Geometry can be taught in an informal manner. The formal method of teaching geometry tends to limit the ability to reason out spatially. According to the Van Hiele method there are five levels of understanding that culminate in geometric thinking. It would be as well to begin by defining geometric thinking before discussing the salient features of teaching Geometry. Geometric thinking is a field of Mathematics where visual thinking dominates. In order to practice it the student has to progress past various levels. It is the tutor’s duty to guide the student along the various zones. Geometric teaching must enable what are known as ‘geometric experiences’. The various five levels that lead to geometric thinking are ‘Visualization’, ‘Analysis’, ‘Informal Deduction’, ‘Deduction’ and ‘Rigour’.

In the first level of ‘visualization’ learners tend to think in images, shapes and patterns. At this stage the tutor guides students to identify shapes, categorize them by sorting them, manipulate shapes in terms of personal identifications, differentiate sizes and shapes based on visual criteria and build, draw, collate and separate shapes. In the second level of ‘analysis’, learners identify properties of shapes and the tutor needs to help them to build vocabulary concerning these properties. Students at this level would be able to describe relationships between shapes and properties.

In the third level, ‘Informal Deduction’ learners tend to recognize relationships between shapes and properties and the aim at this level is to construct logical arguments using their properties. The tutor guides the students to start solving problems which involve dealing with properties of shapes. This will involve using informal deductive language such as “if then”, “what if” etc. It will involve understanding the verbal notion of “converse” as in “the converse is true”. The fourth and the fifth levels are ‘Deduction’ and ‘Rigour’ respectively and they constitute the higher order geometric experiences. In the fourth level of ‘deduction’ the tutor should guide the students to construct proofs with postulates and axioms. The fifth level, ‘Rigour’ is the highest level of thinking in the Van Hiele hierarchy. Here the tutor must guide the students to work with different geometric models which employ more complicated systems of Geometry. Typically if the foundation is not strong and the student ends up dealing with higher level Geometry there will be a tendency to misinterpret concepts and the online tutor can step in and help them find their way out of the geometric maze. Online tutoring can give you the magic wand to play with shapes and patterns.

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