Patterns of Understanding

In this phenomenological study, seven veteran elementary schoolteachers explored how they were able to make accurate and timely decisions about their students' mathematical understanding. These teachers provided rich mathematical environments for their students and made accurate diagnostic judgments about the development of students' knowledge in these environments. Nevertheless, at first it was difficult for them to describe the kinds of evidence they used in assessing their children's thinking. They engaged in a recursive process of watching, individually reacting to, and discussing videotaped sessions of math activities in their classrooms and sharing stories about their observations of children. Through this process, patterns of recognizable phenomena emerged that the teachers had implicitly perceived and interpreted as indicators of children's understanding. These patterns included changes in children's demeanor, extending a concept into new discoveries, making spontaneous modifications rather than simple imitations, using a process in new contexts, feeling confident in using shortcuts, being able to explain ideas to others, and showing high engagement and focus during activities.