ERIC ED410726: Teachers Nurturing Math-Talented Young Children.

120 Pages·1996·2.5 MB·English

by  ERIC

DOCUMENT RESUME
EC 305 784
ED 410 726
Waxman, Barbara; Robinson, Nancy M.; Mukhopadhyay, Swapha
AUTHOR
Teachers Nurturing Math-Talented Young Children.
TITLE
National Research Center on the Gifted and Talented, Storrs,
INSTITUTION
CT.
Office of Educational Research and Improvement (ED),
SPONS AGENCY
Washington, DC.
RM96228
REPORT NO
1996-00-00
PUB DATE
NOTE
117p.
R206R50001
CONTRACT
Research (143)
Reports
Non-Classroom (055)
Guides
PUB TYPE
Tests /Questionnaires (160)
MF01/PC05 Plus Postage.
EDRS PRICE
Classroom Environment; Early Childhood Education; *Early
DESCRIPTORS
Identification; *Educational Strategies; *Gifted;
Kindergarten Children; Learning Strategies; Mathematics
Achievement; *Mathematics Instruction; Preschool Children;
Questionnaires; Resource Materials; *Talent Development;
*Talent Identification
ABSTRACT
This book is an outgrowth of a 2-year study of 284 children
discovered during preschool or kindergarten to be advanced in mathematics. In
addition to psychometric and cognitive testing conducted at the beginning,
middle, and end of the study, half of the children attended biweekly
interventions designed to enrich their experience with mathematics. Results
found the children remained advanced in math over the 2-year period, their
spatial reasoning related more closely to their math reasoning than did their
verbal reasoning, and the math scores of the boys started and remained higher
than those of the girls. The intervention proved effective in enhancing
mathematical reasoning. The book discusses ways of identifying verv-yoiIng
math-advanced children as well as a variety of educational strategies to meet
their needs. Its primary emphasis is on creating an open-ended approach to
teaching mathematics that provides an opportunity for children at differerit
levels of advancement and different personal styles to engage with
mathematical challenges in a playful way, to conceptualize math broadly, to
pose problems, and to make sense of the mathematical system. Appendices
include a questionnaire for parents, jobcards and other activities, annotated
(Contains 84
bibliography, and recommendations for the classroom.
references.)
(CR)
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University of Connecticut
City University of New York, City College
Stanford University
University of Virginia
Yale University
Teachers Nurturing Math-Talented
Young Children
Barbara Waxman
Nancy M. Robinson
Swapna Mukhopadhyay
University of Washington
Seattle, Washington
U.S. DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
EDUCATIONAL
CATIONAL RESOURCES INFORMATION
CENTER (ERIC)
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received from the person or organization
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December 1996
Number RM96228
BEST copy AVAILABLE
2
Teachers Nurturing Math-Talented Young Children
Barbara Waxman
Nancy M. Robinson
Swapna Mukhopadhyay
University of Washington
Seattle, Washington
December 1996
Number RM96228
THE NATIONAL
RESEARCH CENTER
ON THE GIFTED
AND TALENTED
The National Research Center on the Gifted and Talented (NRC/GT) is funded under the
Jacob K. Javits Gifted and Talented Students Education Act, Office of Educational
Research and Improvement, United States Department of Education.
The Directorate of the NRC/GT serves as an administrative and a research unit and is
located at the University of Connecticut.
The participating universities include the City University of New York, City College,
Stanford University, University of Virginia, and Yale University, as well as a research
unit at the University of Connecticut.
University of Connecticut
Dr. Joseph S. Renzulli, Director
Dr. E. Jean Gubbins, Associate Director
City University of New York, City College
Dr. Deborah L. Coates, Site Research Coordinator
Stanford University
Dr. Shirley Brice Heath, Site Research Coordinator
University of Virginia
Dr. Carolyn M. Callahan, Associate Director
Yale University
Dr. Robert J. Sternberg, Associate Director
Copies of this report are available from:
NRC/GT
University of Connecticut
362 Fairfield Road, U-7
Storrs, CT 06269-2007
The work reported herein was supported under the Educational Research and Development Centers
Program, PR/Award Number R206R50001, as administered by the Office of Educational Research
and Improvement, U.S. Department of Education. The findings and opinions expressed in this report
do not reflect the position or policies of the National Institute on the Education of At-Risk Students,
the Office of Educational Research and Improvement, or the U.S. Department of Education.
4
ii
Note to Readers...
All papers by The National Research Center on the Gifted and Talented may be
reproduced in their entirety or in sections. All reproductions, whether in part or
whole, should include the following statement:
The work reported herein was supported under the Educational
Research and Development Centers Program, PR/Award Number
R206R50001, as administered by the Office of Educational Research and
Improvement, U.S. Department of Education. The findings and opinions
expressed in this report do not reflect the position or policies of the
National Institute on the Education of At-Risk Students, the Office of
Educational Research and Improvement, or the U.S. Department of
Education.
This document has been reproduced with the permission of The National
Research Center on the Gifted and Talented.
If sections of the papers are printed in other publications, please forward a copy to:
The National Research Center on the Gifted and Talented
University of Connecticut
362 Fairfield Road, U-7
Storrs, CT 06269-2007
Acknowledgements
The nurturing of young math-talented children could not have been accomplished without
the insight, caring, and energy expressed by the aides and teachers over the two years that
the Saturday Clubs took place. The aides were a dedicated group of volunteers that
included upper-level psychology undergraduates and students pursuing their teacher
certification. The teachers included: Rachel Bukey, Patty Chastain, Julie Cooper, Randy
Katz, Marjorie Lamarre, Joy McBride, Joan O'Connor, Chris Poserycki, Jan Tillotson,
Barbara Waxman, and Paul Williams.
By far, our biggest debt goes to the participating children and their parents. We learned
much from observing and teaching these lively, spirited children. They taught us, too.
Teachers Nurturing Math-Talented Young Children
Barbara Waxman
Nancy M. Robinson
Swapna Mukhopadhyay
University of Washington
Seattle, Washington
ABSTRACT
Talent in mathematical reasoning is highly valued in this society and yet very little is known
about its early course. This book is an outgrowth of a two-year study of children
discovered during preschool or kindergarten to be advanced in their thinking about math. In
addition to psychometric and cognitive testing conducted at the beginning, middle, and end
(Saturday
of the study, half of the children were randomly assigned to biweekly intervention
revealed
Club) for a total of 28 weeks over the two years. Among other findings, the study
that their
that, as a group, the children remained advanced in math over the two-year period,
reasoning
spatial reasoning related more closely to their math reasoning than did their verbal
(although they were ahead in all three domains), and that the math scores of the boys started
and remained somewhat higher than those of the girls. The Saturday Club intervention
proved effective in enhancing mathematical reasoning.
This book discusses ways of identifying very young math-advanced children as well as a
creating an
variety of educational strategies to meet their needs. Its primary emphasis is on
open-ended approach to teaching mathematics that provides an opportunity for children at
different levels of advancement and different personal styles to engage with mathematical
challenges in a playful way, to conceptualize math broadly, to pose problems, and to make
the importance of representing and
sense of the mathematical system. Also emphasized are
communicating mathematical ideas in multiple ways in order to deepen children's
understanding. A variety of engaging activities such as the Fibonacci series, the Vedic
of these activities emanate from "big ideas"
square, and chip-trading are described. Most
such as the nature of numerals and the number system, equivalence, visualizing and
graphing numbers, measurement, estimation, and so on. Job cards for various mathematical
tasks are included, as well as ways to integrate mathematics into other aspects of the
curriculum. The approach to mathematics portrayed in this book is one that creative
teachers can flexibly adapt to meet the needs of math-advanced children in a regular or
specialized classroom.
7
vii
Table of Contents
vii
ABSTRACT
CHAPTER 1: Young Children Advanced in Mathematical Thinking
How Can We Recognize Them?
1
2
Some Common Characteristics of Math-Advanced Children
CHAPTER 2: Math TrekIdentifying and Nurturing Mathematical Precocity
in Young Children
5
5
Research About Very Young Math-Talented Students
6
Research Questions
7
The Research Team
7
The Research Plan
9
Research Findings
CHAPTER 3: Alternatives in Meeting the Needs of Math-Advanced
ChildrenA Smorgasbord
13
13
A Guiding Concept: The Principle of the Optimal Match
14
Fundamental Versus Complementary Components
15
Acceleration
16
Smorgasbord Options Within the Classroom
16
Compacting the Curriculum
16
Working Ahead in the Curriculum
17
Mentoring
Diagnostic Testing Followed by Prescriptive Instruction (DT-PI)
17
17
Learning Contracts
Activities to Extend the Math Curriculum Without Driving
18
the Teacher Crazy
21
Smorgasbord Options Between Classes
21
All-School Math
21
Cluster Grouping
Ability Grouping Within the Classroom for Core Instruction,
21
Especially for High Ability Students
22
Multi-Age Classrooms
22
Trading Students: Subject-Matter Acceleration
23
Early Entry to Kindergarten or First Grade
23
Skipping a Grade
24
Pull-Out Programs and Resource Rooms
24
Special Classrooms
24
Teacher Consultants/Enrichment Specialists
25
Open-Ended Strategies in the Classroom
25
Conclusion
CHAPTER 4: The Math Trek CurriculumPhilosophy and Practice
27
27
Beliefs About Learning
28
Multiple Abilities or "Intelligences"
28
Play and Playfulness
29
Problem-Posing
29
Sense-Making, Model Building, and Understanding
30
Inventing Procedures, Developing Number Sense
ix
8
Table of Contents
Defining the Teacher's Role
30
Teacher as Learner, Too
31
The Role of Good Questions
31
A Word About Manipulatives
31
Importance of the Social Context in Learning
31
Conceptualizing Mathematics Broadly
32
What to Teach? The Power of Big Ideas
32
Teachers' Beliefs About Mathematics
33
Aesthetics, Passion, and Transcendence
33
Open-Ended Curriculum
34
Structure of the Saturday Clubs
34
CHAPTER 5: The Culture of the Classroom
37
How to Set a Climate That Empowers
37
Wait Time as Empowerment
37
Mess-Around Time Is Learning Time
39
Extending Problems and Ideas
39
Leading Questions
39
Individual Differences
40
To Praise or Not to Praise
40
Developing Autonomy
40
Talent Is Not a Guarantee of Immediate and Complete Comprehension
41
CHAPTER 6: CurriculumBig Ideas and Many Extensions
45
What Is a Number?
45
What Is a Number System?
46
Equivalence
47
Chip-Trading
47
Visualizing Numbers: Patterns, Functions, Squares, Rectangles, Golden
Rectangles, and the Fibonacci Sequence
49
Lots of Boxes
50
Other Square Activities
51
Fibonacci Series
52
From Number Patterns to Graphing Patterns Via the Ancient Vedic Square
55
Graphing Vedic Square Patterns
57
Visualization of Function: Magic Number Machines
58
Everything Can Be Measured
58
From Triangular Numbers to the Measurement of Angles
60
Estimation
62
Probability
63
Sampling
65
Conclusion
66
CHAPTER 7: Integration and Assessing Math
67
Math and Literature
67
Math and Science
69
Mental Model Building
69
Math and Writing
72
Recording and Representation: A Problem to Solve and Another Way
to Assess Understanding
73
Assessment
76
3
Table of Contents
79
CHAPTER 8: Character Profiles
79
Creative and Artistic
80
Computational Wizardry
81
Marching to a Different Drummer
82
Anything But Writing!
82
Diligent, Hard-Working, and Perfectionist
83
High-Spirited and High Energy
Epilogue: A Reminder
85
87
References
Appendices
93
Appendix A: Questionnaire for Parents
97
Appendix B: Job Cards and Other Activities
111
Appendix C: Annotated Bibliography
117
Appendix D: Recommendations for the Classroom
xi