Young Mathematicians at Work

by Fosnot, Catherine Twomey

• ISBN13: 9780325003559
• ISBN10: 0325003556
• Format: Paperback
• Copyright: 2002-03-26
• Publisher: Greenwood Heinemann

Summary

In this third volume in a series of three, Fosnot and Dolk focus on how children in grades 58 construct their knowledge of fractions, decimals, and percents.

Table of Contents

Acknowledgments	xiii
Preface	xv
About This Book	xix
``Mathematics'' or ``Mathematizing''?
Learning and Teaching in the Classroom	2	(9)
What Is Revealed	7	(2)
Back to the Classroom	9	(2)
Teaching and Learning as Development	11	(5)
The Math Congress	12	(3)
What Is Revealed	15	(1)
Strategies, Big Ideas, and Models in a Teaching/Learning Framework	16	(2)
Strategies as Schemes	16	(1)
Big Ideas as Structures	16	(1)
Models as Tools for Thought	17	(1)
Walking the Edge	18	(1)
Summing Up	18	(3)
The Landscape of Learning
Describing the Journey	21	(4)
Linear Frameworks	21	(1)
Learning Trajectories	22	(3)
The Role of Context	25	(7)
Word Problems vs. Truly Problematic Situations	25	(4)
Finding Situations for Mathematizing	29	(1)
Building in Constraints	29	(1)
Open vs. Closed Situations	30	(1)
Word Problems vs. Context Problems	30	(1)
Context-Based Investigations and Inquiries	31	(1)
Turning Classrooms into Mathematical Communities	32	(3)
The Edge Between the Individual and the Community	32	(1)
Facilitating Dialogue	33	(1)
Structuring Math Workshop	33	(1)
Investigations	33	(1)
Math Congress	34	(1)
Minilessons	34	(1)
Summing Up...	35	(2)
Equivalence on the Horizon
Fair Sharing: The Historical Roots of Equivalence	37	(3)
Early Number Systems	37	(1)
Unit Fractions	38	(2)
Measurement Contexts	40	(1)
Sexagesimals	40	(1)
The Rise of Common Fractions	40	(1)
The Rise of Decimal Fractions	41	(1)
Expanding Commercial Markets as a Context	41	(2)
Children's Strategies for Equivalence	43	(9)
Math Congress	45	(5)
Extending Equivalence to Decimals and Percents	50	(2)
Summing Up...	52	(3)
Developing Big Ideas and Strategies
Big Ideas on the Horizon	55	(11)
Part/Whole Relation	55	(1)
Equivalency vs. Congruency	56	(1)
Connecting Multiplication and Division to Fractions	56	(1)
The Whole Matters: Making Things Equivalent	57	(1)
Relations on Relations	58	(5)
Decimals and Percentage Equivalents	63	(2)
Place Value	65	(1)
Strategies on the Horizon	66	(4)
Summing Up...	70	(3)
Developing Mathematical Models
What Are Mathematical Models?	73	(1)
Modeling Actions and Situations	74	(1)
Facilitating the Development of Modeling: The Role of Context	74	(24)
Partitive and Quotative Models	75	(4)
Generalizing the Models, Generalizing the Operations	79	(2)
From Models of Thinking to Models for Thinking	81	(1)
Ratio Tables	81	(3)
Double Number Line	84	(1)
Double Number Line and Percentages	85	(4)
The Clock	89	(1)
Arrays	89	(1)
Summing Up...	90	(8)
Algorithms Versus Number Sense
The History of Algorithms	98	(3)
Teaching for Number Sense	101	(4)
Summing Up...	105	(4)
Developing Efficient Computation with Minilessons
Minilessons with Mental Math Strings	109	(3)
Choosing the Strategies, Choosing the Numbers, Choosing the Models	112	(13)
Finding a Familiar, Landmark Whole	112	(3)
Choosing a Common Whole	115	(2)
Multiplying Numerators and Denominators	117	(3)
Swapping Numerators and Denominators	120	(2)
Getting Rid of the Fraction	122	(2)
Developing Strategies for Computation with Decimals	124	(1)
Using Money	124	(1)
Using Fractions and Decimals Interchangeably	125	(1)
Summing Up...	125	(4)
Assessment
Performance-Based Evaluation and Assessment	129	(1)
Portfolio-Based Documentation	130	(1)
Assessing Mathematizing	130	(4)
Capturing Genuine Mathematizing	131	(1)
Linking Contexts to Reality	132	(1)
Providing for Various Levels of Mathematizing	132	(1)
Informing Teaching	133	(1)
Assessment in Reformed Classrooms	134	(6)
Assessing the Landscape of Learning	134	(4)
Assessing in the Moment	138	(2)
Portfolio Assessment	140	(1)
Paper-and-Pencil Assessments	140	(1)
Assessment Results	140	(3)
Mathematizing vs. Traditional Instruction	140	(2)
Standardized Tests	142	(1)
Summing Up...	143	(2)
Teachers as Mathematicians
Teacher Preparation	145	(3)
Learning to Mathematize	148	(11)
Exploring the Edge	148	(5)
Formulating a Vision	153	(4)
Living in a Mathematical World	157	(2)
Summing Up...	159	(2)
References	161	(6)
Index	167