I define spatial perception as the ability to identify a specific figure and its intrinsic
information by isolating it from a complex background, as well as to generate the figure into abstract geometry elements. In his framework, Gutierrez (1996) includes three perceptual
abilities. These are figure-ground perception, perception of spatial positions, and perception of spatial relationships. In this research, the author thinks there is not much educational value in differentiating the perception process in such a detailed way.
According to the van Hiele model of developing geometrical thinking, the first stage (zero level) is visualization, and the second stage (first level) is analysis. For van Hiele, visualization is similar to spatial perception. It seems to be more natural that students first process the information (perception), and then differentiate position and relationship. Although scholars use different terminology, they all assert the importance of the visualization process. In addition to the visualization process, there is subtle thinking involved, so the author chooses to use the term “spatial perception.” It includes the process and the global thoughts, and it
represents the process of visualization in a more intuitive way, but without including too many analytical elements.
When learning skew lines, if teachers give rich real-world examples like column versus edge in their classroom, or sky-bridge versus the road underneath as is shown in Figure 3-1, etc., students can perceive the difference and generate the definition for skew lines. Although students are already very familiar with these examples, the connection might never occur to them until the teacher brings these examples deliberately into their classroom discussion. It also shows that spatial perception is teachable, and it can be enhanced by practice.
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Table 3-1: Summarizes the Six Abilities and Those Relative Similar Terminologies from Other Scholars’ Research
Six spatial abilities
Definition Similar terminologies in other
scholars’ research Hierarch ical levels Spatial perception (McGee, 1979; Kimura, 1999)
The ability to identify a specific figure and its intrinsic information by isolating it from a complex background, and to generate the figure into abstract geometry elements.
Figure-ground perception (Gutierrez, 1996), interpretation of figural information (IFI) (Bishop, 1983)
First level - percepti on Spatial relationships
The ability to determine relationships between different spatial objects (McGee, 1979), and the ability to compare and analyze the relationship between different parts or different elements within an object.
Perception of spatial relationships (Gutierrez, 1996), mental relations (McGee, 1979), spatial relations (Lohman, 1979)
Internal representation
The ability to create a quasi-picture from memory without any physical support, and in mathematics particularly it also includes mental representation of a mathematical concept, property, and other information which are attached to the quasi-picture.
Mental image (Kosslyn 1980;
Gutierrez, 1996), spatial visualization (McGee, 1979; Lohman, 1979; Gutierrez, 1996; Kimura, 1999), visual image (Bishop, 1980)
Second level - represen tation External representation (Gutierrez, 1996)
The ability to create any kind of verbal or graphical representation of 3-D objects, concepts, or properties (including pictures, drawings, diagrams etc.) that helps to create or transform mental images and to carry out visual reasoning (Gutierrez, 1992).
Non-mental representation, graphs, charts, visual convention, spatial vocabulary (Bishop, 1980)
Spatial
transformation
An ability involving transforming a mental image by rotating, translating, scaling, (un)folding, decomposing, or transformation it into another format of images etc. in the mind, but it also can be the transforming of concrete objects.
Mental rotation (McGee, 1979; Gutierrez, 1996) Third level - visualiz ation Spatial reasoning
An ability to reason in a figural context while using the form of internal representation or external representation, or both.
Spatial thinking (Kosslyn, 1980), geometrical thinking (van Hiele, 1985)
Figure 3-1: Skew Lines Examples in the Real World
stimulates the observer to arrive at an intuitive understanding by provoking some related long- term memories and information unconsciously.
2.2. Spatial relationships
I define spatial relationship ability as 1) the ability to determine relationships between different spatial objects; 2) the ability to compare and analyze the relationship between different elements within an object. It is an ability that enables observers to progress from the prior state of relative globality and lack of differentiation (spatial perception) to a stage of increasing differentiation, articulation, and hierarchic integration. This ability is very intuitive and is an informal method of deduction.
Spatial relationships is a critical ability in Bishop’s IFI (Interpretation of Figural Information) because it enables the observer to interpret the visual stimulus, either mental or physical, and get from it any relevant information that could help to solve a problem. Language plays vital role in this ability, because only by defining and naming the perceived relations can one arrive at a stage of discerning and describing a spatial phenomenon. Spatial perception
spatial relationship ability enables one to raise that understanding into active awareness. Thus, one is capable for carrying out some conjunction or informal deduction.
An example of this ability is that when one sees a railway track, one can recognize and name the parallel relationship between the two rails. Or, as in the example of Figure 3-1, one can immediately recognize them as skew lines and can recall all the relationships (like the definition and characteristics) associated with skew lines.
2.3. Internal representation
I define internal representation as an ability to create a quasi-picture from memory without any physical support. In mathematics, it also includes mental representation of a mathematical concept, property, or other information which is attached to the quasi-picture.
This term is quite close to terms like mental representation and visual images. Therefore, research about visual (mental) images, such as that contributed by Presmeg (1986) and Kosslyn (1980), can continually offer further and deeper understanding of this domain. It includes two major components. The first is a surface representation, the quasi-pictorial entity present in the active memory, such as concrete, pictorial images. The second is a deep representation, such as pattern images, images of formulas, kinesthetic images, and dynamic images. An example is given in Figure 3-2.
2.4. External representation
I define external representation as an ability to create any kind of verbal or graphic representation of 3-D objects, concepts, or properties (including pictures, drawings, diagrams etc.) that helps to create or transform mental images and to carry out visual reasoning (Gutierrez, 1996).
Figure 3-2: An Example of Internal Representation of a Solid
Several researchers (Duval, 1999; Pittalis & Christou, 2013; Bishop, 1983) have shown the positive effects of diagrammatic training on pupils of low spatial ability. The effectiveness of these teaching methods is due to their utility in enhancing students’ representation ability. Duval (1999) asserts that external representations include conventional symbolic systems of
mathematics or graphical representation.
Brown and Wheatley (1997) found that the representation of 3-D objects by 2-D nets is directly related to students’ ability to combine and analyze visual images, which involves both spatial relationship and internal representation abilities. Their conclusion indicates the
interconnection between internal representation and external representation and that they can enhance each other. Two examples are given in Figure 3-3.
2.5. Spatial transformation
I define spatial transformation as an ability involving transforming a mental image by rotating, translating, scaling, (un)folding, decomposing, or transforming it into another image in the mind, but it also can be the transforming of concrete objects. One has to use internal or
Figure 3-3: Examples of External Representation
Figure 3-4: Examples of Spatial Transformation
Mental rotation is a typical example which requires this ability. It demonstrates an ability to imagine in the mind, but also the ability to differentiate the relationships of different parts and properties involved in the rotation, and to tell the relative positions of elements of the rotating object before and after the rotation. An example is given in Figure 3-4.
2.6. Spatial reasoning
ability to reason in a figural context while using the form of internal representation, external representation, or both. It includes sub-skills such as measurement, orientation, formal deduction, rigor, etc. It involves cognitive activity such as associating, recognition, generalizing and logical reasoning. It is a connection between spatial abilities and mathematical abilities.
An example is shown in Figure 3-5. Given a cube with six different shapes on each side as follows, what would the relative sides of the cube look like?
Figure 3-5: Example of Spatial Reasoning