A definition of graph sense

The terms graph comprehension and graph sense are not defined robustly in the literature, and are used seemingly interchangeably. Graph comprehension was defined as “graph readers’ abilities to derive meaning from graphs created by others or themselves” (Friel et al., 2001, p. 132), but this definition does not distinguish between direct reading of explicit information presented in a graph, and the more complex and abstract processes involved in inference, evaluation, synthesis and extrapolation that might be required to interpret the same graph fully. Friel et al. extended the notion of number and symbol sense to define graph sense as “a set of behaviours and ways of thinking” (p. 145). This includes the ability to recognise the components within a graph, to understand the relationships and conventions within the graph, to speak the language of the graph, to assess the information within the graph objectively, and to select the

45 most useful form for the graph. This study uses the term graph sense, and the term graph comprehension is applied to the processes involved in comprehending a graph. 2.4.14.2 The process of graph comprehension

The process of graph comprehension has been researched extensively from perspectives as diverse as education (Ainley, 2000; Curcio, 1987; Watson & Moritz, 1999); anthropological tool use (Meira, 1998); symbolism, psychology, cognitive science, and information processing (Carpenter & Shah, 1998; Simkin & Hastie, 1987; Trickett & Trafton, 2006); business and management (Jarvenpaa & Dickson, 1988, cited in Friel et al., 2001, p. 125), mass communication and graphic design (Feliciano, Powers, & Keral, 1963; Kosslyn, 1994). More recently research has focussed on computer visualisation associated with the development of Graphical User Interfaces, computer-based imagery, and dynamic displays (e.g., International Journal of Human – Computer Studies). Research evidence specific to statistics and probability education in classroom-based computer environments (e.g., Abrahamson, 2006; Bakker & Gravemeijer, 2004; Ben Zvi, 2004a) examined graph comprehension, but as a subsidiary activity within other mathematical tasks, such as exploratory data analysis or data simulation.

This previous research has enabled researchers to propose models of graph comprehension based on knowledge use, and the spatial, perceptual and cognitive processes involved in graph comprehension (e.g., Carpenter & Shah, 1998; Peebles & Cheng, 2003; Trickett & Trafton, 2006). Many of the models describe comprehension as a series of steps. Bertin 1993, as cited in Carpenter and Shah, 1998 describes the process of graph comprehension as drawing on a series of three elements: translation, to interpret a graph verbally; integration and interpretation of two or more features of the graph; and extrapolation and interpolation beyond the understanding of the essence of the graph to identify inferences and consequences of the information. From the psychology research perspective Ratwani, Tafton, and Boehm-Davis (2008) examined graphing tasks using a choropleth graph (where colour coding and shading represents magnitude), and described the complex nature of graph comprehension as a combination of interactive perceptual and cognitive processes involving three stages of pattern recognition and visual decoding, identification of conceptual relationships between the features, and relating the graph referents of axes and scale to the visual features within the graph. Such studies have been criticised as narrowly focussed,

46 laboratory-based, and lacking the social context of the education learning environment (e.g., Freedman & Shah, 2002).

Researchers have also criticised these approaches as not truly reflecting the non-linear and iterative nature of learning (Bakker, 2004; Carpenter & Shah, 1998; Konold & Higgins, 2003; Peebles & Cheng, 2003). Graph comprehension is described as a process where the graph reader shifts attention from one aspect of the graph, to another, and back again, in a process that serves to reinforce the information in memory, and mentally construct, and assemble progressively, the component “chunks of features” (Liu & Wickens, 1992, cited in Ratwani et al., 2008) into a cohesive structure. A graph is a sign: something visible that stands for something invisible or abstract. The dual nature of visibility and invisibility is described using an analogy of a window through which the outside world is revealed, but where one remains relatively unaware of the window’s presence (Ainley, 2000; Meira, 1998). Bakker (2004) considers a graph as a diagram that describes a complex relationship between symbols. Symbols are developed for a specific purpose and are capable of refinement through a process of evolution and development variously described as a “cascade of inscriptions” (La Tour, 1987, cited in Roth & McGinn, 1998), where multiple translations of the information are performed until the representation reaches its final form. A graph is also an artefact – a term that is used subsequently as an element of instrumental genesis – that provides access to meaning and significance beyond the artefact (e.g., Ainley, 2000).

Graphing is a process of data translation, reduction, and aggregation. Data translation refers to the situation where data that may have once had number values now have their values defined by positions on a graph, and positions relative to other data points. Data reduction is the process where data are aggregated or recalculated, which has the consequence that the original data become invisible. Konold and Higgins (2003) noted the distinction between non-aggregated data, where each data point retained a one-to- one correspondence with the original data, and aggregated data, where a direct reference to the original data points is lost. Aggregation, whether calculating a statistic, or creating a graph, or a summary table, potentially promotes understanding by revealing the underlying structure of the data, but it also increases the level of abstraction and the risk that the meaning will be lost. To minimise the risk that meaning is lost Bakker (2004) argued that students should, at least for initial tasks, be able to trace the source of the original data, and Abrahamson (2006) and Watson (2006) sought

47 to reinforce the link between the original data and the graph for middle-school students by using iconic representations on the graph.