Error Patterns in Computation: Using Error Patterns to Improve Instruction (8th Edition)

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9780130270931: Error Patterns in Computation: Using Error Patterns to Improve Instruction (8th Edition)

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.

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From the Publisher:

This text teaches students how to uncover the "wrong" patterns behind the computational errors children make. Through its unique approach, and the generous use of actual students' samples, future teachers not only read about error patterns, they actually encounter them...and learn to correct them. They learn to identify typical error patterns; receive feedback on their diagnosis; and gain an understanding of why a child might have adopted an incorrect procedure.

Excerpt. © Reprinted by permission. All rights reserved.:

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 in Principles 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 Computation has 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's Principles 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 procedures
    – Developmental and corrective instruction
    – Models and concrete materials
    – The role of language
    – Understanding and recalling number combinations
    – Graphic organizers
    – Peer tutoring
    – Portfolios
  • Instructional ideas are also included in chapters that illustrate error patterns for particular operations and areas of mathematics.
  • Lists of resources for assessment and instruction, and the summary of research on error patterns (Appendix B) have been carefully updated.

NEW SECTIONS

New in the eighth edition are:

  • A glossary with definitions and illustrations
  • A chapter with error patterns in algebra
  • A section presenting gamelike activities with pattern boards and base blocks for instruction with whole-number algorithms (Appendix C)
  • A section describing and differentiating conceptual learning and procedural learning
  • A section on attaining computational fluency
  • Sections on involving peers in assessment and peer tutoring
  • New material on developing number sense and the role of invented algorithms

ORGANIZATION OF THE TEXT

The book is organized into two parts with appendices. Part One considers the place of computation within our age of calculators and computers, then focuses on various aspects of assessment and instruction for both concepts and skills. In Part Two, sample student papers are presented in chapters that focus on a particular mathematical topic. Within each chapter you have opportunities to identify patterns of error and to suggest corrective instruction, then compare your ideas with those in this book.

Experience has shown that direct involvement through simulation, as provided in this text, helps both preservice and inservice teachers become more proficient at diagnosing and correcting computational procedures. You gain skill by actually looking for patterns making decisions, and planning instruction. If you are to benefit, it is important that you "play the game" and actually take time to respond.

Chapters 4-12 can be read in either of two ways: a chapter can be read in sequence or, within a chapter, one error pattern at a time can be read as directed in the text. Most readers choose to focus on one pattern at a time.

Additional student papers are included in Appendix A, where you have further opportunities to practice identifying error patterns.

ACKNOWLEDGMENTS

I wish to express appreciation for the encouragement of many classroom teachers who have shown great interest in this book over the years, and to acknowledge the help of teachers, former students, and their students. These colleagues have identified many of the error patterns presented. I would also like to thank the reviewers of the manuscript for their insights and comments: Bea Babbitt, The University of Nevada-Las Vegas; Carol Larson, The University of Arizona; Donna Strand, Baruch College CUNY-New York City; and Karen A. Verbeke, The University of Maryland-Eastern Shores.

—Robert B. Ashlock

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