Young mathematicians at work : constructing fractions, decimals, and percents / Catherine Twomey Fosnot, Maarten Dolk.

Format:

Book

Author/Creator:

Fosnot, Catherine Twomey.

Contributor:

Dolk, Maarten Ludovicus Antonius Marie, 1952-

Language:

English

Subjects (All):

Fractions--Study and teaching (Elementary).

Fractions.

Decimal fractions--Study and teaching (Elementary).

Decimal fractions.

Percentage--Study and teaching (Elementary).

Percentage.

Physical Description:

xx, 170 pages : illustrations ; 24 cm

Place of Publication:

Portsmouth, NH : Heinemann, [2002]

Summary:

In our efforts to reform mathematics education, we've learned a tremendous amount about young students' strategies and the ways they construct knowledge, without fully understanding how to support such development over time. The Dutch do. So Mathematics in the City was begun, pooling the best thinking from both countries. In Young Mathematicians at Work, Catherine Fosnot and Maarten Dolk reveal what they learned after several years of intensive study in numerous urban classrooms. In this third volume in their series, Fosnot and Dolk focus on how children in grades 5-8 construct their knowledge of fractions, decimals, and percents. Their book contrasts word problems with problematic situations that support inquiry, provides strategies to turn classrooms into math workshops, defines and gives examples of modeling, discusses calculation using number sense and the role of algorithms, and describes how to strengthen performance and portfolio assessment.

Contents:

Chapter 1 "Mathematics" or "Mathematizing"?

Learning and Teaching in the Classroom 2

What Is Revealed 7

Back to the Classroom 9

Teaching and Learning as Development 11

The Math Congress 12

What Is Revealed 15

Strategies, Big Ideas, and Models in a Teaching/Learning Framework 16

Strategies as Schemes 16

Big Ideas as Structures 16

Models as Tools for Thought 17

Walking the Edge 18

Chapter 2 The Landscape of Learning

Describing the Journey 21

Linear Frameworks 21

Learning Trajectories 22

The Role of Context 25

Word Problems vs. Truly Problematic Situations 25

Finding Situations for Mathematizing 29

Building in Constraints 29

Open vs. Closed Situations 30

Word Problems vs. Context Problems 30

Context-Based Investigations and Inquiries 31

Turning Classrooms into Mathematical Communities 32

The Edge Between the Individual and the Community 32

Facilitating Dialogue 33

Structuring Math Workshop 33

Investigations 33

Math Congress 34

Minilessons 34

Chapter 3 Equivalence on the Horizon

Fair Sharing: The Historical Roots of Equivalence 37

Early Number Systems 37

Unit Fractions 38

Measurement Contexts 40

Sexagesimals 40

The Rise of Common Fractions 40

The Rise of Decimal Fractions 41

Expanding Commercial Markets as a Context 41

Children's Strategies for Equivalence 43

Math Congress 45

Extending Equivalence to Decimals and Percents 50

Chapter 4 Developing Big Ideas and Strategies

Big Ideas on the Horizon 55

Part/Whole Relation 55

Equivalency vs. Congruency 56

Connecting Multiplication and Division to Fractions 56

The Whole Matters: Making Things Equivalent 57

Relations on Relations 58

Decimals and Percentage Equivalents 63

Place Value 65

Strategies on the Horizon 66

Chapter 5 Developing Mathematical Models

What Are Mathematical Models? 73

Modeling Actions and Situations 74

Facilitating the Development of Modeling: The Role of Context 74

Partitive and Quotative Models 75

Generalizing the Models, Generalizing the Operations 79

From Models of Thinking to Models for Thinking 81

Ratio Tables 81

Double Number Line 84

Double Number Line and Percentages 85

The Clock 89

Arrays 89

Chapter 6 Algorithms Versus Number Sense

The History of Algorithms 98

Teaching for Number Sense 101

Chapter 7 Developing Efficient Computation with Minilessons

Minilessons with Mental Math Strings 109

Choosing the Strategies, Choosing the Numbers, Choosing the Models 112

Finding a Familiar, Landmark Whole 112

Choosing a Common Whole 115

Multiplying Numerators and Denominators 117

Swapping Numerators and Denominators 120

Getting Rid of the Fraction 122

Developing Strategies for Computation with Decimals 124

Using Money 124

Using Fractions and Decimals Interchangeably 125

Chapter 8 Assessment

Performance-Based Evaluation and Assessment 129

Portfolio-Based Documentation 130

Assessing Mathematizing 130

Capturing Genuine Mathematizing 131

Linking Contexts to Reality 132

Providing for Various Levels of Mathematizing 132

Informing Teaching 133

Assessment in Reformed Classrooms 134

Assessing the Landscape of Learning 134

Assessing in the Moment 138

Portfolio Assessment 140

Paper-and-Pencil Assessments 140

Assessment Results 140

Mathematizing vs. Traditional Instruction 140

Standardized Tests 142

Chapter 9 Teachers As Mathematicians

Teacher Preparation 145

Learning to Mathematize 148

Exploring the Edge 148

Formulating a Vision 153

Living in a Mathematical World 157.

Notes:

Includes bibliographical references (pages 161-165) and index.

ISBN:

0325003556

OCLC:

49225645