## Thursday, January 17, 2008

### MKG1 - Geometry in Primary Grades (1)

MKG1. Students will correctly name simple two and three-dimensional figures, and recognize them in the environment.
a. Recognize and name the following basic two-dimensional figures: triangles, rectangles, squares, and circles.

All of my posts so far have been on the Number and Operations strand. This semester, I am teaching a geometry course for prospective elementary school teachers, so I want to start 2008 with a post on a geometry standard, MKG1.

As I read this standard, I wonder how students are to recognize figures, in particular rectangles and squares. Two Dutch mathematics educators, Dina and Pierr van Hiele, showed that children’s geometric thinking develops sequentially along 5 different levels. In the first stage, called Visualization, children recognize shapes by looking at them as a whole, not their component parts. Thus, they would say a shape is a square because it looks like a square. As they move to the next level, Analysis, they can start paying attention to the characteristics of shapes. However, it is not until the third level that children can actually think about interrelationship among classes of shapes, for example, rectangles are parallelograms. A typical HS geometry class requires students to be in the fourth level, called Deduction. In the fifth level of geometric thinking, a person can now think about different types of geometry, e.g., Euclidean, Spherical, etc. Although the works of van Hiele showed that these levels are not so much age dependent, it is probably safe to assume that most Kindergarteners are still operating in the first level. From this perspective, the inclusion of rectangles and squares in MKG1 is a bit problematic.

There are many different ways we can classify figures, and as we create classes of geometric figures, we sometimes give them specific names. Some classes of figures are mutually exclusive, while others are overlapping. All four types of figures named in this standard are similar in that they are all closed figures. Therefore, we are clearly expecting students to recognize closed figures from non-closed figures. Triangles, rectangles, and squares are members of a class of figures called polygons, while circles are not. Thus, recognizing circles (and distinguishing them from other shapes such as triangles, rectangles and squares) seems to be a slightly simpler task for children, appropriate for those children who are in the Visualization level. Distinguishing triangles from rectangles and squares also seems to be reasonable for such children. Although they may not be explicitly pay attention the number of sides as characteristics of these figures, triangles will definitely “look differently” from rectangles and squares – or any other four (or more) sided figures.

However, recognizing rectangles and squares seem to require much more sophistication that recognizing circles or triangles. For one thing, rectangles and squares are members of a class of figures called quadrilaterals (4-sided figures). Thus, simply noting the number of sides is insufficient to recognize them. What makes them special is the fact that all of them are 4-sided figures with 4 right angles. Moreover, squares and rectangles are not mutually exclusive, as circles and triangles are. In fact, squares are special type of rectangles. I doubt that the writers of the GPS expected Kindergarten children to understand this relationship – that would be developmentally inappropriate. What would have been a more appropriate standard is “to recognize triangles, quadrilaterals (or quadrangles), and circles,” but the term “quadrilateral” seems to be a bit too much for Kindergarteners. Thus, in order to avoid (perhaps) developmentally inappropriate words, we may have unintentionally introduced (perhaps) developmentally inappropriate mathematics expectations.