Learning Mathematics in Elementary and Middle Schools

by Cathcart, W. George; Pothier, Yvonne; Vance, James; Bezuk, Nadine; Cathcart, W.

• ISBN13: 9780130322746
• ISBN10: 0130322741
• Edition: 2nd
• Format: Paperback
• Copyright: 2000-12-01
• Publisher: Prentice Hall

Summary

Text is appropriate for courses in Mathematics for the Elementary School. Built on the foundation of the new 2000 NCTM Principles and Standards, this major new entry for K-8 math methods has impacted the market because of its point-of-use links to the standards and its emphasis on the importance of a child-centered approachcreating a learning environment that informs teachers how to support children as they build understandings of math concepts. Designed to be neither skimpy nor exhaustive, this text presents theory in an accessible manner and models a wealth of practical activities for teaching. Five videos from the Annenberg/CPB TEACHING MATH video series bring real classrooms to life for teachers and are integrated into the text as four/color, resourceful inserts.

Table of Contents

Teaching Mathematics: Influences and Directions	1	(21)
Influences on Mathematics Education	2	(1)
Physchological Influences	2	(1)
Professional Influences	2	(6)
The Framework of the Principles and Standards for School Mathematics	3	(3)
Content Standards and Expectations in Each Grade Band	6	(1)
Integrating the Content and Process Standards	6	(2)
Integrating State and Local Standards with National Math Standards	8	(1)
Technological Influences	8	(3)
Calculators	8	(2)
Computers	10	(1)
Interactive Multimedia	11	(1)
Language Influences	11	(1)
Societal Influences	12	(1)
State Government	12	(1)
School Districts	12	(1)
Lobby Groups	12	(1)
Bandwagons	12	(1)
Research Influences	12	(1)
TIMSS	13	(1)
NAEP	13	(1)
Learner Influences	13	(1)
What Can Teachers Do?	13	(1)
Teacher Influences	14	(1)
Directions in Mathematics Education	14	(1)
Problem Solving	14	(1)
Communication	14	(1)
Reasoning and Proof	15	(1)
Connections	15	(2)
Integration with Other School Subjects	15	(1)
Integration with Real-World Settings	16	(1)
Representation of Mathematical Ideas	17	(1)
Equity	17	(1)
Technology	17	(1)
Computation and Estimation	17	(1)
Assessment	18	(1)
Parent Involvement	19	(1)
Conclusion	19	(1)
For Your Journal	19	(1)
For Your Portfolio	19	(1)
Resources for Teachers	20	(1)
Links to the Internet	20	(1)
Learning and Teaching Mathematics	21	(23)
Learning Styles	22	(1)
Learning Theories	22	(1)
The Behaviorist Approach	22	(1)
The Cognitive/Constructivist Approach	22	(5)
Cooperative Learning	23	(1)
Jean Piaget	23	(2)
Zoltan Dienes	25	(2)
Basic Principles Reviewed	27	(1)
Begin With Concrete Representation	27	(1)
Develop Understandings	28	(2)
Encourage Communication	30	(1)
Make Connections	31	(1)
Take Time to Motivate Children	32	(1)
Attitudes	32	(1)
Provide Opportunities for Practice	32	(2)
Games	33	(1)
Puzzles and Riddles	33	(1)
Surprises	33	(1)
Novel Algorithms	33	(1)
Novel Formats	33	(1)
The Calculator as Practice Tool	34	(1)
Personal Computers	34	(1)
Thinking About Teaching	34	(1)
Pre-Teaching Activities	34	(2)
Pre-Planning Considerations	34	(2)
Planning	36	(1)
Process of Teaching	36	(4)
Models for Teaching Mathematics	37	(3)
Post-Teaching Activities	40	(1)
Evaluation of Teaching	40	(1)
Reflection on Teaching	40	(1)
Thinking About the Curriculum	40	(1)
The Mathematics	41	(1)
The Activities	41	(1)
Conclusion	41	(1)
For Your Journal	42	(1)
For Your Portfolio	42	(1)
Resources for Teachers	42	(1)
Links to the Internet	42	(2)
Developing Mathematical Thinking and Problem-Solving Ability	44	(20)
Mathematical Considerations	45	(1)
What Is a Problem?	45	(1)
Types of Problems	46	(1)
Process Problems	46	(1)
Translation Problems	46	(1)
Application Problems	46	(1)
Puzzles	47	(1)
Ask Children to Write Problems	47	(1)
Posing Problems	48	(1)
The Problem-Solving Process	48	(1)
Understanding the Problem	48	(1)
Devising a Plan to Solve the Problem	49	(1)
Implementing a Solution Plan	49	(1)
Reflecting on the Problem	49	(1)
Problem-Solving Strategies	49	(2)
Dramatize or Model the Situation and Solution Process	51	(1)
Draw a Picture or Diagram	52	(1)
Construct a Table or Chart	52	(2)
Find a Pattern	54	(1)
Solve a Simpler Problem	54	(1)
Guess and Check	55	(1)
Working Backwards	56	(1)
Consider All Possibilities	56	(1)
Logical Reasoning	56	(1)
Change Your Point of View	57	(1)
Write an Open Sentence	57	(1)
Planning for Instruction	58	(1)
Selecting Appropriate Tasks and Materials	58	(1)
Problems That Are Motivating and Culturally Relevant	58	(1)
Problems with Missing, Extraneous, or Contradictory Information	58	(1)
Problems That Encourage the Use of Calculators, Computers, and Other Technology	58	(1)
Activities That Require the Use of a Variety of Problem-Solving Strategies	58	(1)
Activities That Promote Communication About Mathematical Thinking	58	(1)
Sources of Problems	59	(1)
The Teacher's Role	59	(1)
Before Children Begin to Solve the Problem	59	(1)
While Children Are Solving the Problem	59	(1)
After Children Solve the Problem	59	(1)
Organizing and Implementing Instruction	60	(1)
Classroom Climate	60	(1)
Grouping Children	60	(1)
Allocating Time	60	(1)
Assessing Children's Understanding	60	(1)
Changing the Difficulty of Problems	60	(1)
Problem Context	60	(1)
Problem Structure	61	(1)
Classroom Implications	61	(1)
Other Factors Contributing to Children's Difficulties in Problem Solving	61	(1)
Knowledge Factors	61	(1)
Beliefs and Affective Factors	62	(1)
Control	62	(1)
Sociocultural Factors	62	(1)
A Case to Consider the Children	62	(1)
Benefits of Using a Problem-Solving Approach to Mathematics Instruction	62	(1)
Conclusion	63	(1)
For Your Journal	63	(1)
For Your Portfolio	63	(1)
Resources for Teachers	63	(1)
Links to the Internet	63	(1)
Assessing Mathematics Understanding	64	(13)
The Assessment Standards	65	(1)
What Is Assessment?	65	(1)
Purposes of Assessment	66	(1)
Phases of Assessment	66	(1)
Assessment Choices	66	(1)
Achievement Tests	67	(1)
Standardized Tests	67	(1)
Teacher-Made Tests	67	(1)
Diagnostic Tests	67	(1)
Individualizing Assessment	67	(8)
Observation	67	(1)
Conferences and Interviews	68	(1)
Performance Assessment	69	(3)
Portfolio Assessment	72	(3)
Self-Assessment	75	(1)
Assessing Attitudes Toward Mathematics	75	(1)
Conclusion	76	(1)
For Your Journal	76	(1)
For Your Portfolio	76	(1)
Resources for Teachers	76	(1)
Links to the Internet	76	(1)
Developing Number Concepts	77	(22)
The Foundations of Number	78	(1)
Pre-Number Activities	78	(4)
Classification	78	(2)
Seriation	80	(1)
Patterns	81	(1)
One-to-One Correspondence	81	(1)
Conservation of Number	82	(1)
Number Meanings	82	(1)
Cardinal Use of Numbers	83	(1)
Ordinal Use of Numbers	83	(1)
Nominal Use of Numbers	83	(1)
Counting	83	(1)
Discrete and Continuous Quantities	83	(1)
Rote Counting	83	(1)
Rational Counting	83	(1)
Counting All, Counting On	84	(1)
Counting Back	84	(1)
Skip Counting	85	(1)
Representing Numbers	86	(1)
Concrete Models	87	(1)
Pictorial and Graphic Representation of Numbers	87	(1)
Symbolic Representation of Numbers	87	(3)
Numerals	90	(1)
Number Relationships	90	(1)
Order Relations	90	(1)
More Than, Fewer Than	90	(1)
One Greater Than, One Less Than	91	(1)
Part-Part-Whole Relationships	91	(4)
Relationship to Five and Ten	93	(1)
Using Action Language	94	(1)
Bidirectional Relationship of an Equation	95	(1)
Estimation	96	(1)
Conclusion	97	(1)
For Your Journal	97	(1)
For Your Portfolio	97	(1)
Resources for Teachers	97	(1)
Links to the Internet	98	(1)
Developing Understanding of Numeration	99	(28)
Numeration	100	(1)
Number Systems	100	(1)
Numeration Systems	100	(1)
Hindu-Arabic Numeration System	100	(1)
Base-Ten	100	(1)
Positional or Place Value	101	(1)
Multiplicative Principle	101	(1)
Additive Principle	102	(1)
Zero as a Placeholder	102	(1)
Understanding Place Value	102	(1)
Grouping	102	(2)
Communicating mathematics	104	(1)
Grouping by Tens	104	(1)
Equivalent Representations	104	(1)
Place Value	105	(1)
Developing Two-Digit Numbers	106	(1)
Introducing Base-Ten Blocks	107	(1)
Using Place-Value Mats	108	(1)
Introducing Nonproportional Materials	108	(3)
Using Hundreds Charts	111	(1)
Assessing Place-Value Knowledge	111	(1)
What Research Says About Place-Value Learning	112	(1)
Stages in Place-Value Development	112	(1)
Three-Digit Numbers	113	(1)
Number Meanings: Oral Expressions	114	(1)
Developing Number Relationships	114	(1)
Thinking and Writing About Numbers	115	(1)
Understanding Large Numbers	115	(1)
Number Names	115	(1)
Writing Consecutive Numbers	116	(1)
Number Periods	116	(1)
Magnitude of Numbers	117	(1)
Counting to a Thousand and Beyond	117	(1)
Expanded Notation	118	(1)
Rounding Numbers	118	(2)
Estimating	120	(1)
Consolidating Number Skills	121	(1)
Other Numeration Systems	122	(1)
Babylonian Numeration	123	(1)
Egyptian Numeration	123	(1)
Mayan Numeration	124	(1)
Roman Numeration	124	(1)
Conclusion	125	(1)
For Your Journal	125	(1)
For Your Portfolio	125	(1)
Resources for Teachers	125	(1)
Links to the Internet	126	(1)
Developing Whole-Number Operations: Meaning of Operations	127	(20)
Introduce Operations with Word Problems	128	(1)
A Model for Beginning With Word Problems	128	(1)
Encoding and Decoding Word Problems	129	(1)
Understanding Addition and Subtraction	130	(1)
Types of Addition and Subtraction Word Problems	130	(3)
Examples of Each Problem Type	130	(3)
Using Models to Solve Addition and Subtraction Problems	133	(4)
Direct Modeling	133	(3)
Using Measurement Models	136	(1)
Writing Number Sentences for Addition and Subtraction	137	(1)
Understanding Multiplication and Division	138	(1)
Types of Multiplication and Division Word Problems	138	(4)
Examples of Each Problem Type	138	(2)
A Note on Multiplication and Division Word Problems	140	(2)
Division With Remainders	142	(1)
Using Models to Solve Multiplication and Division Problems	142	(2)
Modeling Equal Groups and Multiplicative Comparison Problems	142	(1)
Modeling Area and Array Problems	143	(1)
Modeling Combination Problems	143	(1)
Other Models for Multiplication	144	(1)
An Instructional Sequence for Modeling Multiplication and Division	144	(1)
Another Word About Notation and Children's Language	144	(2)
Conclusion	146	(1)
For Your Journal	146	(1)
For Your Portfolio	146	(1)
Resources for Teachers	146	(1)
Links to the Internet	146	(1)
Developing Whole Number Operations: Mastering the Basic Facts	147	(16)
What Are Basic Facts?	148	(1)
A Three-Step Approach to Fact Mastery	148	(1)
Step 1: Understanding the Meaning of the Operations	149	(1)
Step 2: Using Thinking Strategies to Retrieve Facts	149	(1)
Step 3: Consolidating Activities for Drill and Practice	149	(1)
Addition and Subtraction Facts	149	(1)
Thinking Strategies for Addition and Subtraction	150	(2)
Counting On	150	(1)
Counting On in Subtraction	150	(1)
Counting Back	151	(1)
One More or One Less Than a Known Fact	151	(1)
Compensation	151	(1)
Using Thinking Strategies to Organize Instruction	152	(1)
Mathematical Properties of Addition and Subtraction	152	(1)
Commutative Property	152	(1)
Associative Property	153	(1)
Addition Property of Zero	153	(1)
Fact Families for Addition and Subtraction	153	(1)
Multiplication and Division Facts	154	(1)
Thinking Strategies for Multiplication and Division	154	(2)
Repeated Addition	154	(1)
Skip Counting	154	(1)
Splitting the Product Into Known Parts	154	(1)
Facts-of-Five	155	(1)
Pattern Strategy	155	(1)
Mathematical Properties of Multiplication	156	(3)
Commutative Property	156	(1)
Associative Property	157	(1)
Distributive Property of Multiplication Over Addition	157	(1)
The Multiplication Property of One	158	(1)
The Role of Zero in Multiplication	158	(1)
A Note on Division by Zero	158	(1)
Fact Families for Multiplication and Division	159	(1)
Consolidating Activities for Drill and Practice	159	(1)
Games	160	(1)
Puzzles and Riddles	161	(1)
Novel Formats	161	(1)
Computer Software	161	(1)
Conclusion	161	(1)
For Your Journal	162	(1)
For Your Portfolio	162	(1)
Resources for Teachers	162	(1)
Links to the Internet	162	(1)
Estimation and Computational Procedures for Whole Numbers	163	(48)
Computational Estimation and Mental Arithmetic	165	(1)
Mental Computation	165	(2)
Strategies for Mental Computation	165	(2)
Estimation	167	(3)
Strategies for Computational Estimation	167	(3)
Paper-and-Pencil Computation	170	(1)
An Instructional Philosophy	170	(1)
Prerequisites	171	(1)
Other Considerations	171	(4)
Extension Facts	172	(1)
Higher-Decade Addition	172	(1)
Column Addition	172	(1)
Types of Models	173	(1)
Language	173	(1)
Role of the Calculator	174	(1)
Error Patterns	174	(1)
Addition	175	(6)
Unstructured Concrete Manipulations	176	(1)
Recording	177	(1)
Symbolic Representation	177	(2)
Other Addition Algorithms	179	(1)
Common Error Patterns in Addition	180	(1)
Subtraction	181	(8)
Unstructured Concrete Manipulations	181	(2)
Recording	183	(1)
Symbolic Representation	183	(1)
Zeroes in the Minuend	184	(1)
Communication	185	(1)
Interpretations of Subtraction and the Algorithms	185	(1)
Other Subtraction Algorithms	186	(2)
Common Error Patterns in Subtraction	188	(1)
Multiplication	189	(7)
One-Digit Multiplier	189	(1)
Two-Digit Multiplier	190	(3)
Three- and More-Digit Multipliers	193	(1)
Zero in the Multiplier	193	(1)
Zero in the Multiplicand	193	(1)
Other Algorithms	193	(2)
Enrichment Extensions	195	(1)
Connections	195	(1)
Common Error Patterns in Multiplication	195	(1)
Division	196	(6)
One-Digit Divisors	197	(1)
Cases with Zero	197	(1)
Not Enough Hundreds to Share	198	(1)
Short Division	198	(1)
Making Sense of Remainders	199	(1)
Two-Digit Divisors	200	(1)
The Measurement Approach	201	(1)
Common Error Patterns in Division	201	(1)
Checking Calculations	202	(3)
Consolidating Skills	205	(1)
Puzzles	205	(1)
Games	205	(1)
Riddles	206	(1)
Computer Software	207	(1)
Other Algorithms	208	(1)
Problem Solving	209	(1)
Conclusion	209	(1)
For Your Journal	209	(1)
For Your Portfolio	209	(1)
Resources for Teachers	210	(1)
Links to the Internet	210	(1)
Developing Fraction Concepts	211	(25)
What Are Fractions?	212	(1)
What Do Children Know About Fractions?	212	(1)
What Should Children Understand About Fractions?	212	(1)
Developing Fraction Concepts and Number Sense	213	(1)
Number Sense With Fractions	213	(1)
Assessing Fraction Number Sense	213	(1)
Developing the Meaning of Half	213	(1)
Fraction Names	214	(2)
Fraction Symbolism	216	(1)
Different Units	216	(1)
Different Interpretations of Fractions	216	(10)
Part-Whole Interpretations	218	(2)
Part-of-a-Set Model	220	(6)
Other Interpretations of Fractions	226	(1)
Developing Comparison and Ordering of Fractions	226	(1)
Comparing and Ordering Fractions	226	(2)
Using a Calculator to Compare Fractions	228	(1)
Relative Size of Fractions	228	(1)
Improper Fractions and Mixed Numbers	228	(2)
Using the Math Explorer Calculator	230	(1)
Understanding Equivalent Fractions	230	(1)
Equivalent Fractions	230	(2)
Renaming and Simplifying Fractions	232	(2)
Renaming and Simplifying Fractions With a Calculator	232	(2)
Conclusion	234	(1)
For Your Journal	234	(1)
For Your Portfolio	234	(1)
Resources for Teachers	234	(1)
Links to the Internet	235	(1)
Developing Fraction Computation	236	(22)
Prerequisites for Fraction Computation	238	(1)
Introducing Computation	238	(1)
Developmental Activities	238	(1)
About Algorithms	238	(1)
Connecting Operations on Whole Numbers With Operations on Fractions	238	(1)
Properties	239	(1)
Addition and Subtraction of Fractions	239	(1)
Developing Addition Procedures	240	(4)
Addition Algorithms	242	(2)
Developing Subtraction Procedures	244	(2)
Early Subtraction Activities	244	(1)
Renaming Fractions	245	(1)
Symbolic Fraction Algorithm	246	(1)
Multiplication and Division of Fractions	246	(1)
General Considerations	246	(1)
Developing Fraction Multiplication	247	(5)
Multiplying a Fraction by a Whole Number	247	(1)
Multiplying a Whole Number by a Fraction	248	(1)
Multiplying a Fraction by a Fraction	249	(1)
Multiplying Mixed Numbers	250	(2)
Other Considerations	252	(1)
Developing Fraction Division	252	(3)
Whole Number Divided by a Fraction: Even Division	252	(1)
Whole Number Divided by a Fraction: Uneven Division	253	(1)
Fraction Divisor and Whole Number Dividend	253	(1)
Fraction Divisor and Dividend	253	(1)
Mixed Number Dividend	253	(1)
Developing a Symbolic Division Algorithm	253	(2)
Computing Fractions With a Calculator	255	(1)
Mental Arithmetic and Estimation	255	(1)
Assessing Fraction Knowledge	255	(1)
Conclusion	256	(1)
For Your Journal	256	(1)
For Your Portfolio	257	(1)
Resources for Teachers	257	(1)
Links to the Internet	257	(1)
Developing Decimal Concepts and Computation	258	(22)
Instructional Considerations	260	(1)
Connections to Familiar Concepts	260	(3)
Whole-Number Connection	260	(3)
Fraction Number Connection	263	(1)
Reading and Writing Decimals	263	(1)
Decimal Point	263	(1)
Decimal Names	263	(1)
Decimal Notation	264	(1)
Developing Decimal Number Sense	264	(1)
Base-Ten Blocks	264	(1)
A ``Tens'' Block as the Unit	264	(1)
A ``Hundreds'' Block as the Unit	265	(1)
A ``Thousands'' Block (Large Cube) as the Unit	265	(1)
Decimal Squares	265	(1)
Measurement Sticks	266	(1)
Graph Paper	266	(2)
Number Line	267	(1)
Money	268	(1)
Other Materials to Model Decimals	268	(1)
Equivalent Decimals	268	(1)
Ordering and Comparing Decimals	268	(1)
Computation	269	(1)
Addition and Subtraction	270	(2)
Multiplication	272	(3)
Division	275	(1)
Estimating With Decimals	276	(1)
Writing Fractions as Decimals	277	(1)
Using the Math Explorer Calculator	277	(1)
Terminating and Repeating Decimals	278	(1)
Writing Decimals as Fractions	278	(1)
Scientific Notation	278	(1)
Conclusion	279	(1)
For Your Journal	279	(1)
For Your Portfolio	279	(1)
Resources for Teachers	279	(1)
Links to the Internet	279	(1)
Understanding Ratio, Proportion, and Percent	280	(13)
Ratio and Rate	281	(1)
Language and Notation	281	(1)
Ratio and Rational Number	282	(1)
Proportion	282	(1)
Proportional Reasoning	282	(1)
Equal Ratios	283	(1)
Comparing Ratios	284	(1)
Cross Products	284	(1)
Scale Drawings	285	(1)
Comparison Shopping	286	(1)
Percent	286	(1)
Meaning and Notation	286	(1)
Number Sense	286	(1)
Fraction and Decimal Equivalents	287	(1)
Decimals as Percents	287	(1)
Fractions as Percents	288	(1)
Percents as Fractions	288	(1)
Finding the Percent of a Number	288	(2)
Finding the Percent	290	(1)
Other Procedures for Solving Percent Problems	290	(1)
The Proportion Method	290	(1)
The Equation Method	290	(1)
Unitary Analysis Method	291	(1)
Assessment and Instruction	291	(1)
Assessing Proportional Reasoning	291	(1)
Everyday Life Problem Settings	291	(1)
Conclusion	292	(1)
For Your Journal	292	(1)
For Your Portfolio	292	(1)
Resources for Teachers	292	(1)
Links to the Internet	292	(1)
Developing Geometric Thinking and Spatial Sense	293	(31)
Development of Geometric Thinking	295	(1)
The van Hiele Levels of Geometric Thought	295	(1)
Comments on the Levels of Thought	296	(1)
Other Instructional Notes	296	(1)
Developing the Language	296	(1)
Method of Instruction	297	(1)
Connecting With the World	297	(1)
Learning About Topology	297	(1)
Things That Change and Things That Do Not Change	297	(1)
Place and Order	298	(1)
Mazes and Networks	298	(1)
Distortion of Figures	298	(1)
Learning About Euclidean Geometry	298	(1)
Three-Dimensional Shapes	298	(4)
Polyhedra	299	(3)
Learning About Three-Dimensional Shapes	302	(1)
Comparing Polyhedra	302	(1)
Constructing 3-D Shapes	302	(2)
Learning About Two-Dimensional Figures	304	(1)
Polygons	304	(6)
Triangles	306	(1)
Quadrilaterals	306	(3)
Circles	309	(1)
Learning About Symmetry, Congruence, and Similarity	310	(1)
Symmetry	310	(2)
Congruence and Similarity	312	(1)
Learning About Transformational Geometry	313	(1)
Rigid Transformations	313	(1)
Learning About Tessellations	314	(1)
Developing Spatial Sense	315	(1)
Tangram Puzzles	316	(2)
Polyominoes	318	(1)
Describing Figures	318	(1)
Dissection Motion Operations	319	(1)
Learning About Coordinate Geometry	320	(1)
Learning About Curve Stitching	321	(1)
Conclusion	322	(1)
For Your Journal	322	(1)
For Your Portfolio	322	(1)
Resources for Teachers	323	(1)
Links to the Internet	323	(1)
Developing Measurement Concepts and Skills	324	(28)
Concepts and Instructional Sequence	326	(1)
What Is Measurement?	326	(1)
Instructional Sequence	326	(4)
Perception and Direct Comparison	326	(1)
Nonstandard Units	327	(1)
Standard Units	328	(1)
Instruments	329	(1)
Formulas	330	(1)
Problem Solving and Applications	330	(1)
Summary of Teaching Sequence	330	(1)
Teaching Strategies and Learning Activities	330	(1)
Length	330	(5)
Length: Perception and Direct Comparison	331	(1)
Length: Nonstandard Units	331	(1)
Length: Standard Units	332	(3)
Area	335	(4)
Perception and Direct Comparison	335	(1)
Nonstandard Units	335	(1)
Standard Units	336	(3)
Volume	339	(2)
Perception and Direct Comparison	339	(1)
Nonstandard Units	339	(1)
Standard Units	340	(1)
Capacity	341	(2)
Perception and Direct Comparison	341	(1)
Nonstandard Units	342	(1)
Standard Units	343	(1)
Mass	343	(2)
Perception and Direct Comparison	344	(1)
Nonstandard Units	344	(1)
Standard Units	344	(1)
Time	345	(2)
Perception and Direct Comparison	345	(1)
Nonstandard Units	345	(1)
Standard Units	346	(1)
Temperature	347	(1)
Perception and Direct Comparison	347	(1)
Standard Units	348	(1)
Angle	348	(3)
Perception and Direct Comparison	348	(1)
Nonstandard Units	349	(1)
Standard Units	349	(2)
Conclusion	351	(1)
For Your Journal	351	(1)
For Your Portfolio	351	(1)
Resources for Teachers	351	(1)
Links to the Internet	351	(1)
Collecting, Organizing, and Interpreting Data	352	(31)
Collecting and Organizing Data	354	(1)
Graphing Data	355	(1)
Early Experiences	356	(6)
Concrete Stage	356	(1)
Concrete-Pictorial Stage	357	(1)
Pictorial-Abstract Stage	358	(2)
Abstract Stage	360	(2)
Coordinate Graphing	362	(1)
Circle or Pie Graphs	363	(3)
Histograms, Line Plots, and Stem-and-Leaf Plots	366	(1)
Histograms and Line Plots	366	(1)
Stem-and-Leaf Plots	366	(1)
Computer-Generated Graphs	367	(1)
Interpreting Data: Statistics	368	(1)
Frequency	368	(1)
Central Tendency	368	(3)
Mode	369	(1)
Median	369	(1)
Mean	370	(1)
Variation	371	(2)
Box-and-Whisker Plots	371	(2)
Interpretation Data: Probability	373	(1)
Overview	373	(1)
General Teaching Considerations	374	(6)
Early Experiences	374	(1)
Upper Elementary	375	(3)
Middle Grades	378	(2)
Technology and Data	380	(1)
Conclusion	381	(1)
For Your Journal	381	(1)
For Your Portfolio	381	(1)
Resources for Teachers	381	(1)
Links to the Internet	382	(1)
Developing Integers and Algebraic Thinking	383	(19)
Integers	385	(1)
Introducing the Integers	385	(1)
Ordering the Integers	385	(1)
Addition	386	(1)
Electric Charges Model	386	(1)
Number Line Model	386	(1)
Subtraction	386	(2)
Number Pattern Approach	386	(1)
Number Line Model	386	(1)
Electric Charges Model	387	(1)
The Rule	387	(1)
Multiplication	388	(1)
Division	388	(1)
Assessment	388	(1)
Practice Settings	388	(1)
Variables	389	(1)
Meanings	389	(1)
Misconceptions	390	(1)
Expressions and Number Sentences	390	(1)
Evaluating Expressions	390	(1)
Order of Operations	391	(2)
Properties of Operations	393	(1)
Formulas	393	(1)
Algebraic Expressions	394	(1)
Solving Equations	394	(1)
Inequalities	395	(1)
Variables and Computer Programs	395	(1)
Functions	396	(1)
What Is a Function?	396	(1)
Patterns and Relationships	397	(1)
Function Machines and Tables	397	(1)
Sequences	397	(1)
Calculator Functions	398	(1)
Graphing	398	(1)
Instruction and Assessment Activities	398	(2)
Conclusion	400	(1)
For Your Journal	401	(1)
For Your Portfolio	401	(1)
Resources for Teachers	401	(1)
Links to the Internet	401	(1)
Appendix: Blackline Masters	402	(17)
References	419	(10)
Index	429