Mathematics Methods and Modeling for Today's Mathematics Classroon

by Dossey, John A.; McCrone, Sharon; Giordano, Frank R.; Mwir, Maurice D.; Dossey, John A.

• ISBN13: 9780534366049
• ISBN10: 053436604X
• Format: Paperback
• Copyright: 2001-08-03
• Publisher: Brooks Cole

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