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Tables of Contents for Mathematics Methods for Elementary and Middle School Teachers
Chapter/Section Title
Page #
Page Count
List of Text Activities
xiv
List of CD-ROM Activities
xx
Preface
xxvii
The Past, Present, and Future of Mathematics Education
1
18
Past
2
1
From Thousands of Years Ago
2
1
From One Hundred and Fifty Years Ago
2
1
From Our Most Recent Past
3
1
Present
3
10
Brain Research Enlarges Our View
3
1
The Standards Movement in Mathematics Education
4
1
The NCTM Standards (2000)
4
2
Support for Teaching the NCTM Standards
6
2
Opposing Views of the Standards Movement
8
1
Other Expert Postions on Standards for Mathematics Learning
9
1
National Council for the Accreditation of Teacher Education (NCATE)
9
1
National Directives
9
1
Timss
10
1
Current Trends in Mathematics Learning
10
1
A General Overview
10
1
Active Learning with Real-World Applications for All Students
10
1
Technology
10
2
Other Current Trends
12
1
Future
13
1
Mathematics in the Twenty-First Century School
13
1
A New Vision of School
13
1
A New Vision of Teachers
13
1
Summary
14
1
Exercises
14
1
Bibliography
15
2
Integrating Technology
17
2
Culturally Relevant Mathematics
19
30
The Giftedness of Many Cultures in Mathematics
20
14
Cultural Contributions: A Sense of Pride
20
2
African American Heritage
22
1
Unique Systems of Mathematics Today
22
1
African Mathematics from the Ancient World
22
1
African Ideas in Elementary Mathematics
22
2
African Ideas in Middle School Mathematics
24
1
Resources for Contributions of African Americans Who Used Mathematics
25
1
Native American Heritage
26
1
Native American Culture in Mathematics Lessons
26
1
Mathematics Resoruces from Native American Educators
27
1
Hispanic American Heritage
27
1
Hispanic American Culture in Mathematics Lessons
27
3
Contributions from Hispanic Americans in Mathematics
30
1
Asian American Heritage
30
1
Asian American Culture in Mathematics Lessons
30
2
Contributions from Asian Americans in Mathematics
32
1
European American Heritage
32
1
A Most Unusual Story for Elementary-Age Children and Middle Schoolers
32
1
Women in the Culture of Mathematics Lessons
33
1
In Celebration of All Mathematicians
34
1
Equity in Mathematics: The Right of All People to Acquire Mathematical Power
34
8
The Impact of Technology
34
1
Equity with Computer Technology
35
1
Equity with Calculator Technology
35
1
Equity with Other Emerging Technologies
36
1
The Development of Problem Solving and Critical Thinking
36
1
The Role of Language in Problem Solving and Critical Thinking
36
2
The Role of Conceptual Thinking in Other Cultures
38
2
Minority Students Can Problem Solve in Authentic Ways
40
1
The Use of Cultural Qualities to Help Mathematics Equity
41
1
Learning Style
41
1
Speed of Performance
42
1
Noncompetitive Settings
42
1
Summary
42
1
Exercises
42
1
Bibliography
43
3
Children's Literature
46
1
Integrating Technology
47
2
How Children Learn Mathematics
49
24
The Changing Scene of How People Learn: The Brain Connection
49
2
Use It or Lose It
50
1
Implications for Mathematics Teachers
50
1
The Sooner, the Better
50
1
Implications for Mathematics Teachers
51
1
Learning Theories Applied to Mathematics
51
13
The Constructivists
51
1
Jean Piaget
52
2
Lev Vygotsky
54
2
Jerome Bruner
56
1
Other Theorists
56
1
Today's Constructivists in the Classroom
56
3
The Behaviorists
59
1
Immediate Feedback
59
1
Programmed Learning
60
1
Information-Processing Theories
61
1
Thought Processing
61
1
Learning Styles
62
2
Planning for Effective Learning of Mathematics
64
4
Classroom Instruction
64
1
The Chicago Mathematics Instructional Process
65
1
The TIMSS Lesson Plan Format
66
1
The Video Vignette: Actual Teachers in Classrooms
67
1
Homework in Light of the New Approaches to Instruction
67
1
Summary
68
1
Exercises
68
1
Bibliography
69
2
Integrating Technology
71
2
Assessment in Mathematics
73
18
History
74
1
Standardized Testing
74
1
National Assessment of Educational Progress
74
1
International Studies
75
1
State Assessment Programs
75
1
Changing Assessment
75
11
Purpose of Asessment
76
1
The Evaluation Efforts of the NCTM
76
1
Assessment Standards
76
1
Performance-Based Assessment
77
1
Interviews
77
1
Observations
78
1
Questioning
79
1
Performance Tasks
80
3
Calculators and Testing
83
1
Student Portfolios
83
1
Self-Assessment
84
1
Scoring Performance-Based Assessments
84
2
New Standards Project
86
1
Summary
86
1
Exercises
87
1
Bibliography
87
2
Integrating Technology
89
2
Problem Solving
91
20
Problem Solving
91
15
The Magnitude of Problem Solving
91
1
The Complexity of Problems Solving
91
1
The Many Facets of Problem Solving
92
1
The Methods, Kinds, and Procedures of Problem Solving
92
1
Methods of Integrating Problem Solving
92
2
Kind of Problem Solving
94
1
Kinds of Problems
95
1
How to Structure Your Mind: The Procedures
95
1
Problem-Solving Strategies
96
1
Estimation and Check
96
1
Looking for Patterns
97
1
Insufficient Information
97
1
Drawing Pictures, Graphs, and Tables
98
1
Elimination of Extraneous Data
98
1
Developing Formulas and Writing Equations
98
1
Modeling
98
1
Working Backward
99
1
Flowcharting
99
1
Acting Out the Problem
100
1
Simplifying the Problem
100
1
The Standards in Problem Solving
101
1
Effective Teaching of Problem Solving
101
1
Effective Techniques for Teaching Problem Solving
101
2
Kinds of Cooperative Grouping Activities
103
1
Assessment to Rate Group Problem Solving
104
1
Assessments to Rate Ourselves and Our Students
104
1
Putting It All Together: A Student's Work with Problem Solving
105
1
Summary
106
1
Exercises
107
1
Bibliography
107
1
Children's Literature
108
1
Integrating Technology
109
2
Geometry and Spatial Reasoning
111
26
Informal Geometry
112
1
Importance of Geometry
112
1
Teaching Strategies
112
20
Van Hiele Levels
113
1
Solid Geometry
114
1
Relating Three-Dimensional to Two-Dimensional
115
2
Plane Geometry
117
1
Geoboards
118
2
Tangrams
120
2
Symmetry
122
1
Line Symmetry
122
1
Rotational Symmetry
123
2
Transformations
125
1
Logo
126
1
Area and Perimeter
127
1
Pentominoes
127
1
Tangrams and Geoboards
128
2
Coordinate Geometry
130
1
Geometric Constructions
131
1
Literature
132
1
Assessment
132
1
Field-Independent Learners
132
1
Field-Dependent Learners
133
1
Correcting Common Misconceptions
133
1
Angle Identification
133
1
Angle Misconceptions with Protractors
133
1
Summary
133
1
Exercises
134
1
Bibliography
134
1
Children's Literature
135
1
Integrating Technology
136
1
Measurement
137
26
Measurement as a Process
138
3
How Measurement Develops in the Curriculum
138
1
Direct Comparison
138
1
Indirect Comprison
138
1
Seriation
138
1
Frames of Reference
138
1
Specific Types of Measurement
138
1
How the Roots of Measurement Develop in the Curriculum
139
1
The Teacher's Role in Cultural Awareness
139
1
Creating Mathematical Connections to History
140
1
Teaching Strategies
141
16
Linear Measurement
141
1
Direct Comparison
141
1
Indirect Comparison
141
1
Seriation
142
3
Frames of Reference
145
1
Mass (Weight) Measurement
145
1
Direct Comparison
145
1
Indirect Comparison
146
1
Frames of Reference
147
1
Standard Weights for Measurement
147
1
Area
147
2
Nonstandard Units
149
1
Standard Units
149
1
Frames of Reference
150
1
The Use of Customary and Metric Units
150
1
Capacity and Volume
150
1
Nonstandard Units
151
1
Standard Units
151
1
Computing Volume
151
1
Comparing Capacity and Volume
152
1
Time
153
1
Interpretation of Time
153
1
Understanding Passage of Time
154
1
Understanding Time Sequence
154
1
Temperature
155
1
Literature
156
1
Assessment
157
1
Common Misconceptions of Time
157
1
Correcting Misconceptions with the Concept of Time
157
1
Creating a Rubric for the Assessment of Time Concepts
158
1
Summary
158
1
Exercises
159
1
Bibliography
159
1
Children's Literature
160
1
Integrating Technology
161
2
Number Readiness---Early Primary Mathematics
163
22
A Child's Understanding of Number
164
8
Task: Classification
165
1
Assessment
165
1
Teaching Ideas
165
1
Task: Class Inclusion
166
1
Assessment
166
1
Task: Number Inclusion
167
1
Assessment
167
1
Teaching Ideas
167
1
Task: Seriation
168
1
Assessment
168
1
Teaching Ideas
169
1
Task: Number Conservation
169
1
Assessment
169
1
Teaching Ideas
170
1
Task: Equivalence of Sets
170
1
Assessment
170
1
Teaching Ideas
171
1
Implications for Curriculum
171
1
Assessment
172
1
Building the Concept of Number
172
9
Patterns
172
1
Developing Patterning Skills
173
1
Number Relationships
173
1
One-to-One Correspondence
173
1
More, Less, and Same
173
1
Rote Counting
174
1
Rational Counting
174
1
Counting Sets
175
1
Numeral-Set Association
175
1
Developing Meaning for Numbers
176
2
Literature
178
1
Writing Numerals
178
1
Teaching Ideas
178
1
Readiness for Operations
179
1
Developing Meaning for Operations
179
1
Building Part-Whole Understanding
180
1
Using Technology
180
1
Summary
181
1
Exercises
181
1
Bibliography
182
1
Children's Literature
183
1
Integrating Technology
184
1
Numeration and Number Sense
185
30
Structure of Numeration Systems
187
4
Decimals
188
2
Number Bases
190
1
Teaching Strategies
191
18
Instructional Sequence
191
1
The Nature of Place Value
192
2
Early Experiences
194
1
Numeration Models
194
2
Number Base Activities
196
1
Counting Activities
197
4
Number Sense
201
1
Understanding and Interpreting Large Numbers
202
2
Reading Numbers
204
1
Rounding
204
2
Literature
206
1
Decimals
206
1
Representing Decimals
207
1
Comparing Decimals
207
1
Estimation
208
1
Assessment
209
2
Field-Dependent Learners
209
1
Field-Independent Learners
209
1
Correcting Common Misconceptions
209
1
Correcting Common Misconceptions with Decimals
210
1
Nonalignment of the Decimal Points
210
1
Annexing Zeros
210
1
Attention to Decimal Point as Place Holder
210
1
Name Value Confused with Place Value
210
1
Summary
211
1
Exercises
211
1
Bibliography
212
1
Children's Literature
213
1
Integrating Technology
214
1
Operations and Number Sense
215
36
Overview of the Operations and Properties
217
24
Addition
217
1
Combining
217
1
Static
217
1
Concrete to Symbolic
217
1
Literature
218
1
Properties
218
1
Subtraction
218
1
Take-Away
218
1
Comparison
219
1
Missing Addend
219
1
Literature
219
1
Properties
220
1
Multiplication
220
1
Repeated Addition
220
1
Combinations
221
1
Arrays
221
1
Literature
222
1
Properties
222
1
Division
222
1
Measurement Division
222
1
Partitive Division
223
1
Literature
223
1
Properties
223
1
Readiness to Learn Basic Facts
224
1
Prerequisites
224
1
Piaget
225
1
Number Meaning
225
1
Thinking Strategies for the Basic Facts of Addition
226
1
Strategy Preferences
226
1
Addition Facts
227
1
Principles
227
1
Adding One
228
1
Counting On
228
1
Near Doubles
228
1
Bridging to 10
228
3
Thinking Strategies for the Basic Facts of Subtraction
231
1
Helpful Devices
231
2
Fact Families
233
1
Using 10
233
1
Part-Whole
233
1
Thinking Strategies for the Basic Facts of Multiplication
234
1
Strategies
234
2
Skip Counting
236
1
Finger Multiplication
237
1
Friendly Facts
238
1
Split a Factor
238
1
Thinking Strategies for the Basic Facts of Division
238
1
Missing Factor
239
1
Mastering the Facts
239
1
Basic Facts---Pros and Cons
239
1
Cluster the Facts
240
1
Build Confidence
240
1
Assess Mastery
240
1
Practice Devices
240
1
Assessment
241
5
Field-Dependent and Field-Independent Learners
241
1
Basic Fact Tables
241
1
Correcting Common Misconceptions
242
1
Faulty Reasoning 1
243
1
Faulty Reasoning 2
243
1
Faulty Reasoning 3
243
1
Faulty Reasoning 4
244
1
Teacher Assessment of Student Work: Steps in Analytic Thinking
245
1
Steps in Analytic Thinking
245
1
Summary
246
1
Exercises
246
2
Bibliography
248
1
Children's Literature
248
2
Integrating Technology
250
1
Operations with Whole Numbers
251
34
The Development of Algorithmic Models
251
4
Proportinal versus Nonproportional Models
253
1
Representational versus Nonrepresentational Models
254
1
Teaching Strategies
255
19
Changing Perspective
255
1
The Pros and Cons of Teaching Algorithms
255
1
Promoting Invented Strategies
256
1
Teaching the Addition Algorithm
257
3
Teaching the Subtraction Algorithm
260
1
Take-Away
260
1
Comparison
260
1
Missing Addend
261
1
Suggestions for Instruction
262
1
Teaching the Multiplication Algorithm
262
1
Repeated Groups
262
2
Area Model
264
2
Teaching the Division Algorithm
266
1
Using Models
266
1
Comparing Models
267
1
Special Problems
267
2
Remainders
269
1
Combining the Operations
270
1
Literature
271
1
Mental Computation
271
1
Estimation
272
1
Front-End Estimation
273
1
Compatible Numbers
273
1
Clustering
274
1
Rounding
274
1
Assessment
274
5
Important Aspects
274
1
Correcting Common Misconceptions
275
1
Special Needs Students
275
1
Addition
275
1
Subtraction
276
1
Multiplication
277
1
Division
278
1
Field-Dependent Learners
279
1
Field-Independent Learners
279
1
Summary
279
1
Exercises
280
1
Bibliography
281
1
Children's Literature
282
1
Integrating Technology
283
2
Common Fractions and Decimals
285
46
Interpretation of Fractions
286
1
Part-Whole Region (Area Model)
286
1
Measure
287
1
Set
287
1
Ratio
288
1
Division
288
1
Teaching Strategies: Fractions
288
26
Whole-to-Part Activities
288
2
Partitioning Activities
290
1
Equivalence Activities
291
2
Multiple Bars
293
1
Multiple Embodiments
294
1
Operations with Fractions
295
1
Addition and Subtraction
296
2
Alternative Solutions
298
1
Estimation
299
1
Mixed Numbers
300
1
Multiplication and Division
300
1
Fraction x Whole Number
301
1
Fraction x Fraction
301
2
Understanding Division
303
3
Invented Strategies
306
1
Reciprocal
307
1
Estimation
307
1
Using Other Manipulatives
308
1
Literature
308
1
Decimals
308
1
Terminating Decimals
309
3
Repeating Decimals
312
1
Linking Fractions to Decimals
313
1
Teaching Strategies: Decimals
314
6
Decimals as Money
314
1
Coin Equivalencies
315
1
Operations with Decimals
316
1
Addition
316
1
Subtraction
316
1
Multiplication
317
2
Division
319
1
Assessment
320
6
Field-Independent Learners
320
1
Field-Dependent Learners
320
1
Correcting Common Misconceptions
321
1
Adding or Subtracting Denominators as Whole Numbers
321
1
Changing Mixed Numbers to Improper Fractions
322
1
Inverting and Multiplying the Incorrect Factors
322
1
Multiplying and Dividing Mixed Numbers
322
1
Regrouping Fractions as Whole Numbers
323
1
Correcting Common Misconceptions with Decimals and Money
324
1
Poor Estimation Skills in Multiplication and Division
325
1
Summary
326
1
Exercises
326
1
Bibliography
327
1
Children's Literature
328
1
Integrating Technology
329
2
Percent, Ratio, Proportion, and Rate
331
24
Teaching Strategies
332
14
Percent
332
1
From Concrete to the Connecting Level
332
1
Money
332
1
Shapes
332
2
From the Connecting Level to the Symbolic
334
2
Ratio
336
1
From Concrete to the Connecting Level
336
3
From the Connecting Level to the Symbolic
339
1
Proportions
340
1
The Part-Part Relationship in Proportions
340
3
Other Relationship Models for Proportion
343
1
Rate
344
1
Students with Special Needs
345
1
Percent, Ratio, Proportion, and Rate in the World of Work
345
1
Assessment
346
5
Field-Independent and Field-Dependent Learners
346
1
Correcting Common Misconceptions
346
1
Writing the Decimal with the Percent
347
1
Percent of a Given Value
347
1
Portfolio Assessments
347
1
Students' Understanding of Percent: Building with Pattern Pieces
347
1
Students' Understanding of Percent: Seeing Percent in Whole Figures
348
3
Summary
351
1
Exercises
351
1
Bibliography
352
1
Children's Literature
352
1
Integrating Technology
353
2
Number Theory, Patterns, and Functions, and Algebra
355
30
Teaching Strategies
356
20
Number Theory
356
1
Odd and Even Numbers
356
2
Prime and Composite Numbers
358
4
Divisibility Rules
362
1
Polygonal or Figurate Numbers
363
1
Square Numbers
363
1
Pythagorean Triples
364
1
Fibonacci Number Sequence
365
1
Pascal's Triangle
366
1
Patterns and Functions
367
1
Activities for Pre-K-2 Grades
367
1
Activities for 3-5 Grades
368
1
Activities for 6-8 Grades
368
1
Teaching Integers
369
1
Postman Stories
369
1
Beginning Algebra Concepts
370
1
Activities for K-4 Grades
371
1
Activities for 5-8 Grades
372
4
Assessment
376
5
Field-Independent Learners
377
1
Field-Depedent Learners
377
1
Correcting Common Misconceptions
377
1
Jumping to Premature Conclusions
377
1
Correcting Unknown Errors... the Reason for Oral Discourse
377
1
Not Seeing the Pattern
377
1
Portfolio Assessments
377
1
Fibonacci Number Sequence
377
2
Functions---Input-Output Tables
379
1
Algebra
379
2
Summary
381
1
Exercises
381
1
Bibliography
382
1
Children's Literature
383
1
Integrating Technology
384
1
Data Analysis, Statistics, and Probability
385
Teaching Strategies
386
Data Analysis---Graphing
386
Creating Real Graphs
387
Creating Picture Graphs
387
Creating Bar Graphs
389
Creating Line Graphs
391
Creating Circle Graphs
392
Creating Scatter Graphs
394
Technology: Creating All the Graphs Together
394
Statistics
395
Mean
396
Mode
397
Median
397
Range
397
Stem and Leaf Plots
398
Box and Whisker Plots
398
Putting It All Together: Statistical Information about Real-World Mathematics
399
Probability
400
Early Experiences: The Pre-K-5 Grades
401
Assigning Probabilities: The 6-8 Middle School Grades
405
Permutations and Combinations
408
Homework Possibilities
411
Assessment
411
Data Analysis---Graphing
411
Bar Graph Stories
411
Continuous Graph Stories
412
Students with Special Needs
413
Averages
413
Summary
415
Exercises
415
Bibliography
416
Children's Literature
417
Integrating Technology
418