The most prominent feature of the model is a five-level hierarchy of ways of understanding spatial ideas. Each of the five levels describes the thinking processes used in geometric contexts. The levels describe how we think, and what types of geometric ideas we think about, rather than how much knowledge we have. A significant difference from one level to the next is the objects of thought—what we are able to think about geometrically.
Level 0: Visualization
The objects of thought at level 0 are shapes and what they “look like.” Students recognize and name figures based on the global, visual characteristics of the figure—a gestaltlike approach to shape. Students operating at this level are able to make measurements and even talk about properties of shapes, but these properties are not abstracted from the shapes at hand. It is the appearance of the shape that defines it for the student. A square is a square “because it looks like a square.” Because appearance is dominant at this level, appearances can overpower properties of a shape. For example, a square that has been rotated so that all sides are at a 45-degree angle to the vertical may now be a diamond and no longer a square. Students at this level will sort and classify shapes based on their appearances—“I put these together because they are all pointy” (or “fat,” or “look like a house,” or are “dented in sort of,” and so on). With a focus on the appearances of shapes, students are able to see how shapes are alike and different. As a result, students at this level can create and begin to understand classifications of shapes.
The products of thought at level 0 are classes or groupings of shapes that seem to be “alike.”
Level 1: Analysis
The objects of thought at level 1 are classes of shapes rather than individual shapes. Students at the analysis level are able to consider all shapes within a class rather than a single shape. Instead of talking about this rectangle, it is possible to talk about all rectangles. By focusing on a class of shapes, students are able to think about what makes a rectangle a rectangle (four sides, opposite sides parallel, opposite sides same length, four right angles, congruent diagonals, etc.). The irrelevant features (e.g., size or orientation) fade into the background. At this level, students begin to appreciate that a collection of shapes goes together because of properties. Ideas about an individual shape can now be generalized to all shapes that fit that class. If a shape belongs to a particular class such as cubes, it has the corresponding properties of that class. “All cubes have six congruent faces, and each of those faces is a square.” These properties were only implicit at level 0. Students operating at level 1 may be able to list all the properties of squares, rectangles, and parallelograms but not see that these are sub classes of one another, that all squares are rectangles and all rectangles are parallelograms. In defining a shape, level 1 thinkers are likely to list as many properties of a shape as they know.
The products of thought at level 1 are the properties of shapes.
Level 2: Informal Deduction
The objects of thought at level 2 are the properties of shapes. As students begin to be able to think about properties of geometric objects without the constraints of a particular object, they are able to develop relationships between and among these properties. “If all four angles are right angles, the shape must be a rectangle. If it is a square, all angles are right angles. If it is a square, it must be a rectangle.” It is at this level that students can appreciate the nature of a definition. With greater ability to engage in “if–then” reasoning, shapes can be classified using only minimum characteristics. For example, four congruent sides and at least one right angle can be sufficient to define a square. Rectangles are parallelograms with a right angle. Observations go beyond properties themselves and begin to focus on logical arguments about the properties. Students at level 2 will be able to follow and appreciate an informal deductive argument about shapes and their properties. Proofs may be more intuitive than rigorously deductive. However, there is an appreciation that a logical argument is compelling. An appreciation of the axiomatic structure of a formal deductive system, however, remains under the surface.
The products of thought at level 2 are relationships among properties of geometric objects.
Level 3: Deduction
The objects of thought at level 3 are relationships among properties of geometric objects. At level 3, students are able to examine more than just the properties of shapes. Their earlier thinking has produced conjectures concerning relationships among properties. Are these conjectures correct? Are they “true”? As this analysis of the informal arguments takes place, the structure of a system complete with axioms, definitions, theorems, corollaries, and postulates begins to develop and can be appreciated as the necessary means of establishing geometric truth. The student at this level is able to work with abstract statements about geometric properties and make conclusions based more
on logic than intuition. This is the level of the traditional high school geometry course.
The products of thought at level 3 are deductive axiomatic systems for geometry.
Level 4: Rigor
The objects of thought at level 4 are deductive axiomatic systems for geometry. At the highest level of the van Hiele hierarchy, the objects of attention are axiomatic systems themselves, not just the deductions within a system. This is generally the level of a college mathematics major who is studying geometry as a branch of mathematical science. The products of thought at level 4 are comparisons and contrasts among different axiomatic systems of geometry.
We have given brief descriptions of all five levels to illustrate the scope of the van Hiele theory. In every grade from 5 to 8, you will certainly see students at levels 0, 1, and 2.
You no doubt noticed that the products of thought at each level are the same as the objects of thought at the next. This object–product relationship between levels of the van Hiele theory . The objects (ideas) must be created at one level so that relationships among these objects can become the focus of the next level. In addition to this key concept of the theory, four related characteristics of the levels of thought merit special attention.
1. The levels are sequential. To arrive at any level above level 0, students must move through all prior levels. To move through a level means that one has experienced geometric thinking appropriate for that level and has created in one’s own mind the types of objects or relationships that are the focus of thought at the next level.
2. The levels are not age dependent in the sense of the developmental stages of Piaget. A third grader or a high school student could be at level 0. Indeed, some students and adults remain forever at level 0, and a significant number of adults never reach level 2. But age is certainly related to the amount and types of geometric experiences that we have. Therefore, it is reasonable to assume that most children in the K–2 range as well as many children in grades 3 and 4 are at level 0.
3. Geometric experience is the greatest single factor influencing advancement through the levels. Activities that permit children to explore, talk about, and interact with content at the next level, while increasing their experiences at their current level, have the best chance of advancing the level of thought for those children. Some researchers believe that it is possible to be at one level with respect to a familiar area of content and at a lower level with less familiar ideas (Clements & Battista, 1992).
4. When instruction or language is at a level higher than that of the student, there will be a lack of communication. Students required to wrestle with objects of thought that have not been constructed at the earlier level may be forced into rote learning and achieve only temporary and superficial success. A student can, for example, memorize that all squares are rectangles without having constructed that relationship. A student may memorize a geometric proof but fail to create the steps or understand the rationale involved (Fuys, Geddes, & Tischler, 1988; Geddes & Fortunato, 1993).
Not every teacher will be able to move children to the next level. However, all teachers should be aware that the experiences they provide are the single most important factor in moving children up this developmental ladder. Every teacher should be able to see some growth in geometric thinking over the course of the year.
The van Hiele theory and the developmental perspective highlight the necessity of teaching at the child’s level of thought. However, almost any activity can be modified to span two levels of thinking, even within the same classroom. For many activities, how we interact with individual children will adapt the activity to their levels and encourage them or challenge them to operate at the next higher level. Explorations help develop relationships. The more students play around with the ideas in activities, the more relationships they will discover. However, students need to learn how to explore ideas in geometry and play around with the relationships in order for ideas to develop and become meaningful.
No comments:
Post a Comment