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In mathematics education, the Van Hiele model is a theory that describes how students learn geometry. The theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht University, in the Netherlands. The Soviets did research on the theory in the 1960s and integrated their findings into their curricula. American researchers did several large studies on the van Hiele theory in the late 1970s and early 1980s, concluding that students' low van Hiele levels made it difficult to succeed in proof-oriented geometry courses and advising better preparation at earlier grade levels. Pierre van Hiele published ''Structure and Insight'' in 1986, further describing his theory. The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels. In the United States, the theory has influenced the geometry strand of the Standards published by the National Council of Teachers of Mathematics and the new Common Core Standards. ==Van Hiele levels==
The best known part of the van Hiele model are the five levels which the van Hieles postulated to describe how children 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 counterexamples, 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. The five van Hiele levels are sometimes misunderstood to be descriptions of how students understand shape classification, but the levels actually describe the way that students reason about shapes and other geometric ideas. Pierre van Hiele noticed that his students tended to "plateau" at certain points in their understanding of geometry and he identified these plateau points as ''levels''. In general, these levels are a product of experience and instruction rather than age. This is in contrast to Piaget's theory of cognitive development, which is age-dependent. A child must have enough experiences (classroom or otherwise) with these geometric ideas to move to a higher level of sophistication. Through rich experiences, children can reach Level 2 in elementary school. Without such experiences, many adults (including teachers) remain in Level 1 all their lives, even if they take a formal geometry course in secondary school.〔 The levels are as follows: 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, square). 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. 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 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.〔 Squares are called "diamonds" and not recognized as squares if their sides are oriented at 45° to the horizontal. 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." 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. For example, they will still insist that "a square is not a rectangle." (They may introduce extraneous properties to support such beliefs, such as defining a rectangle as a shape with one pair of sides longer than the other pair of sides.) 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. 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. 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. 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. 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. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Van Hiele model」の詳細全文を読む スポンサード リンク
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