Question

Determine the student's Level of Van Hiele Model of Geometric Thought based on the response given. For example: Is the student at Level 1 (Basic Level): Visualization, Level 2: Analysis, Level 3: Informal Deduction, Level 4: Formal Deduction, or Level 5: Rigor? Explain your decision.

Miss Gonmez gave her students a paper polygon and asked them to identify the given shape and explain how they decided which polygon it was.

James responded that the shape was a rectangle. He decided this because he folded the polygon in half length-wise and found that it had opposite sides the same length.

He compared the corners of the polygon by placing them on top of each other and found they were all the same. He concluded they must all be right angles.

Answer 1

Before we come to a conclusion ABOUT JAMES' Van Hiele Model student level, let us define these levels in a broad manner

Level 1: (Basic Level): 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. Geometric figures, for example, are recognized by their shape as a whole, that is, by their physical appearance, not by their parts or properties. A person functioning at this level can learn geometric vocabulary, can identlfy specified shapes, and given a figure, can reproduce it.

Level 2: Analysis

At level 2, an analysis of geometric concepts begins. For example, 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 (e.g., in a quadrilateral, opposite sides being parallel necessitates opposite angles being equal) and among figures (a square is a rectangle because it has all the properties of a rectangle). Thus they can deduce properties of a figure and recognize classes of figures. Class inclusion is understood. Delinitions are meaningful. Informal arguments can be followed and given. The student at this level, however, does not comprehend the significance of deduction as a whole or the role of axioms. Empirically obtained results are often used in conjunction with deduction techniques. Formal proofs can be followed, but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.

Level 4 : Deduction

At this level, the significance of deduction as a way of establishing geometric theory within an axiomatic system is understood. The interrelationship and role of undefined terms, axioms, postulates, definitions, theorems, and proof is seen. 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. This last level is the least developed in the original works and has received little attention from researchers. P. M. van Hiele has acknowledged that he is interested in the first three levels in particular.

According to the above synopsis the student is at LEVEL 2; Analysis. Because he has deduced the characteristics of the figure through EXPERIMENTATION and CLOSE OBSERVATION, But he hasn't made any informal deductions or interrelationships.

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