9 THE MATHEMATICAL WORLD
9C) SHAPES
Development of geometrical thinking. Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks like a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction. At the fourth level, students can produce a short sequence of statements to logically justify a conclusion and can understand that deduction is the method of establishing geometric truth.
Progress from one of Van Hiele's levels to the next is more dependent upon instruction than age. Given traditional instruction, middle-school students perform at levels one or two (Clements & Battista, 1992). Despite that, almost 40% of high-school graduates finish high-school geometry below level two (Burger & Shaughnessy, 1986; Clements & Battista, 1992; Suydam, 1985). Further research will help identify what levels of geometric thinking students can attain at different grades given effective instruction that takes account of their difficulties in learning geometry. Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976) and can understand the relations that connect the properties of shapes or make simple deductions by 8th or 9th grade (Clements & Battista, 1992).
Proof. Research on students' development of the ability to construct proofs reflects somewhat conflicting views (Clements & Battista, 1992). Piagetian research suggests that students can reason deductively from any assumptions once they reach the formal operational stage (roughly age 12 and beyond). Other research, however, suggests that the ability to construct proofs depends on the amount and organization of particular knowledge they have. For example, this research indicates that students are not likely to understand and construct geometric proofs before they can see the relationships between classes of figures (Senk, 1989). Still other research suggests that students may need to understand the nature of proof and how it differs from everyday argumentation before they are able to con-struct proofs (Clements & Battista, 1992). Clearly, further research is needed to identify how students can come to understand what it means to prove something in geometry and what such a proof entails.