Students create paths on the floor, in paper mazes, and with Geo-Logo; use the Turtle Turner (protractor) to estimate and measure turns in multiples of 30 and draw triangles; and write procedures to design a face on the computer. Downloadable software. Suggested pacing: 17 sessions.

Turtle Paths (2-D Geometry); Grade Level 3 (Investigations in Number, Data, and Space; TREC)

Education & Teaching

Michael T. Battista

Douglas H. Clements

Joan Akers

Schools & Teaching

Scott,Foresman

Students develop spatial visualization skills. They use interlocking cubes to construct buildings from drawings; investigate the shapes of silhouettes projected by geometric solids; and draw silhouettes of cube buildings from different perspectives. Students also create their own toys out of 5,060 interlocking cubes and write instruction booklets for younger children. Suggested pacing: 14 sessions.

Seeing Solids and Silhouettes (3-D Geometry) (Investigations in Number, Data, and Space)

Students make coordinate mystery pictures and measure distances on a grid; play "sunken ships", identifying points on the grid and using distance feedback to select a next possible location; discuss properties of rectangles; and write rectangle procedures for the computer using Geo-Logo. Downloadable software. Suggested pacing: 15 sessions.

Sunken Ships and Grid Patterns: 2-D Geomentry (Investigations in Number, Data and Space, Grade 4)

Andee Rubin

Virginia Woolley

Through the investigations in this book, children reason about place value by making and using pedometers, understand related changes in two quantities by exploring parade formations of thirty-six marchers, and analyze data by devising rules for earning reading certificates. Students apply mathematical ideas from the five main content areas number, algebra, geometry, measurement, and data analysis. As they work, they infer, generalize, reason by analogy, recognize relationships, and make representations, and they also guess, check, and revise and verify their results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.

Navigating Through Problem Solving and Reasoning in Grade 3 (Principles and Standards for School Mathematics Navigations)

Business & Money

National Council Of Teachers Of Mathematics

Karol L. Yeatts

Sally Mayberry

Judith S. Zawojewski

Deniss R. Thompson

Management & Leadership

National Council of Teachers of Mathematics

This book features investigations in each of the five content areas of Principles and Standards for School Mathematics. In addition to using Eratosthenes' sieve to identify prime numbers, fourth graders evaluate algebraic relationships in contracts between theater owners and movie distributors. They consider geometric possibilities for rectangular puzzles, convert surprising real-world measures into equivalent but more comprehensible measures, and examine tables and graphs to extrapolate missing data. As they work, they infer, generalize, reason by analogy, recognize relationships, and make representations, and they also guess, check, and revise and verify their results. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.

Navigating Through Problem Solving and Reasoning in Grade 4 (Principles and Standards for School Mathematics Navigations)

Denisse R. Thompson

Problem solving and reasoning are students' processes of actively applying what they know to new problems from diverse areas of mathematics. Knowing a variety of mathematical ideas is insufficient unless students can draw on, integrate, and use them to solve challenging, complex problems. The Navigating through Problem Solving and Reasoning books for grades 3-6 present hands-on investigations that nurture reasoning and problem-solving strategies in each of the upper elementary grades. In investigations for grade 5, children reason about place value and divisibility by making a PIN for a security system, explore rates of change by projecting sales at a new pet shop, and determine the best month for a ski trip by analyzing snowfall data. Students apply mathematical ideas from the five main content areas-number, algebra, geometry, measurement, and data analysis. As they work, they infer, generalize, reason by analogy, recognize relationships, and make representations, and they also predict, check, and revise and verify their results. The Navigations Series translates Principles and Standards for School Mathematics into action. Each book includes practical, teacher-tested activities and a supplemental CD-ROM that features applets for students' use and resources for teachers' professional development.

Navigating Through Problem Solving and Reasoning in Grade 5 (Principles and Standards for School Mathematics Navigations)

Karol Yeatts

Peggy House

Denisse Rubilee Thompson

Judith Zawolewski

Bonnie Litwiller

Based on extensive research conducted by the authors, Reasoning and Sense Making in the Mathematics Classroom, Grades 3 5, is designed to help classroom teachers understand, monitor, and guide the development of students reasoning and sense making about core ideas in elementary school mathematics. It describes and illustrates the nature of these skills using classroom vignettes and actual student work in conjunction with instructional tasks and learning progressions to show how reasoning and sense making develop and how instruction can support students in that development. Students who can make sense of mathematical ideas can apply those ideas in problem solving, even in unfamiliar situations, and can use them as a foundation for future learning. Without them, students are reduced to rote learning, often experiencing frustration and failure. But what do reasoning and sense making during learning and teaching look like? Each chapter of Reasoning and Sense Making in the Mathematics Classroom, Grades 3 5 explores a different topic that children encounter in mathematics, demonstrating with actual student work and classroom dialogue how their mathematical knowledge and reasoning ability move through levels of sophistication or learning progressions: After opening with a discussion of the nature of reasoning and sense making and their critical importance in developing mathematical thinking, chapter 1 examines how students attempt to make sense of the concept of length measurement. Chapter 2 focuses on student strategies that exemplify conceptually sound reasoning and sense making in the context of multiplication word problems, and discusses how instruction can support students growth in this reasoning. The critical topic of properties of numbers that underlie reasoning about multiplication is also examined. Chapter 3 describes how students in grades 3 5 extend their understanding of number to include fractions and how they can build reasoning and sense making for fractions through explorations of different representations such as physical materials, pictures, and story contexts. Discussions on the nature of early algebraic reasoning, including research-based descriptions of this reasoning in children, classroom practices that can support this reasoning, and how algebra can be appropriately integrated in elementary mathematical content are provided in chapter 4. Chapter 5 discusses practices and processes connected to reasoning about geometric decomposition and structuring as applied to arrays of squares and cubes and to area and volume problems. A learning progression for the development of such reasoning is examined, and instructional practices that are consistent with this learning progression are considered. Not just a theoretical discussion, the book also provides specific suggestions for related instructional activities for each topic. Reasoning and Sense Making in the Mathematics Classroom, Grades 3 5 will be a valuable and practical addition to your professional library.

Reasoning and Sense Making in the Mathematics Classroom Grades: 3-5

Maria Blanton

Jae Meen Baek

Kathleen Cramer

Reasoning and Sense Making in the Mathematics Classroom, Grades 6-8, based on extensive research conducted by the authors, is designed to help classroom teachers understand, monitor, and guide the development of students' reasoning and sense making about core ideas in middle school mathematics. It describes and illustrates the nature of these skills using classroom vignettes and actual student work in conjunction with instructional tasks and learning progressions to show how instruction can support students in their development of these competencies.Students who can make sense of mathematical ideas can apply those ideas in problem solving, even in unfamiliar situations, and can use them as a foundation for future learning. Without this base of conceptual understanding, students are reduced to rote learning, often experiencing frustration and failure.But what do reasoning and sense making look like in learning and teaching?Each chapter of Reasoning and Sense Making in the Mathematics Classroom, Grades 6-8 explores a different topic that children encounter in mathematics, demonstrating with actual student work and classroom dialogue how their mathematical knowledge and reasoning ability move through "levels of sophistication," or learning progressions:After opening with a discussion of the nature of reasoning and sense making and their critical importance in developing mathematical thinking, chapter 1 examines how students attempt to make sense of the concepts of fractions and geometric properties of shapes.Chapter 2 discusses how reasoning about ratios and proportional relationships involves deep understanding of the multiplicative relationships embedded in the comparisons of two quantities.Chapter 3 focuses on what it means to call algebra a "style of mathematical thinking" and illustrates how students can view it as a reasoning and sense-making activity rather than as an isolated set of concepts to be memorized without understanding and quickly forgotten.Reasoning and sense making are inextricably linked in statistics and probability. Discussion and examples are used in chapter 4 to illustrate pedagogical practices that recognize and address students' development of statistical understanding, including some of the misunderstandings that students display along the way.Chapter 5 examines how students make sense of and reason about decomposing shapes, and discusses the mental processes underlying this reasoning in the context of area, surface area, and volume.Not just a theoretical treatise, the book provides specific suggestions for related instructional activities for each topic. Reasoning and Sense Making in the Mathematics Classroom, Grades 6-8 will be a valuable and practical addition to your professional library.

Reasoning and Sense Making in the Mathematics Classroom: Grades 6-8

National Council of Teachers of Mathematics,U.S.

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Books by Michael T. Battista