students’ learning and achievement especially in mathematics. A recent research by Pittalis and Christou (2010) also revealed and emphasized that spatial ability supports the development of students’ 3D geometry thinking.
They conducted a research involving students from grade 5, 6, 7, 8, and 9 in Cyprus to examine the relation between students’ spatial ability and their 3D geometry thinking. They found that spatial ability is a very important contributor to the development of students’ 3D geometry thinking. There is a direct effect between the spatial ability and 3D geometry thinking. That direct effect suggests that an improvement of spatial ability results in an enhancement of 3D geometry thinking. Therefore, to support students’ 3D geometry thinking, we must support their spatial ability.
According to Pittalis and Christou (2010), 3D geometry thinking consists of 4 different types of reasoning. The types are understanding 3D representations, spatial structuring, conceptualization of mathematical properties, and measurement. These four types of geometry thinking are in line
Measurement
Conceptualization of mathematical properties Spatial structuring
Understanding 3D representations
Figure 2.1 Four types of 3D geometry thinking
with Duval’s theory of 3D geometry thinking. Duval (1998) added that understanding 3D representations is the foundation of the other types of reasoning. Figure 2.1 illustrates how the types of 3D geometry thinking supports each other. Therefore, supporting the development of students’ ability in understanding 3D representation is important since it supports the development of the other types of 3D geometry thinking.
On the other hand, low understanding on 3D representations can lead to some problems such as difficulties in reading two-dimensional representations and determining the blind spot of the objects in the representations since the representations are in 2D drawings (Ben Haim et al., 1985; Revina et al., 2011).
Moreover, students also have difficulties in drawing, representing, and reasoning about 3D objects such as parallel or perpendicular lines in space.
Unfortunately, most of the textbooks and teaching activity use 2D drawings to represent 3D objects such as cubes, boxes, organs, and maps. Therefore, students often confuse to understand the representations (Revina et al., 2011;
Risma et al., 2013). Importantly, low ability in understanding 3D representations can impede the development of their reasoning in 3D geometry thinking.
According to the online dictionary, understanding means “to perceive the meaning” or “to grasp the idea” (dictionary.com, accessed in 17th May 2015).
Understanding is the knowledge about something that somebody have. In mathematics, understanding means connecting between schemes available in mind with new schemes of certain concepts (Sierpinska, 1994). Meanwhile, 3D
representations is a set of representations of 3D objects. Understanding 3D representation is not easy, particularly for young learners. It consists of two types of reasoning: 1) recognizing and 2) manipulating. Recognizing is dealing with mental movement or construction such as determining the shape of a 3D object based on its net. The second type, manipulating, means students are able to identify, interpret, and determine the shape of 3D objects based on the representations (Pittalis & Christou, 2010). In the present study, we focus more on understanding 3D representation in the aspect of manipulation of 3D representations than recognizing them.
Since the ancient time, people always have difficulties in understanding the representations of spatial objects. This due to the difference of the dimension of the objects and its representations (Parzysz, 1988). The only way to represent 3D objects is by making their model or two-dimensional drawings (2D drawings). For instance, a box is originally a 3D object but the drawing is on 2D layout or paper. As a result, some of the shapes changed into another shapes because of the effect of perspective like a rectangle can be a parallelogram or a rhombus. Thus, people have to imagine and create mental images of these shapes of the box based on the drawing. Different from 3D objects, 2D shapes’
representations are easier to recognize since the representations have the same dimension as long as it does not have perspective views. Parzysz (1988) categorized object representations into three different levels. Table 2.1 displays the levels.
Geometry Objects
Levels 2D 3D
Level 0 Shapes on objects Objects Close representation Level 1 Drawing Model
Distant representation Level 2 Drawing
Level 0 means the object itself while level one is called close representations. At level 1, the dimension of the objects and its representations are the same. Thus, the properties of the representations are close to the properties of the real one. The highest level of representations is a drawing of 3D objects or 2D representations of 3D objects which are called distant representations. For instance, if the object is a container, then the close representation of it is a model of the container such as toys, wooden box, or something similar to the container. Meanwhile, the distant representation of the object is a drawing of the container.
Distant representations are the most difficult representations to understand, especially for young learners (Parzysz, 1988). The reason is because there is a lot of information missing when we move from a lower level to a higher level of representations. For instance, a right angle in a cube sometimes will be not equal to 90° in its drawing. Parallel lines sometimes are not parallel in their representations. Drawings of 3D objects also have hidden parts or unseen parts which are usually called as the blind spots of the representations. As a result, most people will have difficulties to understand the
Loss of information
Table 2.1 Levels of 3D representations
object or even to draw it (Parzysz, 1988). Unfortunately, many subjects involve 3D drawings such as mathematics. Lack of ability in understanding 3D representations will impede student’s performance in learning mathematics, particularly in 3D geometry.
Considering the importance of understanding 3D representations for the students, we obviously must pay more attention to it (Clements, 2003; Olkun, 2003; Revina et al., 2011; Risma et al., 2013). Previously, we know that an improvement of students’ spatial ability will lead to an improvement in students’ 3D geometry thinking. Now, the problem is how to support the development of students’ spatial ability to assist their 3D geometry thinking.
Unfortunately, most researches only focus on why spatial ability is related to students’ performance and only few studies examine how to develop it and how to apply it in classroom activities, especially for primary students. The recent research was done by Revina et al. (2011). She designed instructional activities involving spatial visualization tasks with the purpose to support students’
spatial structuring and understanding of volume measurement. Another study, Risma et al. (2013) showed how instructional design supports the development of student’s spatial abilitiy. She created spatial visualization and spatial orientation tasks to help 3rd grade students developing their spatial ability. These tasks are based on the book of Mathematics in Context (MIC). Nevertheless, these activities need to be focused on students’ ability in understanding 3D representations. Therefore, we argued that more research are needed in this area.
Hence, the present study aims to design instructional activities or sequence of lessons to support students’ spatial ability in understanding 3D representation.
In the present study, the designed activities consist of spatial ability tasks which related to the aspects of understanding 3D representations. Pittalis and Christou (2010) explained that some spatial ability tasks have connection to the aspects of understanding 3D representations. Based on their research, only spatial orientation and spatial visualization contribute to the development of students’ ability in understanding 3D representations, spatial structuring, and measurement. Meanwhile, spatial relation only contributes to the development of students’ conceptualization of mathematical properties. Therefore, supporting students’ spatial ability in understanding 3D representations should focus on the aspect of spatial orientation and spatial visualization. Spatial orientation and spatial visualization are related to object perspectives and image perspectives which are about sketches of a solid in different representational modes, manipulation of images of 3D objects, and isometric views of a cube (bird eye view).
In conclusion, understanding 3D representations consists of two types of reasoning which are recognizing and manipulating. The development of students’ ability in understanding 3D representations follows the development of students’ spatial ability, particularly spatial orientation and spatial visualization. Therefore, to support students’ spatial ability in understanding 3D representations, we should support their spatial orientation and spatial visualization.