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They may say F is "upside down". Students at Level 1 will recognize that only E and F are valid triangles. 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...
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These visual prototypes are then used to identify other shapes. A shape is a circle because it looks like a sun; a shape is a rectangle because it looks like a door or a box; and so on. Children view figures holistically without analyzing their properties. If a shape does not sufficiently resemble its prototype, the child may reject the classification. Thus, children at this stage might balk at calling a thin, wedge-shaped triangle with sides 1, 20, 20 or sides 20, 20, 39 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 "triangles" if they bear a holistic resemblance to an equilateral triangle.
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Children at this level often believe something is true based on a single example. 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. 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.
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For example, they will still insist that "a square is not a rectangle. Children begin to notice many properties of shapes, but do not see the relationships between the properties; therefore they cannot reduce the list of properties to a concise definition with necessary and sufficient conditions. They usually reason inductively from several examples, but cannot yet reason deductively because they do not understand how the properties of shapes are related. Level 2. Abstraction: 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.
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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. They recognize that all squares are rectangles, but not all rectangles are squares, and they understand why squares are a type of rectangle based on an understanding of the properties of each. They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus. They understand necessary and sufficient conditions and can write concise definitions. However, they do not yet understand the intrinsic meaning of deduction. They cannot follow a complex argument, understand the place of definitions, or grasp the need for axioms, so they cannot yet understand the role of formal geometric proofs.
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Level 3. 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 definitions are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry.
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Geometric ideas are still understood as objects in the Euclidean plane. 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.
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People can understand the discipline of geometry and how it differs philosophically from non-mathematical studies. American researchers renumbered the levels as 1 to 5 so that they could add a "Level 0" which described young children who could not identify shapes at all. Both numbering systems are still in use. Some researchers also give different names to the levels. Properties of the levels[ edit ] The van Hiele levels have five properties: 1. Fixed sequence: the levels are hierarchical. Students cannot "skip" a level. 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. Distinction: each level has its own linguistic symbols and network of relationships.
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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. 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.
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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. Ideally, the teacher and students need shared experiences behind their language. Attainment: The van Hieles recommended five phases for guiding students from one level to another on a given topic: [7] Information or inquiry: students get acquainted with the material and begin to discover its structure. Teachers present a new idea and allow 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.
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A teacher might say, "This is a rhombus. Construct some more rhombi on your paper. 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 to learn. A teacher might ask, "What happens when you cut out and fold the rhombus along a diagonal?
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Explicitation: students express what they have discovered and vocabulary is introduced. 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. They know the properties being studied, but need to develop fluency in navigating the network of relationships in various situations. This type of activity is much more open-ended than the guided orientation. These tasks will not have set procedures for solving them.
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