Thesis (Ed.M.)--Boston University / The primary purpose of constructing a test on informal geometry is to provide the eighth-grade mathematics teacher with an objective means of measuring the achievement of his students in geometry. A secondary purpose is to explore the possibility that this test alone, or in conjunction with other criteria, might have value in predicting the degree of success of a student in demonstrative geometry.

Early childhood education emphasizes the need of providing high quality early childhood mathematics programs for preschool children. However, there is little research that examines the importance of preschool children’s mathematical knowledge development and teachers’ beliefs about how to teach mathematics to young children. The purposes of this study were to investigate pre-service and in-service preschool teachers’ knowledge of children’s mathematical development and their beliefs about teaching mathematics in the preschool classroom and also to determine how experience differentiates the two groups. This research employed a non-experimental research design with convenient sampling. Ninety-eight pre-service teachers and seventy-seven in-service preschool teachers participated in the research. The Knowledge of Mathematical Development survey (KMD) and the Beliefs survey were used to investigate possible differences between pre-service and in-service preschool teachers’ knowledge of children’s mathematical development and between their beliefs about teaching mathematics. The findings of this study indicate a statistically significant difference between pre-service teachers and in-service preschool teachers in relation to their knowledge of mathematical development. This finding shows that pre-service teachers’ knowledge of children’s mathematical development is somewhat limited; most pre-service teachers have difficulty identifying the process of preschool children’s development of mathematics skills. A second finding reveals a statistically significant difference between pre-service teachers and in-service preschool teachers in relation to their beliefs about (a) age-appropriateness of mathematics instruction in the early childhood classroom, (b) social and emotional versus mathematical development as a primary goal of the preschool curriculum, and (c) teacher comfort with mathematics instruction. No statistically significant difference was found between pre-service teachers’ and in-service preschool teachers’ beliefs regarding the locus of generation of mathematical knowledge. Both groups believe it is the teacher’s responsibility to intentionally teach mathematics to young children. This result suggests that both pre-service and in-service preschool teachers believe that teachers should play a central role in the teaching of mathematics to preschool children. However, both groups would need appropriate education and training to learn how to teach mathematics to young children. Pre-service and in-service preschool teachers’ varying levels of experiences and different levels of education may help explain why there is a significant difference between their knowledge of mathematical development and beliefs about teaching mathematics.