What Successful Math Teachers Do, Grades PreK-5

Edward S. Wall is assistant professor of elementary mathematics education at the City College of the City University of New York. In 1968 he received his MA in mathematics at the University of Maryland and, after a number of years working as an applied mathematician, received a Sloan Foundation Fellowship for the purposes of pursuing a PhD in mathematical biology at the University of Chicago. Along the way, he became intrigued by the very notion of teaching mathematics and, after several years, found himself deeply involved in full-time K-12 mathematics teaching. In 2003 he received a PhD in mathematics education at the University of Michigan and subsequently joined the faculty at City College. He still finds himself intrigued by the very notion of teaching mathematics and his research reflects that fascination. Alfred S. Posamentier is professor of mathematics education and dean of the School of Education at the City College of the City University of New York. He has authored and co-authored several resource books in mathematics education for Corwin Press.

Prologue Acknowledgments About the Authors 1. Numbers and Operations Grades PreK-2 1. Encourage young children's exploration and understanding of relationships among numbers. 2. Encourage young children's understanding of addition and subtraction and how they relate to each other. 3. Encourage young children's fluent computation. Grades 3-5 4. Encourage an understanding of the structure of numbers and relationships among numbers. 5. Encourage an understanding of the meanings of multiplication and division and how they relate to each other. 6. Encourage students to compute fluently and make reasonable estimates. 2. Algebra Grades PreK-2 7. Encourage young children's systematic sorting and classification as they work with a variety of patterns, geometric shapes, and data. 8. Encourage young children's systematic exploration of the general principles and properties of operations such as addition and subtraction. Grades 3-5 9. Encourage the expression and generalization of mathematics relationships. 10. Encourage further understanding of multiplicative structures through application and analysis of the distributivity of multiplication over addition. 3. Geometry Grades PreK-2 11. Encourage young children's exploration of characteristics and properties of two- and three-dimensional shapes. 12. Encourage young children's application of similarity transformations to analyze geometric situations. 13. Encourage young children's use of visualization and spatial reasoning in their exploration of geometric shapes. Grades 3-5 14. Encourage students' exploration and mathematical analysis of the properties of two-dimensional geometric shapes. 15. Encourage students' use of geometric transformations to analyze mathematical situations. 16. Encourage students' use of visualization, spatial reasoning, and geometric modeling to solve problems. 4. Measurement Grades PreK-2 17. Encourage young children's exploration and understanding of measurement concepts and relationships. 18. Encourage young children to accurately apply appropriate tools and techniques to linear measurement. Grades 3-5 19. Encourage students' exploration and understanding of measurement concepts and relationships. 20. Encourage students' appropriate application of conventional measuring tools in varied situations. 5. Data Analysis and Probability Grades PreK-2 21. Encourage young children's collection, display, and organization of objects and data. 22. Encourage the idea in young children that data, graphs, and charts give information. 23. Encourage young children's informal explorations of probability. Grades 3-5 24. Encourage students to explore questions they find personally relevant and that can be addressed by data collection and analysis. 25. Encourage students to become more precise in their mathematical descriptions of data. 26. Encourage students to explore and evaluate issues of representativeness and inference. 27. Encourage students' exploration and quantification of simple probabilistic events. 6. Problem Solving Grades PreK-2 28. Encourage young children to ask mathematical questions and to identify essential mathematical information. 29. Assess young children's abilities to solve problems through examination of student work and conversations. Grades 3-5 30. Encourage students' development and application of problem-solving strategies. 31. Select rich, appropriate, and challenging problems and orchestrate their use. 7. Reasoning and Proof Grades PreK-2 32. Encourage young children to explain their thinking by stating their reasons. 33. Ask questions that encourage young children to make conjectures and to justify their thinking. Grades 3-5 34. Encourage children to reason about the relationships that apply to the numbers, operations, or shapes that they are studying. 35. Focus on general mathematical structures and relationships. 8. Communication Grades PreK-2 36. Encourage young children's verbal and written communication of mathematics concepts and ideas. 37. Expect young children to explain their thinking and give them opportunities to talk with and listen to their peers. Grades 3-5 38. Encourage students to share their thinking, to ask questions, and to justify their ideas. 39. Provide models for student dialogue about mathematics. 9. Connections Grades PreK-2 40. Encourage young children to make connections among mathematical ideas, vocabulary, and representations. 41. Make links between routine school activities and mathematics. Grades 3-5 42. Encourage students to see that mathematics is a web of closely connected ideas. 43. Select tasks that help students explore and develop increasingly sophisticated mathematical ideas. 10. Representation Grades PreK-2 44. Encourage young children to represent their mathematical ideas and procedures in varied ways. 45. Create a learning environment that supports and encourages children's use of multiple representations. Grades 3-5 46. Encourage students to use representations to support, clarify, and extend their mathematical ideas. 47. Choose tasks that embody rich and varied representational structures. Epilogue Index

"This is an excellent book. If every elementary teacher does not have one, it should at least be available in the school's professional library." -- Anne Giddings, Assistant Superintendent "The book encapsulates the standards well and will make a good resource for new teachers who might not necessarily have the time to read the entire NCTM standards book." -- Carol Amos, Teacher Leader/Mathematics Coordinator "The research and vignettes that follow each strategy clearly support why the strategy is important and how it can be effective. Teachers who read this, reflect on the content, and discuss it with other teachers will be on track to implement the strategies in their classrooms." -- Trish Guinee, Mathematics Coordinator "The book is very easy to follow, with practical, research based strategies for the teacher to use. It also provides me some insight to better remediate students that are struggling." -- Allen Stevens, Math/Science Teacher "The 'Precautions and Pitfalls" section is such a welcome feature! This is a powerful book for beginning teachers, or seasoned teachers who want to improve their practice to ensure student learning." -- Rhonda Naylor, Math Teacher/Coordinator "In this book, you'll find out how to build on students' enthusiasm through research-based teaching strategies, vignettes, and supporting discussions, all based on NCTM standards." -- Learning Magazine, October 2006 "The authors methodically define a strategy that meets a NCTM goal, define the NCTM goal specifically, state the research that supports the goal, and provide a compilation of stories that depict one teacher's pedagogy that meets the goal." -- MAA Reviews, December 2006