9780130270931

Summary

This book is the only one of its kind that instructs teachers to identify typical error patterns, to receive feedback on their diagnosis, and to gain insight regarding why a child may have adopted an incorrect procedure. Revised to link content to the new NCTM Standards, this new edition emphasizes the meaning of operations and using the appropriate method of computation. As in previous editions, it retains its focus on placing paper-and pencil instructional activities within the context of problem solving. Chapter topics cover diagnosing misconceptions and error patterns in computation, providing needed instruction in computation, and helping students who have learned error patterns. For mathematics teachers of special education classes or at the middle school level.

PART ONE DIAGNOSIS AND INSTRUCTION

(96)

Computing with Paper and Pencil in an Age of Calculators and Computers

(8)

Instruction in Mathematics Today

(1)

Instruction in Computation Today

(3)

Algorithms

(1)

Conceptual Learning and Procedural Learning

(1)

Error Patterns in Computation

(1)

Paper-and-Pencil Procedures Today

(2)

Diagnosing Misconceptions and Error Patterns in Computation

(34)

Learning Misconceptions and Error Patterns

(5)

Overgeneralizing

(2)

Overspecializing

(1)

Encouraging Self-Assessment

(4)

Involving Peers in Assessment

(1)

Interviewing: Observing, Recording, and Reflecting

(11)

Getting at a Student's Thinking

(2)

Observing Student Behavior

(1)

Recording Student Behavior

(2)

Watching language: Ours and Theirs

(1)

Probing for Key Understandings

(2)

Designing Questions and Tasks

(1)

Using Graphic Organizers for Diagnosis

(4)

Using Tests and Computers for Diagnosis

(2)

Guiding Diagnosis in Computation

(4)

Providing Needed Instruction in Computation

(52)

Developing Number Sense

(1)

Understanding Concepts and Principles

(3)

Numeration

(2)

Equals

(1)

Other Concepts and Principles

(1)

Understanding Operations

(2)

Attaining Computational Fluency

(2)

Teaching Mental Computation

(1)

Teaching Students to Estimate

(2)

Teaching Students to Use Calculators

(2)

Teaching Paper-and-Pencil Procedures

(7)

Instruction in Grades 1 - 2

(1)

The Role of Manipulatives

(1)

Developmental Instruction

(4)

Corrective Instruction

(1)

Acquiring Specialized Vocabulary

(2)

Using Models and Concrete Materials

(2)

Understanding and Recalling Basic Number Combinations

(5)

Talking and Writing Mathematics

(4)

Using Graphic Organizers for Instruction

(5)

Using Alternative Algorithms

(2)

Addition of Whole Numbers: Hutching's Low-Stress Method

(1)

Subtraction of Whole Numbers: The Equal Additions (or European-Latino) Method

(1)

Subtraction of Rational Numbers: The Equal Additions Method

(1)

Using Peer Tutoring

(1)

Helping Individuals through Cooperative Groups

(1)

Using Portfolios to Monitor Progress during Instruction

(2)

Guiding Instruction

(3)

Focus on the Student

(1)

Teach Concepts and Skills

(1)

Provide Instruction

(1)

Use Concrete Materials

(1)

Provide Practice

(1)

Reflecting on Instructional in Computation

(5)

PART TWO HELPING STUDENTS WHO HAVE LEARNED ERROR PATTERNS

(159)

Whole Numbers: Addition and Subtraction

(25)

Identifying Patterns

100

(6)

Error Pattern N-C-1

100

(1)

Error Pattern A-W-1

100

(1)

Error Pattern A-W-2

101

(1)

Error Pattern A-W-3

101

(1)

Error Pattern A-W-4

102

(1)

Error Pattern S-W-1

103

(1)

Error Pattern S-W-2

104

(1)

Error Pattern S-W-3

104

(1)

Error Pattern S-W-4

105

(1)

Describing Instruction

106

(7)

Error Pattern N-C-1

106

(1)

Error Pattern A-W-1

106

(1)

Error Pattern A-W-2

107

(1)

Error Pattern A-W-3

108

(1)

Error Pattern A-W-4

109

(1)

Error Pattern S-W-1

110

(1)

Error Pattern S-W-2

111

(1)

Error Pattern S-W-3

112

(1)

Error Pattern S-W-4

112

(1)

Considering Alternatives

113

(10)

Error Pattern N-C-1

113

(1)

Error Pattern A-W-1

114

(1)

Error Pattern A-W-2

115

(1)

Error Pattern A-W-3

116

(1)

Error Pattern A-W-4

117

(1)

Error Pattern S-W-1

118

(1)

Error Pattern S-W-2

119

(1)

Error Pattern S-W-3

120

(2)

Error Pattern S-W-4

122

(1)

Conclusion

123

(1)

Whole Numbers: Multiplication and Division

124

(21)

Identifying Patterns

124

(6)

Error Pattern M-W-1

124

(1)

Error Pattern M-W-2

125

(1)

Error Pattern M-W-3

126

(1)

Error Pattern D-W-1

127

(1)

Error Pattern D-W-2

128

(1)

Error Pattern D-W-3

129

(1)

Describing Instruction

130

(4)

Error Pattern M-W-1

130

(1)

Error Pattern M-W-2

131

(1)

Error Pattern M-W-3

131

(1)

Error Pattern D-W-1

132

(1)

Error Pattern D-W-2

133

(1)

Error Pattern D-W-3

134

(1)

Considering Alternatives

134

(10)

Error Pattern M-W-1

134

(3)

Error Pattern M-W-2

137

(1)

Error Pattern M-W-3

138

(2)

Error Pattern D-W-1

140

(1)

Error Pattern D-W-2

141

(1)

Error Pattern D-W-3

142

(2)

Conclusion

144

(1)

Fractions: Concepts and Equivalence

145

(15)

Identifying Patterns

145

(4)

Error Pattern F-C-1

145

(1)

Error Pattern F-C-2

146

(1)

Error Pattern F-E-1

147

(1)

Error Pattern F-E-2

148

(1)

Describing Instruction

149

(3)

Error Pattern F-C-1

149

(1)

Error Pattern F-C-2

150

(1)

Error Pattern F-E-1

151

(1)

Error Pattern F-E-2

152

(1)

Considering Alternatives

152

(6)

Error Pattern F-C-1

152

(1)

Error Pattern F-C-2

153

(2)

Error Pattern F-E-1

155

(2)

Error Pattern F-E-2

157

(1)

Conclusion

158

(2)

Fractions and Decimals: Addition and Subtraction

160

(24)

Identifying Patterns

161

(7)

Error Pattern A-F-1

161

(1)

Error Pattern A-F-2

162

(1)

Error Pattern A-F-3

163

(1)

Error Pattern S-F-1

164

(1)

Error Pattern S-F-2

165

(1)

Error Pattern S-F-3

166

(1)

Error Pattern A-D-1

167

(1)

Error Pattern S-D-1

167

(1)

Describing Instruction

168

(6)

Error Pattern A-F-1

168

(1)

Error Pattern A-F-2

169

(1)

Error Pattern S-F-3

169

(1)

Error Pattern S-F-1

170

(1)

Error Pattern S-F-2

171

(1)

Error Pattern S-F-3

171

(1)

Error Pattern A-D-1

172

(1)

Error Pattern S-D-1

173

(1)

Considering Alternatives

174

(9)

Error Pattern A-F-1

174

(1)

Error Pattern A-F-2

175

(1)

Error Pattern A-F-3

176

(1)

Error Pattern S-F-1

177

(1)

Error Pattern S-F-2

178

(1)

Error Pattern S-F-3

179

(1)

Error Pattern A-D-1

180

(2)

Error Pattern S-D-1

182

(1)

Conclusion

183

(1)

Fractions and Decimals: Multiplication and Division

184

(18)

Identifying Patterns

185

(5)

Error Pattern M-F-1

185

(1)

Error Pattern M-F-2

186

(1)

Error Pattern D-F-1

186

(1)

Error Pattern D-F-2

187

(1)

Error Pattern M-D-1

188

(1)

Error Pattern D-D-1

189

(1)

Describing Instruction

190

(4)

Error Pattern M-F-1

190

(1)

Error Pattern M-F-2

191

(1)

Error Pattern D-F-1

192

(1)

Error Pattern D-F-2

192

(1)

Error Pattern M-D-1

193

(1)

Error Pattern D-D-1

193

(1)

Considering Alternatives

194

(7)

Error Pattern M-F-1

194

(1)

Error Pattern M-F-2

195

(1)

Error Pattern D-F-1

196

(1)

Error Pattern D-F-2

197

(1)

Error Pattern M-D-1

198

(1)

Error Pattern D-D-1

199

(2)

Conclusion

201

(1)

Percent Problems

202

(10)

Identifying Patterns

202

(3)

Error Pattern P-P-1

202

(2)

Error Pattern P-P-2

204

(1)

Describing Instruction

205

(2)

Error Pattern P-P-1

205

(1)

Error Pattern P-P-2

206

(1)

Considering Alternatives

207

(4)

Error Pattern P-P-1

207

(2)

Error Pattern P-P-2

209

(2)

Conclusion

211

(1)

Geometry and Measurement

212

(26)

Identifying Patterns

213

(9)

Error Pattern G-M-1

213

(1)

Error Pattern G-M-2

214

(1)

Error Pattern G-M-3

214

(2)

Error Pattern G-M-4

216

(1)

Error Pattern G-M-5

217

(2)

Error Pattern G-M-6

219

(2)

Error Pattern G-M-7

221

(1)

Describing Instruction

222

(6)

Error Pattern G-M-1

222

(1)

Error Pattern G-M-2

223

(1)

Error Pattern G-M-3

224

(1)

Error Pattern G-M-4

225

(1)

Error Pattern G-M-5

226

(1)

Error Pattern G-M-6

226

(2)

Error Pattern G-M-7

228

(1)

Considering Alternatives

228

(9)

Error Pattern G-M-1

228

(2)

Error Pattern G-M-2

230

(1)

Error Pattern G-M-3

231

(1)

Error Pattern G-M-4

232

(1)

Error Pattern G-M-5

233

(1)

Error Pattern G-M-6

234

(2)

Error Pattern G-M-7

236

(1)

Conclusion

237

(1)

Integers

238

(9)

Identifying Patterns

238

(2)

Error Pattern A-I-1

238

(1)

Error Pattern A-I-2

239

(1)

Error Pattern S-I-1

240

(1)

Describing Instruction

240

(2)

Error Pattern A-I-1

240

(1)

Error Pattern A-I-2

241

(1)

Error Pattern S-I-1

241

(1)

Considering Alternatives

242

(4)

Error Pattern A-I-1

242

(1)

Error Pattern A-I-2

243

(2)

Error Pattern S-I-1

245

(1)

Conclusion

246

(1)

Algebra

247

(9)

Identifying Patterns

247

(3)

Error Pattern ALG-1

247

(1)

Error Pattern ALG-2

248

(1)

Error Pattern ALG-3

249

(1)

Error Pattern ALG-4

249

(1)

Describing Instruction

250

(2)

Error Pattern ALG-1

250

(1)

Error Pattern ALG-2

250

(1)

Error Pattern ALG-3

251

(1)

Error Pattern ALG-4

251

(1)

Considering Alternatives

252

(3)

Error Pattern ALG-1

252

(1)

Error Pattern ALG-2

252

(1)

Error Pattern ALG-3

253

(1)

Error Pattern ALG-4

254

(1)

Conclusion

255

(1)

Glossary

256

(5)

Selected Resources

261

(1)

Assessment and Diagnosis

261

(5)

Instruction

266

(8)

Appendices

Appendix A Selected Student Papers and Key

274

(9)

Appendix B Research on Errors in Computation

283

(6)

Appendix C Gamelike Activities with Base Blocks

289

(8)

Appendix D Sample Activities for Cooperative Groups

297

Supplemental Materials

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Excerpts

We want to help oar students learn to think mathematically. Mere skill in computing is not our primary concern. Even so, computation in its many forms--using estimation, mental computation, calculators and computers, and paper-and-pencil procedures--continues to have a significant role in both the learning of mathematics and in solving problems within the world around us. This is made clear inPrinciples and Standards for School Mathematics,published by the National Council of Teachers of Mathematics in 2000. We hope all of our students will understand what they study. We want them to succeed in mathematics and to enjoy solving problems--but students sometimes learn misconceptions. As they learn about operations and methods of computation, they sometimes adopt erroneous procedures in spite of our best efforts. This book was written for those who are willing to listen carefully to what each student says and to make thoughtful analyses of student papers--teachers who want to help each student by discovering error patterns they may be using, and to be enabled, thereby, to focus instruction more effectively. This book was written to help teachers, whatever the level, to look at all student work diagnostically. Error Patterns in Computationhas changed greatly through the years. The first 96 pages of this edition are devoted to instructional issues. These pages provide help with diagnosing misconceptions and error patterns in computation, and they present many ideas for teaching varied methods of computation. Much of the book focuses on detecting the systematic errors many students make when computing with paper and pencil. Reasons students may have learned erroneous procedures are considered, and strategies for helping those students are presented. Of course, many of the instructional strategies described are useful when teaching any student--whether that student has experienced difficulty under previous instruction or not. The erroneous patterns displayed by students are not due to carelessness alone, nor are they due to insufficient practice. Students observe patterns and make inferences during instruction. In the case of an error pattern, that which has been learned does not always produce correct answers. Interestingly, some incorrect procedures produce correct answers part of the time. When this happens, students are reinforced in their belief that they have learned the desired concepts and skills. You will find looking for error patterns to be a very worthwhile assessment activity. You will gain more specific knowledge of each student's strengths upon which to base future instruction. Whenever you observe error patterns in your own classroom, be sure to refrain from assigning practice activities that reinforce incorrect concepts and procedures. THE EIGHTH EDITION This edition reflects many of the concerns of NCTM'sPrinciples and Standards for School Mathematics,including emphasis on concepts as well as skills. In this edition: Understanding the meanings of operations is emphasized, as is knowing when to use each operation. Computation, by whatever method, is distinguished from the operation itself. Different methods of computation are presented with an emphasis on using each method when appropriate. Paper-and-pencil procedures are considered in the context of other methods of computation. Many error patterns are presented within problem-solving contexts. Varied means of assessment are described and illustrated; self-assessment is emphasized. Extensive material on interviewing is included. Instruction in computation is addressed extensively through topics such as: - Concepts and principles - Understanding operations - Computational fluency - Mental computation, estimation, using calculators - Paper-and-pencil

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Error Patterns in Computation : Using Error Patterns to Improve Instruction

0130270938

Summary

Table of Contents

Supplemental Materials

Excerpts

Rewards Program