In order to identify the way in which visual representations used in ILS facilitate students‘ construction of mathematical knowledge, Kidman and Nason (2000) developed a set of seven principles (Table 2.1) that are based on ―the research literature from the fields of mathematics education, cognitive science, computer-aided learning, computer graphic design, and semiotics‖ (p. 179). These principles were applied successfully by Kidman and Nason as a diagnostic tool to determine the effectiveness of dynamic mathematical representations employed within ILS. The principles were used to evaluate 500 ILS, which, for the most part, were found to only partially fulfil the requirements for interactive construction of mathematical knowledge. The principles developed by Kidman and Nason focus on the evaluation of general mathematical content and need to be interrogated further to determine if they are suitable for evaluating EDA software.
Table 2.1
Principles for Analysing Visual Representations (Kidman & Nason, 2000, p. 181) No. Visual representations should:
1 be clearly displayed and explicitly understood by the student. This facilitates the process of stimulating relationships among the problem data and may also help students to recall knowledge and skills by making connections between prior internal representations and new situations.
2 enable the student to focus on the deep structural rather than surface structural aspects of the problems being investigated.
3 provide physical/iconic environments for learners to abstract and understand mathematical concepts or a relationship of information within problems.
4 provide students with an external memory to display information temporarily during the process of problem solving. By doing this, the visual representation can reduce the working memory demands of the problem solving process. 5 facilitate the exploration and construction of understandings about aspects of
mathematical ideas and concepts that cannot be adequately represented in the semantics of natural language.
6 facilitate the process of translating between mathematical expressions and natural language.
7 be used for both interpretative and expressive learning activities.
Principle 1. This principle focuses on the clarity of visual representations and assumes there is a causal link between providing a clear display and the development of thinking and understanding. It also maintains that if the visual representation is explicitly understood, then students may be able to draw on prior learning and translate the learning gained to new situations. The interactive nature of some ILS may facilitate this but it cannot be assumed that the display itself is a main contributor. With this in mind, it appears that Principle 1 as it stands is insufficient. When considering EDA software it is necessary to consider if students can access and use the features of the software as well as understand the display.
Principle 2. Mathematical content and the development of deep learning are the focus of Principle 2. The development of deep learning should be an expectation of any mathematical learning environment, therefore, essential to include in an evaluation. In the case of ILS, such as e-learning objects, the evaluation of mathematical content and the extent to which it is developed is necessary, as the information is usually displayed without any
input from the user. In the case of un-populated software (without any data entered), the display of information is in the control of the user and the software provides a different learning context to those used in ILS resources. With EDA software in particular, deep learning may be facilitated by interacting with the software, engaging with learning activities, and accessing support offered by the teacher. It is, therefore, important to go beyond evaluating what mathematical content is accessed and developed to including what opportunities are provided for students to develop their mathematical understanding further within the context of the learning environment.
Principle 3.This principle takes into account the learning environment and the role it has to play in allowing students to make connections across information within problems. It is, however, quite broad and does not allude to the specific aspects of the ILS environment that may be attributed to the development of understanding. Additionally, it does not consider how students may engage in the learning sequence. Determining if ILS allows learners to have some control over the learning sequence is important. This is particularly relevant to statistics education as students make sense of data through activities, such as constructing and deconstructing graphs, comparing data sets, and representing data in different ways (Watson, 2006). For this to be successful, the learning environment needs to be flexible enough for students to engage with the learning sequence from different entry points.
Principle 4. This is an important principle that recognises the potential of ILS to provide an external memory to display information temporarily. It supports the notion that the ILS environment can display some information thereby reducing the cognitive load when problem solving. This principle also recognises that computer environments should provide students with the tools to reorganise information and support their thinking. Those tools and the external memory may be in various forms. The Help function, for example, provides external memory as do drop-down menus and pop-up annotations. Tools for reorganising information may be graphing functions or calculators as well as text boxes for inserting information.
Principle 5. Principle 5 recognises the value of visual representations for conveying mathematical ideas and concepts that are not easily expressed using natural language. In statistics education graphs are used for this purpose. Often, trends or patterns in data are more easily expressed in a graph than explained from raw data. This principle does maintain that visual representations should facilitate the exploration and construction of mathematical ideas and is related to Principle 3, as the learning environment determines the way in which students engage in the learning process.
Principle 6. This principle makes the assumption that ILS display both natural language and mathematical expressions. As this is not always the case, this principle is only partially appropriate. ILS, such as learning objects, display mathematical representations and explanatory notes in both symbols and written text, whereas EDA software is un-populated with data, and requires input from the user to construct visual representations that are not easily represented by natural language. The value of EDA software lies in the user being able to construct visual representations that are not easily represented in natural language.
Principle 7. This is the only principle in the Kidman and Nason (2000) framework that recognises that students can contribute actively to the learning process. They note the important role visual representations play in the construction of knowledge and describe that they do this through engagement with interpretive and expressive learning activities. Expressive learning activities provide the opportunity for students to construct and/or modify visual representations, such as manipulate an axis of a covariation graph to change the scale. Interpretive learning activities come into play when students engage in activities that require them to reason, think logically, and make decisions. Such activities include comparing data sets with box plots, making inferences about larger populations from the results gleaned from a smaller sample, and making judgements about the inclusion of outliers.