Teaching Elementary and Middle School Mathematics : Raising the Standards

by Sgroi, Laura Shannon

• ISBN13: 9780534508753
• ISBN10: 0534508758
• Edition: 1st
• Format: Paperback
• Copyright: 2000-08-23
• Publisher: Cengage Learning
• More Book Details
• Purchase Benefits

Summary

This book is written to help prospective teachers of preschool through grade 8 understand how children in these grades learn mathematics and how they themselves might teach mathematics. It is based on the National Council of Teachers of Mathematics (NCTM) Principles and Standards 2000 and grounded in the belief that before one can become a confident and competent teacher of mathematics, one must first become a competent and confident learner of mathematics.

Table of Contents

Preface	xv
New Standards for Learning and Teaching Mathematics	1	(30)
The Goals of Mathematics Education	3	(2)
The Nature of the Pre-K--2 Learner	5	(1)
Teaching the Young Child	6	(10)
Teachers of primary-grade children. . . must be aware of the ``whole child''	7	(1)
should integrate their instruction	8	(1)
should engage them in active as opposed to passive activities	9	(1)
should provide small-group activities that allow discussion and interaction	9	(4)
Communication	13	(1)
Teachers of primary-grade children. . . should keep in mind the interests of their students when planning lessons	13	(1)
need to provide a range of activities to meet the various developmental needs of their students	14	(2)
Teaching Children Aged 8--13	16	(9)
Cooperative Learning	16	(2)
Teachers of intermediate-grade children. . . should involve them in logical investigations	18	(1)
should involve them in investigations of other cultures	18	(7)
Summary	25	(6)
Standards for Number and Numeration	31	(44)
Planning for Instruction	32	(4)
Factors Affecting the Mathematical Achievement of Students	36	(3)
Time on Task	36	(1)
Wait Time	36	(1)
Gender Issues	37	(1)
Multicultural Issues	37	(2)
Curricular Issues	39	(1)
Constructivism in the Mathematics Classroom	39	(6)
Cognitively Guided Instruction	40	(3)
Problem Solving	43	(1)
Performance-Based Assessment	43	(2)
Number and Operation for Grades Pre-K---2	45	(16)
Performance-Based Assessment: One-to-One Correspondence	47	(1)
Observing, Describing, Classifying	48	(1)
Use of Number	48	(4)
Performance-Based Assessment: The Cardinal Use of Number	52	(1)
The Hindu-Arabic Number System	53	(5)
Chip Trading in Other Bases	58	(3)
Number and Operation for Grades 3---5	61	(3)
Performance-Based Assessment: Classification	63	(1)
Number and Operation for Grades 6---8	64	(5)
Prime and Composite Numbers	67	(1)
Estimation	67	(1)
Rectangular Arrays for Prime and Composite Numbers	68	(1)
Summary	69	(6)
Standards for Whole Numbers: Addition and Subtraction	75	(36)
Operation for Grades Pre-K--2	76	(4)
Basic Facts of Addition and Subtraction	80	(8)
Properties of Whole-Number Addition	82	(1)
Basic Facts Strategies	82	(1)
Guess My Rule	83	(1)
I Have. . . Who Has. . . ?	83	(1)
Cover-Up	84	(1)
Salute	84	(1)
Add `Em Up	85	(1)
Bingo	85	(1)
Computer Programs	86	(1)
Performance-Based Assessment: Basic Addition and Subtraction Facts	86	(1)
Magic Squares	87	(1)
Addition and Subtraction of Multi-Digit Numbers	88	(18)
Estimation and the Commutative and Associative Properties	89	(1)
Forming Multiples of Ten	89	(2)
Rounding	91	(1)
Front-End Estimation	91	(1)
Powers-of-Ten Blocks and the Standard Addition Algorithm	92	(1)
Alternative Algorithms for Whole-Number Addition	93	(1)
Partial-Sums Algorithm	94	(1)
Powers-of-Ten Blocks with the Standard Subtraction Algorithm	95	(3)
Alternative Algorithms for Whole-Number Subtraction	98	(1)
Equal Addends Subtraction	98	(1)
Counting-Up Subtraction	99	(1)
The Spreadsheet as a Problem-Solving Tool	99	(4)
Problem Solving	103	(2)
Performance-Based Assessment: Multi-Digit Addition and Subtraction	105	(1)
Alternative Subtraction Algorithm	106	(1)
Summary	106	(5)
Standards for Whole Numbers: Multiplication and Division	111	(38)
Operations of Multiplication and Division	111	(1)
Number and Operation for Grades 3--5	112	(3)
Properties of Whole-Number Multiplication	113	(2)
Basic Facts of Multiplication and Division	115	(13)
Basic Facts Strategies	115	(4)
Even-Odd	119	(2)
Two Over	121	(1)
Multiplication Cover-Up	122	(1)
Salute	122	(2)
Bingo	124	(1)
Patterns	124	(1)
Computer Software	125	(1)
Problem Solving	126	(1)
Performance-Based Assessment: Basic Multiplication and Division Facts	127	(1)
Multiplication and Division of Multi-Digit Numbers	128	(11)
Estimation and the Distributive and Associate Properties	128	(1)
Powers-of-Ten Blocks with Standard Multiplication Algorithm	129	(4)
Alternative Algorithms for Whole-Number Multiplication	133	(1)
Lattice Multiplication	134	(1)
Russian Peasant Multiplication	135	(1)
Performance-Based Assessment: Multi-Digit Multiplication	135	(1)
Alternative Algorithms for Multiplication	136	(1)
Manipulatives with the Standard Division Algorithm	136	(1)
An Alternative Algorithm for Whole-Number Division	136	(1)
The Subtraction Method of Division	137	(2)
Number and Operation for Grades 6--8	139	(5)
Estimation	140	(1)
Rounding to Multiples of Ten	140	(1)
Rounding with Compensation	140	(1)
Estimating Quotients	141	(1)
Using Multiplication and Division to Solve Problems	142	(2)
Summary	144	(5)
Standards for Geometry	149	(24)
Geometry and Spatial Sense for Grades Pre-K--2	150	(4)
The Contribution of Research	150	(1)
Visualization	150	(2)
Analysis	152	(1)
Informal Deduction	152	(1)
Formal Deduction	152	(1)
Rigor	153	(1)
Observation, Description, and Classification	154	(4)
Performance-Based Assessment: Combining Shapes	155	(3)
Connecting Geometry to Other Mathematical Ideas	158	(3)
Tangrams	159	(2)
Geometry and Spatial Sense for Grades 3---5	161	(8)
Performance-Based Assessment: Line and Rotational Symmetry	167	(1)
Computer Software	167	(2)
Summary	169	(4)
Continuing the Study of Geometry	173	(34)
Geometry as Measurement	173	(1)
Geometry and Spatial Sense for Grades 6--8	173	(18)
Performance-Based Assessment: Perimeter	176	(1)
Performance-Based Assessment: Area	177	(1)
Angle Measurements	178	(2)
Guided Discovery	180	(6)
The Relationship Between Surface Area and Volume	186	(5)
Geometric Drawing Utilities	191	(8)
LOGO	194	(4)
Other Computer Software	198	(1)
Transformational Geometry	199	(1)
Geometry and Spatial Sense for Grades 6---8	199	(4)
Performance-Based Assessment: Transformations	202	(1)
Conjectures	203	(1)
Summary	203	(4)
Standards for Extending the Number Line	207	(32)
Number and Operation for Grades Pre-K--2 and 3--5	209	(10)
Models and Concepts of Fractions and Decimals	210	(3)
Performance-Based Assessment: Naming Fractions	213	(3)
Performance-Based Assessment: Equivalent Fractions	216	(3)
Adding and Subtracting with Fractions and Decimals	219	(1)
Number and Operation for Grades 3--5	219	(6)
Estimating Sums and Differences of Rational Numbers	225	(3)
Tangrams and Fractions	225	(3)
Linear Measurement	228	(1)
Measurement for Grades Pre-K--2	228	(2)
Measurement for Grades 3--5	230	(5)
Performance-Based Assessment: Linear Measurement	232	(2)
Linear Measurement	234	(1)
Summary	235	(4)
Standards for Multiplication and Division of Fractions and Decimals	239	(28)
Multiplying and Dividing with Fractions and Decimals	239	(1)
Number and Operation for Grades 6--8	239	(11)
Performance-Based Assessment: Multiplying Decimals	244	(4)
Computer Software	248	(1)
Percents	248	(2)
Measurement of Mass and Volume	250	(1)
Measurement for Students in Grades 6--8	251	(8)
Performance-Based Assessment: Identifying Appropriate Measures	257	(2)
Proportional Thinking	259	(4)
Performance-Based Assessment: Proportions	261	(2)
Summary	263	(4)
Standards for Patterns and Relationships	267	(38)
Simple Patterns, Functions, and Algebra	267	(10)
Problem Solving	269	(1)
Performance-Based Assessment: Even and Odd Numbers	270	(1)
Relationships	270	(3)
Performance-Based Assessment: Identifying Relationships	273	(1)
The Relationship Between Variables and Problem Solving	274	(3)
More Complex Patterns, Functions, and Algebra	277	(23)
Computer Software	280	(1)
Number Theory in Context	280	(1)
Performance-Based Assessment: Prime Factors	281	(4)
Divisibility	285	(1)
Divisibility by 2	285	(1)
Divisibility by 4	285	(1)
Divisibility by 8	285	(1)
Divisibility by 3 or 9	286	(1)
Integers	286	(1)
Divisibility by 11	287	(1)
Operations on Integers	288	(1)
Addition	288	(1)
Subtraction	289	(1)
Multiplication and Division	290	(1)
Performance-Based Assessment: Integers	291	(3)
Computer Software	294	(1)
Geometric Patterns	294	(2)
Multiplication of Whole Numbers and Binomials	296	(3)
Solving Equations with One Variable	299	(1)
Summary	300	(5)
Standards for Data Analysis, Statistics, and Probability	305	(28)
Graphing and Data Analysis	305	(1)
Focus Areas for Grades Pre-K---2	306	(12)
Physical Graphs	306	(1)
Bar Graphs	306	(3)
Data Collection and Representation	309	(1)
Combinations	310	(3)
Using Graphs to Assess Children's Knowledge	313	(1)
Performance-Based Assessment: Placing Graphs in Contexts	314	(1)
Using Data to Solve Problems	315	(2)
Line Graphs	317	(1)
Measures of Central Tendency	317	(1)
Introducing Concepts of Chance	318	(2)
Representing Probability as a Fraction	319	(1)
Probability for Grades 3--5	320	(9)
Performance-Based Assessment: Equally Likely Events	323	(2)
Tree Diagrams	325	(2)
Computer Software	327	(2)
Summary	329	(4)
Further Investigations into Data Analysis, Statistics, and Probability	333	(26)
Focus Areas for Grades 6--8	333	(15)
Data Collection	334	(1)
Histograms, Line Graphs, and Circle Graphs	335	(5)
Performance-Based Assessment: Graphs	340	(1)
Graphing	341	(3)
Simulations	344	(1)
Hanging Bar Graph	344	(1)
Plots: Box-and-Whisker and Stem-and-Leaf	345	(2)
Performance-Based Assessment: Plotting	347	(1)
Using Data to Make Decisions	347	(1)
Theoretical and Experimental Probability	348	(7)
Independent Events	349	(2)
Permutations and Combinations	351	(1)
Pascal's Triangle	352	(3)
Summary	355	(4)
Appendix A Masters	359	(8)
Appendix B NCTM Summary	367	(16)
Index	383