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. 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, 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
The authors describe a project that has gone further than most in developing the new knowledge, habits of mind, and individual and social resources that will be needed for reform to persist and prosper. It has been neither easy nor fast - but significant learning rarely is. Few professional development projects would commit themselves to such an ambitious agenda as reforming mathematics education, promoting equity, and developing teacher leadership, for any one of these objectives alone presents a daunting challenge. Learning mathematics is threatening to most teachers, especially elementary teachers whose limited experiences with mathematics have often been anxiety-provoking and uninspiring. Creating new understandings of equity brings out deep affective responses as participants attempt to make sense of emotionally charged issues. Developing leadership requires teachers to learn new skills and abilities while taking on unfamiliar roles and responsibilities. It can also provoke feelings of uncertainty and ambivalence. Almost four years ago a far-sighted mathematics professor whom we will call Charles(1) started the Mathematics Education, Equity, and Leadership (MEEL) project with the intention of meeting all three of these goals simultaneously. But he had not fully envisioned the complexities and challenges inherent in such an effort. This article is about those challenges, about the project that took them on, and about some of the people who faced them together. The Project and Its Leaders Charles has been working for many years to develop a cadre of knowledgeable teacher leaders who will foster change in their schools through their own teaching and through their interaction with others. He grew up in New York City as a Jew attending a school that enrolled mostly Christians. There he experienced firsthand the oppressive effects of inequitable treatment by a majority group. His thinking has been greatly influenced by learning about the Holocaust and by the civil rights movement of the 1960s. In the summer of 1992 Charles started MEEL, which was designed to develop teacher leaders who would work particularly on developing equity in mathematics education. Building on his many years as a director of and participant in a local mathematics project, Charles launched MEEL with nine elementary teachers. By drawing on networks of bilingual teachers, the project has grown from nine teachers to more than 30.(2) Every summer the project holds a two-week institute that includes sessions on mathematics, equity, and leadership. Throughout the academic year, participants convene in two-day retreats and monthly meetings, and ongoing networking and support groups are provided by project leaders. Charles believes that significant educational change entails learning on the part of everyone involved in reform, including himself. Maria, the Latina teacher whom he chose to lead the project with him, has greatly influenced his learning. Maria's knowledge and understanding were partially forged from her everyday experiences as a minority female in a white, male-dominated world and from her extensive classroom experiences working with students of color, who often came from economically disadvantaged situations and spoke little English. In addition, Maria gained understanding from previous experience as a leader. Maria grew up in a Mexican American community in California that consisted of two distinct cultures: Mexican American and white. She learned English because her parents didn't want her to struggle against the sometimes oppressive language barriers in school; she learned Spanish because it was the language of her community. Maria married a man who was half Native American and half Mexican American, and she lived with him on his reservation in Montana. There she worked as a teacher's aide while pursuing a teaching degree, and later she taught first grade. Maria's adopted son is American Indian and has sometimes struggled with mathematics in school. …
For a number of years, re searchers have studied the relationship between teacher behaviors (process) and student achievement (product) with the hope of determining what teacher behaviors will lead to increases in student achieve ment and attitude. At last, this research has borne fruit. Several reviewers of process-product research have recently concluded that effective teaching is characterized by a pattern of teaching behaviors that they have called (See, for ex ample: Gage, 1978; Good, 1979; Medley, 1979; Rosenshine, 1979.) According to Barak Rosenshine (1979), direct in struction has the following characteristics: an aca demic focus; a teacher-centered focus; little student choice of activity; use of large groups rather than small groups for instruction; and use of factual ques tions and controlled practice in instruction. Thomas Good (1979) describes direct instruction as active teaching:
