She made good use of the flexibility available to her in lesson planning as she built on her knowledge of what the students were doing. Notice that Ms. Cariello called on three important kinds of knowledge. First, she knew the mathematics she was teaching, including alternative representations, and could invent tasks that made use of them and establish connections among them. Third, her skill in observing students working on mathematics let her monitor how they were thinking. Ms. Cariello had a commitment to students' developing mathematical power and the knowledge and pedagogical expertise to assist them.
Students can build on their own ideas.
Striving toward this vision has required many years of hard work and support from administrators, supervisors, and parents. The reward for these efforts came recently to the Kenilworth Schools when New Jersey's governor, James Florio, and Lucinda Florio, a former elementary school teacher, visited some classes and did mathematics with the students. Ms. Cariello, her colleagues, and the students will tell you that it is well worth the effort. References Davis, Robert. The Cognitive Science Approach to Mathematics Education.