v >
About the authors x
Preface xi
1 Instructional Activities:
T he Building Blocks for Effective Instruction 1
What Are the Students Learning? 1
Developmental Activities 2
Exploratory Developmental Activities 2
Consolidating Developmental Activities 2
Practice Activities 2
Think-Time Practice Activities 3
Speed-Drill Practice Activities 3
Application Activities 3
Classroom Applications 3
Real-World Problems 4
Assessment Activities 4
Varied Assessment Methods 4
Monitoring and Assessment 4
Level of Involvement 6
Flexible Use of Activities and Materials 7
Exercises and Activities 7
References and Related Readings 8
Websites 9
2 Less on Design:
C reating Lessons That Meet the Needs of a Diverse Classroom 10
Combining Activities into a Lesson 10
What Is a Lesson? 10
A Traditional Lesson Plan 11
The Nature of Standard Traditional Lessons 13
Adapting Lessons for Diverse Learning Needs 13
A Lesson Adapted for Diverse Learners 16
Adapting Another Lesson 19
The Planning Process and “Official” Lesson Plans 21
contents
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vi C o n t e n t s
The Planning Process and Teaching Notes 22
Exercises and Activities 22
References and Related Readings 23
Websites 24
3 Beginnings:
M athematics Learning in Early Childhood 25
A Common Misconception 25
About Young Children 25
Teaching Classification 27
Pattern Recognition 28
Teaching Comparison and Seriation 29
Comparison 29
Seriation 32
Matching and Prenumber Comparisons 33
Matching and Prenumber Seriation 33
The Beginning of Geometric Concepts: Relative Position 34
A Revised Lesson 37
Exercises and Activities 40
References and Related Readings 40
Websites 41
4 Whole Nu mbers and Nu meration:
N aming and Writing Quantity 42
Number Sense 42
Foundations of Algebra 43
Building on What Children Already Know 43
The Big Picture 45
Development of Numbers and Numeration 45
One-Digit Numbers 46
Two-Digit Numbers 51
Three or More Digits 56
Rounding Numbers 59
Adapting a Lesson 61
Adapting the Lesson for a Diverse Group of Students 61
Exercises and Activities 64
References and Related Readings 64
Websites 64
5 Adding and Subtracting Whole Nu mbers:
C ombining and Separating Quantities 65
An Overview of the Development of Computation 65
The Meaning of the Operation 65
The Basic Facts 66
The Algorithm(s) 66
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C o n t e n t s vii
Teaching Addition of Whole Numbers 67
Developing the Meaning of Addition 67
Developing the Easy Basic Addition Facts 69
Activities for Exploring Relationships 73
Developing the Hard Basic Addition Facts 76
Teaching the Addition Algorithm 82
Summary of the Developmental Sequence for Addition 85
Teaching Subtraction of Whole Numbers 86
Developing the Meaning of Subtraction 86
Developing the Easy Basic Subtraction Facts 87
Developing the Hard Basic Subtraction Facts 90
Teaching the Subtraction Algorithm 92
Summary of the Developmental Sequence for Subtraction 93
Adapting a Lesson 94
Teaching Problem Solving Using Addition and Subtraction 96
Exercises and Activities 99
References and Related Readings 100
Websites 100
6 Mu ltiplying and Dividing Whole Nu mbers:
C ombining Equal-Sized Groups and Separating Quantities
into Equal-Sized Groups 101
Teaching Multiplication of Whole Numbers 101
Developing the Meaning of Multiplication 101
Developing the Easy Basic Multiplication Facts 103
Developing the Hard Basic Multiplication Facts 107
Teaching the Multiplication Algorithm 111
Summary of the Developmental Sequence for Multiplication 121
Adapting a Multiplication Lesson 121
Teaching Division of Whole Numbers 127
Developing the Meaning of Division 127
Developing the Easy Basic Division Facts 129
Developing the Hard Basic Division Facts 131
Teaching the Division Algorithm 133
Adapting a Division Lesson 144
Teaching Problem Solving Using Multiplication and Division 147
Exercises and Activities 147
References and Related Readings 148
Websites 149
7 Fractions:
Working with Units Smaller Than One 150
Defining Fractions 150
Three Sides of Fractions 151
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viii C o n t e n t s
Fractional Units 152
Beyond Unit Fractions 154
Fractions of a Set 155
Equivalent Fractions 156
Using the Laboratory Approach 158
Comparison of Fractions 159
Adding Fractions 161
Subtracting Fractions 163
Addition and Subtraction Activities 163
Improper Fractions and Mixed Numbers 165
Adapting a Lesson on Fractions 167
Solving Problems Using Fractions 170
Exercises and Activities 171
References and Related Readings 171
Websites 171
8 Decimals:
Working with Base-Ten Units Smaller Than One 172
Decimals 172
Place Value for Decimals 174
Comparing Decimals 178
Adding and Subtracting Decimals 181
Adapting a Lesson on Decimals 184
Using Decimals to Solve Problems 187
Exercises and Activities 187
References and Related Readings 188
Websites 188
9 Measu rement:
A ssigning a Number to a Quantity 189
Measurement and Geometry 189
Defining Measurement 189
Measuring Length 190
Teaching Area Measurement 199
Teaching Volume Measurement 204
Measuring Time 208
Measuring Weight 210
Measuring Temperature 210
Measuring Value 210
Adapting a Lesson on Volume 211
Using Measurement to Solve Problems 214
Exercises and Activities 214
References and Related Readings 215
Websites 215
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C o n t e n t s ix
10 Geometry:
L earning the Names and Characteristics of Shapes 216
The Big Ideas of Elementary School Geometry 216
Straightness 217
Congruence 217
Similarity 218
Parallelism 218
Perpendicularity 219
Symmetry 220
Using the Big Ideas to Study Geometric Shapes 220
Rectangles in Elementary School 220
Circles in Elementary School 225
Angles in Elementary School 228
Prisms in Elementary School 230
Adapting a Geometry Lesson 231
Exercises and Activities 235
References and Related Readings 236
Websites 236
11 Data Analysis and Probability:
G etting Information from Data and Measuring Likelihood 237
Data Analysis and Probability–Two Distinct but Related Areas of Mathematics 237
Data Analysis 238
Emphasizing the Big Ideas of Data Analysis 238
From Exploratory Experiences toward Conceptual Understanding:
A Typical K—4 Development of Data Analysis 238
Adapting a Data Analysis Lesson 245
Using Data Analysis to Solve Problems 250
Probability 250
Emphasizing the Big Ideas of Probability 250
From Exploratory Experiences toward Conceptual Understanding:
A Typical K—4 Development of Probability 251
Using Probability to Solve Problems 256
Exercises and Activities 256
References and Related Readings 257
Websites 257
A ctivities to Take to Your Class room 258
I ndex 260
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about the authors
Benny F. Tucker earned his Ph.D. at the University of Illinois in 1975. He has authored
or co-authored more than 50 books, on topics ranging from teaching methods for elementary
school mathematics to the use of instructional activities in the mathematics
classroom. He has authored or co-authored more than 20 articles in professional journals
and has made more than 30 presentations at professional conferences.
Ann Haltom Singleton is Associate Dean of the School of Education at Union University
in Jackson Tennessee. She earned her Ed.D. in Special Education from the University
of Memphis. Her research areas include leadership development and mathematics
instruction, especially in inclusive settings. She has contributed to numerous articles and
has made over 30 national presentations. She was recognized as the Union University
2003 Faculty of the Year.
Terry L. Weaver honed his teaching skills in the Miami-Dade County School System.
He received his Ph.D. in Special Education from George Peabody College for Teachers
at Vanderbilt University. Dr. Weaver then shared his teaching skills at Carson-Newman
College and Union University where he continues to teach. Dr. Weaver has served as an
item writer for and participated in the revalidation of the Praxis II Specialty Area Test
in SE (Core Knowledge). He is a co-author of Teaching Mathematics to All Children:
Designing and Adapting Instruction to Meet the Needs of Diverse Learners, has presented
on differentiated instruction and assessment, universal design, inclusion, and adapting
instruction for diverse learners, and recently lead the revision of a chapter on mathematics
in Vaughn’s and Bos’s Strategies for Teaching Students with Learning and Behavior
Problems.
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xi
Why This Book?
The diversity of students in K—4 classrooms is extensive. The children in a typical classroom
are diverse in gender, diverse in race and ethnicity, and diverse in religion and
culture. They are diverse in ability, diverse in interests, and diverse in preferred learning
styles. And they are diverse in family background, and diverse with respect to resources
in the home such as books and technology. In the face of such diversity, how can the
teacher expect to plan for effective instruction?
Although teachers must certainly be aware of student diversity and the need to
accommodate that diversity, it is perhaps more important for K—4 teachers to be aware of
the ways in which their students are alike. For example, almost universally, children are
kinesthetic
learners. It is natural for them to be active and move around. They love classroom
activities that allow (even require) them to be energetic and animated. Children
are also naturally inquisitive. They are interested in what, why, and how. It is the nature
of children to be curious about things. They like to talk to one another, to exchange
ideas, and to discuss the things that they are experiencing and learning. Children are
concrete learners. They enjoy handling things, seeing how things are related. They
like to understand.
In this text, we provide an approach to the planning and teaching of K—4 mathematics
that is based on the nature of children. We believe that the teaching suggestions in
this text will help teachers be more effective as they plan and teach mathematics in diverse
classrooms, grades K—4.
Structure of the Book
The book begins with two introductory chapters that provide a basic understanding
of instructional activities and lesson planning. Then there are nine chapters devoted to
teaching the content that most commonly appears in K—4 mathematics textbooks. We
do not attempt to provide comprehensive coverage of every topic that might appear in
a K—4 textbook. Rather, our intent is to emphasize a way of teaching effectively that
will result in learning, understanding, retention of important concepts and skills, and
an ability to apply those concepts and skills to solve problems. Important to that way of
teaching is effective planning. Therefore, we have made planning for effective teaching
an important part of this text.
preface