GEOMETRICAL THINKING

GEOMETRICAL THINKING (VAN HIELE MODEL OF GEOMETRICAL THOUGHT)

     The student learns by rote to operate with mathematical relations that he does not understand, and of which he has not seen the origin,. Therefore the system of relations is an independent construction having no rapport with other experiences of the child. This means that the student knows only what has been taught to him and what has been deduced from it. He has not learned to establish connection between the system and the sensory world. He will not know how to apply what he hars learned in a new situation.-Pierre van Hiele, 1959

     The best known part of the van Hiele model are the five levels which the van Hiele postulated to describe how childen learn to reason in geometry. Students cannot be expected to prove geometric theorems until they have built up an extensive understanding of the systems of relationships between geometric ideas. These systems cannot be learned by rote, but must be developed through familiarity by experiencing numerous examples and counter examples, the various properties of geometric figures, the relationships between the properties, and how these properties are ordered. The five levels postulated by the van Hieles describe how students advance through this understanding.

LEVEL- 0: VISUALIZATION: At this level, the focus of a child’s thinking is on individual shapes, which the child is learning to classify by judging their holistic appearance. Children simply say,    ’ That is a circle’, usually without further description. Children identify prototypes of basic geometrical figures (triangle, circle, and square). These visual prototypes are then used to identify other shapes.

     A square seems to be a different sort of shape than a rectangle, and a rhombus does not look like other parallelograms, so these shapes are classified completely separately in the child’s mind. Children at this stage might balk at calling a thin, wedge-shaped triangle a triangle, because it’s so different in shape from an equilateral triangle, which is the usual prototype for ‘triangle’. If the horizontal base of the triangle is on top and the opposing vertex below, the child may recognize it as a triangle, but claim it is “upside down”. Shapes with rounded or incomplete sides may be accepted as “triangle.” If they bear a holistic resemblance to an equilateral triangle.

·         Students are aware of space as something that exists around them.

·         Geometric components are viewed as a whole by their shape and appearance, not by their parts and properties.

·         At this level students can learn geometric names (E.g: Square, rectangle, triangle); identify shapes and also draw or reproduce thes4 shapes.

LEVEL -1.ANALYSIS: At this level, the shapes become bearers of their properties. The objects of thought are classes of shapes, which the child has learned to analyze as having properties. A Person at this level might say “A square has 4 equal sides and 4 equal angles. Its diagonals are congruent and perpendicular, and they bisect each other.” The properties are more important than the appearance of the shape. If a figure is sketched on the blackboard and the teacher claims it is intended to have congruent sides and angles, the students accept that it is a square, even if it is poorly drawn. Properties are not yet ordered at this level. Children can discuss the properties of the basic figures and recognize them by these properties, but generally do not allow categories to overlap because they understand each property in isolation from the others.

·         At this stage through experimenting and observing students can begin to analyse the characteristics of a shape. For example they may be able to deduce that all sides of a square are equal, or that the opposite angles of a parallelogram are equal. They can begin to make generalilsations about the properties of shapes.

LEVEL -2.ABSTRACTION/Informal Deduction:  At this level, properties are ordered. The objects of thought are geometric properties, which the student has learned to connect deductively. The student understands that properties are related and one set of properties may imply another property.  Students can reason with simple arguments about geometric figures. A student at this level might say, “Isosceles triangles are symmetric, so their base angles must be equal.” Learners recognize the relationships between types of shapes.

·         At this level students begins to understand the interrelationship between properties within a figures and between two or more figures. For example, a square is a rectangle because it satisfies all the properties of a rectangle.

LEVBEL -3. DEDUCTION/Formal Deduction: Students at this level understand the meaning of deduction. The object of thought is deductive reasoning (simple proofs), which the student learns to combine to form a system of formal proofs (Euclidean geometry). Learners can construct geometric proofs at a secondary school level and understand their meaning. They understand the role of undefined terms, definitions, axioms and theorems in Euclidean geometry. However, students at this level believe that axioms and definition are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry. Geometric ideas are still understood as objects in the Euclidean plane.

·         At this level students can understand axiomatic systems (Eg. Euclidian Geometry); can understand the interrelationships between terms, axioms, theorems, proofs. The student can also prove theorems (not just by memorising), but by developing it on her/his own.

LEVEL-4.RIGOR:  At this level, geometry is understood at the level of a mathematician. Students understand that definitions are arbitrary and need not actually refer to any concrete realization. The object of thought is deductive geometric systems, for which the learner compares axiomatic systems. Learners can study non-Euclidean Geometries with understanding. People can understand the discipline of geometry and how it differs philosophically from non-mathematical studies.

     American researchers renumbered the levels at 1 to 5 so that they could add a “Level0” which described young children who could not identify shapes at all. Both numbering systems are still in use. Some preachers also give different names to the levels.

·         This is the last stage where a person can work with multiple axiomatic systems, compare different systems and think of geometry in the abstract.

PROPERTIES OF THE LEVELS: Van Hiele levels have five properties:

1. Fixed sequence:  The levels are hierarchical. Students cannot “skip” a level.  The van Hieles claim that much of the difficulty experienced by geometry students is due to being taught at the Deduction level whe3n they have not yet achieved the Abstraction level.

2. Adjacency: Properties which are intrinsic at one level become extrinsic at the next. (The properties are there at the Visualization level, but the student is not yet consciously aware of them until the Analysis level. Properties are in fact related at the Analysis level, but students are not yet explicitly aware of the relationships.

3. Distinction:  Each level has its own linguistic symbols and network of relationships. The meaning of a linguistic symbol is more than its explicit definition; it includes the experiences the speaker associates with the given symbol. What may be “correct” at one level is not necessarily correct at another level. At level-0, a square is something that looks like a box. At level-2 a square is a special type of rectangle. Neither of these is a correct description of the meaning of “square” for someone reasoning at level-1. If the student is simply handed the definition and its associated properties, without being allowed to develop meaningful experiences with the concept, the student will not be able to apply this knowledge beyond the situations used in the lesson.

4. Separation:  A teacher who is reasoning at one level speaks a different ‘language’ from a student at a lower level, preventing understanding. When a teacher speaks of a “square” she or he means a special type of rectangle. A student at level-0 or 1 will not have the same understanding of this term. The student does not understand the teacher, and the teacher does not understand how the student is reasoning, frequently concluding that the student’s answers are simply “wrong”. The van Hieles believed this property was one of the main reasons for failure in geometry. Teacher believe they are expressing themselves clearly and logically, but their level 3 or 4 reasoning is not understandable to students at lower levels, nor do the teachers understand their students’ thought processes. Ideally, the teacher and students need shared experiences behind their language.

5. Attainment:  The van Hieles recommended five phases for guiding students from one level to another on a given topic,

          * Information or inquiry:  Students get acquainted with the material and begin to discover its structure. Teacher presents a new idea and allows the students to work with the new concept. By having students experience the structure of the new concept in a similar way, they can have meaningful conversations about it.(A teacher might say, “This is a rhombus. Construct some rhombi on your paper.”)

          * Guided or directed orientation: Students do tasks that enable them to explore implicit relationships. Teachers propose activities of a fairly guided nature that allow students to become familiar with the properties of the new concept which the teacher desires them toi learn. (A teacher might ask, “What happens when you cut out and fold the rhombus along a diagonal?  the other diagonal?” and so on, followed by discussion.)

* Explicitation: Students express what they have discovered and vocabulary is introduced. The students’ experiences are linked to shared linguistic symbols. The van Hieles believe it is more profitable to learn vocabulary after students have had an opportunity to become familiar with the concept. The discoveries are made as explicit as possible. (A teacher might say, “Here are the properties we have noticed and some associated vocabulary for the things you discovered. Let’s discuss what these mean.”)

* Free orientation: Students do more complex tasks enabling them to master the network of relationships in the material. They know the properties being studied, but need to develop fluency navigating the new work of relationships in various situations. These tasks will not have set procedures for solving them. Problems may be more complex and require more free exploration to find solutions. (A teacher might say, “How could you construct a rhombus given only two of its sides?” and other problems for which students have not learned a fixed procedure.)

* Integration: Students summarize what they have learned and commit it to memory. The teacher may give the students an overview of everything they have learned. It is important that the teacher not present any new material during this phase, but only a summary of what has already been learned. The teacher might also give an assignment to remember the principles and vocabulary learned for future work, possibly through further exercise. (A teacher might say, “Here is a summary of what we have learned. Write this in your note book and do these exercises for home work.”)  Supporters of the van Hiele model point out that traditional instruction often involves only this last phase, which explain why students do not master the material