UNIDADES FUNCIONALES DIFUSAS Y REUSO. 65

Visual Reasoning Model by Park and Kim (2007) was originally designed as a mean to assess the types of visual reasoning and the related cognitive activities among architectural students when sketching their designs. Tversky (1999) proposed two ways to go beyond the visual information; one is to transform to visual information according to the predetermined rules and second, to make deduction and conclusion on the visual information. When students solve the given tasks, their visual reasoning can be summarized in terms of three broad categories consisting of eight interrelated types: perception, analysis, and interpretation in the seeing, generation, transformation, and maintenance in the imagining, and internal representation and external representation in the drawing as illustrated in Figure 2.9.

In the seeing process, the activities of visual perception, analysis, and interpretation occur. During the perception activity, basic properties of the visual information and their combinations are recognized and identified. The image of the object is attained as and when it is observed. This is a very selective process and this

selectivity is accountable for the qualitative value in the subsequent visual images to be produced. The difference contexts and purposes in which the visual are perceived and generated play important role in creating the final visual images. During the analysis activity, the observation on the relationships among the properties and the exploration about the characteristics of the visual information occur. During the interpretation activity, the naming, categorization, and giving new meaning to the perceived visual information occur. These activities in the seeing process bring about the extraction of characteristics as required for new visual generation and transformation.

KNOWLEDGE

SCHEMA

Figure 2.9: The Visual Reasoning Model (Park & Kim, 2007)

The imagining process enables the synthesizing of conceptual information for the new visual representation. Imagining process can be classified into the generation, transformation, and maintenance activities. In an earlier study conducted by Kavakli & Gero (2002), they proved that the generation and transformation activities were very

Internal Representation External Representation Perception Analysis Interpretation Maintenance Transformation Generation DRAWING IMAGINING SEEING

from the perceptual input during the seeing process while the other one emerged from the activated knowledge and schema that were stored in the long-term memory (Kosslyn, 1994). Visual transformation can be differentiated into two types: congruent transformation and pattern change transformation (Park & Kim, 2007). Kosslyn (1994) defined congruent transformation as equivalent to the actual perception such as the mental rotation or the resize of visual objects. On the other hand, Oxman (2002) suggested for the pattern change transformation to involve the developing or evolving of visual objects. Following the visual transformation, the maintenance activity takes place to store the internal representations.

The drawing process enables visual objects to be represented through both the internalization and externalization. In internal representation, the transformed visuals are to be confirmed. This drawing process occurs through interactions with imagining and seeing processes. In addition, the external representation serves as external memory, in which ideas are settled as visual tokens, and to be revisited later for inspection, if necessary (Suwa, Purcell, & Gero, 1998). The process of generating the imagined objects might also occur during the process of converting from internal representation to external representation. As a result, the drawing process is important in visual reasoning. In addition, the drawing process also make possible for the visual information to be manipulated and transformed.

Knowledge and schema are engaged in the interaction within the visual reasoning activities. A schema is a collection of objects, processes and actions and other previously constructed schemas that are coordinated and synthesized by the individual to form structures utilized in problems situations (Sabella & Redish, 2005). The retrieval of visual knowledge from long term memory becomes a cue to match between visual input and visual memory for visual perception in seeing process. The visual schema retrieved from the long term memory becomes a rule for the extraction of the

characteristics of the visual information. The iterative process of seeing and imagining make it possible to reorganize, transform and modify the existing visual input in imagining process. Oxman (2002) highlighted the importance on how to transform and to access schema of basic structure in reformulating visuals, since the order and pattern of visuals can cause different types of reasoning. The schema, therefore, plays a critical role to link between the conceptual and perceptual processes in drawing process. As a result, diverse manipulation or interpretation of images can be generated. In the visual reasoning process, seeing, imagining, and drawing processes do not occur independently but interactively with knowledge and schema, together with interaction between perceptual and conceptual knowledge.

Costa (2010) finalized a model to understand four different modes of the visual- spatial thinking: from perception, from mental manipulation of images, from the mental construction of relationships among images and from the exteriorization of thinking. The visual-spatial thinking that resulted from perception used visual information that are represented based on movement. It involved different individual perceptions referred to as concrete images and memory images when images of experiences were recalled. The thinking processes engaged in the process were intuitive inference, visual recognition, construction of visual, recalled visual representation, evaluation of images, identifying of objects and images, recognition of abstraction and concepts generation. Among the mental processes that took place in the visual-spatial thinking that resulted from mental manipulation of images were the secondary and anticipatory stages of intuitions which involved a stable cognitive attitude on understanding reasoning on more common situations. Other processes were mental transformation, constructive and synthesizing, coordinating spatial structure and visual construction.

The visual-spatial thinking that resulted from the mental construction of relationships among images involved the mental construction of how visuals were

related and comparing the models, ideas and concepts. The thinking processes involved include the searching for relationships among images, facts and properties, and continuous evaluation along the process of solving a problem. Lastly, the visual-spatial thinking that resulted from the exteriorization of thinking involved the mental processes of translation, describing the mental dynamics through verbalization and gestures and using the analogies.

The abstract mathematical objects, concepts and processes can best be experienced by students through the use of visual representations. Therefore, there is a need of clear meaning on how visual processing can help to solve mathematical problems. Gorgorio and Jones (1996) described three distinctive components of visualization process that resulted from the ability to mentally manipulate, influence and transform visual images and visual representations. Starting with crude visualization where students are able to draw diagrams with either pencil or pen, or with the help of technological software, visuals were used to represent mathematical objects, concepts, of processes and subsequently to use them to understand and help in the solving of mathematical problems by interpreting the technical rules or mathematical formula. This was then followed by the visualization be regarded as the activity to read the visual information where the interpretation of the relationships among the properties of the visual representations. The final part of visual processing involved the ability to manipulate and transform the visual images and visual representations mentally and conceptually.

Due to the increase in the number of tools that are able to help users interact with mathematical visualization, Sedig (2009) presented three frameworks describing the interaction design of mathematical visualization; the micro-level interaction framework, the micro-level interactivity framework and the macro interaction framework. The micro-level interaction framework characterizes the interaction in the

context of exhibiting low-level cognitive tasks and epistemic behaviours. The interaction framework organized the user activities into basic (conversing, manipulating, and navigating) and task-based (animating, cutting, filtering, rearranging and scoping). The second level of micro-level interactivity framework organized the user activities into factors such as cognitive offloading, constraints, flexibility, focus, scaffolding and transition. Lastly, the third macro-level interaction framework listed the design space into four categories ; access-, annotation- , construction- and combination- based.