Mathematics Content for Elementary Teachers

by Brumbaugh; Douglas K.

• ISBN13: 9780805842470
• ISBN10: 0805842470
• Format: Nonspecific Binding
• Copyright: 2004-08-06
• Publisher: Routledge

Summary

THE book for elementary education mathematics content courses! Designed to help prospective teachers of elementary school mathematics learn content beyond the rote level, this text stimulates readers to think beyond just getting the problem right and fosters their development into thoughtful, reflective, self-motivated, life-long learners. It stresses the what and why of elementary school mathematics content. Hints are provided about how to teach the content but this is mostly left to courses and texts that are dedicated to that purpose. The text is organized around the National Council for Teachers of Mathematics' Principles and Standards for School Mathematics. The Standards dictate the basic sections of the text. Within each section, appropriate specific topics are developed, intertwined with technology, problem solving, assessment, equity issues, planning, teaching skills, use of manipulatives, sequencing, and much more. In addition, major focal points of the Standards are emphasized throughout: effective teachers of mathematics should be able to motivate all students to learn, should understand the developmental levels of how children learn, should concentrate on what children need to become active participants in the learning environment, and should be engaged in ongoing investigations of new mathematical concepts and teaching strategies. Mathematics Content for Elementary Teachers is based on several fundamental premises: *The focus of mathematics education should be on the process, not the answer. *Elementary teachers should know the mathematics content they are teaching, know more than the content they are teaching, and teach from the overflow of knowledge. *It is important for teachers to be flexible in allowing students to use different procedures--teaching from the "overflow of knowledge" implies knowing how to do a given operation more than one way and being willing to examine many different ways. *Teachers need to learn to carefully cover the topics to be taught, to reflect upon them, and to be able to organize them. To help prospective elementary teachers concentrate on the mathematics content they will be expected to teach and begin to build the foundation for the methods they will use, this text includes only elementary mathematics content and does not address middle school concepts. Pedagogical features: *The text is organized according to NCTM Standards. *An informal writing style speaks directly to readers and is geared to pre-service teachers. *Focus is given to multiple methods of problem solving at four developmental levels. *Questions, exercises and activities are interspersed throughout each section rather than gathered at the end of each chapter. *Complete solutions for exercises are provided.

Author Biography

Douglas K. Brumbaugh has taught college, in-service, or K-12 mathematics for close to 45 years. He received his B.S. from Adrian College and master's and doctorate in mathematics education from the University of Georgia. When talking with others about teaching and learning in the K-12 environment, his immersion in teaching is beneficial. The thoughts and examples in this book are based experiences working with garden-variety kids.
Peggy L. Moch decided to become a teacher, following nearly 25 years in the laboratory as a medical technologist mainly specializing in immunohematology, microbiology, quality control, and quality assurance. She obtained her teaching degrees, a bachelor's, master's, and doctorate--all in mathematics education--from the University of Central Florida. Together with her writing colleagues, Moch has endeavored to share that love with readers.
MaryE Wilkinson received her B.S., master's, and doctorate in mathematics education from the University of Central Florida. She has taught high school mathematics courses, college mathematics courses, and mathematics and science methods courses. When thinking and writing about elementary school mathematics, she draws upon all of her experiences and continuing presence in elementary classrooms.

Table of Contents

Preface	xv
To the Student	xv
Focal Points	xvi
To the Instructor	xvii
Overview of Content	xviii
Pedagogical Features	xix
References	xix
Personal Acknowledgments	xxi
About the Authors	xxiii
Guiding Principles	1	(8)
The Equity Principle	2	(1)
The Curriculum Principle	2	(1)
The Teaching Principle	3	(1)
The Learning Principle	4	(1)
The Assessment Principle	5	(1)
The Technology Principle	5	(1)
The Challenge	6	(2)
References for Guiding Principles	8	(1)
Number and Operations	9	(108)
Sets	9	(23)
Set Definitions	9	(6)
Set Operations	15	(1)
Special Sets	16	(1)
Venn Diagrams	16	(2)
Properties	18	(4)
Factors and Multiples	22	(1)
Prime and Composite Numbers	22	(2)
Sieve of Eratosthenes	24	(1)
Divisibility Rules	25	(3)
Greatest Common Factor and Least Common Multiple	28	(4)
Conclusions	32	(1)
Whole Number Addition	32	(8)
Terminology	34	(1)
Standard Algorithm	34	(1)
Partial Sum	35	(1)
Denominate Numbers	36	(1)
Horizontal and Vertical Writing	36	(1)
Expanded Notation	37	(1)
Left to Right Addition	37	(1)
Scratch Method	38	(1)
Any Column First	38	(1)
Low Stress Addition	39	(1)
Conclusions	40	(1)
Bibliography	40	(1)
Whole Number Subtraction	40	(8)
Terminology	41	(1)
Concrete Subtraction	42	(2)
Denominate Numbers	44	(1)
Expanded Notation	44	(1)
Standard Algorithm	44	(1)
Borrow Pay Back Method	45	(1)
Left to Right Subtraction	46	(1)
Scratch Method	46	(1)
Any Column First	47	(1)
Integer Subtraction	47	(1)
Conclusions	48	(1)
Whole Number Multiplication	48	(7)
Terminology	48	(1)
Beginnings	49	(1)
Standard Algorithm	49	(1)
Partial Product Method	50	(1)
FOIL	51	(1)
Lattice Multiplication	51	(2)
Left to Right Multiplication	53	(1)
Horizontal and Vertical Writing	53	(1)
Russian Peasant (Simple Halving/Doubling Method)	54	(1)
Try This	55	(1)
Conclusions	55	(1)
Bibliography	55	(1)
Whole Number Division	55	(8)
Terminology	57	(1)
Division as a Rectangle	58	(1)
Repeated Subtraction Division	58	(2)
Standard Division Algorithm	60	(1)
Technology	61	(1)
Remainders	61	(1)
Say What?	62	(1)
Conclusions	63	(1)
Equivalent Fractions and Multiplication of Fractions	63	(9)
Concrete Beginnings	63	(1)
Equivalent Fractions	64	(2)
Converting a Mixed Number to an Improper Fraction	66	(1)
Converting an Improper Fraction to a Mixed Number	67	(1)
Product of a Whole Number and a Fraction	68	(1)
Product of Whole Number and Mixed Number	68	(1)
Product of Two Fractions	69	(1)
Product of a Fraction and a Mixed Number	70	(1)
Product of Two Mixed Numbers	71	(1)
Conclusions	71	(1)
Addition of Fractions	72	(5)
Concrete Beginnings	72	(1)
Adding Fractions When Denominators Are the Same	72	(2)
Adding Fractions When the Denominators Are Related	74	(1)
Adding Fractions When the Denominators Are Relatively Prime	75	(1)
Adding Fractions When the Denominators Are Not Relatively Prime and One Is Not a Multiple of the Other	76	(1)
Conclusions	77	(1)
Subtraction of Fractions	77	(4)
Same Denominators	77	(1)
Related Denominators	78	(1)
Relatively Prime Denominators	79	(1)
Denominators That Are Not Relatively Prime or Related	80	(1)
Mixed Numbers	80	(1)
Conclusions	81	(1)
Division of Fractions	81	(5)
Concrete Beginnings	82	(1)
Whole Number Divided by a Fraction	82	(1)
Fraction Divided by a Fraction	83	(1)
Mixed Number Divided by Mixed Number	84	(1)
Common Denominator Division	84	(1)
Conclusions	85	(1)
Addition of Decimals	86	(3)
Concrete Beginnings	86	(1)
Denominate Numbers	87	(1)
Adding With the Same Number of Places	87	(1)
Zeros at the End	88	(1)
Lining up the Ones	88	(1)
Expanded Notation	89	(1)
Conclusions	89	(1)
Subtraction of Decimals	89	(2)
Concrete Beginnings	90	(1)
Subtracting With the Same Number of Places	90	(1)
Zeros at the End	91	(1)
Multiplication of Decimals	91	(4)
Concrete Beginnings	91	(1)
Moving Beyond the Concrete	92	(2)
Relating to Old Ways	94	(1)
Conclusions	94	(1)
Division of Decimals	95	(4)
Concrete Beginnings	95	(1)
Whole Number Divided by a Whole Number	96	(1)
Decimal Divided by a Whole Number	97	(1)
Whole Number Divided by a Decimal	98	(1)
Decimal Divided by a Decimal	98	(1)
Conclusions	99	(1)
Addition of Integers	99	(4)
Models	99	(2)
Rules for Adding Integers	101	(1)
Conclusions	102	(1)
Subtraction of Integers	103	(2)
Models	103	(2)
Conclusions	105	(1)
Multiplication of Integers	105	(2)
Multiplying Integers	105	(2)
Conclusions	107	(1)
Division of Integers	107	(2)
Division Using Signed Numbers	108	(1)
Signed Numbers in Inverse Operations	108	(1)
Conclusions	109	(1)
Ratios and Proportions	109	(8)
Rational Numbers as Ratios	109	(2)
Percents	111	(1)
Equivalents	112	(1)
Proportions	112	(2)
Cross Products	114	(1)
Solving Problems Using Proportions	114	(2)
Conclusions	116	(1)
Algebra	117	(16)
Historic Underpinnings	118	(1)
Laying the Foundation for Algebra	119	(1)
Having Fun With Algebra	119	(2)
Integrating Algebra	121	(1)
Patterning	122	(1)
Representing Situations With Algebra	123	(1)
Using Models	124	(2)
Rate of Change	126	(2)
Sequences	128	(2)
Formulas	130	(1)
Conclusions	131	(1)
References	132	(1)
Geometry	133	(18)
Undefined Terms	133	(3)
Angles	136	(2)
Simple Closed Curves, Regions, and Polygons	138	(1)
Polygons	139	(3)
Circles	142	(1)
Constructions	143	(1)
Third Dimension	144	(2)
Coordinate Geometry	146	(1)
Transformations and Symmetry	147	(2)
Conclusions	149	(2)
Measurement	151	(22)
Terminology	152	(1)
Attributes	152	(1)
Systems	153	(2)
Units, Tools (Instruments), and Precision	155	(1)
Linear Measure	155	(1)
Area	156	(5)
Volume	161	(3)
Capacity	164	(1)
Weight	165	(1)
Time	165	(1)
Money	166	(1)
Temperature	167	(1)
Angle Measure	167	(4)
Dimensional Analysis	171	(1)
Conclusions	172	(1)
Data Analysis and Probability	173	(30)
Data Collection and Representations	173	(9)
Data and Where to Get It	173	(1)
Representations of Data	174	(1)
Venn Diagrams	175	(1)
Percentages	175	(1)
Circle Graph	175	(1)
Line Plot	176	(1)
Bar Graph	176	(1)
Line Graph	177	(1)
Histogram	178	(1)
Frequency Polygon	178	(1)
Box and Whisker Plot	179	(1)
Scatter Plot	180	(1)
Stem and Leaf Plot	181	(1)
Conclusions	182	(1)
Data Analysis and Statistics	182	(9)
Measures of Central Tendency and Scatter	183	(1)
Mean	183	(1)
Median	184	(1)
Mode	185	(1)
Range	186	(1)
Variance	186	(1)
Standard Deviation	187	(1)
Quartiles	188	(1)
Statistics	189	(1)
Assumptions	190	(1)
Generalizations	190	(1)
Conclusions	191	(1)
Counting and Probability	191	(12)
Factorial	191	(2)
Permutations	193	(1)
Combinations	194	(1)
Independent	194	(3)
Dependent	197	(2)
Conditional Probability	199	(1)
Lottery	199	(1)
Odds	200	(1)
Conclusions	201	(2)
Problem Solving	203	(6)
What Makes a Problem a Problem?	203	(1)
Polya's Steps	204	(1)
Strategies	205	(2)
Collection of Problems	207	(1)
Conclusions	208	(1)
Bibliography	208	(1)
Reasoning and Proof	209	(10)
And	209	(1)
Or	210	(1)
If, Then	211	(1)
Negations	212	(1)
Tautologies	212	(1)
Logical Equivalence	212	(1)
Informal Proofs	213	(1)
Beyond the Informal Proofs	214	(3)
Two-Column Proof	215	(2)
Paragraph Proof	217	(1)
Indirect Proof	217	(1)
Conclusions	217	(2)
Communication	219	(4)
Connections	223	(6)
Independent or Interconnected Topics	223	(1)
How Are Things Connected?	224	(3)
Conclusions	227	(2)
Representation	229	(2)
Different Ways of Saying the Same Thing	229	(1)
Conclusions	230	(1)
Bibliography	230	(1)
Index	231
Solutions Manual	1