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Games for Early
Number Sense
A Yearlong Resource
CAT H E R I N E T WO M E Y F O S N O T
A N T O N I A CA M E R O N
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firsthand
An imprint of Heinemann
A division of Reed Elsevier, Inc.
361 Hanover Street
Portsmouth, NH 03801–3912
firsthand.heinemann.com
Harcourt School Publishers
6277 Sea Harbor Drive
Orlando, FL 32887–6777
www.harcourtschool.com
Offices and agents throughout the world
ISBN 13: 978-0-15-360561-1
ISBN 10: 0-15-360561-8
ISBN 13: 978-0-325-01009-0
ISBN 10: 0-325-01009-9
© 2007 Catherine Twomey Fosnot
All rights reserved.
Except where indicated, no part of this book may be reproduced in any form or by any electronic or
mechanical means, including information storage and retrieval systems, without permission in
writing from the publisher, except by a reviewer, who may quote brief passages in a review.
The development of a portion of the material described within was supported in part
by the National Science Foundation under Grant No. 9911841. Any opinions, findings,
and conclusions or recommendations expressed in these materials are those of the
authors and do not necessarily reflect the views of the National Science Foundation.
Library of Congress Cataloging-in-Publication Data
CIP data is on file with the Library of Congress
Printed in the United States of America on acid-free paper
11 10 09 08 07
ML
1 2 3 4 5 6
Acknowledgements
Photography
Herbert Seignoret
Mathematics in the City, City College of New York
Schools featured in photographs
The Muscota New School/PS 314 (an empowerment school in Region 10), New York, NY
Independence School/PS 234 (Region 9), New York, NY
Fort River Elementary School, Amherst, MA
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Contents
Overview . . . . . . . . . . . . . . . . . .1
Bear Tracks . . . . . . . . . . . . . . . .6
Part-Whole Bingo . . . . . . . . . .8
Bunk Beds . . . . . . . . . . . . . . . .10
Hide-and-Peek . . . . . . . . . . . .12
Capture Five . . . . . . . . . . . . . .13
Leapfrog . . . . . . . . . . . . . . . . . .15
Fly Capture . . . . . . . . . . . . . . .16
Capture Ten . . . . . . . . . . . . . . .17
The Shoe Game . . . . . . . . . . . .19
Finding Doubles . . . . . . . . . . .21
Doubles More or Less . . . . . .22
Passenger Pairs . . . . . . . . . . .23
Bus Stops . . . . . . . . . . . . . . . . .24
Going to School . . . . . . . . . . .25
Bag It . . . . . . . . . . . . . . . . . . . .26
Hexagons (Version I) . . . . . .28
Hexagons (Version II) . . . . .30
Rekenrek Bingo . . . . . . . . . . .31
Chip Change . . . . . . . . . . . . . .33
Target . . . . . . . . . . . . . . . . . . . .35
Rack Ten . . . . . . . . . . . . . . . . . .36
Rolling for Tens . . . . . . . . . . .37
Closer To (Version I) . . . . . . .38
Closer To (Version II) . . . . . .39
APPENDIXES
A: Bear Tracks game
board . . . . . . . . . . . . . . . . . . . .40
B: Part-Whole Bingo game
boards . . . . . . . . . . . . . . . . . . .41
C: Student recording sheet
for Part-Whole Bingo . . . . . .49
D: Bunk Beds game board . .50
E: Up and Down the Ladder
game cards . . . . . . . . . . . . . . .51
F: Student recording sheet
for Hide-and-Peek . . . . . . . . .52
G: Number cards (Set I) . . . .53
H: Instructions for making
arithmetic racks . . . . . . . . . . .54
I: Leapfrog game board . . . .55
J: Leapfrog game cards . . . .56
K: The Shoe Game game
board . . . . . . . . . . . . . . . . . . . .57
L: Finding Doubles game
board . . . . . . . . . . . . . . . . . . . .59
M: Passenger Pairs game
cards . . . . . . . . . . . . . . . . . . . . .61
N: Bus Stops game cards
(addition) . . . . . . . . . . . . . . . .64
O: Bus Stops game cards
(subtraction) . . . . . . . . . . . . .65
P: Going to School game
board . . . . . . . . . . . . . . . . . . . .66
Q: Student recording sheet
for Bag It . . . . . . . . . . . . . . . . .67
R: Hexagons game
board . . . . . . . . . . . . . . . . . . . .68
S: Number cards (Set II) . . .69
T: Rekenrek Bingo game
boards . . . . . . . . . . . . . . . . . . .70
U: Student recording sheet
for Chip Change . . . . . . . . . . .73
V: Target game board . . . . . .74
W: Student recording
sheet for Target . . . . . . . . . . .75
X: Student recording sheet
for Rolling for Tens . . . . . . . .76
Y: Closer To game board . . . .77
Z: Student recording sheet
for Closer To (Version II) . . .78
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Overview
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n contrast to the other units in this series, which consist of two-week sequences of investigations and related minilessons, this unit is meant to be
used as a resource throughout the year. As such, it includes 24 games that
you can choose from as you consider the needs of your students. The unit includes notes for each game describing the mathematical landscape—the possibilities and openings for learning that can occur as children play. Sample
classroom episodes (titled“Inside One Classroom”) are interspersed throughout this unit to help you anticipate what learners might say and do, and to
provide you with images of teachers and children at work.
Games as a Context for Learning
Play is a natural behavior that is characteristic of all young children, across all
cultures. They wake in the morning and begin playing. They explore their surroundings and the physical nature of objects in those surroundings. They construct with cans, stones, or blocks; engage in sociodramatic and language play;
and eventually (around the age of four or five) become intrigued with games
that have rules, such as running games (e.g. Tag, Hide-and-Seek, Giant Steps)
and board and card games. Where commercially marketed games are unavailable, children make their own. Often they even incorporate their own rules into
their play for novelty, structure, and challenge (Piaget 1962). Play serves, in fact,
as young children’s lived experience, providing many opportunities for mathematizing, activity, and reflection. It is a powerful context for learning.
The value of play for young children has been researched and written
about extensively. Play provides a medium for exploration and creativity (Garvey 1990), for bridging sensory-motor schemas to conceptual schemas (Piaget
1962), for developing autonomy and reasoning (Kamii and Devries 1980), for
constructing early number sense (Kamii 1982, 1985), and even for developing
early algebraic reasoning about functions, change, and exchange (Forman and
Hill 1980).
Emergence vs. Reinforcement
Although games have been used for years in many early childhood programs
as a context for learning, often the guiding purpose underlying their use has
been to make practice and reinforcement of skills fun. The approach we have
taken in this resource unit is quite different. We think of games as openended—providing potential for several strategies and rich conversation around
them. Play is a powerful medium precisely because it allows for experimentaA YEARLONG RESOURCE
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tion. To harness the generative power of play as a
medium for learning, games should support the emergence of new strategies and new insights, rather than
simply providing a context to practice and reinforce
skills.
Maximizing Openings and
Possibilities for Learning
Each of the games described within this resource unit
has been crafted to allow for experimentation—to
provide openings for a variety of mathematical strategies to emerge. At the same time, they have been designed to maximize the potential for certain landmark
strategies and ideas on the landscape of learning to
develop. Some traditional games (such as Bingo, Go
Fish, and Dominoes) have been used but with new
twists, to maximize the potential for learning. Other
games in this unit were designed to be included in
specific context-based units in this series. Still others
were developed and refined by several of the Mathematics in the City staff and the creative teachers with
whom we have the opportunity to work with in classrooms across New York City (Mathematics in the City,
located at City College of New York, is a national inservice provider for mathematics educators, K–8.) Most
specifically, they were designed to support the development of several of the strategies and big ideas on
the landscape of learning for early number sense.
A simple card game in which children compare
numbers to ten can be turned into a mathematically
rich experience for young children. Let’s listen in as
this unit’s co-author, Cathy Fosnot, changes the rules
slightly from other versions on the market to maximize the openings and possibilities. She is teaching
the game to Julia and Susie (each age five). This version is called Bag It. (See page 26.) Cathy distributes
one card to each girl in turn from a deck of number
cards (see Appendix G, page 53). She says while passing out the cards,“One for Julia, one for Susie, one for
Julia, one for Susie,”until all forty cards have been distributed. When she finishes, she ponders,“Hmm, do
you think it’s fair? Do you each have the same number of cards?”Julia begins to count hers.
Cathy asks her to wait:“Before you count, what do
you think?”
“I think yes,”Susie posits.“Because you gave one
to Julia, and one to me, one to Julia . . . like that. Unless you made a mistake.”
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“What do you think, Julia, about what Susie
said?”Cathy asks, attempting to get Julia to consider
one-to-one correspondence in relation to the resulting numbers. Julia just shrugs. “Do you want to
check?” Both girls check by counting. Julia struggles
after fourteen, and Susie helps her. They report that
they each have twenty cards.
“See,” Susie says with a toss of her head,“I was
right. And you didn’t make a mistake!”She smiles at
Cathy.
“Okay, so let’s play.” Cathy begins to explain the
game.“Here’s how you play. Julia, you turn over a card,
and then, Susie, you do too. And we see which is
more.” Julia turns over a six; Susie, a five. “Which is
more?”Cathy inquires.
Both girls respond in chorus,“Six.”(Even children
who cannot read the numerals can play this game.
They can count the objects on the cards and compare
them. Even if they double-tag or do not count synchronously, they can match one for one. Also, they
often correct each other.)
“How do you know that six is more than five?”
Cathy probes.
Susie shows with her finger how there is one
symbol on the “6” card for every one on the “5” card,
plus one additional symbol. Julia agrees and explains,
“When you count, you go past five to get to six. One,
two, three, four, five, six. So six is more.”
Cathy continues to explain the game.“So the six
captures the five, and we’ll put both cards in the six
bag.”Cathy has provided ten small paper bags, labeled
with the numerals from one through ten. In the traditional version, Julia, the winner, would keep the two
cards herself. Using the bags lessens the sense of competition, but it also maximizes the potential for the
idea of hierarchical inclusion to emerge: each bag
should contain only numbers contained within the
amount specified by the numeral on the bag.
Play continues. Julia turns over a ten.“Wait,”Cathy
says, stopping Susie from turning over her next card.
“Before you go, do you think you can beat that?”Cathy
wants to encourage the children to think about hierarchical inclusion, not just to count.
Both girls think. Julia then says,“No, because ten
is the biggest in here!”Susie agrees and turns over a
six. Both cards are placed in the bag labeled 10.
Play continues until all the cards are played. Then
the bags are examined to see what numbers were captured by the numeral labeled on the bag.
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“Hey, the one bag has no cards!”Julia exclaims.
“Because one is the littlest. It can’t beat anybody,”
Susie explains.
Julia continues exploring the bags.“Four doesn’t
have any cards either.”
“Could it?”Cathy asks.
“It could have captured three,”Julia responds.
Susie completes the inclusion sequence.“And two
and one! How come it didn’t?”She opens other bags.
“Oh, look, here’s some fours. There’s one in the five
bag, two in the eight bag, and one in the ten bag.”
Questions asked during these games can facilitate the development of particular ideas and strategies. In the vignette above, Cathy focuses her
questions on hierarchical inclusion and uses the bags
to provide an opening for that idea to emerge in conversation. Tweaking the traditional game rules and
crafting new materials and structures (like using the
bags) can turn a ho-hum game into a powerful learning medium. Bag It can also be turned into Capture
Ten: instead of turning over one card each, each
player turns over two cards and determines if the sum
can capture a ten. For example, 9 and 7 can capture 10
and 6 (see page 17).
Bingo is another traditional game that has been
used frequently in early childhood classrooms. Usually
it is played to practice numeral recognition (a number
is drawn and the corresponding numeral is covered),
sometimes to reinforce addition facts (when two numbers are drawn, the sum must be covered). In our version, called Part-Whole Bingo, we have added twists
to maximize openings and possibilities for learning.
Each pair of children has a game card with tracks of
various lengths drawn on it (see Appendix B, page 41).
Two regular number cubes are rolled and the total is
called out. Players must use that number of connecting cubes either to cover a single track that matches
the total or to cover two or more smaller tracks that
add up to the total.Tracks must be covered completely
in a single turn. For example, if a child rolls 6 and 4, totaling 10, ten cubes must be used. They can be used to
cover the 10 track or the 2 and 8 tracks or perhaps four
short tracks: 2, 2, 3, and 3. The objective is to cover the
complete card, as in Bingo. Although young children
often begin by counting out the cubes and covering up
only the track matching the total, they quickly begin to
see that if they break up the whole into parts, they will
be able to cover more tracks. Part-whole relationships
begin to emerge—numbers can be decomposed in a
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variety of ways and the quantity is still conserved.
Even children who struggle to count can play, because
after they count the connecting cubes they must lay
them on the corresponding tracks. The tracks match
the size of the connecting cubes, and children help
and correct each other. Even after children are quite
familiar with the game, questions emerge: if you roll 2
sixes and you have a 12 track on your board, is it better to cover the 12 track or to cover several smaller
tracks whose sum equals 12? How many ways can you
get 12? Which numbers will be the hardest to get?
Which are the easiest? Why?
Even simple games like Hide-and-Seek can be
tweaked to become a rich context for learning. This
version is called Hide-and-Peek (see page 12). Ten
connecting cubes are used. One child closes his eyes
while the other takes some of the ten in each hand.
When the first child opens his eyes, he sees some of
the ten cubes still on the table, and he can peek into
one hand to see what is in it. Using this information,
he must try to determine the amount in the second
hand.
Cooperative Play vs. Competitive Play
All our games are played cooperatively rather than
competitively. There are several reasons for this.
First, young children often have little understanding
of the role of chance in games and they experience
losing as a personal failure. Tears, anger, and arguments are often the result, as well as cheating and
lying to win. All these behaviors can be avoided simply by making the end goal dependent on collaboration at the start. Instead of “the first player out of
cards wins,”the goal is“play continues until all cards
are used.” Or, in Part-Whole Bingo, instead of “first
player to cover a card wins,” it’s “to win, all boards
must be covered.”In Bag It, the bags (instead of the
players) are the winners.
Second, when competition is replaced by collaboration, the result is often an increase in conversation
on the math involved. Players want to help each
other use good strategies. They offer helpful suggestions and explanations.They justify their reasoning to
convince others of their thinking. In contrast, in competitive games players keep their strategies to themselves. Games used during math workshop should
be opportunities for learning, not opportunities for
winning.
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Using Games in the Math Workshop
Usually the games are introduced in the meeting area.
Children can sit in a circle and you can sit in the center with one or two children and model how the game
is played. Most of the games are played in pairs. After
introducing the game, pass out materials, and ask children to find a place in the room to play. Move around
and confer with children to support and challenge
them. Observe their strategies and comment when
helpful to focus reflection on a specific strategy. Engage children in conversation about their strategies.
Encourage them to help each other.
As you observe the strategies being tried and the
ideas being discussed, consider the landscape of learning. Do you see any landmarks emerging? These will be
important to discuss in a math congress. To help children prepare for the congress, you can ask them to consider one or two helpful strategies they tried and come
to math congress prepared to explain and discuss them.
Use the math congress discussions to support development. Invite children to consider the strategies shared
and ponder with them whether they work, and if so,
why. Encourage children to share interesting things
they have noticed as they played. Patterns noticed can
often lead to interesting inquiries, and some important
mathematical ideas usually underlie those patterns.
Storing the Games
The games in this resource can be made quite durable
by gluing cards and boards onto oaktag and laminating them (or using clear adhesive paper). Lamination
allows you to use overhead markers on them and
wipe off the writing at the end of the session. Some
teachers use manila file folders as game boards. They
glue or redraw the boards onto the inside of the file
folder, lying open and flat.They write the name of the
game on the tab, and they use small plastic zippered
bags for game pieces, attaching it to the folder by staples or a paper clip. When closed, the file folder is
easy to store. Other teachers have used small storage
boxes covered with colorful adhesive paper. Whatever
method you use to make the game equipment
durable, think about a way (and place) to store the
games that will be easily accessible to children. Low
shelves or plastic bins or milk crates work well. Having to find and distribute math materials before each
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math workshop can be time-consuming. Making the
materials accessible will allow children to get what
they need without your help.
Differentiating Instruction
After many games have been taught and children are
accustomed to playing them, you may decide to use
them for differentiating instruction. For example, you
may assign different children to play different games
at the same time. Choose games that will provide the
most helpful openings and possibilities for learning to
the children you assign to them. The notes provided
for each game will help you make good choices. Before
math workshop, you can post a list of math partners
and the game they will be playing. When math workshop begins, you need only go over the list with the
children, who can then collect the accessible equipment and go off to play. No time is wasted. You will
also need to think about which children to pair. Will
the conversation be rich enough for learning? Will
they challenge and support each other? Doise, Mugny,
and Perret-Clermont (1975) found that slight differences in levels was the most beneficial to rich conversation and learning—more beneficial than two
children at the same level and even more beneficial
than the teacher and a child!
At other times you might decide to provide a
menu and allow children to choose the game they
wish to play. Just as you would help children choose
an appropriate book to read, you will need to support
children in selecting an appropriate game.
A Few Words of Caution
As you work with the games in this resource book, it
is very important to remember two things. First, the
purpose of the games is to support development.
Honor children’s strategies. Accept alternative solutions and explore why they work. Do not try to get
every child to use your strategy.The intent is not to get
all learners to use the same strategy at the end of the
game or to practice a skill. That would simply be rote
learning. The games are crafted to encourage children
to take risks and try out new strategies. Look for rich
moments when you can maximize discussion, reflection, and inquiry.
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Second, do not assume that the rules of the games
cannot be altered. Consider the landscape of learning
for early number sense. Are there variations to the
games that might be helpful in supporting further
learning? Try them out and see. Eleanor Duckworth
(1987), an education researcher from Harvard, once
wrote that good teachers are like good researchers.
They try things out and study the effect on learning.
Look for moments and ways to maximize learning.
References and Resources
Doise, Willem, G. Mugny and A. Perret-Clermont.
1975. Social interaction and the development of
cognitive operations. European Journal of Social Psychology 5(3), 367–383.
Dolk, Maarten and Catherine Twomey Fosnot. 2004.
Fostering Children’s Mathematical Development,
Grades PreK–3: The Landscape of Learning. CD-ROM
with accompanying facilitator’s guide by Sherrin B.
A YEARLONG RESOURCE
Hersch, Antonia Cameron, and Catherine Twomey
Fosnot. Portsmouth, NH: Heinemann.
Duckworth, Eleanor. 1987. The Having of Wonderful
Ideas and Other Essays on Teaching and Learning.
New York, NY: Teachers College Press.
Forman, George E. and Fleet Hill. 1980. Constructive
Play: Applying Piaget in the Pre-school. Monterey, CA:
Brooks/Cole Publishing Company.
Garvey, Catherine. 1990. Play. Cambridge, MA: Harvard University Press.
Kamii, Constance. 1982. Number in Preschool and
Kindergarten. Washington, DC: National Association for the Education of Young Children.
Kamii, Constance. 1985. Young Children Reinvent Arithmetic. New York, NY: Teachers College Press.
Kamii, Constance and Rheta Devries. 1980. Group
Games in Early Education: Implications of Piaget’s Theory. Washington, DC: National Association for the
Education of Young Children.
Piaget, Jean. 1962. Play, Dreams, and Imitation in Childhood. NewYork, NY: WW Norton and Company, Inc.
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