Strategies for teaching mathematics

321 Pages·2010·3.897 MB·English

by  Deborah V Mink, Janis K Drab Fackler, Linda H Pickett, Shell Education

Author
Deborah V. Mink, Ph. D.
Foreword
Earlene J. Hall, Ed. D.
Author
Deborah V. Mink, Ph. D.
Contributing Authors
Janis K. Drab Fackler, Ph. D.
Linda H. Pickett, Ph. D.
Foreword
Earlene J. Hall, Ed. D.
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.S.Ed. Print Production
Juan Chavolla
Editorial Manager
Gisela Lee, M.A. Mathematics Consultants
Lori Barker
Donna Erdman, M.Ed.
Publisher
Corinne Burton, M.A.Ed.
Learning Standards
Copyright 2004 McREL. www.mcrel.org/standards-benchmarks.
Shell Education
5301 Oceanus Drive
Huntington Beach, CA 92649-1030
http://www.shelleducation.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.
22
SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education
Table of Contents
Introduction Using Counters for Solving Linear
Equations .................................................89
Foreword ...............................................................5
Using Linking Cubes for Data Analysis ....92
Research ................................................................7
Using Linking Cubes for Understanding
How to Use This Book .......................................20 Area and Perimeter .................................95
Using Linking Cubes for
Correlation to Standards ..................................22
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
Using Pattern Blocks for Transformations
Alike and Different .....................................30
in Quadrant I .........................................114
Total Physical Response .............................36
Using Pattern Blocks for Rotational
Math Hunt ....................................................41
Symmetry ..............................................119
Root Word Tree.............................................46
Sharing Mathematics .................................53 Strategies for Teaching Procedures
Vocabulary Flip Book ..................................58
Teaching Procedures Overview ......................125
Content Links ..............................................64
Teaching Number Sense Procedures
Alternative Algorithm for Addition ........136
Strategies for Using Manipulatives
Alternative Algorithm for
Manipulatives Overview ...................................71
Subtraction ............................................139
Common Questions About Alternative Algorithm for
Manipulatives ..........................................71 Multiplication ........................................142
Using Manipulatives Effectively ................73 Alternative Algorithm for Division .........145
Strategies for Acquiring Manipulatives ...74 Adding Fractions .......................................148
Strategies for Organizing Teaching Algebra Procedures
Manipulatives ..........................................74
Missing Addends .......................................151
Strategies for Individual Use of
Collecting Like Terms ...............................154
Manipulatives ..........................................77
Linear Equations .......................................157
Strategies for Free Exploration of
Teaching Geometry Procedures
Manipulatives ..........................................77
Calculating Area ........................................160
Strategies for Modeling While Using
Manipulatives ..........................................78 Calculating Perimeter ...............................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 .....82 Measuring Length .....................................172
Sample Lessons Converting Measurements ......................175
Using Counters for Skip Counting Teaching Data Analysis Procedures
by 2s and 5s .............................................83 Creating Graphs.........................................178
Using Counters for Multiplication Finding Central Tendencies .....................181
with Arrays ..............................................86
33
© Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics
Table of Contents
(cont.)
Strategies for Understanding Strategies for Assessing
Problem Solving Mathematical Thinking
Problem-Solving Overview .............................187 Assessment Overview .....................................265
Problem-Solving Diffi culty Factors .........188 Strategies
Teaching About Diffi culty Factors ...........195 Interviews...................................................268
Problem-Solving Process ..........................195 Observation ................................................272
12 Strategies for Problem Solving ...........198 Performance Tasks ....................................275
Strategies Self- and Peer-Evaluations .......................278
Drawing a Diagram ...................................199 Graphic Organizers ...................................282
Acting It Out or Using Mathematical Journals .............................286
Concrete Materials. ...............................203 Rubrics ........................................................289
Creating a Table .........................................207 Portfolios ....................................................292
Looking for a Pattern ................................210
Appendices
Guessing and Checking ............................214
Creating an Organized List ......................217 Appendix A: References Cited .......................296
Working Backwards ..................................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 ..........................229 Vocabulary Lists...........................................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 ...............................247
MATHO Games ..........................................249
Matching Games .......................................251
Game Rules and Templates ............................253
44
SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education
Foreword
Earlene J. Hall, Ed.D.
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 defi ne the content of which students must demonstrate
understanding. Having an expected outcome defi ned 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 fl uency,
• 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 refl ect a need
for instructional resources for planning lessons.
55
© Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics
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 profi ciency in the following:
• mathematical vocabulary
• conceptual understanding through the use of manipulatives
• procedural profi ciency
• 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.
66
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 specifi c 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.
77
© Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics
Strategies for Teaching Mathematics Introduction
Research
(cont.)
The Importance of Teaching Mathematics in a
Balanced Approach
As early as 1989, educators identifi ed 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 fi nal 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.S. 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 benefi ts
of conceptual understanding, procedural fl uency, and automatic (i.e., quick and effortless) recall
of facts” (U.S. 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 fl uency. 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. Specifi c 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. Because of these differences, a one-size-fi ts-all
approach to mathematics instruction will not meet the needs of all students and ensure that
they have conceptual understanding. However, using a differentiated approach to mathematics
instruction will allow teachers to meet these needs. In their research, Strong, Thomas, Perini,
and Silver (2004) note that “recognizing different mathematical learning styles and adapting
differentiated teaching strategies can facilitate student learning.”
88
SEP 50249 (i3423)—Strategies for Teaching Mathematics © Shell Education
Strategies for Teaching Mathematics Introduction
Research
(cont.)
Differentiating Mathematics
(cont.)
Differentiation is the modifi cation of what is taught, how it is taught, and the product that
students create based on instruction. This is commonly referred to as differentiating content,
process, and product. However, differentiation can look very different depending on the learning
outcome, the needs of the learners, and the structure of the classroom environment (Pettig 2000).
Strategies for Teaching Mathematics provides suggestions for differentiation throughout each section
of the notebook. There are general suggestions for instructional techniques as well as specifi c
suggestions that pertain to specifi c lessons. The differentiation suggestions will enable all
students to connect with the content at a level that is appropriate for them.
Vocabulary Development
Vocabulary has long been overlooked in mathematics instruction. Yet, mathematics has more
vocabulary and diffi cult text than other content areas (Schell as cited in Monroe 1998). Teaching
vocabulary in mathematics is especially important for English language learners. Too often,
students do not have an accurate understanding of mathematical terms. Researchers have
identifi ed several barriers that inhibit their learning as well (Thompson and Rubenstein 2007).
Many vocabulary words have different meanings in other content areas. For example, the term
solution has different meanings in mathematics and science. Other terms are also commonly used
in everyday language, but they have more precise meanings in mathematics. Still other words
are only used in mathematics. These terms must be explicitly taught to students. Vocabulary
words, such as quotient, that are unique to the subject are diffi cult to learn. In this case, students
are essentially learning a new language, and they need instructional strategies that will help them
become aware of the new terms and apply them to problem-solving situations.
The National Reading Panel Report (2000) also identifi ed academic vocabulary as essential in
the development of students’ reading skills. Academic vocabulary includes terms that are used
throughout schooling. English teacher and author, Jim Burke (no date) identifi ed almost 400
academic vocabulary words that students must know by the time they enter middle school.
Terms such as solve, quotient, and sum are commonly understood by profi cient students, but those
who only have a limited understanding of these terms may not be able to complete the simplest
mathematical problems. Further, a variety of teaching strategies is necessary to teach academic
vocabulary. The National Reading Panel (2000) found that vocabulary is learned both indirectly
and directly, and that dependence on only one instructional method does not result in optimal
vocabulary growth.
99
© Shell Education SEP 50249 (i3423)—Strategies for Teaching Mathematics