Error Patterns in Computation: Using Error Patterns to Improve Instruction

by Ashlock, Robert B.

• ISBN13: 9780131198869
• ISBN10: 0131198866
• Edition: 9th
• Format: Paperback
• Copyright: 2006-01-01
• Publisher: Pearson College Div

Summary

Reflecting many of the concerns of NCTM's Principles and Standards for School Mathematics, the ninth edition of Error Patterns in Computation is the only book available that assists teachers with identifying typical error patterns, receiving feedback on their diagnosis, and gaining insight regarding why a child may have adopted an incorrect procedure. Two major causes of error patterns - over-generalizing and over-specializing - are discussed. Simple and well organized, this book explains common errors students make in computation in every math operation and with whole numbers, rational numbers, geometry and algebra. This book is an ideal resource for teachers of mathematics in education or special education at the elementary or middle school level.

Table of Contents

PART ONE DIAGNOSIS AND INSTRUCTION	1	(98)
Number and Operations: Computation	3	(9)
Instruction in Mathematics	3	(1)
Computational Fluency	4	(3)
Algorithms	7	(1)
Conceptual Learning and Procedural Learning	7	(2)
Error Patterns in Computation	9	(1)
Paper-and-Pencil Procedures Today	9	(3)
Diagnosing Misconceptions and Error Patterns in Computation	12	(33)
Learning Misconceptions and Error Patterns	15	(6)
Overgeneralizing	18	(2)
Overspecializing	20	(1)
Encouraging Self-Assessment	21	(2)
Interviewing	23	(12)
Getting at a Student's Thinking	26	(2)
Observing Student Behavior	28	(1)
Recording Student Behavior	29	(1)
Watching Language: Ours and Theirs	30	(2)
Probing for Key Understandings	32	(1)
Designing Questions and Tasks	33	(2)
Using Graphic Organizers for Diagnosis	35	(2)
Using Tests	37	(4)
Guiding Diagnosis in Computation	41	(1)
Reflecting on Diagnosis of Misconceptions and Error Patterns	42	(3)
Providing Data-Driven Instruction in Computation	45	(54)
Developing Number Sense	46	(1)
Helping Students Understand Big Ideas	47	(4)
Numeration	48	(2)
Equals and Equivalent	50	(1)
Other Concepts and Principles	51	(1)
Using Models and Concrete Materials	51	(2)
Helping Students Understand Operations	53	(2)
Attaining Computational Fluency	55	(1)
Teaching Mental Computation	56	(1)
Teaching Students to Estimate	57	(2)
Teaching Students to Use Calculators	59	(2)
Teaching Paper-and-Pencil Procedures	61	(8)
Instruction in Grades 1--2	61	(2)
The Role of Manipulatives	63	(1)
Developmental Instruction	63	(4)
Corrective Instruction	67	(1)
Students with Learning Disabilities	67	(2)
Developing Mathematical Vocabulary	69	(2)
Talking and Writing Mathematics	71	(3)
Using Graphic Organizers for Instruction	74	(5)
Understanding and Recalling Basic Number Combinations	79	(5)
Using Alternative Algorithms	84	(2)
Addition of Whole Numbers: Hutchings's Low-Stress Method	85	(1)
Subtraction of Whole Numbers: The Equal Additions (or European-Latino) Method	85	(1)
Subtraction of Rational Numbers: The Equal Additions Method	86	(1)
Involving Peers	86	(3)
Using Portfolios to Monitor Progress During Instruction	89	(1)
Working with Parents	90	(1)
Guiding Instruction	91	(3)
Focus on the Student	91	(1)
Involve Parents	92	(1)
Teach Concepts and Skills	92	(1)
Provide Instruction	92	(1)
Use Concrete Materials	93	(1)
Provide Practice	94	(1)
Reflecting on Instruction in Computation	94	(5)
PART TWO HELPING STUDENTS WHO HAVE LEARNED ERROR PATTERNS	99	(128)
Whole Numbers: Addition and Subtraction	101	(19)
Identifying Patterns	101	(7)
Planning Instruction	108	(11)
Conclusion	119	(1)
Whole Numbers: Multiplication and Division	120	(17)
Identifying Patterns	120	(6)
Planning Instruction	126	(10)
Conclusion	136	(1)
Fractions and Decimals: Concepts and Equivalence	137	(14)
Identifying Patterns	137	(5)
Planning Instruction	142	(7)
Conclusion	149	(2)
Fractions and Decimals: Addition and Subtraction	151	(20)
Identifying Patterns	151	(8)
Planning Instruction	159	(10)
Conclusion	169	(2)
Fractions and Decimals: Multiplication and Division	171	(15)
Identifying Patterns	172	(5)
Planning Instruction	177	(8)
Conclusion	185	(1)
Percent and Proportion	186	(8)
Identifying Patterns	186	(3)
Planning Instruction	189	(4)
Conclusion	193	(1)
Geometry and Measurement	194	(19)
Identifying Patterns	194	(10)
Planning Instruction	204	(8)
Conclusion	212	(1)
Integers	213	(7)
Identifying Patterns	213	(2)
Planning Instruction	215	(4)
Conclusion	219	(1)
Algebra	220	(7)
Identifying Patterns	220	(3)
Planning Instruction	223	(3)
Conclusion	226	(1)
Glossary	227	(5)
Selected Resources	232	(40)
Assessment and Diagnosis	232	(4)
Instruction	236	(8)
Appendices
Appendix A Selected Student Papers and Key	244	(9)
Appendix B Research on Errors in Computation	253	(6)
Appendix C Game-Like Activities with Base Blocks	259	(7)
Appendix D Sample Activities for Cooperative Groups	266	(6)
Appendix E A Diagnostic Interview	272