Metacognitive strategieswhich essentially involve "thinking about thinking"increase students' learning.25 By helping students to reflect on and communicate what they learn and what is still unclear, teachers have a unique opportunity to aid their students. There are a number of ways to help manage or help students to manage their thinking by:
One way of accomplishing this kind of reflection is through the use of student-created portfolios. Even the process of selection that goes into making portfolios helps the student to build self-awareness and ultimately gives the student more control over her or his own learning. By shifting the focus of the teacher's role from instructor to facilitator of learning, the process gives students the responsibility to determine what they need to know. These portfolios support students in looking at mathematics in different ways and in seeing value in mathematics.
Writing is another way to encourage students to analyze, communicate, discover, and organize their growing knowledge. Facilitating classroom dialogues and other interactions and helping students to develop reasoning skills make it possible for students to evaluate their own and others' thinking.
During a class discussion of the relationship between perimeter and area, several students in Janet's class stated that increasing the perimeter would have to increase the area. None of the other students questioned these statements. That evening, Janet reflected on why her students developed this misconception. In reviewing the activities that she had asked the students to do prior to the discussion, she discovered that in every instance when the student increased the perimeter the area also increased. The students were justified in assuming this would always happen. She had decided that during the discussion she needed to do more than state that increasing the perimeter did not always increase the area. The students' belief structure had to be challenged by examples that did not fit their misconception.
After looking at several options, Janet decided to have the students work in groups using a geoboard. She would ask students to use Pick's theorem26 to find as many triangles as they could with area equal to one. For each triangle, the students would measure the sides and calculate the perimeter. She would then ask them to create a table showing the perimeter and area of each triangle found, arranging the table from smallest perimeter to largest perimeter. She would then ask them to reflect on the relationship between perimeter and area, asking, "Does increasing the perimeter always yield a larger area?"
At first my team did not see how we could make any triangle with area equal to 1, except for a right triangle with legs of length 2 and 1. When the teacher pointed out that the key to finding the area using Pick's theorem was the number of geoboard pins on the boundary of the triangle and the number of geoboard pins interior to the triangle, we focused on the pins instead of the triangle.
We were now able to find many triangles with area equal to 1. To my surprise, the perimeters of some of these triangles were much more than the perimeter of the first triangle. When we were working with rectangles, if we increased one of the sides of the rectangle, both the perimeter and the area increased. Maybe this only works with rectangles. At least I now know it does not work with triangles. If I get time tomorrow I may try to find some quadrilaterals with area equal to 1 and see if all of them have the same perimeter.
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