Error Patterns in Computation Using Error

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.

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.

Preface	p. xi
Misconceptions and Error Patterns	p. 1
Computaion, Misconceptions, and Error Patterns	p. 2
Instruction in Mathematics	p. 2
Computational Fluency	p. 3
Algorithms	p. 6
Conceptual Learning and Procedural Learning	p. 7
Paper-and-Pencil Procedures Today	p. 8
Learning Misconceptions and Error Patterns	p. 8
Overgeneralizing	p. 11
Overspecializing	p. 13
Error Patterns in Computation	p. 13
Further Reflection	p. 16
References	p. 16
Error Patterns: Addition and Subtraction with Whole Numbers	p. 18
Identifying Patterns	p. 18
Planning Instruction	p. 25
Conclusion	p. 36
Further Reflection	p. 37
Additional Practice	p. 37
References	p. 38
Error Patterns: Multiplication and Division with Whole Numbers	p. 39
Identifying Patterns	p. 39
Planning Instruction	p. 45
Conclusion	p. 54
Further Reflection	p. 55
Additional Practice	p. 55
Misconceptions and Error Patterns: Concepts and Equivalence with Fractions and Decimals	p. 56
Identifying Patterns	p. 56
Planning Instruction	p. 61
Conclusion	p. 68
Further Reflection	p. 69
Refrence	p. 69
Error Patterns: Addition and Subtraction with Fractions and Decimals	p. 70
Identifying Patterns	p. 70
Planning Instruction	p. 78
Conclusion	p. 88
Further Reflection	p. 89
Additional Practice	p. 89
Refrences	p. 90
Error Patterns: Multiplication and Division with Fractions and Measurement	p. 97
Identifying Patterns	p. 91
Planning Instruction	p. 97
Conclusion	p. 104
Further Reflection	p. 105
Additional Practice	p. 105
Introduction to Misconceptions and Error Patterns: Geometry and Measurement	p. 107
Identifying Patterns
Planning Instruction	p. 117
Conclusion	p. 124
Further Reflection	p. 125
Introduction to Misconceptions and Error Patterns: Percent, Proportion, Integers, and Algebra	p. 126
Identifying Patterns
Planning Instruction	p. 132
Conclusion	p. 142
Further Reflection	p. 143
Additional Practice	p. 144
Diagnosis and Instruction	p. 145
Diagnosing Misconceptions and Error Patterns in Computation and in Other Mathematical Topics	p. 146
Assessing for Varied Purposes	p. 147
Using Formative Assessment: Diagnosing	p. 148
Using Open-Ended Assessment	p. 149
Encouraging Self-Assessment	p. 150
Interviewing	p. 153
Getting at a Student's Thinking	p. 154
Observing Student Behavior	p. 156
Recording Student Behavior	p. 158
Watching Language: Ours and Theirs	p. 159
Probing for Key Understandings	p. 160
Designing Questions and Tasks	p. 162
Using Graphic Organizers for Diagnosis	p. 164
Using Tests for Diagnosis	p. 168
Using Problem Writing for Diagnosis	p. 169
Assessing Dispositions	p. 170
Guiding Diagnosis in Computation
Conclusion	p. 172
Further Reflection	p. 173
Refrences	p. 173
Providing Data-Driven Instruction in Computation	p. 175
Developing Number Sense	p. 176
Helping Students Understand Big Ideas	p. 177
Many Names for a Number	p. 178
Numeration	p. 178
Equals and Equivalent	p. 181
Operations	p. 182
Other Concepts and Principles	p. 183
Making Connections	p. 183
Understanding and Recalling Basic Number Facts	p. 184
Attaining Computational Fluency	p. 189
Teaching Mental Computaion	p. 191
Teaching Students to Estimate	p. 192
Teaching Students to Use Calculators	p. 193
Teaching Paper-and-Pencil Procedures	p. 195
Instruction in Grades 1-2	p. 196
Developmental Instruction	p. 197
Corrective Instruction	p. 200
Students with Special Needs	p. 201
Conclusion	p. 203
Further Reflection	p. 203
References	p. 204
Enriching Instruction in Computation and Other Mathematical Topics	p. 206
Teaching So Students Can Use What They Learn	p. 206
Using Representations	p. 207
The Role of Representations in Learning	p. 207
Using Representations When Teaching	p. 208
Developing Mathematical Vocabulary	p. 210
Talking and Writing Mathematics	p. 212
Using Graphic Organizers for Instruction	p. 215
Using Classroom Discourse	p. 221
Using Portfolios to Monitor and Encourage Progress	p. 221
Conclusion	p. 223
Focus on the student	p. 223
Involve Parents	p. 223
Teach Concepts and Skills	p. 223
Provide Instruction	p. 224
Use Concrete Materials	p. 225
Provide Practice	p. 225
Further Reflection	p. 226
References	p. 226
Glossary	p. 228
Key For Additional Practice	p. 233
Selected Resources	p. 234
Assessment and Diagnosis	p. 234
Instruction	p. 237
Appendixes
Using Alternative Algorithms	p. 245
Addition of Whole Numbers: Hutchings's Low-Stress Method	p. 246
Subraction of Whole Numbers: The Equal Additions (or European-Latino) Method	p. 246
Subraction of Rational Numbers: The Equal Additions Method	p. 247
Involving Peers	p. 248
Working with Parents	p. 251
Game-Like Activities with Base Blocks or the Equivalent	p. 254
Activities for Cooperative Groups	p. 261
Introducing Total-and-Parts Meanings for Operations	p. 267
A Diagnostic Interview	p. 269
A Thematic Unit Can Make Connections Clear	p. 273
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Error Patterns in Computation Using Error Patterns to Help Each Student Learn

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