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

**by**Ashlock, Robert B.

### 9780135009109

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## 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,

**, which is currently in its tenth edition. Although retired, Ashlock continues to teach a few classes.**

*Error Patterns in Computation: Using Error Patterns to Improve Instruction*## Table of Contents

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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