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Abstract
Four elementary preservice teachers participated in a field-based mathematics methods course about childrens mathematics with respect to numbers and operations and discussed an Experiential Model of Childrens Counting and Whole Number Development proposed by Steffe, Von Glasersfeld, Richards, and Cobb (1983). Each participant worked one-on-one with a child for eight weeks. The purpose of the study was to investigate the working model that each preservice teacher constructed of her childs mathematics and how her model informed her instructional decisions during the field experience. Data were collected in the form of interviews, videos and observations of the field experience sessions, and course products. The data were analyzed using qualitative case study methods, including micro-analysis (Strauss & Corbin, 1998). The activities in which the preservice teachers engaged during the course were specifically designed so that the preservice teachers would learn to listen to and learn from children and think about how the childrens mathematics informed their instructional decisions. Thus, this course provided the necessary context for my study in that it made it likely that the preservice teachers would attempt to construct and use models of childrens mathematics. I found that the participants engaged in a Mathematics Teaching Cycle similar to that described by Simon (1997). Each participant constructed a model of the childs mathematics based on her interpretation of the experiential model from class. They then used their working models of their childrens mathematics to determine learning goals and eventually designed or chose activities to meet those learning goals. In order to continuously inform the preservice teachers working model and to extend her childs mathematics she asked many questions, which were classified as probing, prompting, and prodding. Through learning the mathematics of children, the participants also learned mathematics for themselves, including learning to reason strategically rather than rely on traditional algorithms. The participants also saw the need to redefine their conception of what it means to teach mathematics.