Transnumerative thinking: finding and telling stories within data

Abstract

Abstract A critical component in the development of students' statistical thinking and reasoning is transnumerative thinking; that is, changing representations of data to engender an understanding of observed phenomena. Examples from Years 6 to 9 New Zealand students' and Australian students' representations of data from a given multivariate dataset are described. Their representations are discussed in terms of their developing abilities to explore data and unlock the stories contained therein. The implications of changing the focus of statistics instruction and the curriculum from merely teaching students how to construct graphs to exploring and representing patterns and relationships in data are presented. Introduction What do you think the graph in Figure 1 is telling us? Is it helpful to know that the children on the left are girls and that the group on the right is made up of boys? [FIGURE 1 OMITTED] Does Table 1 contain the same data? Does it tell the same story? Flow does it differ from the graph? Which representation better tells the story: the graph or the table? Why? What might the data have looked like before they were turned into a graph or a table? Are there other ways of showing the data? Would these tell the same story? How can we tell the clearly? Collecting and exploring data in order to answer questions of interest is an important component of statistical learning. Given a dataset, then, what can we do with it in order to reveal the answers or stories that are hidden it? Messages are not always easy to see in raw data, and so strategies for making those messages visible are important. Such strategies involve analysing and representing the data in ways that show the outcomes clearly. The two examples above demonstrate that there are different ways of representing data, but that some approaches may be better than others for revealing the stories them. The graph, for instance, allows a visual comparison of the two groups--the striking contrast between its two halves clearly shows the difference between the girls' and the boys' fast food consumption. The table, on the other hand, also presents the contrast between the boys' and girls' data, but here this contrast is not as visually obvious as in the graph. Nevertheless, the table summarises the data better, and would be well suited to displaying larger datasets. The process of deciding what to do with a dataset in order to represent it is critical. There has been considerable emphasis on ensuring that students can interpret data in an already existing representation, often focusing on students' ability to read data and read beyond the data, as suggested by Curcio (2001). In contrast, it appears to be more difficult to create successful representations that reveal stories data (Chick & Watson, 2001). Even for adults, producing good representations of data is difficult. There are numerous examples in the media of poorly designed representations, including some that are actually wrong or misleading. The process of going from a raw dataset to a representation that reveals and provides evidence for the story within is, apparently, challenging. Part of the problem is that the curriculum has emphasised univariate datasets and the construction of conventional statistical graphs, but without emphasising the actual purpose of statistical exploration. For many students, graphs are illustrations rather than reasoning tools to detect patterns and unlock the information contained in the data. Furthermore, the emphasis on univariate datasets has prevented students from observing differences and relationships between variables and realising that the purpose of statistical investigations is to seek explanations, to make predictions, and to explore new contextual knowledge. Instruction has focused on how to draw graphs. It now needs to focus on how to represent, explore, and think with data. …