## Tuesday, October 21, 2008

### M3G1 - Geometry in Primary Grades (3)

M3G1. Students will further develop their understanding of geometric figures by drawing them. They will also state and explain their properties.

In Kindergarten, students are expected to "recognize" certain geometric figures. In Grade 1, students are expected to classify geometric figures by comparing their structures. In M3G1, the GPS mentions "properties" of geometric figures for the first time. By the end of Grade 3, students are expected to state and explain properties of geometric figures. The GPS does not clearly spell out what properties are to be explored. However, based on the types of figures students have previously explored, properties such as the equality of the base angles of isosceles triangles are within the reach for 3rd graders. Of course, we need to keep in mind that equality of angles at this point is based on the fact that two angles can be made to overlap each other completely as measuring angles is a Grade 4 expectation.

The general flow of the geometry standards in the GPS is very much consistent with the developmental levels of geometric thinking identified by Dina and Pierre van Hiele. According to the van Hiele's model, children can only treat geometric figures as wholes. Their thinking is based on how figures appear. In the second stage, children develop the ability to identify and use various components of geometric figures as objects of their thought. It is in this stage, they can start identifying figures based on relationships among their component parts. As children move into the 3rd level, typically called Informal Deduction, they can now start focusing on those relationships as the objects of their thought. One characteristic of students in this stage is that they can now start using some logical statements, like if ... then .... In order for students to be successful a typical high school geometry class, students must be at this stage at the beginning of the course. Since geometry is discussed throughout Math 1 ~ 4 in the GPS, and since geometry proof happens in Math 1, this means students must be at this stage by Grade 9 the latest.

As children begin to explore properties of geometric figures in Grade 3, it is essential that teachers distinguish properties from definitions. The van Hiele theory suggests that children in the second stage can identify relationships among various components of a given figure. However, as children move toward the third stage, they can identify the minimum amount of relationships that will be sufficient to define a shape. Those relationships become the definition of the shape, while other relationships are now treated as properties. For example, children in the second stage might be able to identify the following relationships in isosceles triangles:
• two sides are equal in length
• two angles are equal
• has a line of symmetry (you can fold it in half - symmetry is a Grade 6 topic)
• two angles that are equal are both acute angles
• third angles can be acute, right or obtuse
• etc.
As children move to the third stage, they can go from this laundry list of relationships to the realization that only the first relationship is necessary to define an isosceles triangle. A useful activity to help children go from a laundry list to the minimum set of defining characteristics is to have children actually draw shapes. As children are asked to draw isosceles triangle, they will realize that they don't need to use all of those characteristics to draw one - indeed if you draw a triangle with two equal length sides that's enough.

Although most 3rd graders are still in the second van Hiele stage, it is important that teachers' communication (with students and with parents) clearly distinguish definitions and properties. The parent letter for the third grade geometry unit have the following "terminology":
• Parallelogram: A quadrilateral with opposite sides that are parallel and of equal length and with opposite angles that are of equal measure.
• Rectangle: A quadrilateral with four right angles and two pairs of opposite, equal parallel sides.
• Rhombus: A parallelogram with four equal sides and equal opposite angles.
All of these are "laundry lists" and not definitions. Although at times it is perfectly fine to be less rigorous about our language use, when we do use language loosely, we should do so with clear understanding that we are indeed being less rigorous.