“It is fun to figure out the puzzle of how children go about making sense of mathematics and then how to help teachers help kids.” John A. Van de Walle, Late of Virginia Commonwealth University
This is the philosophy behind Elementary and Middle School Mathematics: Teaching Developmentally. John A. Van de Walle wrote this book to help students understand mathematics and become confident in their ability to teach the subject to children in kindergarten through eighth grade. Although he could not have foreseen the changes in mathematics teaching over the last three decades, he was at the forefront of the movement towards a constructivist view of teaching, or teaching developmentally. Constructivism says that children construct their own knowledge. They are not blank slates waiting to absorb whatever the teacher tells them. Teachers must understand both mathematics itself and how students learn mathematics in order to teach it effectively.
Learning through problem solving is another major theme of this book. Students solve problems not just to apply mathematics, but also to learn new mathematics. Effective problems will take into account where students are, the problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn and not be diluted by non-mathematical activities such as cutting or pasting, and the problem must require justifications and explanations for answers and methods. Learning then becomes an outcome of the problem solving process.
The book also addresses in more detail than any other book on the market the effect that the trends of standards-based education, increased pressure to test, and increased teacher accountability have had on teaching mathematics. He addresses the 2000 NCTM Standards in depth, in Chapter 1 on Teaching Mathematics in the Era of the NCTM Standards, through the NCTM icon that appears in the margins throughout the text, and in two appendices in the back of the book. Chapter 5 on Building Assessment into Instruction has also been heavily revised to focus on increased testing pressure, creating more explicit links between objectives and assessment, and including assessments for students with special needs.
Elementary and Middle School Mathematics: Teaching Developmentally is a book for doing math today–for both students who want to become teachers, and the students they will eventually teach.
New To This Edition:
NEW! Revises Chapter 5 on assessment--Discusses increased testing pressure and accountability, adds more information on equitable assessments, creates more explicit links between objectives and assessment, and includes assessments for students with special needs.
NEW! Updates the Literature Connections feature to remove all out of print children’s literature and include more non-fiction, poetry, and other types of readings.
NEW! Weaves the Focal Points throughout the chapters as well as links them with the Big Ideas feature–Focal Points have also been added to the Appendix.
NEW! Includes expanded coverage of working with diverse learners.
NEW! Gives greater emphasis on dealing with math anxiety.
Part II Resources Organized by Content vii
NCATE Standards ix
Design of Field Experience Guide x
PART I Menu of Field Experiences 1
Field Experience Cover Sheet 2
Field Experience Activity Log 3
CHAPTER 1 Professional and Pedagogical Knowledge: Classroom Environment 4
Standards Alignment 4 • Rubric 5
1.1 Observation: Physical Environment 6
1.2 Observation: NCTM Professional Teaching Standards 7
1.3 Teacher Interview: An Environment for Learning 8
1.4 Student Interview: Attitudes and Environment 9
1.5 Teaching: Establishing Your Environment 10
1.6 Project: Assessing School Environment 12
CHAPTER 2 Professional and Pedagogical Knowledge: Planning 13
Standards Alignment 13 • Rubric 14
2.1 Teacher Interview: Selecting Goals and Objectives 15
2.2 Observation: Evidence of Higher-Level Thinking 16
2.3 Teaching: Mathematics Task Analysis 17
2.4 Teaching: Worthwhile Task Evaluation 18
2.5 Teaching: Planning a Problem-Based Lesson 19
2.6 Teaching: Using Children’s Literature in Math Teaching 21
CHAPTER 3 Content Knowledge 23
Standards Alignment 23 • Rubric 25
3.1 Observation: Conceptual and Procedural Knowledge 26
3.2 Observation: Mathematical Proficiency 27
3.3 Teacher Interview: Selecting a Worthwhile Task 28
3.4 Student Interview: Conceptual and Procedural Knowledge 29
3.5 Teaching: Create a Web of Ideas 31
3.6 Teaching: Design (and Teach) a Lesson 32
3.7 Reflection: Student Understanding 33
CHAPTER 4 Pedagogical Content Knowledge: Instruction 34
Standards Alignment 34 • Rubric 35
4.1 Observation: The Process Standards 36
4.2 Observation: Classroom Discourse 38
4.3 Observation: Cooperative Groups 39
4.4 Teacher Interview: Teaching for Understanding 41
4.5 Student Interview: Learning Mathematics Developmentally 42
4.6 Teaching: Teaching a Small Group 43
4.7 Teaching: Teaching a Standards-Based Lesson 44
4.8 Reflection: Reflecting on Teaching and Learning 45
4.9 Feedback on Teaching 46
CHAPTER 5 Pedagogical Content Knowledge: Technology 47
Standards Alignment 47 • Rubric 48
5.1 Project: Learning Online 49
5.2 Project: Evaluating Mathematics Software or Websites 50
5.3 Teaching: Using the Calculator to Support Learning 51
5.4 Teaching: Develop a Calculator Learning Center 52
5.5 Reflection: Reactions to Learning with Technology 54
CHAPTER 6 Dispositions 55
Standards Alignment 55 • Rubric 56
6.1 Teacher Interview: Communicating with Parents 57
6.2 Parent Interview: What Is Important to You? 59
6.3 Teaching: Prepare a Family Math Take-Home Activity 60
6.4 Reflection: Professional Growth 61
CHAPTER 7 Student Learning 62
Standards Alignment 62 • Rubric 63
7.1 Classroom Observation: Assessing to Inform Instruction 64
7.2 Diagnostic Interview: Assessing for Understanding 65
7.3 Student Observation and Interview: Learning Through Problems 69
7.4 Teaching: Assessing Student Understanding with Rubrics 70
7.5 Teaching: Creating and Using Rubrics 71
7.6 Teaching: Using Anecdotal Notes to Assess Students 72
7.7 Teaching: Using a Checklist to Assess Students 74
CHAPTER 8 Diversity 76
Standards Alignment 76 • Rubric 77
8.1 Observation: One Child’s Experience 78
8.2 Teacher Interview: Student Diversity 80
8.3 Teacher Interview: Environment and Students 81
8.4 Teaching: Lesson Adaptations 82
8.5 Teaching: Sheltering a Lesson for English Language Learners 84
8.6 Reflection: Meeting the Needs of Individuals 86
PART II Resources for Teaching 89
CHAPTER 9 Expanded Lessons 91
9.1 Exploring Subtraction Strategies 93
9.2 Close, Far, and in Between 95
9.3 Two More Than/ Two Less Than 97
9.4 Learning about Division 99
9.5 Estimating Groups of Tens and Ones 101
9.6 Dot-Paper Equivalences 103
9.7 Multiplication of Fractions 105
9.8 Friendly Fractions to Decimals 107
9.9 Division of Fractions Using the Partition Concept 109
9.10 How Close Is Close? 111
9.11 Comparing Ratios 113
9.12 One Up and One Down 115iv
9.13 Predict How Many 117
9.14 Create a Journey Story 119
9.15 Crooked Paths 121
9.16 Fixed Areas 123
9.17 Shape Sorts 125
9.18 Diagonal Strips 127
9.19 Triangle Midsegments 129
9.20 Using Data to Answer a Question 131
9.21 Create a Game 133
9.22 Bar Graphs to Circle Graphs 135
9.23 Testing Bag Designs 137
9.24 Toying with Measures 139
CHAPTER 10 Mathematics Activities 141
10.1 The Find! 143
10.2 Odd or Even? 145
10.3 Factor Quest 147
10.4 Interference 149
10.5 Target Number 151
10.6 Fraction Find 153
10.7 Illustrating Ratios 155
10.8 Building Bridges 157
10.9 Compensation Decision 159
10.10 Solving the Mystery 161
10.11 Cover All 163
CHAPTER 11 Balanced Assessment Activities 165
11.1 Magic Age Rings 167
11.2 Grocery Store 171
11.3 Bolts and Nuts! 174
PART III Blackline Masters 179
Suggestions for Use and Construction of Materials 179
More-or-Less Cards 181
Number Cards 182
Dot Cards 3—8 183
Five-Frame 189
Ten-Frame 190
Double Ten-Frame 191
10 × 10 Multiplication Array 192
Missing-Part Worksheet 193
Base-Ten Materials Grid 194
Little Ten-Frames 15—16 195
Place-Value Mat (with Ten-Frames) 197
Base-Ten Grid Paper 198
Addition and Subtraction Recording Charts 199
Multiplication and Division Recording Charts 200
Blank Hundreds Chart (10 × 10 Square) 201
Hundreds Chart 202
Four Small Hundreds Charts 203
Circular Fraction Pieces 24—26 204
10 × 10 Grids 207
Rational Number Wheel 208
10,000 Grid 209
Look-Alike Rectangles 210
Look-Alike Rectangles Recording Sheet 211
Degrees and Wedges 212
Clock Faces 213
2-cm Square Grid 214
1-cm Square Grid 215
0.5-cm Square Grid 216
1-cm Square Dot Grid 217
2-cm Isometric Grid 218
1-cm Isometric Dot Grid 219
1-cm Square/Diagonal Grid 220
Assorted Shapes 42—47 221
Coordinate Grid 228
Geoboard Pattern 229
Geoboard Recording Sheets 230
Tangrams and Mosaic Puzzle 231
Motion Man 54—53 232
Property Lists for Quadrilaterals 54—57 234
Assorted Triangles 238
Woozle Cards 239
Design a Bag 240
What Are the Chances? 241
at Collections 242
2 More Than 243
2 Less Than 244
How Long? 245
Fraction Names 246
Solving Problems Involving Fractions 247
It’s a Matter of Rates 248
Predict How Many 69—70 249
Create a Journey Story 251
Crooked Paths 252
Rectangles Made with 36 Tiles 253
Fixed Area Recording Sheet 254
Properties of Quadrilateral Diagonals 255
Toy Purchases 256
Toying with Measures 257