## Saturday, September 20, 2008

### M2G1 - Geometry in Primary Grades (2)

M2G1. Students will describe and classify plane figures (triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes) according to the number of edges and vertices and the sizes of angles (right angle, obtuse, acute).

Well, it has been a long time since I wrote an entry here. I apologize for those who posted their comments for not responding. I will try to keep updating this blog a bit more frequently.

Last time, I started writing on the geometry standards, and I will continue the discussion of geometry in this entry. In M2G1, students are now expected to classify plane figures "according to the number of edges and vertices and the sizes of angles (right angle, obtuse, acute)." The GPS includes the following figures: triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes. Last time, I questioned the appropriateness of the expectation that kindergarteners to distinguish squares and rectangles. In Grade 2, students are learning about different types of angles, right, acute and obtuse. Therefore, it is in Grade 2 when it is appropriate for students to learn that rectangles are quadrilaterals with 4 right angles, and squares are rectangles with all sides equal.

However, let's think about how students can understand right angles. When I ask my adults, including some teachers and teacher candidates, what right angles are, they almost always respond by saying "90-degree angles." Although it is true that right angles measure 90 degrees, measuring angles using "degree" as the unit is a Grade 4 standard. Thus, how are second grade students to understand what right angles are? A Japanese elementary math textbook by Hironaka and Sugiyama has an interesting approach to this topic. They define a right angle to be the angle you obtain when you fold a piece of paper as shown in the figures below:

Note that the piece of paper can be any shape to start with. The second fold is made in such a way that the first fold line will be folded onto itself. Although it might also be helpful to point out to children that the corners of note papers, index cards, etc. are right angles, we cannot always be sure that corners of any piece of paper are right angles.

Interestingly, this definition of a right angle is very much comparable to Euclid's definition of right angles in his book The Elements. He defines that the angles you obtain by equally dividing a straight angle are right angles. When the second fold is made so that the first fold line will be folded onto itself, we are indeed dividing the straight line (the first fold line) into two equal angles.

I want to end this entry by raising another issue with the standard, however. This standard expects students to describe and classify trapezoid. However, in order to describe trapezoids, children need to concept of parallelism. The Grade 2 Geometry unit of Math Frameworks define trapezoids as "quadrilaterals with two parallel sides." Unfortunately, parallelism is a Grade 4 topic. Therefore, it is very strange that we should expect students in Grade 2 to describe and classify trapezoids.

To make the matter even worse, Grade 4 Geometry unit of Math Frameworks defines trapezoids as quadrilaterals "with only one pair of parallel sides." This definition is different from the Grade 2 definition, which does not say anything about the other two sides of quadrilaterals. In a recent publication, Zalman Usiskin and his collaborators document how these two definitions of trapezoids has existed in US mathematics textbooks. Therefore, the fact that the definitions are different isn't too surprising. However, it is rather unfortunate that a document that emphasizes coherence will not try to be consistent in their definition of a geometric figure.