Where in the Neighborhood Has “Polygon”?

Driving Question:  Polygons in the real world:  how do they relate to us and to each other?

Background:

 Rich and varied experiences with shape and spatial relationships, when provided consistently over time, can and do develop spatial sense. Without geometric experiences, most people do not grow in their spatial sense or spatial reasoning. The van Hiele theory describes how we think and what types of geometric ideas we think about. In grades K-3, students primarily function in the first two levels.  As deductive reasoning develops, students move into Informal Deduction or Abstraction in grade 4 and 5.

 At the first level, Visualization, students think about what shapes are and what they “look like”. They will sort and classify shapes based on their appearances. With a focus on the appearance of shapes, students are able to see how shapes are alike and different. As a result, students at this level can create groupings of shapes that seem to be “alike”.  At the second level, Analysis, students think about classes of shapes rather than individual shapes. They will consider all shapes within a class rather than a single shape. By focusing on a class of shapes, students are able to think about what makes a rectangle a rectangle (four sides, opposite sides of the same length, four right angles (square corners), etc.). and not be confused by size or orientation. They can state that a collection of shapes goes together because of properties.  Ideas about an individual shape can now be generalized to all shapes that fit that class. Students may be able to list all of the properties of shapes but not see them as subclasses of one another, such as that all squares are rectangles and all rectangles are parallelograms (that would move the students into Level 3 thinkers which make informal deductions).  Without many varied experiences with shapes, many people have difficulty going beyond the second level of analysis.  (Van de Walle, p. 186-193)

In the intermediate grades (4-5), students work beyond the Analysis level and apply deductive connections (i.e. – knowledge that all squares are rectangles, but not all rectangles are squares) to attain Abstraction.  As the simple proofs on their connections become more formalized, some learners will advance to the Deductive level, but this is more commonly achieved in secondary school.

The following are recommendations for teaching to advance students through the levels within a given geometric concept: 

     · Information

E.g. "This is a rhombus."

• Guided orientation: students do tasks that enable them to explore implicit relationships.

"What happens when you fold the rhombus along a diagonal?"

• Explicitation: students express what they have discovered and vocabulary is introduced.

"Here are the properties we have noticed and some associated vocabulary."

•  Free orientation: students do more complex tasks enabling them to master the network of relationships in the material.

"Use those properties to figure out solutions to these problems."

• Integration: students summarize what they have learned and commit it to memory.

"Write a summary of what we have learned."

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