search for books and compare prices
Tables of Contents for Mathematics Methods and Modeling for Today's Mathematics Classroon
Chapter/Section Title
Page #
Page Count
Part I The Context for Teaching Mathematics
1
123
What Is Mathematics and Why Do We Teach It?
3
26
Overview
4
1
Focus on the NCTM Standards
4
1
Focus on the Classroom
5
1
What Is Mathematics?
5
5
The Three Views of Mathematics
7
1
Current Views of Mathematics
8
2
Mathematical Literacy and Modeling with Discrete Systems
10
5
Modeling Change
10
1
A Savings Certificate
11
2
Mortgaging a Home
13
1
Exercises 1.2
14
1
Project: Modeling Exploration
15
1
Approximating Change with Difference Equations
15
10
Discrete Versus Continuous Change
16
1
Approximating Change
16
1
Growth of a Yeast Culture
16
3
The Spotted Owl Population
19
2
Spread of a Contagious Disease
21
1
Warming of a Cooled Object
21
1
Exercises 1.3
22
3
Projects: Modeling Explorations
25
1
Further Reading
25
1
Why Do We Study Mathematics?
25
4
Exercises 1.4
26
1
Projects: Find Out for Yourself
27
1
References
27
2
A Model for Mathematics Education
29
36
Overview
30
1
Focus on the NCTM Standards
30
1
Focus on the Classroom
31
1
The Intended Curriculum
31
14
The Standards and the Principles and Standards
32
1
Background
32
2
Comparisons with Others
34
3
Learning in Math Classes
37
3
Middle School Mathematics Programs
40
2
High School Math Programs
42
2
Exercises 2.1
44
1
Projects: Find Out for Yourself
45
1
Implemented Curriculum
45
3
Trends in Student Performance
46
1
Instructional Programs
47
1
Calculator Usage
47
1
Exercises 2.2
47
1
Projects: Find Out for Yourself
48
1
Attained Curriculum
48
8
NAEP Assessment Problem
50
3
Sample Extended Response
53
1
Sample Satisfactory Response
53
1
Sample Partial Response
54
1
Sample Minimal Response
54
1
Sample Incorrect Response
55
1
Exercises 2.3
55
1
Project: Find Out for Yourself
56
1
Student Attitudes Toward Mathematics
56
3
Exercises 2.4
58
1
Projects: Find Out for Yourself
59
1
Planning for Change in the Teaching of Mathematics
59
6
Exercises 2.5
62
1
Projects: Find Out for Yourself
62
1
References
63
2
Doing Mathematics: Living the Standards
65
58
Overview
66
1
Focus on the NCTM Standards
66
1
Focus on the Classroom
66
1
Principles That Guide the Teaching of Mathematics: More Than a Wish List
67
4
The Equity Principle
67
1
The Mathematics Curriculum Principle
68
1
The Teaching and Assessment Principles
69
1
The Learning Principle
70
1
The Technology Principle
70
1
Exercises 3.1
71
1
Doing Mathematics
71
1
Mathematics as Problem Solving
71
16
Mathematics as Reasoning and Proof
73
6
Mathematics as Communication
79
2
Mathematics as Connections
81
2
Mathematics as Representations
83
2
Exercises 3.2
85
1
Projects: Find Out for Yourself
86
1
The Mathematical Behavior of Dynamical Systems
86
1
Long-Term Behavior
87
11
Linear Dynamical Systems an+1 = ran, for r Constant
87
4
Finding Equilibrium Values
91
1
Dynamical Systems of the Form an+1 = ran + br Where r and b Are Constants
91
1
Prescription for Digoxin
91
1
An Investment Annuity
92
1
A Checking Account
93
1
Finding Equilibrium Values
94
1
Nonlinear Systems
95
1
Exercises 3.3
95
2
Projects: Modeling Explorations
97
1
Further Reading
98
1
Systems of Difference Equations
98
14
A Car Rental Company
99
2
Competitive Hunter Model-Spotted Owls and Hawks
101
4
Predator-Prey Model: Owls and Mice
105
1
Voting Tendencies
106
3
Summary
109
2
Exercises 3.4
111
1
Projects: Modeling Explorations
112
1
Further Reading
112
1
Curriculum Content for Grades 6-12
112
1
The Modeling Process
113
10
Mathematical Models
114
1
The Construction of Models
115
3
Iterative Nature of Model Construction
118
1
Exercises 3.6
119
2
Projects: Modeling Explorations
121
1
References
122
1
Part II Middle and High School Mathematics Curricula: Fitting Mathematics to Reality
123
360
Number and Operations
125
32
Overview
126
1
Focus on the NCTM Standards
126
1
Focus on the Classroom
127
1
Number and Operations in the Curriculum
127
1
Number Systems, Their Properties, and Operations
128
4
Number Systems
128
1
Properties of Number Systems
129
1
Operations
130
1
Exercises 4.2
131
1
Project: Find Out for Yourself
131
1
Number Theory in Problem Solving
132
3
Exercises 4.3
134
1
Counting Theory and Techniques
135
8
Addition and Multiplication
135
1
Selecting from Disjoint Sets
135
1
Selecting from Overlapping Sets
136
1
Counting Meal Choices
136
1
Permutations and Combinations
137
1
Arranging Paintings
138
1
More Paintings
138
1
Two-Topping Pizzas
139
1
Combinations Algorithm
140
1
Other Counting Techniques
141
1
Exercises 4.4
142
1
Estimation and Algorithms
143
5
Estimation
144
1
Strategies for Estimating
145
1
Algorithms
146
1
Exercises 4.5
147
1
Sequences and Series
148
1
Using Technology to Teach Number and Operations
148
2
Exercise 4.7
150
1
Project: Find Out for Yourself
150
1
Number and Operations Lesson Plans
150
7
Number and Operation (Middle School Level)
150
2
Number and Operation (Secondary Level)
152
3
References
155
2
Patterns, Functions, and Algebra
157
42
Overview
158
1
Focus on the NCTM Standards
158
1
Focus on the Classroom
159
2
Patterns, Functions, and Algebra in the Curriculum
161
4
Exercises 5.1
163
1
Projects: Find Out for Yourself
164
1
Patterns as the Basis of Mathematics
165
4
Exercises 5.2
166
2
Project: Find Out for Yourself
168
1
Variables, Expressions, and Equations
169
4
Steve's Collection of Sports Cards
170
1
Javier's Money
171
1
Exercises 5.3
172
1
Projects: Find Out for Yourself
172
1
Functions: Their Representations-Their Properties
173
4
Exercises 5.4
175
1
Projects: Find Out for Yourself
176
1
Solving Equations, Inequalities, and Systems of Equations
177
8
Solving Equations
177
2
Example Graphing Functions
179
1
Solving Inequalities
180
1
Solving a System of Equations
181
2
Exercises 5.5
183
1
Project: Find Out for Yourself
184
1
Modeling Using Proportion: Transformations That ``Linearize the Data''
185
7
Proportion
186
2
Kepler's Third Law
188
1
A Spring-Mass System
189
1
Exercises 5.6
190
1
Projects: Modeling Explorations
191
1
Algebra Lesson Plans
192
7
Solving a Linear Equation (Middle School Level)
192
2
Locating Solutions for Linear Programming Problems (Secondary Level)
194
3
References
197
2
Geometry and Measurement
199
56
Overview
200
1
Focus on the NCTM Standards
200
1
Focus on the Classroom
201
1
Geometry and Measurement in the Curriculum
201
3
Geometry and Spatial Sense
202
1
Measurement
202
1
Teaching and Learning Geometry and Measurement
203
1
Exploring Geometric Relationships
204
5
Relationships Among Shapes
204
2
Relationships Between Dimensions
206
1
Exploring Right Triangles
206
1
Exploring Geometric Relationships
207
1
Transformational Relationships
207
1
Relationships to Other Topics
208
1
Exercises 6.2
208
1
Projects: Find Out for Yourself
209
1
Geometric Similarity
209
8
Example A Bass Fishing Derby
211
3
Exercises 6.3
214
1
Projects: Exploring Modeling
214
3
Transformational Geometry
217
9
Isometries
218
1
Other Transformations
219
1
Connections to Other Concepts
220
2
Applications of Transformational Geometry
222
3
Exercises 6.4
225
1
Geometric Representations
226
6
Representing Algebra, Data, and Functions
226
2
Fractals and Graph Theory
228
2
A Different Look at Geometry
230
1
Example Smalltown's Committees
230
1
Exercises 6.5
231
1
The Importance of Justification
232
2
Project: Find Out for Yourself
233
1
Using Technology to Understand Geometry
234
5
``Constructing'' Figures with Dynamic Software
234
2
Explorations and Problem Posing with Dynamic Software
236
1
Example Exploring Deductive Arguments
236
1
Other Technology Tools for Geometry
237
1
Exercises 6.7
238
1
Project: Find Out for Yourself
238
1
Spatial Reasoning
239
3
Spatial Visualization in the Classroom
240
1
Building Spatial Abilities
241
1
Exercises 6.8
241
1
Geometry and Measurement Lesson Plans
242
13
Geometry and Measurement (Middle School Level)
242
6
Geometry and Measurement (Secondary Level)
248
4
References
252
3
Data Analysis and Probability
255
60
Overview
256
1
Focus on the NCTM Standards
256
1
Focus on the Classroom
257
1
Exploring Data
257
10
Exploratory Data Analysis
258
6
Thinking About Collecting Data
264
1
Exercises 7.1
265
2
Projects: Find Out for Yourself
267
1
Exploring Bivariate Data
267
8
Scatterplots
267
2
Correlation
269
1
Trend Lines
269
4
Exercises 7.2
273
2
Project: Find Out for Yourself
275
1
Chance and Probability
275
6
Basic Probability
276
1
Example Covering the Dot
277
2
Independence and Conditional Probability
279
1
Exercises 7.3
279
1
Projects: Find Out for Yourself
280
1
Simulating Deterministic Behavior
281
4
Introduction
281
1
Area Under a Curve
282
2
Exercises 7.4
284
1
Generating Random Numbers
285
3
Middle-Square Method
285
1
Linear Congruence
286
1
Exercises 7.5
287
1
Projects: Exploring Modeling
287
1
Simulating Probabilistic Behavior
288
9
A Fair Coin
289
1
Roll of a Fair Die
290
2
Roll of an Unfair Die
292
1
Exercises 7.6
293
1
Projects: Exploring Modeling
294
3
Queuing: A Harbor System
297
11
Unloading Ships
297
6
Model Refinement
303
2
Exercises 7.7
305
3
Projects: Exploring Modeling
308
1
Data Analysis and Probability Lesson Plans
308
7
Simulation (Middle School Level)
308
3
Line of Best Fit (Secondary Level)
311
3
References
314
1
Advanced Topics
315
38
Overview
316
1
Focus on the NCTM Standards
316
1
Focus on the Classroom
316
1
Trigonometry
317
8
Exercises 8.1
324
1
Projects: Find Out for Yourself
325
1
Discrete Mathematics
325
14
A Question
330
5
Example Modeling a Timber Stand
335
1
Exercises 8.2
336
2
Projects: Find Out for Yourself
338
1
Advanced Placement Programs: Statistics and Calculus
339
6
Exercises 8.3
342
3
Organizations and Competitions
345
3
Lesson Plan for Discrete Mathematics
348
5
Lesson 9: Examining Change with Recursive Relations
348
3
References
351
2
Model Fitting and Empirical Model Construction
353
50
Overview
354
1
Introduction to Model Fitting
354
3
The Relationship Between Model Fitting and Interpolation
355
1
Sources of Error in the Modeling Process
356
1
Fitting Models to Data Graphically
357
8
Visual Model Fitting with the Original Data
358
1
Transforming the Data
359
3
Exercises 9.2
362
3
Analytical Methods of Model Fitting
365
4
Chebyshev Approximation Criterion
365
1
Minimizing the Sum of the Absolute Deviations
366
1
Least-Squares Criterion
367
1
Exercises 9.3
368
1
Applying the Least-Squares Criterion
369
6
Fitting a Straight Line
369
1
Fitting a Power Curve
370
1
Transformed Least-Squares Fit
371
2
Choosing a Best Model
373
1
Exercises 9.4
373
2
Projects: Exploring Modeling
375
1
Further Reading
375
1
Fitting One-Term Models
375
13
Introduction
375
1
Harvesting in the Chesapeake Bay and Other One-Term Models
376
4
Harvesting Bluefish
380
1
Harvesting Blue Crabs
381
1
Verifying the Models
382
2
Exercises 9.5
384
4
Projects: Exploring Modeling
388
1
Further Reading
388
1
Smoothing with Polynomial Models
388
15
Elapsed Time of a Tape Recorder
389
2
The Lagrangian Form of the Polynomial
391
1
Advantages and Disadvantages of High-Order Polynomials
391
1
Data Smoothing
392
1
Elapsed Time of a Tape Recorder Revisited
393
2
Divided Differences
395
3
Elapsed Time of a Tape Recorder Revisited
398
1
Exercises 9.6
398
5
Linear Programming and Numerical Search Methods
403
48
Overview
404
1
Focus on the NCTM Standards
404
1
Focus on the Classroom
405
1
Linear Programming I: Geometric Solutions
405
11
Introduction
405
1
Defining a Linear Program: The Carpenter's Problem
406
1
Interpreting a Linear Program Geometrically
407
1
The Carpenter's Problem
407
2
A Data Fitting Problem
409
6
Exercises 10.1
415
1
Linear Programming II: Algebraic Solutions
416
4
Solving the Carpenter's Problem Algebraically
418
1
Computational Complexity: Intersection Point Enumeration
419
1
Exercises 10.2
420
1
Linear Programming III: The Simplex Method
420
10
Solving the Carpenter's Problem Using the Simplex Method
426
2
Another Maximization Problem
428
2
Exercises 10.3
430
1
Project: Exploring Modeling
430
1
Linear Programming IV: Sensitivity Analysis
430
7
Testing the Optimal Solution for Sensitivity to Change
431
2
Changes in the Amount of Resource Available
433
3
Economic Interpretation of a Unit Change in a Resource
436
1
Exercises 10.4
436
1
Numerical Search Methods
437
14
The Search Method Paradigm
437
4
Using the Dichotomous Search Algorithm
441
1
Golden Section Search Method
441
3
Using the Golden Search Algorithm
444
2
Model Fitting Criterion Revisited
446
1
Optimizing Industrial Flow
447
1
Exercises 10.5
448
1
Projects: Exploring Modeling
449
1
Further Reading
450
1
Modeling Using Calculus
451
32
Overview
452
1
Modeling Discrete and Continuous Behaviors
452
4
The Derivative as a Rate of Change
454
1
The Derivative as the Slope of the Tangent Line
455
1
Modeling Population Growth and Analytic Solutions
456
12
Separation of Variables
463
1
Exercises 11.2
464
3
Projects: Exploring Modeling
467
1
Further Reading
468
1
Graphical Solutions
468
6
Graphical Interpretation of the Derivative
468
2
Graphical the Derivative
470
1
Exponential Growth
470
1
The Logistic Growth Model
471
1
Stable and Unstable Rest Points
472
1
The Phase Line
472
1
Exercises 11.3
473
1
Project: Exploring Modeling
474
1
Numerical Approximation Methods
474
9
First-Order Initial-Value Problems
475
1
Approximating Solutions to Initial-Value Problems
476
1
Example Interest Compounded Continously
476
4
Exercises 11.4
480
1
Projects: Modeling Explorations
481
2
Part III Decisions in Organizing for Mathematics Instruction
483
132
Getting Ready for Teaching
485
32
Overview
486
1
Focus on the NCTM Standards
486
1
Focus on the Classroom
486
1
The Importance of Planning
487
2
Exercises 12.1
489
1
Focus on Key Ideas: Setting Goals
489
12
Long-Range Planning
489
2
Example Adjusting to Student Needs
491
2
Unit Planning
493
1
Daily Lesson Planning
494
4
Lesson 10: Sample Daily Lesson from Symmetry Unit
498
2
Exercises 12.2
500
1
Projects: Find Out for Yourself
501
1
Strategies for Effective Teaching
501
9
Cooperative Learning
502
2
Long-Term Group Activities
504
1
Short-Term Group Activities
504
1
Problem Solving and Discovery Activities
505
1
Student Involvement
506
1
Other Resources for Teachers
507
2
Exercises 12.3
509
1
Projects: Find Out for Yourself
509
1
Focus on the Students
510
4
Assessing Prior Knowledge
510
2
Motivating Student Learning
512
1
Monitoring Student Progress
513
1
Exercises 12.4
513
1
Projects: Find Out for Yourself
514
1
Reflecting on and Evaluating the Plan
514
1
Creating an Effective Lesson Plan
515
2
Elements of a Lesson Plan
515
1
References
516
1
Providing for Individual Differences
517
32
Overview
518
1
Focus on the NCTM Standards
518
1
Focus on the Classroom
518
1
Differences in How Students Approach Learning
519
3
Multiple Intelligences
519
1
Learning Styles
520
2
Exercises 13.1
522
1
Differences in Student Ability
522
11
Addressing Interest and Motivation
523
1
What Is Fair?
524
1
Global Warming
524
1
Addressing Students' Special Needs
525
1
Gifted Students
526
2
Students with Learning Difficulties
528
5
Exercises 13.2
533
1
Project: Find Out for Yourself
533
1
Cultural Differences and the Role of Culture in Learning
533
8
Addressing Cultural Backgrounds
534
2
Second Language Learners
536
2
Gender Issues
538
1
Resources
539
1
Exercises 13.3
540
1
Projects: Find Out for Yourself
541
1
Accommodating Differences
541
3
Teachers' Beliefs in Student Ability
541
1
Students' Self-Confidence
542
1
Easing Math Anxiety
543
1
Working with Parents and the Community
544
5
References
546
3
Assessing and Evaluating Mathematics Achievement
549
40
Overview
550
1
Focus on the NCTM Standards
550
1
Focus on the Classroom
550
1
Assessing and Evaluating
551
5
Assessing-Collecting the Information
551
2
Evaluating-Ranking and Valuing Student Work
553
1
Two Concepts: Validity and Reliability
553
1
Exercises 14.1
554
2
Student-Constructed Work
556
15
Short-Answer or Regular Student-Constructed Response Items
557
1
Regular Student-Constructed Response Item
558
1
Interpreting Rubric Codes
559
1
Extended Student-Constructed Response Items
560
1
Extended Student-Constructed Response Item for Grades 9-12
560
1
NAEP Extended Student-Constructed Response Item Rubric
561
1
Rubric for Tile Pattern Test Item
562
3
Exercises 14.2
565
6
Projects and Portfolios
571
5
Projects
571
1
A Mini-Project Investigation
572
1
A Highway Design Project
572
1
Portfolios
573
1
Methods of Development
573
1
Methods of Portfolio Management and Review
574
2
Exercises 14.3
576
1
Multiple Choice and Other Forms of Assessment
576
6
Multiple Choice
577
1
A 1986 Multiple-Choice Item for Grade 8
577
1
A 1992 Multiple-Choice Item for Grade 8
577
2
True-False
579
1
Role in Reliability
580
1
Exercises 14.4
581
1
Student Evaluation and Grades
582
7
Keeping Daily Grades
582
1
More Major Summative Assessments
583
2
Deciding Final Grades
585
1
Dealing with Questions on Grading
585
1
Dealing with Cheating
586
1
Exercises 14.5
587
1
References
587
2
Classroom Management in Mathematics
589
22
Overview
590
1
Focus on the NCTM Standards
590
1
Focus on the Classroom
590
1
Developing a Classroom That Works
591
12
Before Classes Begin
591
1
Getting a Scope of the Year
592
1
Organizing Your Room
592
3
Developing Your Routine
595
1
Beyond the Classroom
596
1
Beginning the School Year
597
2
Homework
599
1
Discipline
600
1
Summary
601
1
Exercises 15.1
601
1
Projects: Find Out for Yourself
602
1
Organizing Your Classroom to Improve Focus
603
4
Efficient Filing and Records Systems
603
1
Maintaining an Orderly Classroom
604
2
Exercises 15.2
606
1
Projects: Find Out for Yourself
606
1
Motivating Students by Involving Them
607
4
Exercises 15.3
609
1
Projects: Find Out for Yourself
610
1
References
610
1
Futures--Mathematics Education in the 21st Century
611
4
Reference
614
1
Index
615
14
Credits
629