own a children's shoe store. We need to know whether to have more cloth or more leather shoes for sale in our store. What could we do to decide?

The children might decide to make a floor graph with one shoe from each child as a way of determining the number of cloth and leather shoes in their class. Questions to guide students' activities can include these: Are there more cloth shoes or leather shoes? Are the two numbers close? Should we have about the same number of cloth and leather shoes in our store?

Children should learn that data can be displayed in different ways and that depending on the question being asked, one type of display might be more appropriate than another. A variety of early experiences helps children build a foundation for creating conventional graphs. See figure 11.1.

Fig. 11.1

Which display would we use to find out the kind of ice cream that Molly likes? Which display would we use to find out which flavor is the most popular?

A class or group project conducted over time enables the students to make predictions and modify them as more data are collected.

Suppose, for example, that children are interested in comparing the temperatures in their hometown with the temperatures in two other cities. They can obtain pertinent data from such sources as newspapers or television. They can participate in making decisions about what questions to ask; what data to collect; and how to collect, organize, and display them for others to see and interpret. See figure 11.2.

Fig. 11.2

Again, teachers might find that questions like the following are helpful in guiding students' efforts.

What patterns do you notice? Did anything about the data surprise you? What temperature do you predict for each city on 24 March? Will these temperature trends continue through 24 December? Why do you think New York is getting warmer and Sydney is getting colder?

Children of this age will also enjoy and profit from the exploration of chance. This pursuit should have the same investigative flavor as that recommended for statistics, as illustrated by the following activity involving spinners (fig. 11.3).

Fig. 11.3

Is red or blue more likely? How likely is yellow? How likely is getting either red or blue? If we spin twelve times, how many blues might we expect to get?

The following game combines an exploration of probability with data analysis:

Cut a hole slightly smaller than the size of a craft bead in the lid of an opaque bottle. Secretly place ten beads of two colors in the bottle (e.g., three red and seven blue). Two teams of children alternate turns; each team uses a number line that goes from 0 to 10. The starting point is 5. On each turn, the team decides whether red or blue will indicate movement toward 10. The members of each team take turns shaking the bottle upside down until one bead falls out. If the bead is of the color predicted, the team moves forward one space. If not, it moves back one space. The first team to reach either 0 or 10 wins. Children should be encouraged to keep a record of the colors that appear for later analysis and discussion.

At the conclusion of the game, each team guesses the number of beads of each color. The first team to identify correctly the number of each color gets to change the distribution of the two colors in the bottle for the next round.

This game allows students to explore many aspects of probability and gather and analyze data in a problem-solving atmosphere. Discussions following the game can include the concepts of events that are likely, events that are certain, and common perceptions of "luck."