A new approach to teaching and curriculum takes seriously the knowledge children have when they enter school. Teachers use the knowledge each child has to make instructional decisions so that the child learns mathematics with understanding, how to solve problems, and the computational skills. Research concerning the problem-solving strategies actually used by children has led to the development of the Cognitively Guided Instruction (CGI) project, in which the use of such knowledge has been studied. Forty first-grade teachers in Madison (Wisconsin) were randomly assigned to treatment or control groups. The 20 experimental group members received extensive training in children's solution strategies during a training workshop in the summer of 1986, but were allowed to plan for themselves how they would use the knowledge. The other 20 teachers served as a comparison/control group in 1986 and took part in a similar workshop in the summer of 1987. Observations after one year showed that experimental group teachers adapted CGI ideas according to their own styles. However, the following three key elements were recognized: (1) multiple solution strategies were recognized and encouraged; (2) there was a focus on problem solving; and (3) teachers had an expansive view of the children's knowledge and thinking. When teachers know about children's mathematical thinking and problem solving, they can facilitate the development of mathematical abilities for children from disadvantaged backgrounds. Two tables, one graph, and a 24-item list of references are included. The paper's discussant is Judith Johnson Richards in a training section entitled "Appreciating Children's Mathematical Knowledge and Thinking in Ethnically, Linguistically, and Economically Diverse Classrooms". (SLD)