Using Children's Mathematical Knowledge.

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-J.tem 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) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. *********************************************************************** USING CHILDREN'S MATHEMATICAL KNOWLEDGE Penelope L. Peterson Michigan State University Elizabeth Fennema Thomas Carpenter University of Wisconsin, Madison The research reported in this paper was funded in part by a grant to Elizabeth Fennema, Thomas Carpenter, and Penelope Peterson from the National Science Foundation (Grant No. MDR-8550263). The opinions expressed do not necessarily reflect the position, policy, or endorsement of the National Science Foundation. BEST COPY AVAILABLE '2 U.S. DEPARTMENT OF EDUCATION Office of Educational Rearterch and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) *lot document has ben reproduced as received from die demon of Organization originating it CI Minor changes have been made to improve reproduction quality Points ol view or opinions stated in tnis document do not nesearily represent official OEM position or policy USING CHILDREN'S MATHEMATICAL KNOWLEDGE Marla has 4 peanuts. Her mother gave her some more. Now she has 11 peanuts. How many peanuts did her mother give her? Most firstand second-grade teachers, and probably most adults, see the above problem as a subtraction problem that will be difficult tor young children to solve. However, consider what Elissa (a four-year-old) did when asked to solve this problem. First, Elissa counted out four counters. Then she added more counters until she had 11. With her hand she separated out the original four counters, then she pointed to the group she had lett and said to the interviewer, "This many. " The interviewer asked her, "How inany is that? ' Elissa counted, "One, two, three, four, five, six, seven. Turning to the interviewer, Elissa announced firmly, "Seven peanuts! " Elissa did what most young children do. She invented a way to solve the problem that was based on how she thought about the problem, not on any procedure that had been taught to her. She recognized that the problem involved joining some things together, and she did that. Elissa is not unusual. In fact, we have learned from research that all children come to school knowing a great deal about mathematics. If adults take children's mathematical knowledge seriously, they can help children use their knowledge to solve problems and learn more mathematics. Aduits have not alwaws taken children's knowledge seriously. Typically, parents and teachers have assumed ,hat children begin school with little or no knowledge of mathematics. This assumption was not unreasonable when the primary goal of the elementary mathematics curriculum was to develop skill in computation (e.g., to learn the basic facts and the algorithms of addition and subtraction). Children did not come to school with much knowledge of formal algorithms, so it made sense to assume that children did not have much mathematical knowledge. Although most educators knew that computational skills were not sufficient, they presumed that before children could understand the algorithms and use them to solve problems, children needed to have mastered computational skills. Thus, primary school instruction has focused on the practice of these skills to attain mastery. This emphasis is even more pronounced in th instruction of children in less advantaged socioeconomic areas, who spend more time in computational tasks than children in schools with more resources (Zucker, 1990). The tacit assumption is that once children have learned to compute with a reasonable level of facility, they can be taught to understand why the various procedures work and to apply the procedures to solve problems. Findings from the National Assessment of Educational Progress and other research programs have documented, however, that this heavy emphasis on computation has been misplaced (Dossey, Mullis, Lindquist, & Chambers, 1988). Although children in the