Mathematics Methods and Modeling for Today's Mathematics Classroon,9780534366049

Mathematics Methods and Modeling for Today's Mathematics Classroon

by ; ; ; ;
ISBN13: 9780534366049
ISBN10: 053436604X
Format: Paperback
Pub. Date: 2001-08-03
Publisher(s): Brooks Cole
List Price: $183.70

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Summary

Written by a team of nationally known mathematics educators and mathematicians who have experience in teaching at the public school and university levels, this book builds on the recommendations of NCTM's Principles and Standards. It provides guidance in teaching content and developing processes in readers from the middle school through high school. Its emphasis on mathematical modeling as an approach focuses on the development of the competencies needed to teach contemporary curricular materials in today's schools.

Table of Contents

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

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