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Research in mental computation: Multiple perspectives

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Product Description

By Rosemary A Callingham, Jane M Watson

Published: 2008
ISBN: 978-1-921214-36-3
Pages: ii+122
Imprint: Post Pressed

Overview

It’s only relatively recently (in relation to the history of teaching mathematics) that teachers and researchers have become interested in students’ mental computation skills and strategies. after all, if written algorithms are the main means of calculating, then all that is of interest is whether or not students have learned the addition and subtraction facts to 20 and the times tables. Now many curriculums not only expect mental computation to be taught but many also specify particular methods to focus on. Most of these expectations are based on beliefs rather than evidence: although there has been much research on school students’ difficulties with written computation algorithms, little systematic research has considered similar issues in the mental computation arena. This book makes a valuable and important contribution to developing the evidence base for mental computation by drawing together several varied research projects in Australia that provide different perspectives on the issues surrounding students’ successes and difficulties with mental computation.

Where other research has been carried out on calculation strategies it has usually been small scale – qualitative interviews with a limited number of students. And the research has usually provided a snapshot of student understanding at a moment in time. Large scale, quantitative analyses of students’ mental computations over time are rare. This book presents one of those rarities. It considers the relative difficulty of whole and part-whole number problems using the four operations in relation to student ability using Rasch measurement; as far as I am aware this type of research has not been carried out before. The resulting eight levels of mental computation competence provide benchmarks for teachers and curriculum planners. The collection of longitudinal data has also allowed student development to be monitored for individuals, illustrating the diversity of student progress.

Student interviews have further highlighted the strategies employed in solving problems mentally by students across the school years from grade 3 to grade 10; again such longitudinal detail has not been reported previously. Together the quantitative and qualitative analysis on fractions, decimals, and percents surpasses any previous research in the area of mental computation.

Finally, the descriptions of interventions with teachers and suggestions for implementation of programs to develop computation skills and understanding reflect the work of the authors, and especially their colleague, Alistair McIntosh, over many years in Australia. This book’s comprehensive reporting on the many research perspectives employed to characterise the issues encountered by school students with mental computation is a fitting and valuable closure for the team’s work together. I hope the readers find this book as engaging and interesting as I did.

Table of Contents

Foreword
Acknowledgements

Chapter 1: Introduction

  • Mental Computation in the Mathematics Curriculum
  • Research in Mental Computation
  • Strategies for Mental Computation
  • Teaching Mental Computation
  • Mental Computation with Part-whole Numbers
  • This Publication

Chapter 2: Measuring Mental Computation Competence

  • The Rasch Measurement Model
  • Interpreting the Rasch Model
  • Instruments
  • Schools and Students
  • A Developmental Scale of Mental Computation Competence
    • Description of the Identified Levels of Mental Computation Competence
    • Discussion of the Levels Identified
    • Distribution of Students Across Levels
  • Sub-domains of Mental Computation
  • Summary

Chapter 3: Performance and Growth in Mental Computation

  • Growth and Performance across the Grades of Schooling
    • Performance by Grade
    • Performance by Grade within Schools
  • Longitudinal Study of Individual Students
    • Students Starting in Grade 3
    • Students Starting in Grade 5
    • Students Starting in Grade 7
  • Summary

Chapter 4: Error Analysis from Mental Computation Tests

  • Analysis of Level B Mental Computation Items
    • Whole Number Single-digit Addition and Subtraction
    • Two-digit Addition and Subtraction
    • Whole Number Single-digit Multiplication and Division
  • Analysis of Level D Mental Computation Items
    • Whole Number Single-digit Multiplication and Division
    • Multi-digit Addition and Subtraction
    • Two-digit Multiplication and Division
    • Fractions Addition and Subtraction
    • Fractional Operators
  • Analysis of Level F Mental Computation Items
    • Whole Number Single-digit Multiplication and Division
    • Two-digit Addition and Subtraction
    • Two-digit Multiplication and Division
    • Decimals Addition and Subtraction
    • Decimals Multiplication and Division
    • Fractions Addition and Subtraction
    • Fractional Operators
    • Percent
  • Analysis of Level H Mental Computation Items
    • Two-digit Multiplication and Division
    • Decimals Multiplication and Division
    • Fractions Addition and Subtraction
    • Fractions Multiplication and Division
    • Percent
  • Links among Errors

Chapter 5: Strategy Use in Whole Number Mental Computation

  • Whole Number Two-digit Addition and Subtraction
    • Addition
    • Subtraction
  • Whole Number Multiplication and Division
    • Multiplication
    • Division
    • Halving and Doubling
  • Discussion

Chapter 6: Mental Computation with Fractions, Decimals and Percents

  • A Scale of Mental Computation Competence with Part-whole Numbers
    • Interpretation of the Part-whole Competence Scale
    • Misfitting Items
  • Performance across Grades
  • Error Analysis from the Tests
  • Strategies for Mental Computation with Part-whole Numbers
    • Strategy Use with Fraction Problems
    • Strategy Use with Decimal Problems
    • Strategy Use with Percent Problems
    • Summary

Chapter 7: Developing Mental Computation Programs

Key Teachers and Resource Development

  • Structure for Working with Teachers
  • Outcomes for Teachers
    • Organisation and Curriculum
    • Impact on Teachers
    • Factors that Made a Difference
  • Focus on the Interface between Mental and Written Computation
    • Outcomes from the Project
  • Implementing a Whole School Approach in One Primary School
    • Outcomes from the Whole School Program
  • Common Features of the Projects
    • Teacher Ownership
    • Underpinning Framework
    • Links between Research and Practice
    • Systemic Support
  • Implications for Computation in the Classroom

Chapter 8: Implications for Schooling and Further Research

  • Implications for Curriculum Development
    • Standards and Benchmarks of Performance
    • Development of Strategies for Part-whole Mental Computation
  • Implications for Schools and Classrooms
    • School organisation
    • Developing a Mental Computation Program
  • Implications for Future Research
  • Final Remarks

References

Reviews

This book makes a valuable and important contribution to developing the evidence base for mental computation by drawing together several varied research projects in Australia that provide different perspectives on the issues surrounding students' successes and difficulties with mental computation... Where other research has been carried out on calculation strategies it has usually been small scale-qualitative interviews with a limited number of students. And the research has usually provided a snapshot of student understanding at a moment in time.

Large scale, quantitative analyses of students' mental computations over time are rare. This book presents one of those rarities...

I love finding research that surprises me and surprises were in store here, as different student strategies emerge from those that might be expected in written computation, for both whole and part-whole numbers. Together the quantitative and qualitative analysis on fractions, decimals, and percents I think surpasses any previous research in the area of mental computation.

Dr Mike Askew
Professor of Mathematics Education
King's College London


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