In a paper entitled ―Do Tangibles support learning?”, Marshall (2007), summarises the different approaches that have been taken to identify the benefits of this form of technology. These approaches are summarised in Figure 1.8 below.
93 Figure 1.8: Analytical framework for Tangibles for learning (Marshall, 2007)
Marshall identifies many lines of research not focused upon in this thesis, such as the potential for Tangibles to support collaboration (Africano et al., 2004; Price, Rogers, Scaife, Stanton, & Neale, 2003; Stanton, Bayon, Abnett, Cobb, & O'Malley, 2002) and accessibility and enjoyment (Price et al., 2003; Xie, Antle, & Motamedi, 2008). However, one dimension identified which is particularly relevant is the ‗effects of physicality‟.
Unfortunately, considering the broad coverage within a relatively short paper, Marshall is only able to indicate research in this area, such as the possible role of embodiment.
Nevertheless, the conclusions reached are similar to those made in the debates surrounding manipulatives:
“Thus, despite the common view that the physical materials used in tangible interfaces are particularly suitable for learning tasks, there is only limited evidence to support this claim. This
94 suggests that intuitions about the benefits of physical manipulation should be abandoned. Instead, empirical research is required to investigate in which (if any) domains and situations physical manipulation will be of benefit to the learner.” (p.168)
A more thorough analysis of the possible learning benefits of Tangibles was presented by O'Malley & Stanton-Fraser (2004). The review examines both the theoretical and empirical arguments surrounding physical manipulation and learning, and links this to frameworks as well as case studies of Tangibles. The main focus of the analysis is centred on children‘s ability to map between physical representations and the domain they are intended to represent. This focus reflects how, in contrast to analogue materials, digital manipulatives (such as computer representations) present a separation between input and output. The authors go on to describe at least three levels to such interactive learning environments (p. 23):
1) Representation of the learning domain
2) Representation of the learning activity
3) Representation embodied in the tools themselves
These three levels help describe some of the issues surrounding manipulatives and the role of technology.
95 1.4.4.1 Traditional Manipulatives
With analogue manipulatives like plastic cubes, the learning representation is also the tool – input is also output. This direct relationship reflects many of the arguments put forward in the previous section for the advantages of manipulatives, such as embodiment and tactile feedback. However, it was also suggested that these materials have no implicit link to more formal symbolic mathematics: this relationship needs to be created through the activities presented by the teacher.
1.4.4.2 Virtual manipulatives
According to Moyer (2002), a virtual manipulative is defined as ―an interactive web-based visual representation of a dynamic object the presents opportunities for constructing mathematical knowledge‖. In this thesis, virtual manipulatives will be defined as on-screen objects that can be manipulated using a graphical user interface, but which are not necessarily accessed through the internet (i.e. not necessarily web-based).
Kaput (1992) identifies several key advantages of virtual representations that address limitations of physical materials: the potential to link representations, provide feedback and provide a trace of past actions. These advantages are echoed by others such as Moyer et al (2002) who add more pragmatic factors to the list such as: adaptability, availability, ease of setting up and clearing away, and ability to print. Moyer et al also highlight how the materials may overcome the stigma that is sometimes associated with the use of concrete materials for younger, less able children.
Clements (1999) makes the case that the emphasis on the use of physical materials results from a desire to make learning concrete, but argues that the benefits of concreteness are not simply due to physicality so much as to how well the materials
96 connect ideas to the real world. Using this definition, Clements describes ways in which computer based representations can achieve this more effectively, referring to various advantages such as how the materials can be designed to help externalise mathematical ideas and processes, thus helping to reinforce the link between concrete and symbolic representations.
Despite the purported benefits of virtual manipulatives, their advantages have yet to receive much empirical support. However, the aim here is not so much to evaluate the different learning opportunities presented as to focus on certain aspects that help examine representational differences with physical manipulatives. Three key aspects are discussed below: how the materials can help map to symbolic representations; provide a record of representational change; and the different forms of manipulation.
Mapping to symbolic representations
One key argument in support of virtual manipulatives is that, unlike analogue materials, they can provide a means to link learning representation to symbolic representations (Clements, 1999; Kaput, 1992; P. W. Thompson, 1992). Materials can be designed to create a dynamic link between the learning representation and more symbolic notation.
For example, in an activity provided by the National Library of Virtual Manipulatives (NLVM, 2007), children are able to manipulate virtual Dienes‘ blocks to try to match a written number (Figure 1.9).
97 Figure 1.9: Tens and units activity with Virtual Manipulatives (NLVM, 2007)
With virtual manipulatives, it is also possible to manipulate symbols and explore the resultant changes on the learning representation (P. W. Thompson, 1992).
Unfortunately, it is not clear whether such a transparent link between representations is desirable. As argued by O‘Malley (1992), such transparency does not require the learner to reflect upon their actions. Recognising this, designers may opt to incorporate some degree of opacity to foster more reflection on the mapping between levels of representation. A key challenge might therefore be to design technologies in a way to augment representations that draw children‘s attention to important numerical concepts without making the link so explicit as to limit reflection.
Virtual manipulatives present various ways in which the representation of the learning activity can be augmented. Objects can be designed to emulate familiar physical materials such as cubes or rods but can also be extended - allowing them, for example, to change colour or make sounds. The materials are not constrained by physical laws – they can be designed to change shape and size, or be made to appear and disappear
98 instantaneously. On the other hand, certain physical aspects are more difficult to emulate – for example, creating the illusion of three dimensional structures and movement.
However, it is still not clear what design of learning representation is most effective for building children‘s understanding of number; including certain features may only serve to distract (McNeil & Jarvin, 2007; Uttal et al., 1997).
Record of Representational change
Kaput (1992) identified a further limitation of physical materials that can be addressed by virtual manipulatives. Unlike physical materials, it is possible to keep a log of actions with virtual materials so that a record can be presented of the changes made to a representation, thereby facilitating a review of these changes. According to Kaput, this ability is important in maths where changes in representational state reflect key numerical processes, such as how two objects have been combined in different ways to create a whole. This particular feature may be highly relevant for activities such as the partitioning task where comparing a list of different solutions may be beneficial (Clements, 2009). It is unclear, however, how easily young children can identify and then reflect on such symbolic relationships.
Manipulation
With virtual manipulatives, representations on screen are typically mouse controlled.
There is therefore a physical separation between the tool and the learning activity representation. In this set up manipulation is indirect, and the designer needs to decide what physical actions with the mouse relate to actions on screen. For some actions, such as moving on-screen objects, this mapping is quite simple. Indeed, Donker & Reitsma
99 (2007) showed that even 4 year old children were proficient at ‗drag and drop‘ actions (although less so than 5 year olds, which indicates possible difficulties for younger children and children with physical disabilities). However, other actions, such as attaching or detaching objects or moving groups of objects, may require less obvious mouse actions, or combinations of mouse and keyboard actions (such as pressing the shift key to select multiple objects). The actions can obviously be learnt, but this does raise questions about how seamless they will be to young children. Importantly, the indirect relationship between actions with the tool (mouse) and the learning representation may limit many of the benefits of physical actions (e.g., embodiment, tactile feedback). If these learning mechanisms are important, the potential advantages of virtual manipulatives may be greatly limited by the form of interaction. Certain limitations of mouse-controlled virtual manipulatives may be addressed by emerging interfaces such as tabletops where multiple virtual representations can be manipulated through touch, but it is possible that there are still limitations presented by the inability to physically interact with representations. Digital manipulative may address this limitation by providing a tangible interface.
1.4.4.3 Digital manipulatives (including Tangibles)
Digital manipulatives allow designers to create a tight coupling between physical actions and the learning representation. Indeed, the learning representation may actually be
100 embodied in the object being manipulated5. It is thereby possible to augment the learning representation with digital technology, and to explore the effects of so doing through direct physical manipulation. Nevertheless, considering the comparative ease and availability of virtual manipulatives, it is important to consider what benefits are provided by such physical interaction.
Several studies have attempted to identify the possible benefits of physical manipulation by comparing performance between physical and virtual representations.
Of these, relatively few have attempted to limit confounding variables, and those that have tend to report no significant differences (Klahr, Triona, & Williams, 2007; Triona &
Klahr, 2003; Zacharia & Constantinou, 2008). Unfortunately, this indicates a key difficulty in comparing physical and virtual representations: by controlling variables to examine effects, it is easy to ‗design out‘ many of the advantages of either medium. This point is exemplified in a study by Triona & Klahr (2003), who compared the effects of using virtual and physical materials (springs) on children‘s ability to design experiments.
No differences were found between the physical and virtual materials although, in order to balance conditions, the authors noted that they focused on the length and width of the spring rather than the weight of an attached object as ―the effect of the mass of the weight used on the springs is not visually discernable‖ (p. 159). In other words, to balance conditions, possible unique advantages of either medium may have been eliminated.
5 Various taxonomies have been created to describe the range of couplings between interface and digital representation in Tangibles. These will not be discussed here but the reader is directed toward work such as Fishkin (2004) , Koleva et al (2003), and Price (2008).
101 It is possible that studies comparing physical and virtual representations have not been designed specifically to detect certain benefits of physical manipulation as identified in the previous section. Some mechanisms, such as the embodiment of motor actions, may be hard to detect. Others, such as the use of tactile information to reduce cognitive demands, may be easier. Clearly, the extent to which these processes play a role, and consequently the potential benefit of a physical interface, will depend on the nature of the task.