Students need opportunities to build models and representations that suit particular explanatory and communicative purposes. They need experience refining and improving models and representations, experience that can be facilitated by critically examining the qualities of multiple models or representations for a given purpose.

In the following example we visit a fifth-grade classroom in which students are studying species variation. Having tracked the growth of Wisconsin Fast Plants over a period of 19 days, they are grappling with the best way to represent their data. Hubert Rohling, the teacher, has posted a list of unordered measures that the students had taken over the previous 18 days on chart paper at the front of the class. He has asked them to consider two questions: (1) how they might organize the data in a way that would help them consider typical height on the 19th day and (2) how to characterize how spread out the heights were on this day. He chose to have the students focus on these qualities of their representation in order to draw their attention to critical aspects of representing data sets.

Mr. Rohling understood that his students would need to grapple with how best to portray data and to practice doing so as a purposeful activity. Rather than assigning children particular data displays to use in capturing data, he asked them to invent displays. He introduced additional uncertainty into the assignment by asking students to identify typical values. Often the approach to learning about typical values is to teach children different measures of central tendency and to assign children to calculate means, or identify the modal or median values in a data set. Mr. Rohling’s interest, however, was to push children to wrestle with the notion of typicality and articulate their understanding through creating and critiquing data displays.

In the process students would be forced to grapple with the value of maintaining regular intervals between data points (thus providing a visual cue as to the quantitative relationship among points) and sampling distribution. (What aspect of the data provides a fair sense of the overall shape of the data set?) Students would confront the same kinds of problems that scientists do in the course of their work. They must find meaningful ways to organize information to reveal particular characteristics of the data.

The students had previously been assigned to seven working teams of three to four students each. The students in each group worked to construct a data display that they believed would support answers to Mr. Rohling’s two questions. Mr. Rohling encouraged each group to come up with its own way to arrange the data, explaining that it was important that the display, standing alone, make apparent the answer to the two questions about typicality and spread of heights.

The students’ solutions were surprisingly varied. From the seven groups, five substantively different representational designs were produced. Over the next two days, students debated the advantages and trade-offs of their representational choices; their preferences shifted as the discussion unfolded. To encourage broad participation in critical discussion of displays, Mr. Rohling assigned pairs of students to present displays that their classmates had developed. And following this he facilitated discussions which drew in display authors, presenters, and other classmates. Despite the opportunity to exchange ideas with their peers, students did not easily or simply adopt conventions suggested by others. Instead, there was a long process of negotiation, tuning, and eventually convergence toward a shared way of inscribing what students came to refer to as the shape of the data.