Error Patterns in Computation Using Error Patterns to Help Each Student Learn

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  • Edition: 10th
  • Format: Paperback
  • Copyright: 2009-01-13
  • Publisher: Pearson

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As students learn about mathematical operations and methods of computation, they may adopt erroneous procedures and misconceptions. This engaging book was written to model how teachers can make thoughtful analyses of their students work and in so doing, discover patterns in errors made.

Author Biography

Robert Ashlock began his career in 1957 as a fourth and fifth grade teacher with a bachelor’s degree in elementary education. He later received a master’s degree in elementary school administration from Butler University and a doctorate in elementary education from Indiana University. He became a graduate assistant and teaching associate at Indiana University in 1964 and went on to teach at several institutions including the University of Maryland, Reformed Theological Seminary, Belhaven College, and finally, Covenant College in Lookout Mountain, Georgia.  Ashlock came to Covenant in 1988 to fill the need for a specialist in elementary education who could also teach on the graduate level.  He directed the Master of Education Program, taught both undergraduate and graduate education courses, and coordinated the process necessary for the teacher education program to be approved by the Georgia Professional Standards Commission.


He is one of only two professors ever to receive the title Professor Emeritus at Covenant.  He is known throughout the education community for his book, Error Patterns in Computation: Using Error Patterns to Improve Instruction , which is currently in its tenth edition. Although retired, Ashlock continues to teach a few classes.

Table of Contents

Prefacep. xi
Misconceptions and Error Patternsp. 1
Computaion, Misconceptions, and Error Patternsp. 2
Instruction in Mathematicsp. 2
Computational Fluencyp. 3
Algorithmsp. 6
Conceptual Learning and Procedural Learningp. 7
Paper-and-Pencil Procedures Todayp. 8
Learning Misconceptions and Error Patternsp. 8
Overgeneralizingp. 11
Overspecializingp. 13
Error Patterns in Computationp. 13
Further Reflectionp. 16
Referencesp. 16
Error Patterns: Addition and Subtraction with Whole Numbersp. 18
Identifying Patternsp. 18
Planning Instructionp. 25
Conclusionp. 36
Further Reflectionp. 37
Additional Practicep. 37
Referencesp. 38
Error Patterns: Multiplication and Division with Whole Numbersp. 39
Identifying Patternsp. 39
Planning Instructionp. 45
Conclusionp. 54
Further Reflectionp. 55
Additional Practicep. 55
Misconceptions and Error Patterns: Concepts and Equivalence with Fractions and Decimalsp. 56
Identifying Patternsp. 56
Planning Instructionp. 61
Conclusionp. 68
Further Reflectionp. 69
Refrencep. 69
Error Patterns: Addition and Subtraction with Fractions and Decimalsp. 70
Identifying Patternsp. 70
Planning Instructionp. 78
Conclusionp. 88
Further Reflectionp. 89
Additional Practicep. 89
Refrencesp. 90
Error Patterns: Multiplication and Division with Fractions and Measurementp. 97
Identifying Patternsp. 91
Planning Instructionp. 97
Conclusionp. 104
Further Reflectionp. 105
Additional Practicep. 105
Introduction to Misconceptions and Error Patterns: Geometry and Measurementp. 107
Identifying Patterns
Planning Instructionp. 117
Conclusionp. 124
Further Reflectionp. 125
Introduction to Misconceptions and Error Patterns: Percent, Proportion, Integers, and Algebrap. 126
Identifying Patterns
Planning Instructionp. 132
Conclusionp. 142
Further Reflectionp. 143
Additional Practicep. 144
Diagnosis and Instructionp. 145
Diagnosing Misconceptions and Error Patterns in Computation and in Other Mathematical Topicsp. 146
Assessing for Varied Purposesp. 147
Using Formative Assessment: Diagnosingp. 148
Using Open-Ended Assessmentp. 149
Encouraging Self-Assessmentp. 150
Interviewingp. 153
Getting at a Student's Thinkingp. 154
Observing Student Behaviorp. 156
Recording Student Behaviorp. 158
Watching Language: Ours and Theirsp. 159
Probing for Key Understandingsp. 160
Designing Questions and Tasksp. 162
Using Graphic Organizers for Diagnosisp. 164
Using Tests for Diagnosisp. 168
Using Problem Writing for Diagnosisp. 169
Assessing Dispositionsp. 170
Guiding Diagnosis in Computation
Conclusionp. 172
Further Reflectionp. 173
Refrencesp. 173
Providing Data-Driven Instruction in Computationp. 175
Developing Number Sensep. 176
Helping Students Understand Big Ideasp. 177
Many Names for a Numberp. 178
Numerationp. 178
Equals and Equivalentp. 181
Operationsp. 182
Other Concepts and Principlesp. 183
Making Connectionsp. 183
Understanding and Recalling Basic Number Factsp. 184
Attaining Computational Fluencyp. 189
Teaching Mental Computaionp. 191
Teaching Students to Estimatep. 192
Teaching Students to Use Calculatorsp. 193
Teaching Paper-and-Pencil Proceduresp. 195
Instruction in Grades 1-2p. 196
Developmental Instructionp. 197
Corrective Instructionp. 200
Students with Special Needsp. 201
Conclusionp. 203
Further Reflectionp. 203
Referencesp. 204
Enriching Instruction in Computation and Other Mathematical Topicsp. 206
Teaching So Students Can Use What They Learnp. 206
Using Representationsp. 207
The Role of Representations in Learningp. 207
Using Representations When Teachingp. 208
Developing Mathematical Vocabularyp. 210
Talking and Writing Mathematicsp. 212
Using Graphic Organizers for Instructionp. 215
Using Classroom Discoursep. 221
Using Portfolios to Monitor and Encourage Progressp. 221
Conclusionp. 223
Focus on the studentp. 223
Involve Parentsp. 223
Teach Concepts and Skillsp. 223
Provide Instructionp. 224
Use Concrete Materialsp. 225
Provide Practicep. 225
Further Reflectionp. 226
Referencesp. 226
Glossaryp. 228
Key For Additional Practicep. 233
Selected Resourcesp. 234
Assessment and Diagnosisp. 234
Instructionp. 237
Using Alternative Algorithmsp. 245
Addition of Whole Numbers: Hutchings's Low-Stress Methodp. 246
Subraction of Whole Numbers: The Equal Additions (or European-Latino) Methodp. 246
Subraction of Rational Numbers: The Equal Additions Methodp. 247
Involving Peersp. 248
Working with Parentsp. 251
Game-Like Activities with Base Blocks or the Equivalentp. 254
Activities for Cooperative Groupsp. 261
Introducing Total-and-Parts Meanings for Operationsp. 267
A Diagnostic Interviewp. 269
A Thematic Unit Can Make Connections Clearp. 273
Table of Contents provided by Ingram. All Rights Reserved.

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