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  1. Visualisation → Relationship → Deduction → Analysis → Axiomatic
  2. Relationship → Visualisation → Analysis → Axiomatic → Deduction
  3. Visualisation → Analysis → Relationships → Deduction → Axiomatic
  4. Analysis → Relationship → Visualisation → Deduction → Axiomatic
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Option 3 : Visualisation → Analysis → Relationships → Deduction → Axiomatic

Geometry includes Euclidean geometry, Projection, Analytical geometry, Trigonometry, Combinatorial geometry, differential and Algebraic Geometry, etc. 

  • According to two Dutch educators, Dina van Hiele-Geldof and her husband, Pierre Marie van Hiele, behaviors such as these reflect a student's level of geometric maturity.
  • The model consists of five levels of understanding. The levels, labeled "visualization," "analysis," "informal deduction," "formal deduction," and "rigor" describe characteristics of the thinking process.  

The sequence of development of geometrical understanding:

Level 1: Visualization

  • At this initial stage, students are aware of space only as something that exists around them.
  • Geometric concepts are viewed as total entities rather than as having components or attributes.

Level 2: Analysis

  • At this level, an analysis of geometric concepts begins, through observation and experimentation, students begin to discern the characteristics of figures.
  • These emerging properties are then used to conceptualize classes of shapes.
  • Thus figures are recognized as having parts and are recognized by their parts. 

Level 3: Informal Deduction.

  • At this level, students can establish the interrelationships of properties both within figures and among figures.
  • Thus they can deduce the properties of a figure and recognize classes of figures.  

Level 4: Formal Deduction

  • At this level, the significance of deduction as a way of establishing geometric theory within an axiomatic system is understood.
  • A person at this level can construct, not just memorize, proofs; the possibility of developing a proof in more than one way is seen; the interaction of necessary and sufficient conditions is understood; distinctions between a statement and its converse can be made.

Level 5: Rigor

  • At this stage the learner can work in a variety of axiomatic systems, that is, non-Euclidean geometries can be studied, and different systems can be compared.
  • Geometry is seen in the abstract.

Hence, we can conclude that option 3 is correct regarding the correct sequence of development of geometrical understanding.

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