What This Means for Instruction

Here are some tips to help educators planning instruction in geometry:

• Encourage young children to describe why figures belong or do not belong to shape categories. Accept visual descriptions. Encourage students to address properties of shapes (Clements, Swaminathan, Hannibal, & Sarama, 1999).
• Use a wide range of examples and nonexamples of the geometric shapes to help children build rich concepts of the geometric shapes. For example:
• vary the size, material and color of materials;
• for rectangles and triangles, vary their orientation;
• for triangles, vary their type including scalene (no sides of equal length) and obtuse (one angle larger than 90 degrees);
• compare examples and nonexamples to focus attention on key attributes.

It is especially important for students to understand that a shape is still a triangle or rectangle regardless of its orientation and that a "diamond" shape is really a square with a 45 degree rotation.

• To help children understand hierarchical relationships among geometric shapes, introduce the class of the geometric shape first and then provide individual examples. For example, introduce quadrilaterals as a class of geometric shapes with four straight sides. Next introduce special types of quadrilaterals such as rectangles. Introduce squares as a special type of rectangles (Clements & Sarama, 2000).
• Encourage the active manipulation of objects to develop students' spatial sense. Use an instructional process that involves predictions, exploration, comparison of prediction to actual result and discussion to help children develop spatial sense (Clements & Battista, 1992).
• According to van Hiele, children need specific instruction to progress from level to level. Engage in these instructional steps to encourage this progress:
• Information: Introduce content, clarify questions and provide information;
• Guided Orientation: Guide students in structured sequenced tasks designed so they encounter specific geometric concepts and procedures;
• Explication: Invite students to describe the targeted geometric concepts in their own language, and learn some of the mathematical language;
• Free Orientation: Let students solve problems whose solution requires the synthesis and utilization of those targeted concepts and relations;
• Integration: Ask students to summarize learning and integrate new knowledge with previous knowledge (Clements & Battista, 1992).
• Use manipulatives effectively; they are particularly important for young or struggling students. Guide students to relate models to their informal concepts and to reflect on manipulative use (Fuys, Geddes, & Tischler, 1988).
• Use technology to develop geometric thinking. Appropriately designed software (Logo and the Geometric Supposer are just two examples) can result in higher levels of geometric thinking (Clements & Battista, 1992).