Thinking in Arithmetic.

This paper discusses a radically different set of assumptions to improve educational outcomes for disadvantaged students. It is argued that disadvantaged children, when exposed to carefully organized thinking-oriented instruction, can acquire the traditional basic skills in the process of reasoning and solving problems. The paper is presented in three sections. The first discusses the intuitive basis for early mathematical reasoning, describing the reasoning about amounts and sizes of material that preschool children do without measurement or exact numerical quantification as "protoquantitive" reasoning. The integration of protoquantitative schema and counting, the first step in making quantitative judgements, is discussed. The second section discusses six principles for a reason-based arithmetic program: (1) develop children's trust in their own knowledge; (2) draw children's informal knowledge, developed outside school, into the classroom; (3) use formal notations as a public record of discussions and conclusions; (4) introduce key mathematical structures as quickly as possible; (5) encourage everyday problem finding; and (6) talk about mathematics, don't just do arithmetic. The final section presents the results of the program. Two cohorts of first- and second-graders were evaluated by the California Achievement Test (CAT) on reading and mathematics achievement with an experimental group and a control group at each level. The intervention group in mathematics rose from about the 25th percentile to the 70th percentile and maintained that level into the second year of the program. The paper concludes that an interpretation- and discussion-oriented mathematics program is an effective instructional approach, suitable for children not socially favored, and provides mathematics classroom activities that exercise reasoning skills. (MDH)