Fases de creación

To inform the inquiry, a comprehensive consideration of the relevant literature was undertaken to identify theoretical models of statistical thinking and reasoning that were directly related to data analysis and in particular, graphing. Models developed by Friel et al. (2001); Jones, Thorton, Langrall, Mooney, Perry, and Putt (2000); Mooney (2002); and Moritz (2004) were considered. In addition, suggestions by Shaughnessy (2007) for an

additional level to be added to Friel et al.‘s levels of thinking and the notion of transnumeration presented by Pfannkuch and Wild (2004) were reviewed to provide an extensive view of the development of graphing and graph sense-making.

The model developed by Friel et al. (2001) was conceptualised around a focus on understanding students‘ development of graph comprehension. The term ―graph sense‖ was introduced and described as what ―develops gradually as a result of one‘s creating graphs and using already designed graphs in a variety of problem contexts that require making a sense of data‖ (p. 145). This description placed the emphasis for graph comprehension on the use of graphs as tools for making sense of information. In line with this view, they identified six behaviours associated with graph sense: recognising components of graphs; speaking the language of graphs; understanding relationships among tables, graphs, and data; making sense of a graph; interpreting a graph and answering questions about it; and recognizing appropriate graphs for a given data set and its context.

In 2007, Shaughnessy extended the work of Friel et al (2001) further and suggested two additional behaviours in recognition of the influence of context on the data analysis process and to provide a wider view of data handling. The additional behaviours were: look for possible causes of variation, and look for relationships among variables in the data. When considered together, the additional behaviours suggested by Shaughnessy gives the Friel et al. model greater depth in terms of the way students may develop graph sense. The behaviours do not, however, recognise explicitly the impact that constructing graphs may have on the development of statistical thinking and reasoning.

Prior to the work presented by Friel et al. (2001) and influenced by the work of Kosslyn (1989), Curcio (1989) considered school students‘ interpretation of graphs from three perspectives, Read the data, Read within the data, Read beyond the data (cited in Shaughnessy, 2007). Shaughnessy suggested that each of the six behaviors identified by Friel et al. (2001) fits with one of Curcio‘s three levels of graph reading. Shaughnessy went on to suggest extending the categories to include the two behaviours he identified under the level of Reading behind the data.

The levels of graph interpretation suggested by Curcio (1989) (as cited in Shaughnessy, Garfield, & Greer, 1996) have been used as a foundation for exploring primary

students‘ interpretation of graphs (Jones et al., 1997, 2000; Mooney, 2002). Although applied successfully by Jones et al. (1997, 2000) to describe and assess students‘ levels of statistical reasoning and by Mooney to develop a cognitive model of graph interpretation, these studies did not extend the levels of graph interpretation further. The cognitive model put forward by Mooney can be seen as a hierarchy but like Friel et al. (2001) did not include consideration of the thinking processes associated with constructing graphs.

An extensive four-dimensional model proposed by Wild and Pfannkuch (1999) and elaborated on by Pfannkuch and Wild in 2004 includes behaviours associated with constructing graphs. The model relates to the way statisticians work and think statistically and applies to the way in which students engage in statistical investigations. It includes four dimensions:

Dimension 1: The investigative cycle Dimension 2: Types of thinking Dimension 3: The interrogative cycle Dimension 4: Dispositions

Dimension 1 is related to the thinking processes employed when working through a statistical investigation. This involves planning an investigation, collecting data, analysing data, and drawing conclusions. Dimension 2 is related to the types of problem solving strategies applied when working through a statistical problem. Wild and Pfannkuch (1999) posit that the types of thinking in this dimension are ―the foundations on which statistical thinking rests‖ (p. 227). Dimension 3 adopts a cyclical process of data interrogation that involves thinking critically about the data in order to distil and encapsulate ideas and information. Dimension 4 includes the personal qualities, dispositions, and habits of mind employed when working with data.

Dimension 2: Types of Thinking of the Statistical Thinking Model (Pfannkuch & Wild, 2004; Wild & Pfannkuch, 1999) is particularly useful when considering the way in which students work with data when creating graphs. The section most relevant to this inquiry is transnumeration. Pfannkuch and Wild (2004) describe transnumeration as changing data representations to engender understanding, capturing the characteristics of a real situation, and communicating messages in data. The notion of transnumeration is

extremely important as new technologies that incorporate interactive and dynamic commands as a way of working within software environments foster the manipulation of both data and data representations – an element not included in other models of graphing sourced in this inquiry. Transnumeration also encompasses the application of data summary EDA strategies, such as the box-and-whisker plot.

The translation processes when reasoning about covariation detailed in the Moritz model (2004, p. 523) (Figure 1.6) are graph production, graph interpretation, and speculative data generation. The arrows on the model in Figure 1.6 indicate processes of translating among numerical data, graphical representations, and verbal statements of covariation. Unlike the models developed by Mooney (2002), Jones et al. (1997, 2000) and Friel et al. (2001), Moritz did not construct his model as a hierarchy. He proposed that students could enter the graph interpretation process from multiple entry points. The starting point could be constructing a graph from raw data or a verbal statement. It could also be extracting data from a graph or making informal inferences based on the trend in a graph. The model does not, however, incorporate elements associated with summarising data other than displaying the data in graphs. Of all the models accessed for this inquiry, the Moritz model is the only one to address covariation specifically.

"More people in the classroom cause a lower level of noise" Causal Inference

Graphical Representation Raw Numerical Data

Graph Production

Verbal Statement of Covariation "Level of noise is related to number of people", or

"Classrooms with more people make less noise" Numerical Graph Interpretation Speculative Data Generation _Verbal Graph Interpretation Verbal Data Interpretation Causal Statement

Figure 1.6. Translating processes involved with reasoning about covariation. (Reproduced from Moritz, 2004, p. 523)