Helping Children Learn Mathematics

by Reys, Robert E.; Lingquist, Mary M.; Lambdin, Diana V.; Smith, Nancy L.; Suydam, Marilyn N.

• ISBN13: 9780471367857
• ISBN10: 0471367850
• Edition: 6th
• Format: Paperback
• Copyright: 2000-07-01
• Publisher: John Wiley & Sons Inc

Summary

A best-selling activity-oriented approach to methods of teaching elementary and middle school mathematics that assists elementary school preservice/inservice teachers in helping children learn mathematics meaningfully. The new Sixth Edition has been designed to meet the needs of the changing elementary school mathematics curriculum:

Table of Contents

Preface	iv
School Mathematics	1	(8)
Introduction	1	(1)
What Is Mathematics?	1	(1)
What Determines the Mathematics Being Taught?	2	(3)
Needs of the Subject	2	(1)
Needs of the Child	3	(1)
Needs of Society	4	(1)
Where Can You Turn?	5	(2)
National Guidelines for School Mathematics	5	(1)
State Guidelines	5	(1)
Research	5	(1)
History	5	(1)
Textbooks and Other Materials	6	(1)
Electronic Materials	6	(1)
Testing	6	(1)
Professional Organizations	7	(1)
Professional Development	7	(1)
Other Teachers	7	(1)
What Is Your Role Now?	7	(1)
A Glance at Where We've Been	7	(1)
Things to Do: From What You've Read	7	(1)
Things to Do: Going Beyond This Book	8	(1)
Selected References	8	(1)
Historical References	8	(1)
Learning Mathematics with Understanding	9	(18)
Introduction	9	(1)
How Do Children Learn Mathematics?	9	(4)
Building Behavior	10	(1)
Constructing Understanding	11	(2)
What Mathematical Knowledge Should Be Learned?	13	(1)
Implications of What We Know about Learning Mathematics	14	(9)
Actively Involve Students	14	(1)
Learning Is Developmental	14	(1)
Build on Previous Learning	15	(1)
Communication Contributes Significantly to Understanding	16	(1)
Good Questions Facilitate Learning	17	(1)
Manipulatives Aid Learning	17	(2)
Metacognition Affects Learning	19	(1)
Teacher Attitudes Are Vital	19	(1)
Experiences Influence Anxiety	20	(1)
Gender Aptitudes Are Equal	21	(1)
Retention Can Improve	22	(1)
A Glance at Where We've Been	23	(1)
Things to Do: From What You've Read	23	(1)
Things to Do: Going Beyond This Book	24	(1)
Book Nook for Children	25	(1)
Selected References	25	(2)
Teaching: Your Role	27	(23)
Snapshot of a Lesson Plan	27	(1)
Introduction	28	(1)
Recommendations for Teaching from What We Know about Learning Mathematics	28	(2)
Expectations	29	(1)
Tasks	29	(1)
Discourse	29	(1)
Tools	29	(1)
Learning Environment	29	(1)
The Professional Standards for Teaching Mathematics	30	(2)
The Importance of Planning	32	(7)
Levels of Planning	34	(1)
Components of a Lesson Plan	34	(1)
Planning Different Types of Lessons	35	(3)
Using the Strategic Moment	38	(1)
Adapting Textbook-Based Lessons	38	(1)
Using Children's Literature to Motivate Mathematics Lessons	39	(1)
Points to Consider in Planning and Teaching	39	(7)
Maintaining a Focus on the Mathematics	39	(1)
Meeting the Needs of All Students	40	(2)
Grouping for Instruction	42	(1)
The Importance of Questioning	43	(1)
The Use of Manipulative Materials and Models	44	(1)
The Role of Drill and Practice	44	(1)
Incorporating Technology	45	(1)
Integrating Assessment and Instruction	46	(1)
A Glance at Where We've Been	46	(1)
Things to Do: From What You've Read	47	(1)
Things to Do: Going Beyond This Book	48	(1)
Selected References	48	(2)
Assessment: Enhanced Learning and Teaching	50	(26)
Snapshot of a Teacher's Thoughts on Assessment	50	(1)
Introduction	51	(2)
Making Instructional Decisions	52	(1)
Monitoring Student Progress	52	(1)
Evaluating Student Achievement	52	(1)
Phases of Assessment	53	(1)
Ways to Gather and Make Judgments about Evidence	53	(14)
Observation	54	(2)
Questioning	56	(1)
Interviewing	57	(1)
Performance Tasks	58	(2)
Self-Assessment and Peer Assessment	60	(2)
Work Samples	62	(1)
Portfolios	63	(1)
Writings	64	(1)
Teacher-Designed Written Tests	64	(1)
Achievement Tests	65	(2)
Keeping Records and Communicating about Assessments	67	(5)
Recording the Information	67	(4)
Communicating the Information	71	(1)
A Glance at Where We've Been	72	(1)
Things to Do: From What You've Read	73	(1)
Things to Do: Going Beyond This Book	73	(1)
Selected References	74	(2)
Five Mathematical Processes	76	(31)
Snapshot of a Classroom Investigation	76	(1)
Introduction	77	(1)
The Five Mathematical Processes	77	(10)
Problem Solving	77	(4)
Reasoning and Proof	81	(2)
Communication	83	(2)
Connections	85	(1)
Representations	86	(1)
Teaching Mathematics via Problem Solving	87	(1)
What Is a Problem and What Is Problem Solving?	88	(1)
How Can Problem Solving Be Taught Effectively?	89	(6)
Time	91	(1)
Planning	91	(1)
Resources	92	(1)
Technology	93	(1)
Class Management	93	(1)
Problem Posing	94	(1)
What Problem-Solving Strategies Should Be Taught?	95	(6)
Act It Out	96	(1)
Make a Drawing or Diagram	97	(1)
Look for a Pattern	97	(1)
Construct a Table	97	(1)
Identify All Possibilities	98	(1)
Guess and Check	99	(1)
Work Backward	99	(1)
Write an Open Sentence	100	(1)
Solve a Simpler or Similar Problem	100	(1)
Change Your Point of View	101	(1)
The Importance of Looking Back	101	(1)
Generalize	101	(1)
Check the Solution	102	(1)
Find Another Way to Solve It	102	(1)
Find Another Solution	102	(1)
Study the Solution Process	102	(1)
Using Problem-Solving Opportunities	102	(1)
How Can Problem Solving Be Assessed?	102	(2)
Observations	103	(1)
Interviews	103	(1)
Inventories and Checklists	103	(1)
Paper-and-Pencil Tests	104	(1)
A Glance at Where We've Been	104	(1)
Things to Do: From What You've Read	104	(1)
Things to Do: Going Beyond This Book	104	(1)
Book Nook for Children	105	(1)
Selected References	105	(2)
Beginning Number Sense: Counting and Early Benchmarks	107	(25)
Snapshot of a Lesson	107	(1)
Introduction	108	(2)
Prenumber Concepts	110	(7)
Classification	110	(3)
Patterns	113	(1)
Comparisons	114	(1)
Conservation	115	(1)
Group Recognition	116	(1)
Counting	117	(5)
Counting principles	118	(1)
Counting Stages	118	(2)
Counting Strategies	120	(1)
Counting Practice	121	(1)
Early Number Development	122	(4)
Developing Number Benchmarks	123	(1)
Making Connections	124	(2)
Cardinal, Ordinal, and Nominal Numbers	126	(2)
Writing Numerals	128	(1)
A Glance at Where We've Been	129	(1)
Things to Do: From What You've Read	129	(1)
Things to Do: Going Beyond This Book	130	(1)
Book Nook for Children	130	(1)
Selected References	131	(1)
Extending Number Sense: Place Value	132	(26)
Snapshot of a Lesson	132	(1)
Introduction	133	(1)
Our Numeration System	134	(4)
Thinking Place Value	134	(1)
Grouping or Trading	134	(1)
Nature of Place Value	135	(2)
Modeling	137	(1)
Developing Place Value	138	(10)
A Place to Start	138	(7)
Counting and Patterns	145	(2)
Regrouping and Naming	147	(1)
Reading and Writing Numbers	148	(3)
Estimation and Rounding	151	(4)
Estimation	152	(1)
Rounding	153	(2)
A Glance at Where We've Been	155	(1)
Things to Do: From What You've Read	155	(1)
Things to Do: Going Beyond This Book	156	(1)
Book Nook for Children	156	(1)
Selected References	157	(1)
Operations: Meanings and Basic Facts	158	(31)
Snapshot of a Lesson	158	(1)
Introduction	159	(1)
Helping Children Develop Number Sense and Computational Fluency	160	(1)
Facility with Counting	160	(1)
Experience with a Variety of Concrete Situations	160	(1)
Familiarity with Many Problem-Solving Contexts	160	(1)
Experience in Using Language to Communicate Mathematical Ideas	161	(1)
Developing Meanings for the Operations	161	(6)
Addition and Subtraction	162	(1)
Multiplication and Division	163	(4)
Mathematical Properties	167	(1)
Overview of Basic Fact Instruction	167	(5)
Get Ready: Starting Where Children Are	169	(1)
Get Set: Building Understanding of the Basic Facts	169	(1)
Go: Mastering the Basic Facts	170	(2)
Thinking Strategies for Basic Facts	172	(13)
Thinking Strategies for Addition Facts	172	(6)
Thinking Strategies for Subtraction Facts	178	(1)
Thinking Strategies for Multiplication Facts	179	(4)
Thinking Strategies for Division Facts	183	(2)
A Glance at Where We've Been	185	(1)
Things to Do: From What You've Read	185	(1)
Things to Do: Going Beyond This Book	186	(1)
Book Nook for Children	186	(1)
Selected References	187	(2)
Operation Sense: Calculators, Mental Computation, and Estimation	189	(22)
Snapshot of a Lesson	189	(1)
Introduction	190	(1)
Computational Tools	190	(18)
Calculators	192	(2)
Mental Computation	194	(5)
Computational Estimation	199	(9)
A Glance at Where We've Been	208	(1)
Things to Do: From What You've Read	208	(1)
Things to Do: Going Beyond This Book	209	(1)
Selected References	209	(2)
Operations: Standard and Alternative Algorithms	211	(26)
Snapshot of a Lesson	211	(1)
Introduction	212	(1)
Role of Materials in Learning Algorithms	213	(1)
Importance of Place-Value Ideas	213	(1)
Addition	213	(4)
Column Addition	215	(1)
Higher-Decade Addition	216	(1)
Subtraction	217	(3)
Zeros in the Sum	219	(1)
Multiplication	220	(5)
Multiplication with One-Digit Multipliers	220	(1)
Multiplication with Two-Digit Multipliers	221	(2)
Multiplying by 10 and Multiples of 10	223	(2)
Multiplying with Zeros	225	(1)
Multiplication with Large Numbers	225	(1)
Division	225	(6)
Division with One-Digit Divisors	226	(3)
Division with Two-Digit Divisors	229	(1)
Making Sense of Division with Remainders	230	(1)
Checking	231	(1)
Choosing Appropriate Ways	232	(1)
A Glance at Where We've Been	233	(1)
Things to Do: From What You've Read	233	(1)
Things to Do: Going Beyond This Book	234	(1)
Book Nook for Children	235	(1)
Selected References	235	(2)
Fractions and Decimals: Meanings and Operations	237	(26)
Snapshot of a Lesson	237	(1)
Introduction	238	(1)
Conceptual Development of Fractions	238	(11)
Three Meanings of Fractions	239	(1)
Models of the Part-Whole Meaning	239	(2)
Making Fractions Meaningful	241	(3)
Ordering Fractions and Equivalent Fractions	244	(3)
Benchmarks	247	(2)
Mixed Numbers and Improper Fractions	249	(1)
Operations with Fractions	249	(6)
Addition and Subtraction	250	(1)
Multiplication	251	(3)
Division	254	(1)
Development of Decimals	255	(3)
Relationship to Common Fractions	255	(2)
Relationship to Place Value	257	(1)
Ordering and Rounding Decimals	258	(1)
Decimal Operations	258	(2)
Addition and Subtraction	258	(1)
Multiplication and Division	259	(1)
A Glance at Where We've Been	260	(1)
Things to Do: From What You've Read	261	(1)
Things to Do: Going Beyond This Book	261	(1)
Book Nook for Children	262	(1)
Selected References	262	(1)
Ratio, Proportion, and Percent: Meanings and Applications	263	(16)
Snapshot of a Lesson	263	(1)
Introduction	264	(1)
Ratios	265	(2)
Proportions	267	(3)
Percents	270	(6)
Understanding Percents	271	(3)
Applying Percents	274	(2)
A Glance at Where We've Been	276	(1)
Things to Do: From What You've Read	276	(1)
Things to Do: Going Beyond This Book	277	(1)
Book Nook for Children	278	(1)
Selected References	278	(1)
Patterns, Relationships, and Algebra	279	(16)
Snapshot of a Lesson	279	(1)
Introduction	280	(1)
Patterns	280	(4)
Equality	280	(1)
Change	280	(1)
Generalizations	281	(1)
Repeating Patterns	281	(1)
Growing Patterns	282	(2)
Number Patterns	284	(1)
Relationships	284	(2)
Number Theory	286	(7)
Number Theory	286	(1)
Specific Number Theory Topics	287	(6)
A Glance at Where We've Been	293	(1)
Things to Do: From What You've Read	293	(1)
Things to Do: Going Beyond This Book	294	(1)
Book Nook for Children	294	(1)
Selected References	294	(1)
Geometry	295	(24)
Snapshot of a Lesson	295	(1)
Introduction	296	(2)
Geometry in School Mathematics	296	(1)
Geometric Characteristics and Properties	296	(1)
Location and Spatial Relationships	297	(1)
Transformations	297	(1)
Visualization and Spatial Reasoning	298	(1)
Solid Geometry	298	(7)
Describing and Classifying Objects	298	(3)
Constructing to Explore and Discover	301	(1)
Polyhedron Models	301	(1)
Relating Three Dimensions to Two	302	(3)
Plane Geometry	305	(10)
Properties of a Shape	305	(6)
Names of Geometric Shapes	311	(1)
Relationships between Shapes	312	(1)
Classification Schemes	313	(2)
A Glance at Where We've Been	315	(1)
Things to Do: From What You've Read	316	(1)
Things to Do: Going Beyond This Book	316	(1)
Book Nook for Children	317	(1)
Selected References	317	(2)
Measurement	319	(25)
Snapshot of a Lesson	319	(1)
Introduction	320	(2)
Teaching Measurement	322	(12)
Identifying Attributes	323	(3)
Units of Measure	326	(3)
Instruments for Measuring	329	(2)
Formulas for Measuring	331	(3)
Comparing Measurements	334	(3)
Equivalences	335	(1)
Conversions	336	(1)
Estimating Measurements	337	(2)
Connecting Attributes	339	(1)
Area and Shape	339	(1)
Volume and Shape	339	(1)
Perimeter and Area	339	(1)
Volume and Surface Area	339	(1)
Perimeter and Dimensions	340	(1)
Metric Relations	340	(1)
A Glance at Where We've Been	340	(2)
Things to Do: From What You've Read	342	(1)
Things to Do: Going Beyond This Book	342	(1)
Book Nook for Children	343	(1)
Selected References	343	(1)
Data Analysis, Statistics, and Probability	344	(30)
Snapshot of a Lesson	344	(2)
Introduction	346	(1)
Data Analysis	347	(2)
Posing a Question and Collecting Data	347	(1)
Surveys	348	(1)
Experiments	349	(1)
Simulations	349	(1)
Displaying Data	349	(7)
Quick and Easy Graphing Materials	350	(1)
Line Plots and Stem-and-Leaf Plots	350	(2)
Picture Graphs	352	(1)
Bar Graphs	353	(1)
Pie Graphs	353	(1)
Line Graphs	354	(1)
Box Plots	355	(1)
Graphical Roundup	356	(1)
Analyzing and Communicating Results	356	(3)
Data Sense	356	(1)
Analysis	356	(2)
Misleading Graphs	358	(1)
Communicating Results	359	(1)
Descriptive Statistics	359	(5)
Measure of Variation	359	(1)
Measures of Central Tendency or Averages	360	(2)
Choosing Averages	362	(2)
Probability	364	(6)
Probability of an Event	365	(2)
Randomness	367	(1)
Independence of Events	368	(2)
A Glance at Where We've Been	370	(1)
Things to Do: From What You've Read	370	(1)
Things to Do: Going Beyond This Book	371	(1)
Book Nook for Children	372	(1)
Selected References	372	(2)
Appendix A Standards and Expectations, NCTM 2000	374	(7)
Appendix B Blackline Masters	381	(25)
Appendix C Publishers and Distributors	406	(1)
Index	407