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Error Patterns in Computation : Using Error Patterns to Improve Instruction

by
Edition:
8th
ISBN13:

9780130270931

ISBN10:
0130270938
Format:
Paperback
Pub. Date:
1/1/2002
Publisher(s):
Prentice Hall

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    Error Patterns in Computation: Using Error Patterns to Improve Instruction





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.

Table of Contents

PART ONE DIAGNOSIS AND INSTRUCTION 1(96)
Computing with Paper and Pencil in an Age of Calculators and Computers
3(8)
Instruction in Mathematics Today
3(1)
Instruction in Computation Today
4(3)
Algorithms
7(1)
Conceptual Learning and Procedural Learning
8(1)
Error Patterns in Computation
9(1)
Paper-and-Pencil Procedures Today
9(2)
Diagnosing Misconceptions and Error Patterns in Computation
11(34)
Learning Misconceptions and Error Patterns
14(5)
Overgeneralizing
17(2)
Overspecializing
19(1)
Encouraging Self-Assessment
19(4)
Involving Peers in Assessment
23(1)
Interviewing: Observing, Recording, and Reflecting
24(11)
Getting at a Student's Thinking
26(2)
Observing Student Behavior
28(1)
Recording Student Behavior
29(2)
Watching language: Ours and Theirs
31(1)
Probing for Key Understandings
32(2)
Designing Questions and Tasks
34(1)
Using Graphic Organizers for Diagnosis
35(4)
Using Tests and Computers for Diagnosis
39(2)
Guiding Diagnosis in Computation
41(4)
Providing Needed Instruction in Computation
45(52)
Developing Number Sense
46(1)
Understanding Concepts and Principles
47(3)
Numeration
47(2)
Equals
49(1)
Other Concepts and Principles
50(1)
Understanding Operations
50(2)
Attaining Computational Fluency
52(2)
Teaching Mental Computation
54(1)
Teaching Students to Estimate
55(2)
Teaching Students to Use Calculators
57(2)
Teaching Paper-and-Pencil Procedures
59(7)
Instruction in Grades 1 - 2
60(1)
The Role of Manipulatives
61(1)
Developmental Instruction
62(4)
Corrective Instruction
66(1)
Acquiring Specialized Vocabulary
66(2)
Using Models and Concrete Materials
68(2)
Understanding and Recalling Basic Number Combinations
70(5)
Talking and Writing Mathematics
75(4)
Using Graphic Organizers for Instruction
79(5)
Using Alternative Algorithms
84(2)
Addition of Whole Numbers: Hutching's Low-Stress Method
84(1)
Subtraction of Whole Numbers: The Equal Additions (or European-Latino) Method
85(1)
Subtraction of Rational Numbers: The Equal Additions Method
85(1)
Using Peer Tutoring
86(1)
Helping Individuals through Cooperative Groups
86(1)
Using Portfolios to Monitor Progress during Instruction
87(2)
Guiding Instruction
89(3)
Focus on the Student
89(1)
Teach Concepts and Skills
90(1)
Provide Instruction
90(1)
Use Concrete Materials
91(1)
Provide Practice
92(1)
Reflecting on Instructional in Computation
92(5)
PART TWO HELPING STUDENTS WHO HAVE LEARNED ERROR PATTERNS 97(159)
Whole Numbers: Addition and Subtraction
99(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

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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