In prekindergarten through grade 2, students will have learned that data can give them information about aspects of their world. They should know how to organize and represent data sets and be able to notice individual aspects of the datawhere their own data are on the graph, for instance, or what value occurs most frequently in the data set. In grades 35, students should move toward seeing a set of data as a whole, describing its shape, and using statistical characteristics of the data such as range and measures of center to compare data sets. Much of this work emphasizes the comparison of related data sets. As students learn to describe the similarities and differences between data sets, they will have an opportunity to develop clear descriptions of the data and to formulate conclusions and arguments based on the data. They should consider how the data sets they collect are samples from larger populations and should learn how to use language and symbols to describe simple situations involving probability.
Investigations involving data should happen frequently during grades 35.
These can range from quick class surveys to projects that take several
days. Frequent work with brief surveys (How many brothers and sisters
do people in our class have? What's the farthest you have ever been from
home?) can acquaint students with particular aspects of collecting, representing,
summarizing, comparing, and interpreting data. More extended projects
can engage students in a cycle of data analysisformulating questions,
collecting and representing the data, and considering whether their data
are giving them the information they need to answer their question. Students
in these grades are also becoming more aware of the world beyond themselves
and are ready to address some questions that have the potential to influence
decisions. For example, one class that studied playground injuries at
their school gathered evidence that led to the conclusion that the bars
on one piece of playground equipment were too large for the hands of most
students below third grade. This finding resulted in a new policy for
At these grade levels, students should pose questions about themselves and their environment, issues in their school or community, and content they are studying in different subject areas: How do fourth graders spend their time after school? Do automobiles stop at the stop signs in our neighborhood? How can the amount of water used for common daily activities be decreased? Once a question is posed, students can develop a plan to collect information to address the question. They may collect their own data, use data already collected by their school or town, or use other existing data sets such as the census or weather data accessible on the Internet to examine particular questions. If students collect their own data, they need to decide whether it is appropriate to conduct a survey or to use observations or measurements. As part of their plan, they often need to refine their question and to consider aspects of data collection such as how to word questions, whom to ask, what and when to observe, what and how to measure, and how to record their data. When they use existing data, they still need to consider and evaluate the ways in which the data were collected.
Students should become familiar with a variety of representations such as tables, line plots, bar graphs, and line graphs by creating them, watching their teacher create them, and observing those representations found in their environment (e.g., in newspapers, on cereal boxes, etc.). In order to select and interpret appropriate representations, students in grades 35 need to understand the nature of different kinds of data: categorical data (data that can be categorized, such as types of lunch foods) and numerical data (data that can be ordered numerically, such as heights of students in a class). Students should examine classifications of categorical data that produce different views. For example, in a study of which cafeteria foods are eaten and which are thrown out, different classifications of the types of foods may highlight different aspects of the data.
As students construct graphs of ordered numerical data, teachers need to help them understand what the values along the horizontal and vertical axes represent. Using experience with a variety of graphs, teachers should make sure that students encounter and discuss issues such as why the scale on the horizontal axis needs to include values that are not in the data set and how to represent zero on a graph. Students should also use computer software that helps them organize and represent their data, including graphing software and spreadsheets. Spreadsheets allow students to organize and order a large set of data and create a variety of graphs (see fig. 5.20).
When students are ready to present their data to an audience, they need to consider
aspects of their representations that will help people understand them:
the type of representation they choose, the scales used in a graph, and
headings and titles. Comparing different representations helps students
learn to evaluate how well important aspects of the data are shown.
In prekindergarten through grade 2, students are often most interested in individual pieces of data, especially their own, or which value is "the most" on a graph. A reasonable objective for upper elementary and middle-grades students is that they begin to regard a set of data as a whole that can be described as a set and compared to other data sets (Konold forthcoming). As students examine a set of ordered numerical data, teachers should help them learn to pay attention to important characteristics of the data set: where data are concentrated or clumped, values for which there are no data, or data points that appear to have unusual values. For example, in figure 5.21 consider the line plot of the heights of fast-growing plants grown in a fourth-grade classroom (adapted from Clement et al. [1997, p. 10]). Students describing these data might mention that the shortest plant measures about 14 centimeters and the tallest plant about 41 centimeters; most of the data are concentrated from 20 to 23 centimeters; and the plant that grew to a height of 41 centimeters is very unusual (an outlier), far removed from the rest of the data. As teachers guide students to focus on the shape of the data and how the data are spread across the range of values, the students should learn statistical terms such as range and outlier that help them describe the set of data.
Much of students' work with data in grades 35 should involve comparing related data sets. Noting the similarities and differences between two data sets requires students to become more precise in their descriptions of the data. In this context, students gradually develop the idea of a "typical," or average, value. Building on their informal understanding of "the most" and "the middle," students can learn about three measures of centermode, median, and, informally, the mean. Students need to learn more than simply how to identify the mode or median in a data set. They need to build an understanding of what, for example, the median tells them about the data, and they need to see this value in the context of other characteristics of the data. Figure 5.22 shows the results of plant growth in a third-grade classroom (adapted from Clement et al. [1997, p. 10]). Students should compare the two sets of data from the fourth-and third-grade classrooms. They may note that the median of the fourth-grade data is 23 centimeters and the median of the third-grade data is 28 centimeters. This comparison provides information » that, overall, the set of third-grade plants grew taller than the set of fourth-grade plants. But it is also important to look at the distributions of the data, which tell an even more dramatic story: Although the ranges of the two data sets are about the same, most of the third graders' plants grew taller than all but a few of the fourth graders' plants.
Data can be used for developing arguments that are based on evidence
and for continued problem posing. As students discuss data gathered to
address a particular question, they should begin to distinguish between
what the data show and what might account for the results. For example,
a fourth-grade class investigating the sleep patterns of first graders
and fifth graders found that first graders were heavier sleepers than
fifth graders, as shown in the graphs in figure 5.23 (Russell, Schifter,
and Bastable 1999). They had predicted that first graders would be lighter
sleepers and were surprised by their results. After describing their data,
they developed a hypothesis: First graders have a higher activity level
because they play outside more, and this higher activity level leads to
deeper sleep. They realized they would need to collect data about a typical
day for first and fifth graders in order to investigate their hypothesis.
This example demonstrates how students can be encouraged to develop conjectures,
show how these are based on the data, consider alternative explanations,
and design further studies to examine their conjectures.
With appropriate experiences, students should begin to understand that many data sets are samples of larger populations. They can look at several samples drawn from the same population, such as different classrooms in their school, or compare statistics about their own sample to known parameters for a larger population, for example, how the median family size for their class compares with the median family size reported for their town. They can think about the issues that affect the representativeness of a samplehow well it represents the population » from which it is drawnand begin to notice how samples from the same population can vary.
Students in grades 35 should begin to learn about probability as a measurement of the likelihood of events. In previous grades, they will have begun to describe events as certain, likely, or impossible, but now they can begin to learn how to quantify likelihood. For instance, what is the likelihood of seeing a commercial when you turn on the television? To estimate this probability, students could collect data about the number of minutes of commercials in an hour.
Students should also explore probability through experiments that have only a few outcomes, such as using game spinners with certain portions shaded and considering how likely it is that the spinner will land on a particular color. They should come to understand and use 0 to represent the probability of an impossible event and 1 to represent the probability of a certain event, and they should use common fractions to represent the probability of events that are neither certain nor impossible. Through these experiences, students encounter the idea that although they cannot determine an individual outcome, such as which color the spinner will land on next, they can predict the frequency of various outcomes.
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