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FOR LEMtNGLUS SHOOTTHE PHOTOS FORTHE COVER AND COVERSTORY.

36 SCHOLASTIC EARLY CHILDHOOD TODAY 0JANUARY/FEBRUARY 2003

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7777--- 0 X Children possess and build mathematical

competencies from their first year and

keep on learning mathematical ideas

throughout their preschool years and

beyond.This is not surprising. Mathematics

helps children make more sense of their

k physical and social worlds."That doesn't fit

me-I grew too big!" "No fair! She has

more than I do!"

CREATIVE PATHWAYS TO

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: D by Douglas.H. Clements,__-with Julie Sarama, Ph.D.

photos byJames Levin

Young children invent mathematical

ideas and strategies.For example, take 5-

year-old Alex-

Alex's brother, Paul, is 3. Alex bounds into

the classroom and announces,"When Paul

is 6, I'll be 8; when Paul is 9, I'll be I I; when

Paul is 12, I'll be 14:'

, Teacher: My word! How on earth did you

figure all that out?

Alex: It's easy.You just go "three-FOUR-

five" [saying the "four" very loudly andN

.lapping hands at the same time], you go

"six-SEVEN [clap]-eight, you go "nine-

1 TEN [clap!]-eleven".....

This small but remarkable dialogue reflects

the potential all young children have to

_ learn-and even to create-mathematics.

SCHOLASTIC EARLY CHILDHOOD TODAY * JANUARYIFEBRUARY 2003 37

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CREATIVE PATHWAYS TO

MI1ATIIBEFORE THEY ENTER SCHOOL, MANY CHILDREN DEVELOP EARLYabilities in number and geometry, from accurate countingof objects to finding their way through their environmentto making shapes. They use mathematical ideas in everydaylife and develop informal mathematical knowledge that issurprisingly complex and sophisticated. With your guidance,children can become more acutely aware of this knowl-edge-an awareness that is crucial for mathematicalunderstanding and learning.

Zachary's grandmother saw this awareness when she waswalking him out of preschool. He stopped, pointed, andexclaimed, "Look, grandma! Hexagons! Hexagons all overthe walk. You can put them together with no spaces!"

As such examples show, youngchildren like doing mathematics.They all do. Boys and girls alike, r V--in all socio-economic situations'all 1 X 0 Eexhibit spontaneous interest'in m

mathematical ideas.Young children can and should

engage in mathematical thinking. IiAt eAll young children possess aninformal knowledge of mathe-matics. Instruction should build a

upon and extend children's daily activities, interests, andquestions, bringing the mathematics in such activities to thefore. This approach ensures that mathematical content willbe meaningful for very young children.

You Hold the Key!You can structure the classroom environment so that thepotential for mathematics surrounds children. Show themthe math in their everyday activities and plan special activ-ities that focus on mathematics. Support their curiosity andoffer appropriate challenges. You can:O1 provide lots of unit blocks, along with time to use them.EO ask a child to get just enough scissors for every child who

iLearning About Math-PLAYFULLY!Play is often about mathematics. Nearly half ofalltheepisodes of children's natural play observed byresearchers included mathematics.This included.

l classification (putting away blocks in categories)

l magnitude ("This isn't big enough to cover the table.')

l enumeration (a boy says,"Look! I got 100!" and he and afriend count to check that estimate)

X dynamics (child stretches dough with her hand and makes' 7 ?< :} -i7} Xa flat, circular shape)

0 pattem and shape (a boy puts a double unit block on therug, two unit blocks on the double block, and continues to

bul ymetrical structure)spatial relations (telling a location or direction)

Children use math in everyday life and develop informed math knowledge.

is in the group.El challenge children to guess and checkhow many steps it is to the playground.O sit down with children in large and smallgroups to pose, solve, and discuss mathe-matical problems.

It's also important to make sure math-ematically oriented materials such as blocksare readily available. Notice that momentwhen building mathematical language andconcepts requires intervention. For exam-ple, when two children each claim that hisbuilding is the largest, you might discusshow one is "taller" but the, other is "wider"(or "contains more blocks"). You maydecide to add materials after observing chil-dren. For example, when you see childrencomparing the length of two rugs, makesure that connecting cubes, stdng, and otherobjects that might be used for measuringare close by.

Math Around the RoomYou can help children connect their infor-mal knowledge to their budding explicitknowledge of mathematics. For example,children might be able to manipulate blocksto find that adding one block to a groupof three blocks results in a group of fourblocks. Later, they can be asked to do sim-ilar problems even when the three blocksare hidden. Eventually, they will be able to"count on." Asked what two more thanthree is, they might say, "Threeeeee ... four... five. Five!"

Children should also be encouraged toconnect mathematics topics to each other.For example, children connect number to

Circle 24 on Reader Service Card

CREATIVE PATHWAYS TO

geometry by counting the sides of shapes, usingrows and columns to understand number combi-nations, or measuring the length of a rug. This helpsstrengthen concepts in these areas as well as beliefsabout mathematics as a coherent system.

Our world can be better understood with math-ematics. Early childhood is a good time for children A to become interested in counting, sorting, buildingshapes, patterning, measuring, and estimating.Quality preschool mathematics is not elementaryarithmetic pushed down onto younger children.Instead, it invites children to experience mathe-matics as they play in, describe, and think abouttheir world.

Literacy Links to MathLinking mathematics to literacy and other areasstrengthens both. Most good mathematics activi-ties also develop language and vocabulary. Forexample, when children are lining up, teachers can -w_ibuild in many opportunities to develop an under- >standing of mathematics. Children wearingsomething red can be asked to get in line first, those uwearing blue to get in line second, and so on. Or,children wearing both something red and sneakerscan be asked to head up the line.

Understanding stories involves mathematicalunderstandings, such as conditionals (if/then), clas-sification, patterning, order, and number. Think ofthe numbers, size relationships, sequences; and rep-etitious patterns in "Goldilocks and the ThreeBears" and other favorite stories. Its no wonderthat research shows that early mathematics expe-riences, especially geometric ones, result in laterimprovements in language and literacy, as well asgeneral intelligence.

ConnectingWith FamiliesHere are some ways you can involve families inchildren's math learning:El Feature math nights. During these events, youmight want to:V Talk with families about your mathematics curriculum,including the wide range of mathematical concepts (seeChart: Development of Mathematical Concepts, page 42)and mathematical thinkmg that children will be involved in.V Engage families in making some of the mathematics mate-rials you'll be using (for example, cutting out colorful paperpattern blocks).V Give families the opportunity to solve mathematical prob-lems themselves, such as shape puzzle problems, so that theycan experience the learning firsthand.El Have a "mathematics show" in which children share some

of the mathematics projects they've been involved with. E(T

Douiglas H. Clements, Ph.D., is a professor of nathematics and compltereducation at the State Universit; of Netw York at Buffalo. He has devel-oped an innovative ctrriiclum for early childhood education finding themathematics in, and developing mathematics from, children's everydayactivities, using old and new technologies.Julie Sarama, Ph.D., is an assistant professor of mathematics educationat the University at Buffalo (SUNY). She conducts research on the imple-mentation and effects of her own softwvare environments in mathematicsclassrooms, youing children's development of mathematical concepts andcompetencies, implementation of educational reform, and professionaldevelopment.

40 SCHOLASTIC EARLY CHILDHOOD TODAY * JANUARY/FEBRUARY 2003

((OmEATIVE WAYS TO TEACH MATHHe are some activities for your classroom to add a bit of sparide and creativity. As children work

: as ~~itical questions such as "Did you try this?" "What would have happened if ... ?" "Do you think

could ...?" to enhance children's understanding of mathematical ideas and vocabulary.

Ui Use dramtiii o6niis. Invite childrein Use a variety of strategies. Bring lu-mplaiingiand communicadonwith parents.Ilto pretend to be in a ball or box, feel- mathematics everywhere you go in Use computers to mathematize situations and

; ing the faces, edges,and corners and to LY your classroom, from counting chil- provide individualized instrucdon.

dramatize simple arithmedc problems such as: dren at morning meeting to settng the table, Use assessments to measure

Three frogs jumped in the pond, then one to asking children to clean up a given number children's mathematics learn-l

more. How many are there in all? of items. c w ing. Observations, discussions

children show how many feet, mouths, to record children's mathematical work you learn about children's mathemaical think-

and so on they have.Then invite chil- in their play and activities.Then use the ing and to make informed decisions aboutwhat

.dren to show numbers with fingers, starting photographs to aid class discussions, curricu- each child can learn from future experiences.

with the familiar,"How old are you?" to show-ing numbers in different ways.

3 Use children's play. Engage childrenin block play that allows them to domathematics in numerous ways, indud-

,in sordng, seriating, creadng symmetric designsand buildings, making patterns, and so forth.Then introduce a game of Dinosaur Shop. oSuggest that children pretend to buy and selltoy dinosaurs or other small objects.

Use children's toys. Encouragechil- r/L tdren to use "scenes" and toys to act

L out situations such as three cars ofcially under control!the road, or,later in the yeartvwo monkeRstMtsinconi'the trees and two on the ground.

Use children's stories.Share books are teseaed :2)5 with children that address mathemat- l using our radio frequencyicsbut are also good stories (see Book %dny) welding technology

Box, p.45). Later, help children see mathemat-'ics in any book

Use children's natural creativity.Children's ideas about mathematicsshould be discussed with all children.

Here's a"mathematical conversation" betveen 0

two boys, each 6 year-s of age:"Think of the U.Ptnbiggst nmbe youcan Nowaddfive.Then,632A1Bimagne f yu ha tht mny upcakes.'

"Wow tha's fve ore han henbgg - Our patent ed Infctron Confol Rest Matsber you could come up with!" Foohbales ihe conirdI of crss infection7 Use children's problem-solving whndeoned wit bleach and vme

abilities. Ask children to describehow they would figure outproblems ..

'Such as getting just enough scissors forterWx*,table or how many snacks they would need if ............

Lague twere joining the group. .......j' i)

Circle 25 on Reader Service Card:

CREATIVE PATHWAYS TO

DEVELOPMENT OF MATHEMATICAL CONCEPTSIAt about what age can children develop specific mathematics concepts? jThis chart outlines what children are capable of understanding at 3,4, and 5 years

NUMBER CONCEPT

Verbal Counting. Learningthe standard sequences of num-ber words

Object Counting. Creating aone-to-one correspondencebetween a number word andan item

AT 3YEARSChildren may:

Count I to 10

AT 4YEARSChildren may:Count one to 30, with empha-sis on counting patterns; for

I instance, knowing that "2 1,2 22 ..." is parallel to"l,2 ..."

L, un on o1tes nwCount one to four items, main- Count one tO 1 D items, know-taining one-to-one I ing that the last counting word

Icorrespondence tells "how many'

"Seeing" Numbers. Instantly 1 See groups of one to three"seeing how many" supportscounting,comparing,andadding

Comparing Numbers.Comparing and ordering builion nonverbal knowledge andexperience with collecdons

' See groups of one to five

Identify whether collecions are ;Use counting or matching tothe "same" number or which is compare two collectons one"more" visually to five, despite appearances

Adding and Subtracting. Use nonverbal adding and sub-Solving problems using informal tracdng with very smallstrategies in math learning numbers of objects

GEOMETRYAND MEASUREMENT

Shapes. Geometric shapes can Match shapes, first with samebe used to represent and size and orientation,thenwithunderstand objects different sizes and orientatdon

I', Solve and make word problem!iusing concrete modeling withI sums to fiveI

I Recognize and name some variations of the circle, square,

i triangle, rectangle,!- - - -

PuttingTogether Shapes. ; Use shapes in isolation to make Cover an outline with shapesShapes can be decomposed and i a picture w without leaving gaps by trial-composed into other shapes ; Iand-errorand structures II

Locations, Directions, andCoordinates. Mathematicscan precisely specify diredctons,routes,and locations

Symmetry. Symmetry can beused to understand and createshapes in geometry and art

Measurement Measuring canbe used to specify and compare"how much"

Patterns. Patterns weavethrough all other topics inmathematics

Understand and use ideas suchas over, under, above, on, beside,next io, between

! Show awareness of symmetry! in block buildings

I Develop language such as big-! ger, longer, and taller

, i

! Notice simple repeating pat-tems, such as a wall of blockswith long, shor long, short ..

Learn a simple route from amap placed in direct relation tothe space

Informally create 2-D shapesand 3-D buildings that havesymmetry

Discuss and compare attributeslinformally, including comparinggross differences

Copy simple repeating patterns

AT 5YEARSChildren may.

Count one to 100, with empha-sis on patterns (e.g.,"60, 70" is

ii parallel to "6,7;'''I4'' to" '19"t parallel to "4" to "9")

CountXone to 20 itetms; tit00 ti; t: ftlftt;tg: 7 : i: E : : i : :::;:::E:;:E :Wi00 i WE i i iSi:y l

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E g E \E i: tE ELi:;i;i)EEER ;;EjjiLj:Si:jijijii:S::iiTEj:757 S 7 f: f E:: L 7: E i E tE igiLiji if i iDji: iS ECijEjifiCi:ijEjELi i: fiEE7: W:i:F | l

:: i: :::SiS ::::::: ::::: i:: ::iiCl: i::E T::::::SE:#jEDiStHiD:fi:ii:ijiCi:EjtjEjERiiiE:i:iLi:E::7::;:::: ;: 5 i i: 7: : :: i i i: L i: ly i:i70 i i:E E iE :EELiLi if i iiiiitiE i f (fiE:::::::: ::: E 2 :::1

7 r . r r r 7 i

See groups of one to six; regularI patterns up to 10

Use counting to compare twocollections one to 10, usingwords "equal,' 4more' 44less;'oand-"fewer"

Pose and solve word problemsusing counting-based strategiessuch as counting on,sums to 10

' I Recognize and name circle,square, triangle, rectangle, in anysize or orientation

Cover an outline with shapeswithout leaving gaps by usingforesight. Make a picture bycombining shapes

I Place toy objects in correct rte-ative position to make a map ofthe classroom

Identify and create shapes thathave line or rotational symmetry

Compare length using anotherobject Measure with multiplecopies of a unit (such as block)

Notice and discuss patterns inarithmetic (such as adding oneto any number results in thenext"counting numbere)

42 SCHOLASTIC EARLY CHILDHOOD TODAY 0 JANUARYIFEBRUARY 2003

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MATERIALSAND-RES OJU4JRCES. ______ ____

Here are some of the latest resources to help support your mathematics program:

,BOOKS FORTEAC HERS ChildrenN AEYC)and- t lonal CounciI MathematicaIThi nki ng PreKinder-EngagingYoung Children In of Teachers of Mathematics (NCTM). garten to Grade 2The first productsMathematics: Findings of the 2000 national standardsn orgThe new math standar include a comprehensive preschool cur-conference on standards for preschool and Principles and Standards for School riculum and integrated software (whichkindergarten mathematics education by Mathematics,and many activities,Web-based can be obtained separately) (Clements &

D.H. Clements,J.Sarama,&A.M. DiBiase, eds. software environments,and videos. Sarama, 2003; Schiller et al., 2003); Pre-K toIn press. Mahwah, NJ: Lawrence Erlbaum grade 2 products are being developed.|Associate. clements@buffolo.edu,jsarama@buffalo.eduAssociates. RESEARCH-BASED CURRICULA or wwwvzgse.buffaloedulorgIbuildingblocks.

'Mathematics in the EarlyYears by Big Math for Little KidsTm (Ginsburg, Creative Publications MathJuanita Copley (National Council ofTeachers Green-es,& Balfanz, 2003). Comprehensive Curriculum Resources A group of sup-of Mathematics, 1999; $33.95) mathematics curriculum for preschool chil- plementary books with stories that engageTheYoung Child and Mathematics by dren. cgreenes@bu.edu children in math problem solving.Juanita Copley (NAEYC, 2000; $15) Building Blocks-Foundations for caseyb@bc.edu.

IARTICLES FORTEACHERSClements, D.H., & J. Saramna. 2000.'Standards for preschoolers. Teaching Children

Mathematics 7(l):38-41.CIements, D.H. 2001. Mathematics inthepreschool. Teaching Children Mathematics 7:270-75.Clements, D.H., & J. Sarama. 2002.The ZThey'll have hous of fun!role of technology in early childhood leaming.Teaching Children Mathematics 8:34D-43. learning with Educubes!Clements, D.H.,J. Sarama, & A.M.DiBiase.2002. Preschool and kindergarten!mathematics:A nation'al conference. Teachn'Children Mathematics 8: 51 0-14.Geist, E. 200 1. Children are bom mate .................

maiin:Promoting the construction o!early mathematical concepts in children

under five. Young Children 5,6 (4): 12-19. .......

Murray,A. 200 1. Ideas on manipulativemath for young children. Young Children 5

(4): 28-29.Sararnal. (2002). Listening to teachersPlanning for professional development.W.

Tecing Children Mathemaitics, 9,36-39.

E a r l C h i d h o o C o r e r . T e sN T h e

"corner" to teachers of children before.....~first gr-ade.

WEBSITESwwnctm.org National Council ofTeachemFi taan-I' al! M

ofMathematics

wwnaeyc.orglresourceslposition_state-ments,psmath.htm Early ChildhoodMathematics: Promoting Good BegininsIA joint position statement of the Natina'Association for the Education ofYoung

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MATHEMATICS WITH MANAP LATIVEManipulatives such as those listed below can be used in many ways to teach math skills.

DESCRIPTIONShapes that come in logic sets, such as having all com-binations of three colors,two sizes, two thicknesses,and six shapes

Similar to playing cards, these might have dots, numer-als, or both

Cubes in different colors that connect on one end

ACTIVITIES

O Sorting

C1 Matching

C) Ordering the cardsO Games such as war

o Counting building a set of stairsO Showing different ways to make a number(for instance,two red and two blue; three redand one blue)

CountersI

Pattern Blocks

¢ Puppet

Any small objects or two-sided beans

A specific set of shapes that are easy to put togetherto make other shapes and patterns

a

i any puppet I

C Countingo AddingO Patterning

C Making pictures

0 Extending and creating pattemsID Shape puzzlesCl Making"number pictures"

n Play"Mr. Mixup" in which the puppet makescoundng mistakes for the child to correct

"Unite or kdndergarten blocks ; O Counting; measuring, making symmetrybuildings; patterns

I Feely Bo x

Tiles or othersquares

A good feely box is made of tough cardboard,withtwo holes on opposite sides into which is sewn a tubesock with the toes cut off (making a "tunnel" open onboth ends)

The tiles/squares can be ceraamic

I C0 Identifying shapes by touchC 'i 1) Countng objects only by touch

Q Making"number pictures"0 Patterning

44 SCHOLASTIC EARLY CHILDHOOD TODAY * JANUARYIFEBRUARY 2003

MATERIALS

AttributeShapes

1 Cards

I I

Connectingcubes

Blocks

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ISBOOK-B-OXjl Hunter by Pat Hutchins (WilliamMorrow, 1986;$16.95)lAnno's Counting Book byIMitsumasaAnno (HarperCollins, 1992;$22.99)Bat jamboree by KathiAppelt (ScottForesman, 1998; $5.95)Color Zoo by Lois Ehlert(HarperCollins, 1992; $7.99)

Count and See byTana Hoban (Simon&Schuster, 1972;$17)

!Fish Eyes:A BookYou Can CountOn by Lois Ehiert (Harcourt,2001;$6.95)

1The Grouchy Ladybug by Eric Cade(Scott Foresman, 1996;$7.99)

!How Many Bugs in a Box? by DavidA. Carter (Little Simon, 1988; $13.95)

How Many Snails? by Paul Giganti,Jr.(Scott oresman, 1994;$5.95)

SpyTwo Eyes: Numbers in Art byLucy Micldethwait (Mulberry, 1998;$10.99)iThe Icky Bug Counting Book byJerry Pallotta (Charlesbridge, 1992;j$16.95)

,Inch by Inch by Leo Lionni (ScottiForesman, 1995;$5.99)

MyVery First Book of Shapes byEric Carle (HarperCollins, 1985; $4.95)

on the Stairs byJulie Hofstrand Larios(Front Street, 1999; $15.95).

OneWas Johnny by Maurice Sendak(HarperCollins, 1991; $5.99)

Over, Under andThrough andOther Spatial Concepts byTanaHoban (Simon & Schuster, 1973; $17)

iThe Right Number of Elephants byJeff Sheppard (HarperCollins, 1992;$6.99)

rThe Shape ofThings by DayleAnnDodds (Scott Foresman, 1996; $5.99)

Ten Black Dots by Donald Crews(William Morrow, 1986;$15.99)

Ten, Nine, Eight by Molly Bang,(Tupelo, 1998;$6.99)

rheiVery Hungry Caterpillar* byjEric arle (Scholastic Inc.; $5.95)

*To order cal 800-SCHOLASTIC.

Circle 27 on Reader Service uarcl

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COPYRIGHT INFORMATION

TITLE: Creative Pathways to MathSOURCE: Scholastic Early Childhood Today 17 no4 Ja/F 2003

WN: 0300100774014

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