Geometric Thinking

Through this exercise, I recalled some geometry concepts that I have learned back in secondary school. To get the interior angle of pentagon, one has to divide the shape into basic triangle. For pentagon, it can be separated into 3 basic triangles. Each triangle has the interior angle of 180°, so the total interior angle for the pentagon is 3 x 180° = 540°.

Similarly the same concept can be used to work out the interior angle of other polygons. Try finding the interior angles of the following polygons.
 

I realized that good spatial sense is needed. "Spatial sense includes the ability to mentally visualize objects and spatial relationship" (Van De Walle et al, 2009, p.400). Contrary to typical belief, spatial sense has proven to be more of a nurture skill rather than nature. According to Van de Walle (2009), "We are all capable of growing and developing in our ability to think and reason in geometric contexts." Research done by Solder & Wearne has shown that improvement in the geometric reasoning can be achieved by proper training since young. I believe in learning through exploration. Teachers should introduce the basic shapes such as circle, triangle, rectangle and square and provide exploration opportunities for children to "get to know" these shapes. 

Van Hiele's theory suggested the 5 levels of geometric thinking; ranging from level 0 to level 4 (Page 400). 

Visualization Level 0: Students can name and recognize shapes by their appearance, but cannot specifically identify properties of shapes. This level of geometric thinking can be observed in nursery children. I believe one of the most effective instructions of visualization is for children to manipulate with physical models.

Analysis Level 1: Students begin to identify properties of shapes and learn to use appropriate vocabulary related to properties, but do not make connections between different shapes and their properties. A simple instruction using analysis is to classify shapes according to their properties. 

Informal Deduction Level 2: Students are able to recognize relationships between and among properties of shapes and classes of shapes and are able to follow logical arguments using such properties. A suggested instruction using informal deduction can be using properties to define a shape. 

Deduction Level 3: Students can go beyond identifying characteristics of shapes and are able to construct proof. A secondary student should be at this level. 

Rigor Level 4: This is the highest level of Van Hiele's theory. Students who are at this level can work different geometric systems, where they can understand that there are properties within geometry. This group of students would most likely be in university.