Data Analysis and Probability Standard for Grades Pre-K–2

The main purpose of collecting data is to answer questions when the answers are not immediately obvious. Students' natural inclination to ask questions must be nurtured. At the same time, teachers should help them develop ways to gather information to answer these questions so that they learn when and how to make decisions on the basis of data. As children enter school and their interests extend from their immediate surroundings to include other environments, they must learn how to keep track of multiple responses to their questions and those posed by others. Students also should begin to refine their questions to get the information they need.

Organizing data into categories should begin with informal sorting experiences, such as helping to put away groceries. These experiences and the conversations that accompany them focus children's attention on the attributes of objects and help develop an understanding of "things that go together," while building a vocabulary for describing attributes and for classifying according to criteria. Young students should continue activities that focus on attributes of objects and data so that by » the second grade, they can sort and classify simultaneously, using more than one attribute.

Students should learn through multiple experiences that how data are gathered and organized depends on the questions they are trying to answer. For example, when students are asked to put a counter into a bowl to indicate whether they vote for a class trip to the zoo or to the museum, the responses are organized as the data are gathered (see fig. 4.20). To address a particular question such as "What is your favorite beverage served in the school cafeteria?" real objects such as containers for chocolate milk, plain milk, or juice can be collected, organized, and displayed. At other times, pictures of objects, counters, name cards, or tallies can be contributed by students, organized, and then displayed to indicate preferences.

Fig. 4.20. Students can contribute counters to bowls to vote.

Methods used by students in different grades to investigate the number of pockets in their clothing provide an example of students' growth in data investigations during the period through grade 2. Younger students might count pockets (Burns 1996). They could survey their classmates and gather data by listing names, asking how many pockets, and noting that number beside each name. Together the class could create one large graph to show the data about all the students by coloring a bar on the graph to represent the number of pockets for each student (see fig. 4.21). In the second grade, however, students might decide to count the number of classmates who have various numbers of pockets (see fig. 4.22). Their methods of gathering the information, organizing it, and displaying the data are likely to be different because they are grouping the data—three students have two pockets, five students have four pockets, and so on. They will have to think carefully about the meaning of all the numbers—some represent the value of a piece of data and some represent how many times that value occurs.

Fig. 4.21. A bar graph illustrating the number of pockets in kindergarten students' clothes

Fig. 4.22. A line-plot graph of the number of students in a second-grade class who have from one to ten pockets

Students do not automatically refine their questions, consider alternative ways of collecting information, or choose the most appropriate way to organize and display data; these skills are acquired through experience, class discussions, and teachers' guidance. Take, for example, the following episode drawn from a classroom experience: »

The students had become interested in the question of whether more families had cars with two doors or four doors. As they planned, the students had to decide if trucks should be included. What about vans with four doors or station wagons with five doors? After the class had settled on common categories, different groups of students kept track of the data in different ways. One group put cubes in different cups that represented the different categories. Another group recorded the data using tallies. A third group of students made a list of families with cars with two doors and those with cars with four doors without attempting to organize the information or agree on the results of their data collection. The teacher used the students' work for a class discussion about which groups were able to answer the question they had posed.

Students' representations should be discussed, shared with classmates, and valued because they reflect the students' understandings. These representations afford teachers opportunities to assess students' understandings and to initiate class discussions about important issues related to representing data. Misconceptions that arise because of students' representations of data offer situations for new learning and instruction. A teacher asked first-grade students to fold a piece of paper in half and cut out a heart (adapted from University of North Carolina Mathematics and Science Education Network [1997, p. 19]). When the students sorted their hearts into three columns according to size (see fig. 4.23a), some of them stated that the large hearts represented the most popular choice because that column was the tallest. A teacher could use a class discussion of the difference between the sizes and the numbers of hearts as an early experience with scale and as an opportunity for the students to plan how to revise the graph to convey the data more accurately. By pasting their hearts on equal-sized pieces of paper, the students could create a new graph, shown in figure 4.23b. »

Fig. 4.23. A misleading data display and its subsequent revision (source of (a): University of North Carolina Mathematics and Science Education 1997, p. 19).