Susan Jo Russell
In our work on the TERC project, Used Numbers: Real Data in the Classroom, we have consistently encountered problematic characteristics of data analysis activities which limit learning and do not address the issues of content, context, and implementation. Students may see the use of data analysis, as they often see the rest of their mathematics curriculum, as "school math," unrelated to the world they know. In particular, our research has shown that most classroom time is spent collecting data and making graphs. Little time is spent discussing the data, posing questions, or building theories. Also, graphs are used primarily as a reporting mechanism at the completion of the data analysis activity, rather than as a means for viewing the data during the process of analysis. In order for students to become truly intelligent users of quantitative data, they must experience data analysis as a process -- and graphs as a tool -- for illuminating the reality represented by the data.
The Problem with Classroom Graphs
We initially chose the area of data analysis for curriculum development in part because available computer tools seemed to offer promise for elementary students in this area. Graphing software, spreadsheets, and data bases all provide means of storing, organizing, and displaying data. Just as the word processor can free students from mechanical constraints in writing, so, we thought, computer tools can open up mathematics as a process in which students work as mathematical thinkers, not just calculators.
For many students, graphing by hand is tedious, difficult, and time-consuming. Making graphs by hand, therefore, often exacerbates the problems associated with teaching data analysis. Many students become so caught up in the problem of making the graph that they lose sight of the meaning of the data. Because of its time-consuming nature, making graphs by hand usually limits students to one representation of the data so that students do not look at their data in a variety of ways. Making graphs by hand also discourages revision.
Typically, the amount of energy put into creating a simple, hand-drawn graph is out of proportion to the complexity of the nature of the data. This imbalance leads, in many classrooms we have observed, to a feeling of anticlimax when the graphs are completed. Students who report about their finding have little to say. Either they knew what they had to say long before the graph was completed and have lost interest in it or the classroom time used to make the graphs supplanted time which could have been devoted to considering the shape of the data and generating questions and theories from the data.
Does the Computer Help?
In all classrooms students have been excited and engaged by the use of the computer and have been willing to cope with the sometimes cumbersome and limited features of much of the graphing software available to young users (awkward data entry, limited number of data points handled by the software, inflexible scaling, etc.). However, simply using graphing software in a classroom, while it addresses some of the issues listed above, introduces its own problems -- and certainly does not lead automatically to richer experiences in data analysis, any more than the introduction of a word processor necessarily leads to better writing experiences.
Almost all graphing software, whether created for naive or sophisticated users, has several flaws. First, data must be entered initially into a table; it is then transferred by the computer to a graph. The user is unable to view the table and the graph simultaneously and cannot easily link a piece of data on the graph to its counterpart in the table, or vice versa. The young user easily loses track of the connection between table and graph.
Second, most graphing software treats all frequency distributions as if the data were nominal. This problem is most serious for continuous numerical variables, but is also apparent for discrete numerical data. Therefore, many kinds of data in which students are interested -- the heights of their classmates, the number of seconds people can hold their breath, the sizes of families -- cannot be graphed easily using these packages. In fact, such a graph can only be made by treating scalar values as if they are categories, leading to student graphs in which points in the scale are simply ignored. Since histograms are not available in these packages, students must categorize their data in awkward ways or "trick" the software into the kind of display needed.
In addition to such limitations of the software, by its very nature graphing software leads to another problem. The computer can be used to make effective graphs using a limited number of standard representations, yet students cannot design idiosyncratic, but effective, graphs. For example, in a fourth grade classroom, a group of students produced a wonderfully communicative graph using paper and colored markers to depict a comparison of the heights of first and fourth graders (Figure 1). Middle grade students can easily deal with two variables if they can use differently colored or shaped icons on the same graph. The only option graphing software offers for making this kind of comparison is a multiple bar graph (and even this option is not available in all graphing packages), a representation which, in this case (Figure 2), was far less illustrative of the comparison of the distribution of heights in these two groups than the fourth graders' invented graphs.
Despite these problems, we have seen glimmers of the potential of the power of graphing software placed in student hands. For example, students who were using the computer to make graphs showing data about pets for each grade level. They produced graphs for all six grade levels in the time it took other students to produce, by hand, one graph for one grade level, and they were able to make clear comparisons among the grades because the order of the categories was the same across graphs. And in a classroom studying TV-watching habits, students graphed the number of hours each student had watched television during one 48- hour period. At first, the software automatically set a scale of 0-10 for the number of hours (the data ranged from 0 to 8 hours). By rescaling this axis form 0 to 48, students were able to see TV-watching time in relationship to the total period in which it occurred.
It is unlikely that any of these episodes would have occurred if the participants had been creating hand-drawn graphs. In all cases, the ability to easily generate two or for representations led to a greater depth of analysis than could have occurred with a single representation.
Different Graphs for Different Purposes: Building a Repertoire of Tools
Ideally, we want students to develop a large repertoire of tools for representing their daft, with the tool and technique matched to the use of the representations. We must distinguish, for example, between graphs used in the process of analysis and representations used to present findings. Students need many experiences with each of these -- the first simply to get a sense of the shape and characteristics of the data, the second in which students' creativity and artistry can be tapped in making beautiful and effective presentations. In particular, we see three types of representation with which students need to be fluent:
- Quick sketches, used for an initial look at the data. Line plots, stem-and-leaf diagrams, and tallies are examples of techniques with which students should become facile. These can often be done easily by hand, even for relatively large data sets. Our research has shown that when students acquire fluency in making quick sketches of their data, they are more likely to be able to describe the shape, features, and patterns in their data.
- Interactive representations, used for creating multiple views of the data during analysis. Computer tools allow students to compare different representations and to manipulate features of these representations, such as scale, so they can illuminate different aspects of the data.
- Presentation graphs, used for communicating findings to an audience. While the computer can provide neat, accurate, formal graphs, students can also be encouraged to create and construct more idiosyncratic representations, which may be more beautiful, more communicative, and more complex than those available on the computer.
As an adjunct to our research in this area, we are currently experimenting with use of a Dynamic Bargrapher, written in Logo, which allows students to enter their data directly onto the graph, enabling them to watch the shape of the graph develop as data are entered. Also, in TERC's Hands On Data project, we are investigating the use of a highly visual, interactive software environment called the Tabletop which allows students to manipulate data as icons into many different representations (see "Getting our Hands on the Data," in this issue). Continuing to develop and exploit these kinds of graphing software tools will be a key component in enabling students to use numbers in the real world.
Acknowledgements:
Portions of this article appeard as a paper presented at the International Conference on Mathematics Education, Hungary, July 1988. The work reported was developed jointly by the Used Numbers Project staff. In particular, William Barclay III, Rebecca B. Corwin, Susan N. Friel, Janice R. Mokros, and Antonia Stone contributed substantially to the development and training on which this work is based.
Author Info:
Dr. Susan Jo Russell is Co-Director of the Mathematics Center at TERC.
Other Articles by Susan Jo Russell:
Changing the Elementary Mathematics Classrooms: Obstacles and Challenges
Children's Concepts of Average and Representativeness
Developing Computational Fluency with Whole Numbers in the Elementary Grades
Mathematics Curriculum Implementation: Not a Beginning, Not an End