Mathematics Methods for Elementary and Middle School Teachers

by Mary M. Hatfield

• ISBN13: 9780471365440
• ISBN10: 0471365440
• Edition: 4th
• Format: Paperback
• Copyright: 2000-01-01
• Publisher: John Wiley & Sons Inc

Summary

Innovative techniques and approaches for modern mathematics teaching. Always on the cutting-edge of mathematics teaching, "Mathematics Methods for Elementary and Middle School Teachers, Fifth Edition continues to integrate technology with hands-on experience and the latest research and standards. Highlights: Praxis II -Style Test Questions: This is the only textbook that gives you Praxis II -style questions, which will help you prepare for the high-stakes test used for teacher certification. Additional questions are available for instructors. National Research Council' s New Research on Mathematical Proficiency: Includes Student Assessments, Lesson Plans, and Family Math Packets to support all five strands. Students with Special Needs: A new section in each of the concept chapters features alternative teaching strategies especially for these students. Eighteen Video Vignettes: The CD packaged with this text features videos with guiding questions to analyze real teacher-student interaction in the hard to teach math concepts. Use the videos as a motivator for your own K-8 students. Online Study Guide: These online resources will help you achieve better math scores. Available through the website: www.wiley.com/college/hatfield Colored Patterns for Download: 15 pages are ready to use with elementary and middle school students and to practice hands-on manipulations as you prepare for interactive test items.

Author Biography

Mary M. Hatfield is Professor Emeritus, Arizona State University.

Table of Contents

List of Text Activities	xiv
List of CD-ROM Activities	xx
Preface	xxvii
The Past, Present, and Future of Mathematics Education	1	(18)
Past	2	(1)
From Thousands of Years Ago	2	(1)
From One Hundred and Fifty Years Ago	2	(1)
From Our Most Recent Past	3	(1)
Present	3	(10)
Brain Research Enlarges Our View	3	(1)
The Standards Movement in Mathematics Education	4	(1)
The NCTM Standards (2000)	4	(2)
Support for Teaching the NCTM Standards	6	(2)
Opposing Views of the Standards Movement	8	(1)
Other Expert Postions on Standards for Mathematics Learning	9	(1)
National Council for the Accreditation of Teacher Education (NCATE)	9	(1)
National Directives	9	(1)
Timss	10	(1)
Current Trends in Mathematics Learning	10	(1)
A General Overview	10	(1)
Active Learning with Real-World Applications for All Students	10	(1)
Technology	10	(2)
Other Current Trends	12	(1)
Future	13	(1)
Mathematics in the Twenty-First Century School	13	(1)
A New Vision of School	13	(1)
A New Vision of Teachers	13	(1)
Summary	14	(1)
Exercises	14	(1)
Bibliography	15	(2)
Integrating Technology	17	(2)
Culturally Relevant Mathematics	19	(30)
The Giftedness of Many Cultures in Mathematics	20	(14)
Cultural Contributions: A Sense of Pride	20	(2)
African American Heritage	22	(1)
Unique Systems of Mathematics Today	22	(1)
African Mathematics from the Ancient World	22	(1)
African Ideas in Elementary Mathematics	22	(2)
African Ideas in Middle School Mathematics	24	(1)
Resources for Contributions of African Americans Who Used Mathematics	25	(1)
Native American Heritage	26	(1)
Native American Culture in Mathematics Lessons	26	(1)
Mathematics Resoruces from Native American Educators	27	(1)
Hispanic American Heritage	27	(1)
Hispanic American Culture in Mathematics Lessons	27	(3)
Contributions from Hispanic Americans in Mathematics	30	(1)
Asian American Heritage	30	(1)
Asian American Culture in Mathematics Lessons	30	(2)
Contributions from Asian Americans in Mathematics	32	(1)
European American Heritage	32	(1)
A Most Unusual Story for Elementary-Age Children and Middle Schoolers	32	(1)
Women in the Culture of Mathematics Lessons	33	(1)
In Celebration of All Mathematicians	34	(1)
Equity in Mathematics: The Right of All People to Acquire Mathematical Power	34	(8)
The Impact of Technology	34	(1)
Equity with Computer Technology	35	(1)
Equity with Calculator Technology	35	(1)
Equity with Other Emerging Technologies	36	(1)
The Development of Problem Solving and Critical Thinking	36	(1)
The Role of Language in Problem Solving and Critical Thinking	36	(2)
The Role of Conceptual Thinking in Other Cultures	38	(2)
Minority Students Can Problem Solve in Authentic Ways	40	(1)
The Use of Cultural Qualities to Help Mathematics Equity	41	(1)
Learning Style	41	(1)
Speed of Performance	42	(1)
Noncompetitive Settings	42	(1)
Summary	42	(1)
Exercises	42	(1)
Bibliography	43	(3)
Children's Literature	46	(1)
Integrating Technology	47	(2)
How Children Learn Mathematics	49	(24)
The Changing Scene of How People Learn: The Brain Connection	49	(2)
Use It or Lose It	50	(1)
Implications for Mathematics Teachers	50	(1)
The Sooner, the Better	50	(1)
Implications for Mathematics Teachers	51	(1)
Learning Theories Applied to Mathematics	51	(13)
The Constructivists	51	(1)
Jean Piaget	52	(2)
Lev Vygotsky	54	(2)
Jerome Bruner	56	(1)
Other Theorists	56	(1)
Today's Constructivists in the Classroom	56	(3)
The Behaviorists	59	(1)
Immediate Feedback	59	(1)
Programmed Learning	60	(1)
Information-Processing Theories	61	(1)
Thought Processing	61	(1)
Learning Styles	62	(2)
Planning for Effective Learning of Mathematics	64	(4)
Classroom Instruction	64	(1)
The Chicago Mathematics Instructional Process	65	(1)
The TIMSS Lesson Plan Format	66	(1)
The Video Vignette: Actual Teachers in Classrooms	67	(1)
Homework in Light of the New Approaches to Instruction	67	(1)
Summary	68	(1)
Exercises	68	(1)
Bibliography	69	(2)
Integrating Technology	71	(2)
Assessment in Mathematics	73	(18)
History	74	(1)
Standardized Testing	74	(1)
National Assessment of Educational Progress	74	(1)
International Studies	75	(1)
State Assessment Programs	75	(1)
Changing Assessment	75	(11)
Purpose of Asessment	76	(1)
The Evaluation Efforts of the NCTM	76	(1)
Assessment Standards	76	(1)
Performance-Based Assessment	77	(1)
Interviews	77	(1)
Observations	78	(1)
Questioning	79	(1)
Performance Tasks	80	(3)
Calculators and Testing	83	(1)
Student Portfolios	83	(1)
Self-Assessment	84	(1)
Scoring Performance-Based Assessments	84	(2)
New Standards Project	86	(1)
Summary	86	(1)
Exercises	87	(1)
Bibliography	87	(2)
Integrating Technology	89	(2)
Problem Solving	91	(20)
Problem Solving	91	(15)
The Magnitude of Problem Solving	91	(1)
The Complexity of Problems Solving	91	(1)
The Many Facets of Problem Solving	92	(1)
The Methods, Kinds, and Procedures of Problem Solving	92	(1)
Methods of Integrating Problem Solving	92	(2)
Kind of Problem Solving	94	(1)
Kinds of Problems	95	(1)
How to Structure Your Mind: The Procedures	95	(1)
Problem-Solving Strategies	96	(1)
Estimation and Check	96	(1)
Looking for Patterns	97	(1)
Insufficient Information	97	(1)
Drawing Pictures, Graphs, and Tables	98	(1)
Elimination of Extraneous Data	98	(1)
Developing Formulas and Writing Equations	98	(1)
Modeling	98	(1)
Working Backward	99	(1)
Flowcharting	99	(1)
Acting Out the Problem	100	(1)
Simplifying the Problem	100	(1)
The Standards in Problem Solving	101	(1)
Effective Teaching of Problem Solving	101	(1)
Effective Techniques for Teaching Problem Solving	101	(2)
Kinds of Cooperative Grouping Activities	103	(1)
Assessment to Rate Group Problem Solving	104	(1)
Assessments to Rate Ourselves and Our Students	104	(1)
Putting It All Together: A Student's Work with Problem Solving	105	(1)
Summary	106	(1)
Exercises	107	(1)
Bibliography	107	(1)
Children's Literature	108	(1)
Integrating Technology	109	(2)
Geometry and Spatial Reasoning	111	(26)
Informal Geometry	112	(1)
Importance of Geometry	112	(1)
Teaching Strategies	112	(20)
Van Hiele Levels	113	(1)
Solid Geometry	114	(1)
Relating Three-Dimensional to Two-Dimensional	115	(2)
Plane Geometry	117	(1)
Geoboards	118	(2)
Tangrams	120	(2)
Symmetry	122	(1)
Line Symmetry	122	(1)
Rotational Symmetry	123	(2)
Transformations	125	(1)
Logo	126	(1)
Area and Perimeter	127	(1)
Pentominoes	127	(1)
Tangrams and Geoboards	128	(2)
Coordinate Geometry	130	(1)
Geometric Constructions	131	(1)
Literature	132	(1)
Assessment	132	(1)
Field-Independent Learners	132	(1)
Field-Dependent Learners	133	(1)
Correcting Common Misconceptions	133	(1)
Angle Identification	133	(1)
Angle Misconceptions with Protractors	133	(1)
Summary	133	(1)
Exercises	134	(1)
Bibliography	134	(1)
Children's Literature	135	(1)
Integrating Technology	136	(1)
Measurement	137	(26)
Measurement as a Process	138	(3)
How Measurement Develops in the Curriculum	138	(1)
Direct Comparison	138	(1)
Indirect Comprison	138	(1)
Seriation	138	(1)
Frames of Reference	138	(1)
Specific Types of Measurement	138	(1)
How the Roots of Measurement Develop in the Curriculum	139	(1)
The Teacher's Role in Cultural Awareness	139	(1)
Creating Mathematical Connections to History	140	(1)
Teaching Strategies	141	(16)
Linear Measurement	141	(1)
Direct Comparison	141	(1)
Indirect Comparison	141	(1)
Seriation	142	(3)
Frames of Reference	145	(1)
Mass (Weight) Measurement	145	(1)
Direct Comparison	145	(1)
Indirect Comparison	146	(1)
Frames of Reference	147	(1)
Standard Weights for Measurement	147	(1)
Area	147	(2)
Nonstandard Units	149	(1)
Standard Units	149	(1)
Frames of Reference	150	(1)
The Use of Customary and Metric Units	150	(1)
Capacity and Volume	150	(1)
Nonstandard Units	151	(1)
Standard Units	151	(1)
Computing Volume	151	(1)
Comparing Capacity and Volume	152	(1)
Time	153	(1)
Interpretation of Time	153	(1)
Understanding Passage of Time	154	(1)
Understanding Time Sequence	154	(1)
Temperature	155	(1)
Literature	156	(1)
Assessment	157	(1)
Common Misconceptions of Time	157	(1)
Correcting Misconceptions with the Concept of Time	157	(1)
Creating a Rubric for the Assessment of Time Concepts	158	(1)
Summary	158	(1)
Exercises	159	(1)
Bibliography	159	(1)
Children's Literature	160	(1)
Integrating Technology	161	(2)
Number Readiness---Early Primary Mathematics	163	(22)
A Child's Understanding of Number	164	(8)
Task: Classification	165	(1)
Assessment	165	(1)
Teaching Ideas	165	(1)
Task: Class Inclusion	166	(1)
Assessment	166	(1)
Task: Number Inclusion	167	(1)
Assessment	167	(1)
Teaching Ideas	167	(1)
Task: Seriation	168	(1)
Assessment	168	(1)
Teaching Ideas	169	(1)
Task: Number Conservation	169	(1)
Assessment	169	(1)
Teaching Ideas	170	(1)
Task: Equivalence of Sets	170	(1)
Assessment	170	(1)
Teaching Ideas	171	(1)
Implications for Curriculum	171	(1)
Assessment	172	(1)
Building the Concept of Number	172	(9)
Patterns	172	(1)
Developing Patterning Skills	173	(1)
Number Relationships	173	(1)
One-to-One Correspondence	173	(1)
More, Less, and Same	173	(1)
Rote Counting	174	(1)
Rational Counting	174	(1)
Counting Sets	175	(1)
Numeral-Set Association	175	(1)
Developing Meaning for Numbers	176	(2)
Literature	178	(1)
Writing Numerals	178	(1)
Teaching Ideas	178	(1)
Readiness for Operations	179	(1)
Developing Meaning for Operations	179	(1)
Building Part-Whole Understanding	180	(1)
Using Technology	180	(1)
Summary	181	(1)
Exercises	181	(1)
Bibliography	182	(1)
Children's Literature	183	(1)
Integrating Technology	184	(1)
Numeration and Number Sense	185	(30)
Structure of Numeration Systems	187	(4)
Decimals	188	(2)
Number Bases	190	(1)
Teaching Strategies	191	(18)
Instructional Sequence	191	(1)
The Nature of Place Value	192	(2)
Early Experiences	194	(1)
Numeration Models	194	(2)
Number Base Activities	196	(1)
Counting Activities	197	(4)
Number Sense	201	(1)
Understanding and Interpreting Large Numbers	202	(2)
Reading Numbers	204	(1)
Rounding	204	(2)
Literature	206	(1)
Decimals	206	(1)
Representing Decimals	207	(1)
Comparing Decimals	207	(1)
Estimation	208	(1)
Assessment	209	(2)
Field-Dependent Learners	209	(1)
Field-Independent Learners	209	(1)
Correcting Common Misconceptions	209	(1)
Correcting Common Misconceptions with Decimals	210	(1)
Nonalignment of the Decimal Points	210	(1)
Annexing Zeros	210	(1)
Attention to Decimal Point as Place Holder	210	(1)
Name Value Confused with Place Value	210	(1)
Summary	211	(1)
Exercises	211	(1)
Bibliography	212	(1)
Children's Literature	213	(1)
Integrating Technology	214	(1)
Operations and Number Sense	215	(36)
Overview of the Operations and Properties	217	(24)
Addition	217	(1)
Combining	217	(1)
Static	217	(1)
Concrete to Symbolic	217	(1)
Literature	218	(1)
Properties	218	(1)
Subtraction	218	(1)
Take-Away	218	(1)
Comparison	219	(1)
Missing Addend	219	(1)
Literature	219	(1)
Properties	220	(1)
Multiplication	220	(1)
Repeated Addition	220	(1)
Combinations	221	(1)
Arrays	221	(1)
Literature	222	(1)
Properties	222	(1)
Division	222	(1)
Measurement Division	222	(1)
Partitive Division	223	(1)
Literature	223	(1)
Properties	223	(1)
Readiness to Learn Basic Facts	224	(1)
Prerequisites	224	(1)
Piaget	225	(1)
Number Meaning	225	(1)
Thinking Strategies for the Basic Facts of Addition	226	(1)
Strategy Preferences	226	(1)
Addition Facts	227	(1)
Principles	227	(1)
Adding One	228	(1)
Counting On	228	(1)
Near Doubles	228	(1)
Bridging to 10	228	(3)
Thinking Strategies for the Basic Facts of Subtraction	231	(1)
Helpful Devices	231	(2)
Fact Families	233	(1)
Using 10	233	(1)
Part-Whole	233	(1)
Thinking Strategies for the Basic Facts of Multiplication	234	(1)
Strategies	234	(2)
Skip Counting	236	(1)
Finger Multiplication	237	(1)
Friendly Facts	238	(1)
Split a Factor	238	(1)
Thinking Strategies for the Basic Facts of Division	238	(1)
Missing Factor	239	(1)
Mastering the Facts	239	(1)
Basic Facts---Pros and Cons	239	(1)
Cluster the Facts	240	(1)
Build Confidence	240	(1)
Assess Mastery	240	(1)
Practice Devices	240	(1)
Assessment	241	(5)
Field-Dependent and Field-Independent Learners	241	(1)
Basic Fact Tables	241	(1)
Correcting Common Misconceptions	242	(1)
Faulty Reasoning 1	243	(1)
Faulty Reasoning 2	243	(1)
Faulty Reasoning 3	243	(1)
Faulty Reasoning 4	244	(1)
Teacher Assessment of Student Work: Steps in Analytic Thinking	245	(1)
Steps in Analytic Thinking	245	(1)
Summary	246	(1)
Exercises	246	(2)
Bibliography	248	(1)
Children's Literature	248	(2)
Integrating Technology	250	(1)
Operations with Whole Numbers	251	(34)
The Development of Algorithmic Models	251	(4)
Proportinal versus Nonproportional Models	253	(1)
Representational versus Nonrepresentational Models	254	(1)
Teaching Strategies	255	(19)
Changing Perspective	255	(1)
The Pros and Cons of Teaching Algorithms	255	(1)
Promoting Invented Strategies	256	(1)
Teaching the Addition Algorithm	257	(3)
Teaching the Subtraction Algorithm	260	(1)
Take-Away	260	(1)
Comparison	260	(1)
Missing Addend	261	(1)
Suggestions for Instruction	262	(1)
Teaching the Multiplication Algorithm	262	(1)
Repeated Groups	262	(2)
Area Model	264	(2)
Teaching the Division Algorithm	266	(1)
Using Models	266	(1)
Comparing Models	267	(1)
Special Problems	267	(2)
Remainders	269	(1)
Combining the Operations	270	(1)
Literature	271	(1)
Mental Computation	271	(1)
Estimation	272	(1)
Front-End Estimation	273	(1)
Compatible Numbers	273	(1)
Clustering	274	(1)
Rounding	274	(1)
Assessment	274	(5)
Important Aspects	274	(1)
Correcting Common Misconceptions	275	(1)
Special Needs Students	275	(1)
Addition	275	(1)
Subtraction	276	(1)
Multiplication	277	(1)
Division	278	(1)
Field-Dependent Learners	279	(1)
Field-Independent Learners	279	(1)
Summary	279	(1)
Exercises	280	(1)
Bibliography	281	(1)
Children's Literature	282	(1)
Integrating Technology	283	(2)
Common Fractions and Decimals	285	(46)
Interpretation of Fractions	286	(1)
Part-Whole Region (Area Model)	286	(1)
Measure	287	(1)
Set	287	(1)
Ratio	288	(1)
Division	288	(1)
Teaching Strategies: Fractions	288	(26)
Whole-to-Part Activities	288	(2)
Partitioning Activities	290	(1)
Equivalence Activities	291	(2)
Multiple Bars	293	(1)
Multiple Embodiments	294	(1)
Operations with Fractions	295	(1)
Addition and Subtraction	296	(2)
Alternative Solutions	298	(1)
Estimation	299	(1)
Mixed Numbers	300	(1)
Multiplication and Division	300	(1)
Fraction x Whole Number	301	(1)
Fraction x Fraction	301	(2)
Understanding Division	303	(3)
Invented Strategies	306	(1)
Reciprocal	307	(1)
Estimation	307	(1)
Using Other Manipulatives	308	(1)
Literature	308	(1)
Decimals	308	(1)
Terminating Decimals	309	(3)
Repeating Decimals	312	(1)
Linking Fractions to Decimals	313	(1)
Teaching Strategies: Decimals	314	(6)
Decimals as Money	314	(1)
Coin Equivalencies	315	(1)
Operations with Decimals	316	(1)
Addition	316	(1)
Subtraction	316	(1)
Multiplication	317	(2)
Division	319	(1)
Assessment	320	(6)
Field-Independent Learners	320	(1)
Field-Dependent Learners	320	(1)
Correcting Common Misconceptions	321	(1)
Adding or Subtracting Denominators as Whole Numbers	321	(1)
Changing Mixed Numbers to Improper Fractions	322	(1)
Inverting and Multiplying the Incorrect Factors	322	(1)
Multiplying and Dividing Mixed Numbers	322	(1)
Regrouping Fractions as Whole Numbers	323	(1)
Correcting Common Misconceptions with Decimals and Money	324	(1)
Poor Estimation Skills in Multiplication and Division	325	(1)
Summary	326	(1)
Exercises	326	(1)
Bibliography	327	(1)
Children's Literature	328	(1)
Integrating Technology	329	(2)
Percent, Ratio, Proportion, and Rate	331	(24)
Teaching Strategies	332	(14)
Percent	332	(1)
From Concrete to the Connecting Level	332	(1)
Money	332	(1)
Shapes	332	(2)
From the Connecting Level to the Symbolic	334	(2)
Ratio	336	(1)
From Concrete to the Connecting Level	336	(3)
From the Connecting Level to the Symbolic	339	(1)
Proportions	340	(1)
The Part-Part Relationship in Proportions	340	(3)
Other Relationship Models for Proportion	343	(1)
Rate	344	(1)
Students with Special Needs	345	(1)
Percent, Ratio, Proportion, and Rate in the World of Work	345	(1)
Assessment	346	(5)
Field-Independent and Field-Dependent Learners	346	(1)
Correcting Common Misconceptions	346	(1)
Writing the Decimal with the Percent	347	(1)
Percent of a Given Value	347	(1)
Portfolio Assessments	347	(1)
Students' Understanding of Percent: Building with Pattern Pieces	347	(1)
Students' Understanding of Percent: Seeing Percent in Whole Figures	348	(3)
Summary	351	(1)
Exercises	351	(1)
Bibliography	352	(1)
Children's Literature	352	(1)
Integrating Technology	353	(2)
Number Theory, Patterns, and Functions, and Algebra	355	(30)
Teaching Strategies	356	(20)
Number Theory	356	(1)
Odd and Even Numbers	356	(2)
Prime and Composite Numbers	358	(4)
Divisibility Rules	362	(1)
Polygonal or Figurate Numbers	363	(1)
Square Numbers	363	(1)
Pythagorean Triples	364	(1)
Fibonacci Number Sequence	365	(1)
Pascal's Triangle	366	(1)
Patterns and Functions	367	(1)
Activities for Pre-K-2 Grades	367	(1)
Activities for 3-5 Grades	368	(1)
Activities for 6-8 Grades	368	(1)
Teaching Integers	369	(1)
Postman Stories	369	(1)
Beginning Algebra Concepts	370	(1)
Activities for K-4 Grades	371	(1)
Activities for 5-8 Grades	372	(4)
Assessment	376	(5)
Field-Independent Learners	377	(1)
Field-Depedent Learners	377	(1)
Correcting Common Misconceptions	377	(1)
Jumping to Premature Conclusions	377	(1)
Correcting Unknown Errors... the Reason for Oral Discourse	377	(1)
Not Seeing the Pattern	377	(1)
Portfolio Assessments	377	(1)
Fibonacci Number Sequence	377	(2)
Functions---Input-Output Tables	379	(1)
Algebra	379	(2)
Summary	381	(1)
Exercises	381	(1)
Bibliography	382	(1)
Children's Literature	383	(1)
Integrating Technology	384	(1)
Data Analysis, Statistics, and Probability	385
Teaching Strategies	386
Data Analysis---Graphing	386
Creating Real Graphs	387
Creating Picture Graphs	387
Creating Bar Graphs	389
Creating Line Graphs	391
Creating Circle Graphs	392
Creating Scatter Graphs	394
Technology: Creating All the Graphs Together	394
Statistics	395
Mean	396
Mode	397
Median	397
Range	397
Stem and Leaf Plots	398
Box and Whisker Plots	398
Putting It All Together: Statistical Information about Real-World Mathematics	399
Probability	400
Early Experiences: The Pre-K-5 Grades	401
Assigning Probabilities: The 6-8 Middle School Grades	405
Permutations and Combinations	408
Homework Possibilities	411
Assessment	411
Data Analysis---Graphing	411
Bar Graph Stories	411
Continuous Graph Stories	412
Students with Special Needs	413
Averages	413
Summary	415
Exercises	415
Bibliography	416
Children's Literature	417
Integrating Technology	418