Carolyn A. Maher, Robert B. Davis, and Alice Alston
In the course of teaching a typical mathematics lesson, a teacher must make a myriad of decisions as he or she is attempting to meet the needs of the various individual students in the class. Little realtime information is available to the teacher to guide these decisions, and there is little time for deciding. What is involved here is a micro level of assessment, done in seconds or fractions of a second. Despite the necessarily hasty context in which this assessment occurs, it is precisely this level of evaluation of a student's work and needs that ultimately has the greatest impact on the student's progress.
In this chapter we report a study made by the authors, in cooperation with Linda, a sixth grade teacher, as she worked to help her students learn about fractions. During mathematics classes, Linda had the students work in small groups. Avideo camera recorded the discussion in one of these groups. In subsequent analysis of these videotapes, we focused on three main questions: (i) What representations did the students make for each mathematical situation, and how did these representations help them (or hinder them) in dealing with the situation? (ii) How did these representations change over time, as a result of conversations among students, experience with concrete materials (or other forms of experience), and teacher interventions of various sorts? (iii) How successful was the teacher in making correct identifications of student representations and in helping