Mathematics for Young Children Douglas H. Clements, State University of New York at Buffalo Douglas H. Clements is Professor of Mathematics, Early Childhood, and Computer Education at the State University of New York at Buffalo. He has published over 60 research studies, four books, 25 chapters, and 160 additional publications. Through a National Science Foundation (NSF) grant, he co-developed an elementary geometry curriculum, Logo Geometry. He collaborated with colleagues in creating a K-5 mathematics curriculum, Investigations in Number, Data, and Space, which includes a software package that received the 1995 Technology and Learning Software of the Year award. Clements is an author and consultant to a number of publishers, including Computer Curriculum Corp., Steck-Vaughn, and Macmillan-McGraw-Hill. He is editor and author of the National Council of Teachers of Mathematics’ (NCTM) Addenda (to the Standards) Materials and was Chair of the Editorial Panel of NCTM’s research journal, Journal for Research in Mathematics Education. Clements was a member of NCTM’s Standards 2000 PreK-2 Writing Group. He also spent six years teaching preschool and kindergarten. "We need to provide kids with a deeper knowledge of broader topics related to the concepts of numbers and geometry." — Doug Clements Presentation Highlights Preschoolers can and should engage in mathematical thinking. All young children possess informal mathematical skills and can learn more. This presentation discusses strategies Head Start professionals can use to develop children's mathematical abilities, and provides an overview of major recommendations from the Conference on Standards for Preschool and Kindergarten Mathematics Education. Strategies for Building Children’s Mathematical Skills To develop children’s mathematical power, successful teachers of young children: Build on children’s everyday activities, including their cultural background, language, and mathematical ideas and strategies; Provide a variety of instructional strategies, meaningful child-related contexts, and opportunities for active participation to help children learn pre-mathematical and mathematical ideas and develop positive beliefs about mathematics; and Ensure appropriate and ongoing use of technology. Mathematics as Sense-Making Teachers should build upon and extend the use of mathematics in children’s daily activities and interests. They should attempt to see children’s points of view and use their interpretations to plan their interactions with children and their curriculum. This approach—mathematics as sense-making—ensures that mathematic content will be meaningful for young children. Mathematics should be planned and experienced every day. A combination of providing an environment conducive to mathematical explorations, appropriate observations and interventions, and specific mathematical activities helps preschoolers build informal, everyday, pre-mathematical knowledge and connect it to explicit mathematical knowledge. This knowledge should be based on the "big ideas" of mathematics. The notion of "big" implies both the broader topics of mathematics (e.g. geometry, measurement) and a deeper understanding of all topics. Mathematical Standards in Early Education There is a substantial and critical difference between standards as a vision of excellence and standards as rigid requirements for mastery. Only the former is appropriate for early childhood mathematics education. Knowledge of what children are capable of learning and doing at each age, as well as specific learning goals, are necessary for teachers to realize a vision of quality early childhood education. Major Recommendations from the Conference on Standards The following key points are presented in the final report for the Conference on Standards for Preschool and Kindergarten Mathematics Education: Very young children have the motivation and ability to engage in pre-mathematical and mathematical thinking. This thinking can be fostered through child-centered play activities and other appropriate activities introduced by adults. Equity is a major concern in mathematics education for young children. Children from different sociocultural backgrounds have different levels of preparedness for school mathematics. The most important standards for early childhood are those for programs and for teaching. These are built on flexible, developmental guidelines for young children’s mathematical learning that focus on the big ideas of mathematics. Mathematics for young children should be an integrated whole. Connections—between topics, between mathematics and other subjects, and between mathematics and everyday life—should permeate children’s mathematical experiences. Mathematical processes and habits of mind are as important as mathematical content. Processes include problem solving, reasoning and proof, communication, making connections, and representation. Habits of mind include curiosity, imagination, inventiveness, persistence, willingness to experiment, and sensitivity to patterns. Curriculum and Teaching The following points regarding the teaching of mathematics and the development of mathematics curriculum are important for all Head Start professionals to consider: Young children’s mathematics skills are usually not acquired by sitting down in a group lesson, but brought forth by the teacher through the children’s own self-directed, intrinsically-motivated activities. Teachers should build upon and extend mathematics in children’s daily activities, interests, and questions. Children benefit from a variety of instructional approaches. Successful preschool teachers build on children’s informal knowledge and everyday activities, considering children’s cultural background, language, and mathematical ideas and strategies. Children benefit from the use of appropriate and ongoing technology. Especially useful are computer tools that enrich and extend mathematical experiences. Teachers should attempt to see and understand children’s mathematical ideas and strategies and use their interpretations to plan their interactions with children and their curriculum. Teachers should help children develop strong concepts to provide an anchor for skills. Skill learning should build directly on children’s strategies, with teachers encouraging children to create and describe their own solutions. Interview tasks and ongoing observational forms of assessment are more valid, reliable, and predictable than standardized tests in assessing young children’s mathematical learning. Teachers should ask children to record their experiences in increasingly varied and abstract representations, both for learning and for assessment purposes. Curriculum designers and teachers can assume that all children have informal mathematics on which they can build. Not only should all children be exposed to challenging mathematics, but poor children and those from groups underrepresented in mathematics should be provided more support, if necessary, to connect their informal mathematical activity to school mathematics and thus be on par with other children by first grade. Children who experience high-quality mathematics education from their earliest years have a better chance of learning and performing successfully than those who do not. The Head Start community should support and insist on curricula that are developed and tested scientifically. Implementation and Professional Development Professional development demands considerable attention, given the diversity of teacher populations. It demands multiple strategies, including university courses, on-site assistance, satellite downlinks, and distance education. Professional development educators will need to learn new mathematics, new ideas about children’s thinking and learning, new curricula, and new forms of teaching. Deep knowledge of the mathematics to be taught is critical to improving. An understanding of how children’s mathematical thinking and knowledge develop is often a neglected component of reform and professional development efforts. Resources The following curricula and Web sites provide more information about items discussed in the presentation. Building Blocks. Building Blocks, funded by the National Science Foundation (NSF) is a new pre-school—2nd grade software-based mathematics curriculum. It is the first to comprehensively address the 2000 Standards of the National Council of Teachers of Mathematics, Principles and Standards for School Mathematics. Building Blocks’ approach is to find the mathematics in, and develop mathematics from, children’s everyday activities. Activities are based on children’s experiences and interests, with an emphasis on supporting the development of mathematical activity. www.gse.buffalo.edu/org/buildingblocks/ Conference on Standards. The Conference on Standards for Preschool and Kindergarten Mathematics Education, funded by the Exxon and Mobil Foundation, was held to facilitate early communication between, and coordination of efforts among, the educational leaders and agencies who are developing standards and curricula for young children. This was an historic event—the first conference ever to have brought together such a comprehensive range of experts in the diverse fields relevant to the creation of educational standards. The expertise brought to and produced by that conference has resulted in the creation of a book. The Web site provides information about the publication of this book. www.gse.buffalo.edu/org/conference/ Planning for Professional Development in Preschool Mathematics: Meeting the Challenge of Standards 2000. This project, directed by Julie Sarama, includes a massive survey and, based on expert opinion, research, and the results of that survey, the development of a plan for teacher enhancement and professional development for preschool mathematics and computer education. www.gse.buffalo.edu/org/profdev/ References The following references provide more information about research and ideas discussed in the presentation. Clements, D. H. (1997). (Mis?)Constructing constructivism. Teaching Children Mathematics, 4, 198-200. Clements, D. H. (1999). Playing math with young children. Curriculum Administrator, 35(4), 25-28. Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400-405. Clements, D. H. (1999). Young children and technology. In G. D. Nelson (Ed.), Dialogue on early childhood science, mathematics, and technology education (pp. 92-105). Clements, D. H. (1999, October). The geometric world of young children. Early Childhood Today, 34-43. Clements, D. H., & Battista, M. T. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), 34-35. Clements, D. H., & McMillen, S. (1996). Rethinking "concrete" manipulatives. Teaching Children Mathematics, 2(5), 270-279. Clements, D. H., & Sarama, J. (2000). Standards for preschoolers. Teaching Children Mathematics, 7(1), 38-41. Clements, D. H., & Sarama, J. (2000). The earliest geometry. Teaching Children Mathematics, 7(2), 82-86. Clements, D. H., & Swaminathan, S. (1995). Technology and school change: New lamps for old? Childhood Education, 71, 275-281. Kamii, C. K., & Dominick, A. (1998). The harmful effects of algorithms in grades 1-4. In L. J. Morrow & M. J. Kenney (Ed.), The teaching and learning of algorithms in school mathematics (pp. 130-140). Reston, VA: National Council of Teachers of Mathematics. 1  Time to prepare this material was partially provided by three National Science Foundation (NSF) Research Grants, ESI-9730804, "Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development," ESI-9814218: "Planning for Professional Development in Pre-School Mathematics: Meeting the Challenge of Standards 2000," and ESI-98-17540: "Conference on Standards for Preschool and Kindergarten Mathematics Education"; the Conference was co-sponsored and funded by the ExxonMobil Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the NSF or the ExxonMobil Foundation. 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