Contributing Authors Janis K. Drab Fackler, Ph. Editor Creative Director Sara Johnson Lee Aucoin Associate Editor Illustration Manager James Anderson Timothy J. Bradley Editorial Assistant Print Production Manager Kathryn R.
Kiley Don Tran Editor-in-Chief Interior Layout Designer/ Sharon Coan, M. Print Production Juan Chavolla Editorial Manager Gisela Lee, M. Mathematics Consultants Lori Barker Donna Erdman, M. Publisher Corinne Burton, M.
Learning Standards Copyright 2004 McREL.org/standards-benchmarks. Shell Education 5301 Oceanus Drive Huntington Beach, CA 92649-1030 http://www.com ISBN 978-1-4258-0249-3 © 2010 Shell Education Publishing, Inc. The classroom teacher may reproduce copies of materials in this book for classroom use only. The reproduction of any part for an entire school or school system is strictly prohibited.
No part of this publication may be transmitted, stored, or recorded in any form without written permission from the publisher. 2 SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education Table of Contents Introduction Using Counters for Solving Linear Equations .5 Using Linking Cubes for Data Analysis .7 Using Linking Cubes for Understanding How to Use This Book .20 Area and Perimeter.95 Correlation to Standards .22 Using Linking Cubes for Understanding Probability.98 About the Authors .24 Using Base Ten Blocks for Addition .101 Using Base Ten Blocks for Division .104 Strategies for Vocabulary Using Algebra Tiles for Collecting Development Like Terms .107 Vocabulary Development Overview .27 Using Pattern Blocks for Spatial Strategies Visualization.111 Alike and Different .30 Using Pattern Blocks for Transformations in Quadrant I .114 Total Physical Response .36 Using Pattern Blocks for Rotational Math Hunt .119 Root Word Tree.53 Strategies for Teaching Procedures Vocabulary Flip Book .58 Teaching Procedures Overview .64 Teaching Number Sense Procedures Alternative Algorithm for Addition .136 Strategies for Using Manipulatives Alternative Algorithm for Manipulatives Overview .139 Common Questions About Alternative Algorithm for Manipulatives .142 Using Manipulatives Effectively .73 Alternative Algorithm for Division .145 Strategies for Acquiring Manipulatives .148 Strategies for Organizing Teaching Algebra Procedures Manipulatives .151 Strategies for Individual Use of Collecting Like Terms .157 Strategies for Free Exploration of Manipulatives .77 Teaching Geometry Procedures Strategies for Modeling While Using Calculating Area .163 Strategies for Grouping Students Calculating Volume .166 While Using Manipulatives .79 Finding Missing Angle Measures .169 The Manipulative Survival Guide .81 Teaching Measurement Procedures Preventing Manipulative Dependency .172 Sample Lessons Converting Measurements .175 Using Counters for Skip Counting Teaching Data Analysis Procedures by 2s and 5s .178 Using Counters for Multiplication Finding Central Tendencies .86 © Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics 3 Table of Contents (cont.) Strategies for Understanding Strategies for Assessing Problem Solving Mathematical Thinking Problem-Solving Overview .265 Problem-Solving Difficulty Factors .188 Strategies Teaching About Difficulty Factors .268 Problem-Solving Process .272 12 Strategies for Problem Solving .275 Strategies Self- and Peer-Evaluations .282 Acting It Out or Using Mathematical Journals .292 Looking for a Pattern .210 Guessing and Checking .214 Appendices Creating an Organized List .217 Appendix A: References Cited .220 Appendix B: Mathematics-Related Creating a Tree Diagram .223 Children’s Literature List .300 Using Simpler Numbers .226 Appendix C: Grade-Level Using Logical Reasoning .305 Analyzing and Investigating .232 Appendix D: Answer Key .311 Solving Open-Ended Problems .235 Appendix E: Contents of Teacher Resource CD .317 Strategies for Using Mathematical Games Games Overview .241 Strategies Concept-Based Games .243 Who Has? Games .245 Category Quiz Games .251 Game Rules and Templates .253 4 SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education Foreword Earlene J. Curriculum Developer Detroit, Michigan “Tell me, and I’ll forget. Show me, and I may not remember.
Involve me, and I will understand.” —Native American proverb Traveling throughout the country working with groups of teachers, my typical road trip often begins with the address of my ending destination. Sitting down with a road map to chart the route, I realize there are circumstances that will directly impact my arrival at the destination. This process mirrors the challenges of teachers when planning instruction. Each state has developed grade-level standards, which are used to formulate grade-level content assessments.
These objectives define the content of which students must demonstrate understanding. Having an expected outcome defined does not adequately prepare you for the journey to the destination. And although state assessments are not the curriculum, they certainly impact the design of instruction. The assessments require students to demonstrate their understanding by skillfully processing the mathematics content.
Students are bombarded with numerous questions from various content strands in which they must demonstrate • conceptual understanding of mathematics content through modeling or interpretation of representations, • computational fluency, • and problem solving through application of the content. As teachers, we navigate our students through the maze of learning objectives in search of understanding—understanding that is measurable on state assessments. This can be challenging considering many of the circumstances that we encounter when planning instruction that is aligned to these standards. For example, our students are compilations of various skill levels and abilities.
Each student represents a unique set of learning styles and prior experiences. Our challenge is in making the learning accessible for all students. The design of our daily instructional plan must be inclusive of strategies that develop student understanding. Teachers often search for the answers to questions such as, “What does it mean for students to understand,” and “How do we support the development of their understanding?” Most educators can agree that understanding is being able to carry out a variety of actions that demonstrate one’s knowledge of a topic, and apply it in new ways.
Survey responses by teachers also reflect a need for instructional resources for planning lessons. © Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics 5 Foreword (cont.) Strategies for Teaching Mathematics is a toolkit that provides support much like that of a navigation system in your car for a trip as opposed to a traditional road map. An instructional resource for K–8 teachers, this resource provides step-by-step, research-based strategies that represent a balanced approach to learning mathematics. This is a resource that provides comprehensive instructional plans that can be used as “tools” for intervention and the development of critical supporting skills.
Sample lessons provide a sequential guide in developing student proficiency in the following: • mathematical vocabulary • conceptual understanding through the use of manipulatives • procedural proficiency • problem-solving strategies • mathematical games • assessing mathematical thinking Each section establishes key instructional connections. Additionally, differentiated activities ensure adaptability for students of all ability levels. The strategies are structured in a format that provides teachers with the following tools: • an alignment to mathematical standards • vocabulary terms • elementary and secondary applications • step-by-step procedures • differentiated instruction • reproducible student pages Mathematics is constantly developing and becoming ever more specialized. It is a challenge to effectively implement a curriculum that yields understanding for the wide range of student abilities represented in our classrooms today.
As we move forward with our quest to increase student understanding, the learning experiences we provide must actively engage student thinking in the kind of rigorous, relevant learning experiences that are provided in this resource. 6 SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education Strategies for Teaching Mathematics Introduction Research Mathematics is one of the most feared subjects in school, yet it is a subject students will need for the rest of their lives. Students often struggle with learning mathematics, and teachers have long sought more effective methods for teaching it. In their landmark book, Classroom Instruction that Works, Robert Marzano, Debra Pickering, and Jane Pollock (2001) note that teaching has become more a science than an art.
Research in the last two decades has helped educators develop a common understanding of effective instruction. This is especially evident in teaching mathematics. Strategies for Teaching Mathematics includes proven approaches to teaching mathematics at all levels. Strategies for Teaching Mathematics provides educators with background information on effective instructional strategies with sample lesson plans and student reproducibles.
These materials support students in truly understanding mathematical concepts, rather than just memorizing procedures. This kind of deep conceptual understanding has never been more important than it is today. Numerous educational leaders have stressed the importance of mastering 21st-century skills for today’s students. Frank Levy and Richard Murnane, researchers at MIT and Harvard, found that the changing workplace has strong implications for students (2005).
Computerization and globalization in today’s world means that students need to be problem solvers who can think critically. No longer can students graduate from high school and expect to succeed without these qualities. Levy and Murnane note that occupations requiring higher-level problem solving have seen dramatic increases in average salary in the past 30 years, while more traditional blue-collar positions have seen even greater drops in average pay. Even more importantly, these blue-collar positions require more complex skills than similar positions of the past.
Clearly, 21st-century skills need to be mastered by today’s students. So what are 21st-century skills, and what do they look like in mathematics? Students today need to understand both the procedural and conceptual foundations of mathematics. They also need to understand the “language” of mathematics. Mathematics is full of important vocabulary that has specific meanings in this context.
Finally, students need to solve complex problems by making connections between prior learning and new situations. The need for students to learn these skills also means that teachers need to use new ways of teaching. Manipulatives are effective teaching tools that can be used throughout the K–12 curriculum to help students understand key mathematical concepts. New strategies for teaching essential mathematical procedures are also necessary for helping students identify real-world connections to mathematics.
Finally, new methods of assessing student understanding help teachers form a stronger picture of students’ skills and tailor instruction to their needs. © Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics 7 Strategies for Teaching Mathematics Introduction Research (cont.) The Importance of Teaching Mathematics in a Balanced Approach As early as 1989, educators identified the need for a balanced approach to teaching mathematics (Porter 1989). International comparisons of mathematics achievement were often made lamenting the performance of American students (Stigler and Hiebert 1997). As a result, the final decade of the 20th century was spent focusing on teaching problem-solving skills in addition to basic computation.
However, new research has led to a revision of these earlier recommendations. The Mathematics Advisory Panel (U. Department of Education 2008), convened by the federal government, has reviewed the literature on mathematics instruction and sought the advice of key mathematics researchers. A representative study completed by Thomas Good (2008) at the University of Arizona revealed that the current mathematics curriculum in schools is too full, leading to a lack of depth of instruction and failure of students to master important concepts.
Consequently, the Mathematics Advisory Panel recommended “the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic (i., quick and effortless) recall of facts” (U. Department of Education 2008). This is different than earlier practices, because it focuses on all three areas of mathematics instruction without excluding any. Balanced instruction includes more than just procedural and conceptual fluency.
Instruction needs to be balanced to meet the needs of diverse learners as well. Phillip Schlechty (2002), in his book Working on the Work, suggests, “the key to school success is to be found in identifying or creating engaging schoolwork for students.” Students become engaged in learning when they are taught using methods that motivate them. One way to differentiate instruction for learners is to teach with multiple intelligences in mind. Howard Gardner’s (1983) groundbreaking work has helped teachers engage students who do not learn through traditional auditory methods.
Strategies included in this resource such as using mathematical games and developing vocabulary meet the needs of kinesthetic and linguistic learners. Specific student activities such as Vocabulary Flip Books also engage artistic learners. Balancing mathematics instruction supports all learners in the classroom. Differentiating Mathematics As mathematical concepts are introduced, students learn them at different rates.
Additionally, students bring different skill sets, learning styles, and prior knowledge with them to the classroom, which also cause differences in their learning.