My fourth-grade class had just completed an exploration of pentominoes (polygonal shapes with an area of five square units). Finding all twelve shapes gives children valuable geometric problem-solving practice by highlighting transformations (flips, slides, and turns) and congruence (shapes can be differently oriented, yet congruent). Before moving on to another lesson, I realized that the students might use the same twelve shapes to examine perimeter and area. Eleven of the shapes have a perimeter of twelve units. Only one shape yields a different perimeter, ten units (see fig. 1). The children had limited experience with perimeter and area; I doubted that they understood that shapes with a fixed area could have perimeters of different lengths. Because they were so familiar with the pentominoes, I felt that this material would give them a good opportunity to address these concepts in more detail. Although I did expect them to calculate the perimeters and areas of the twelve shapes, I did not foresee that the children's follow-up discussion would open an opportunity for problem-posing explorations. This article describes my evolving curricular decision making, the children's investigations, and what I learned from this unanticipated experience.
The Role of Value Orientations in Curricular Decision Making: A Rationale for Teachers' Goals and Expectations