The theoretical framework, Model of Learning Behaviour in EDA Graphing Environments, developed for this inquiry draws on the work of the following commentators. Friel et al. (2001), Moritz (2004), Pfannkuch and Wild (2004), and Shaughnessy (2007) in relation to graphing and graph sense-making, Kosslyn (1989) in relation to the characteristics of graphs, and Alessi and Trollip (2000) in relation to the features of technological learning environments. The framework, presented in Figure 1.7, incorporates the key behaviours of graph creation and interpretation extracted from each of the models of graphing, takes into account the characteristics of graphs and affordances of technologies from the literature into four interconnected dimensions: Being creative with data, Understanding data, Thinking about data, and Generic knowledge. The dimensions characterise the behaviours employed when using EDA strategies to analyse data.
Dimensions Key Behaviours
Generic knowledge
Being creative with data
Reducing data to graphical representations. Summarising data.
Constructing different forms of graphs. Describing data from graphs.
Speaki ng t he langu age o f d at a and g raph s. R ec ogni si ng t h e c h ar act er is ti cs of dat a a n d gr aphs . U nder st and ing h ow t o u se t he f eat u res of sof tw ar e a nd tec hn ol og y env ir onm ent s. Understanding data
Making sense of data and graphs.
Understanding the relationship among tables, graphs, and data.
Identifying the messages from the data. Answering questions about the data.
Recognising appropriate use of different forms of graphs.
Thinking about data
Asking questions about the data. Recognising the limitations of the data. Interpreting data.
Making causal inferences based on the data. Looking for possible causes of variation.
Looking for relationships among attributes in the data.
Figure 1.7. Model of Learning Behaviour in EDA Graphing Environments. (Adapted from Fitzallen, 2006; Fitzallen & Brown, 2006b, 2007)
The intention of the model introduction is to recognise that each of the dimensions is an independent, yet functionally related, set of behaviours that can be accessed in isolation, in any combination, or collectively to answer questions about the data. How the dimensions of the model come together when working with software packages depends on the complexity of the statistical question asked and what needs to be done to answer the question. For example, a question that requires the time a particular event occurs to be determined from a time series graph only employs behaviours from the Generic knowledge dimension to answer the question. As another example, an answer to a more complex question that requires collecting the data, representing the data using EDA strategies, summarising the data, and making inferences about the data based on the representations and the variation evident to answer a question, will incorporate all the dimensions of the model in an interrelated fashion.
The Model of Learning Behaviour in EDA Graphing Environments (Figure 1.7) is not offered as a hierarchical construct. Like the Moritz (2004) model (Figure 1.6), it recognises that reasoning about covariation can occur from different entry points. At times, questions about data are directly related to a given graph and there is no need to create new representations, yet summarising data in another way may be useful. In this instance, entry into the data analysis process may utilise the behaviours in the Being creative with data dimension. On other occasions, questions about the data may be directly related to determining the trend evident in graphs. In these circumstances, access to the data analysis process may utilise the behaviours in the Thinking about data dimension only.
An important feature of the framework is that it recognises that there are some generic understandings that are inherent in all aspects of data analysis, graphing, and graph sense-making. These are included in the Generic knowledge dimension. The Generic knowledge dimension is situated in the model so that it stands as an independent set of behaviours, which then supports the other three dimensions that encompass behaviours associated with creating and interpreting graphs. As well as the models of graphing discussed in the previous section, this Generic knowledge dimension was informed by Kosslyn‘s (1989) schema for describing the constituent parts of a typical graph. It is important not only to understand how to read data values from a graph but also to understand how to read the constituent parts that make up the structure of a graph, such as the scale on an axis.
The Generic knowledge dimension also considers the technical skills required to use technology and software environments. Bakker (2004) contends that ―the software itself needs to be learned before it can effectively mediate between the learner and what is to be learned‖ (Bakker, 2004, p. 279). This infers that influencing the effective use of learning activities that use technology is reliant on the student‘s knowledge of the features of the technology. Bakker‘s comment is important and should be taken into consideration when thinking about how students use graphing software packages to construct and interpret graphs.
To address the issues raised by Bakker (2004), the work of Alessi and Trollip (2001) in relation to the behaviours associated with using technology were incorporated into the Generic knowledge dimension. Alessi and Trollip developed a checklist for evaluating online
learning resources for the teaching of mathematical concepts. The check list includes the usability of the learning resource interface, ease of navigation, access to invisible features via drop-down menus, and access to supplementary materials, such as the ―Help‖ functions. Alessi and Trollip contend that the interface of technologies impacts on how students access the features of the technologies and the way in which they navigate around the learning environment. Including these features of educational technologies into the Model of Learning Behaviour in EDA Graphing Environments (Figure 1.7) recognises the potential for graphing software to influence student behaviour and development of understanding of statistical concepts.
The Model of Learning Behaviour in EDA Graphing Environments (Figure 1.7) developed for this inquiry is presented again in Table 1.1. The right-hand column of the table details the contributors from the literature that informed the development of the dimensions of the model. In this thesis, when referring to the Model of Learning Behaviour, the words ―model‖ and ―framework‖ are used interchangeably and no particular connotation is intended by the use of either word.
Table 1.1
Contributors to the Model of Learning Behaviour in EDA Graphing Environments Dimensions
Dimensions Key Behaviours Contributors
Generic knowledge Speaking the language of data and graphs.
Friel et al., (2001), Reading the data Moritz (2004), Verbal graph interpretation
Kosslyn (1989), Constituent parts of graphs
Understanding how to use the features of software and technology environments.
Alessi & Trollip (2001), Navigating the interface,
Accessing the features, Supplementary materials Recognising the characteristics of
data and graphs.
Friel et al., (2001), Reading the data Moritz (2004), Numerical and verbal graph interpretation
Being creative with data Reducing data to graphical representations.
Pfannkuch & Wild (2004), Transnumeration
Summarising data. Pfannkuch & Wild (2004), Transnumeration
Constructing different forms of graphs.
Pfannkuch & Wild (2004), Transnumeration
Moritz (2004), Graph production Describing data from graphs. Moritz (2004), Verbal graph
interpretation
Understanding data Making sense of data and graphs. Friel et al., (2001), Reading within the data
Understanding the relationship among tables, graphs, and data.
Friel et al., (2001), Reading within the data
Identifying the messages from the data.
Pfannkuch & Wild (2004), Transnumeration
Answering questions about the data. Friel et al., (2001), Reading beyond the data
Recognising appropriate use of different forms of graphs.
Pfannkuch & Wild (2004), Transnumeration
Friel et al., (2001), Reading beyond the data
Thinking about data Asking questions about the data. Moritz (2004), Verbal graph interpretation
Recognising the limitations of the data.
Shaughnessy (2007), Reading behind the data
Interpreting data. Friel et al., (2001), Reading beyond the data
Making causal inferences based on the data.
Shaughnessy (2007), Reading behind the data
Moritz (2004), Causal inference Looking for possible causes of
variation.
Shaughnessy (2007), Reading behind the data
Looking for relationships among attributes in the data.
Shaughnessy (2007), Reading behind the data