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Developingyoungchildren'smathematicalthinkingandunderstanding

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DouglasH.Clements

UniversityofDenver

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JulieSarama

UniversityofDenver

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28 DEVELOPING YOUNG

CHILDREN'S MATHEMATICAL THINKING AND

UNDERSTANDING

Douglas H. Clements and Julie Sarama

Introduction

In contrast to the view that nuthcrnatics for very young children is developmentally inappropriate and that only simple number tasks are appropriate for the prirnary

grades (see Balfwz, 1999; Hughes, 1986; Sun Lee and Ginsburg, 2009), research has shown that young children can think and underst;md mathcrmtics content th:1t is

surprisingly broJd and deep. Recent research and developmental work has suggested that leamil~:< trajerton'cs can help early childhood educators respect children's develop­

mental processes and constraints, and their potential fix thinking about and under­standing mathematical ideas (Bobis et al., 2005; Clarke, 2008; Clements and Sarama,

2009; Sar,nm and Clements, 2009b; Wright, 2003). in this chapter we briefly discuss young children's natural mathematical thinking. Then, we give two concrete exJ.m­

ples of learning trajectories, illustnting how they can be used to enhance teaching and learning.

Young children's natural mathematical thinking

It seems probable that little is gained by using any of the child's time for arith­

metic before grade 2, though there are many arithmetic flcts dut he [sic] can learn in grade 1.

(Thorndike, 1922, p. 198)

Children have their own preschool arithmetic, which only myopic psychologists could ignore.

!VxRotsky, 193511978, p. 84)

33!

Doug/us I!. Clements arul julie Snmma

For more than 100 years, vi('\VS of young children's mathematics lnve dif-fered

vvidciy, as tl1tse contradictory quotes hom two psychologists sbmv. Across that time,

many have reponed observations of children enjoying pre-mathematical activities.

However, others have expressed fears of the iruppropriatcness of mathemJtics f()r

young cbildrcn, ;ilthough these opinions are based on broad social theories or trends.

not obscrv:ttion (Balf:mz, 1999). The institutionalization of early dJildhood. edw:a·­

tion often extinguished prormsi11!; nl:ttllcnuticll movements.

Consider Edward Thorndike, quoted above. l-Ie \Vished to emphasize lwalth,

rcpbccd the first: gift ofFroebcl's giti:s (rnanipuLnives, which were small spheres) with

a toothbruslJ and the i-lr:sr JlJatheJuariol occupario11 with 'sleep' (Hrosrennan, 1 0')7).

Similarly, .~omc argue cllildrcn :should be playing vv·ith building blocks, rather than

le:tming mathematics. But tbc origin:ll inventor, Caroline Pratt (194g), created

toclay's unit blocks to reach mathematics! She tells of children making enough room

f()r a horse ro fir inside a stable. Tht' teacher told Diam th~tt she could have the horse

when she h:1d llLJde a sublc for it. Dian:1 and Elizabeth bcg;m to build a SllJall con­

struction, but the horse did not fit. Diana had made a large stable with a lm.v rooC

After several unsuccessful ;lttcmpts to gcr the horse in, she removed the roof, added

blocks to the w:1lls ro make the roof higher, ami rcpbccd the roof She then tried to

put into words vvhat she had done. 'Roof too smJJl.' The tc;Jcher gave her new

words, 'high' and 'lovv-', and she gave a ne\.v exphnarion to the other children. Just

building \Vith blocks, children form important ideas. 'Ic~H:her:; such as Diana's bclp

children explicate and fun:hcr develop these intuitive ideas by discussing thelll, giving language to their actions.

Like Pratt, we bdicve that 'doing mathcmdtics' is natural and appropriate fix chil­

ciren of all ages- i( engendered :md supported wclJ. First, to he educative (Dewey.

193S/1997), m:lthematical experiences should involve rnathcmaticaJ procc~ses or

practiL·es such as problem-solving, lT<lsoning, connnunicating, modeling, and con­

necting (discussion of these in depth is b(~yond the scope of this chapter, but the rcfi.:r­

ences contain m_any clabor:1tions :mcl cx<unplcs). That is, every educative experience

should involve hdping children mathcmatize (De Lange, 1YS7; Kaartincn and

KurnpulaiJlcn, 2012) thL'ir world: n'prcsenting :mel claboratiJJg their world n1atlJcn l<l rically creating models of everyday situations with m;Jthcmatical ohjcrts. such as nurnbers and shapes; with llWthemarical actions, such :1s nnmting or trans­

forming shapes; and with strurtura! rclarionships, such as 'one more' or 'cquaJ length'­

and using those rnodels to solve problems. Second, to avoid being miscducative

(J )cwey, 1 1J3B/ 1907), experiences shouid not ill elude imppropriatc and harmfi.1l rou­

tines, such as flash cards and timed tests to promote 'memorization' or basic t:n~ts (especially hefmc thinking strategies are well established) (Henry and Brown, :2001-l) or

dull calendar exercises in which one child performs routine actions while others pas­

sively wait fi)r it to he over (National Research Council, 2009, note this are misedu­

cltivc for self-regulation as well as mathematics competencies). Third, to be educative,

experiences should he ch:ll.lenging but ::tchievable, generative of future learning, :md

consistent with young children's 'mtunl' \vays of thinking and lcaruing (Clements

ct a!., 2004; Tnmdley, 200N). We bclicw research-based lcarnitz~; u~~fcctorics are mcf-i.1l fcw en.mring thJt experiences ue lllaximally eduotive.

332

Developing young children's mathematical thinking and understanding

Learning trajectories: paths for successful learning

Why learning trajectories?

Children generally follow certain developmental paths in leJ.rning mathematics. When teachers understand the progression of levels of thinking along these paths, and sequence and individualize activities based on them, they can build eHCctive mathematics learning environments. Research has suggested learning trajectories are effective in this way (Clements Jnd Sararna, in press; Sararna and Clernents, 2009b). Simihrly, several recent efforts have based their reports on learning trajectories (e.g. Horne and R.owley, 2001; Nes, 2009). The National Research Council report on early m~1thematics (2009) is subtitled, 'Learning paths toward excellence and equity'. The Early Nm11cTacy REsearch Project (ENRP) in Victoria, Australia was built around using 'growth points' to inform planning and teaching (e.g. Clarke, 200B; F-·Iorne and Rowley, 2001; Perry et al., 2008). The authors of the Common Core (CCSSO/NCA, 201 0) started by writing learning trajectories for each rn<ljor topic. These were used to determine what tbe sequence would be and were 'cut' into grJde-level specific standards. Similar approacht~s :11T used in the New Zeahnd Numcr:lcy Development Project, the Victorian Early Numeracy Research Project and the Count Me In Too program in New South Wales, AustraliJ. (Dobis er al., 2005) as well as Mathematics Recovery (Wright, 2003).

Each learning trajectory as we define it has three parts: a goal, a developmental progression, and instructional activities (Sararna and Clements, 2009b). To develop a certain mathematical competence (the goal), children construct each level of think­ing in turn (the developmental progression), aided by tasks and teaching (instruc­tional activities) designed to build the mental actions-on--c)~jects that enable thinking at each higher level (Clements and Sarama, in press; Sarama and Clem.cms, 2009b).

As an initial example, take the goal of measuring length, ~l common goal for rnath­ernatics (e.g. MacDonald et al., 2012), but one that challenges children (e.g. in the iteration of standard units, Nunes et al., 2009). A typical goal is fi)r children to learn, by the end of second grade (ages 7~8 years), to rneasure the length of objects using appropriate tools, relate the size of the unit to the number of units, determine how much longer one object is than another, and so forth. That is the iong-~rangc goal.

Children develop through a series of levels of thinking as they achieve that goal; that is, as they learn the ide;1s and skills that constitute accurate and meaningfitl mea­surement of length. At each level, children can solve a new type of problem. These levels form a dctJeloprnental proy_rr:ssion (cf MacDonald and Lowrie, 2011; McDonough and Sullivan, 2011). The second column in Figure 28.1 describes several levels of thinking in the counting learning tr<~ectory. Th.is includes the narne of each level, a description, and a brief concrete example of a behavior indicative of that level of thinking (the first, or leftmost, column is the approximate age Jt which children achieve each level of thinking. These are present-day averages ::md not the goal- with good education; children ofien develop these levels earlier).

The rigbt~most column in Figure 28.1 provides examples of instructional tasks that ;tre designed to develop each of the levels of thinking. That is, educators can use

333

i

4

6

Length quantity n~cognizcr

!dcntilie~ !ength/dimmce H.> al.irihute_ May unde1·stand knglh as an absolute descriptor (e.jl. all adults are tali). but not as a companlltve (e.g. one person is tallc:·than another)

'I'm wll, sec':'"

Leuglh direct comparcr

Physically ::digns two to detennine

is longer or iflhcy are the same length.

Stands lwo sticks up next to each other on a !able and says, 'Til is one's bigger.'

111:

indirect length com parer

Compares tl1e length of two object~ hy reprcscming them with a third ubjcct

Compares length of two objects with a piece or ~Iring.

May be able lo measure wltll a ruler, hut otten lacks understanding 01 skill (e.g. igmm:o starting point)

Measures two objec1s with a ruler w check if they arc thc same length, but docoH not accurately set the "zero poinl· 0Jr one of the 1tems

!nstnu.·Honal tasks

! Teucl1er~ and other (;arcgivers listen for and extend convcrSfltions al,out things that are "lOilg,' 'tall," 'high,' und so J'onh

i In 'Cmnparc Lengths,' \cacher~ encourage chiidren to compare lengths lhrOuf!houtthc day_ such as the lengths of block towerS or mado. hciglm offurnilllre, and oo forth

\n 'As Long As My i\rm, · children cut a ribh011 tile lenglh of their arms aml find things in the classroom that are the same kn:,;lh.

!n 'Co1npa:·isons,· children simply click on tht object I hat is longer (or wider, CIC.)

ln 'Line Up By Height,' children order themselves (with teal;hcr·s assic;tance) by lie1ght in groups of live du1·ing

Children solve everyday tasks thai 1·cquire indit·cct COlll]J<Irison, such as whctllcl· a doorway is wide cnout~h for a 1ablc to go through

Children oficn !'u1-·er the objects to be compared, so that indirect comparison is actually not possible. Give them a I ask V•.'ith objct:L~ such as felt strips so that, if they cover them with !he third objcc! such as a (widcJ) st1·ip of paper (and therd"i:Jre have lo visually guess), they can he cn::nuragcd w then directly co:np~n: then. !rthcy arc nol correct, ask 1hcm how they could have u~cd the paper \o bc11cr compare Modci laying it next to the objects i C necessary.

ln 'Deep Sea Cumparc.' childi·cn move the cmal to compare the ienglh~ of two !ish, lhcn ciiek on the longer llsh

E ~~~-~~-;~~~,-~~- ;~-;l-;1 ~-;~-~-~~;;-~~~-] ·-

Lays units enJ.to·end_ M~y not rccogmze the need for cqual-lenglh units rhc ability to apply resulting mGasures to comparison situations develops late1· in this level.

Lays 9 inch cubes in a line beside a hook lo mca~w·c how long it is

1 'Length Riddles' asks questions such ao, ·You writc with rue and ! am 7 cubes long. What am I?'

Work in" on the Rui!rood. ln this compukt ac1ivi!y, children iay units end to end to rcp~i1 ~railroad bridge.

1 Age

7

Developing young children's mathematical thinking and understanding

Length unit relater and t·epealer

Measures by t·epeated use or a unit Relates size and number of units explicitly

'If you measure wiih centime(ers instead of inches. you'll need more of them, because each one is smaller.'

Can add up two lengths to obtain the length of a whole.

'This is 5 long and this one is 3 long, so they are S long wgether.'

Iterates a singie unil to

Measure with physical or drawn units. Pocus on long, thin units such ~.s lootbpicb cut to I inch sections. Explicit emphasis should be given to the linear nilfure of the unit. That is, children should learn lhat when measuring with, say, centimeter cubes, it is the length ofone edge lhal is the linear unit- not the area of a t"acc or volume ol"lhe cube.

I Repeat 'Length Riddles' (sec ::~.hove) hut provide fewer , cues (e.g. only the length) and only one unit per ch1ld so ' they have to iterate (repeatedly 'lay down') a single unit

to measure.

'Mr. !vlixup's Measuring Mess' can be used al several levels, adapled li)r the level~ before and alter this one. For example, have the puppet !eave gaps between units Ltsed to measure an object (for I he End-to-End Length Measurer level, gaps are be1wcen multiple units, for this level, gaps would be botwe~;n iterations of one unit). Other errors include overlapping units and not aligning atlhc starting point.

'Draw a line.' Use line--drawing ae1ivities 10 ernphasi:;,e how you start at the 0 (zero point) and discuss how, to measure objects, you have to align the object to that point. Similarly, explicitly discuss whM lhe intervals and the number represent, connecting these to end-to­end length measuring with physicalunils

measure. Uses rulers · Di ITcrent Units.' Confront children with measurements with minimal guidance. using diHCrent units and discuss how many of each unil

Measures n book's length will fill a linear space. llelp children make an ex:plicil

·f········+··········'c'.c'c"'••"c''c'·'··w··'·•'··h:"c· ·•"·••d•·e••t·• i······' .. ' .. " ... ~ ... '." .. ' .. " .. ' ... " .. '." .. ' .. " .. '.' ... '." .. " .. cg.'.' ... ' .. h .. ' ... " .. " .. " ... ' .. " .. '.· ... '.' ... '.' .. '." .. t· .. " .. ~ .. · ... " .. ' .. ' .. d ... ' ... d ..... i Length me:~surer

Measures, knowing need tOr identical units, relationship between different units, partitions of unit, zero point on rulers, and accumulation or distance.

F(r;urc 28. 1

i Begins to estimate.

Children should be able to usc a physical unit and a niler to measure line segments and objects I hat require both an iteration and subdivision of the unit. In leurning to subdivide units, children may fold a unit in hlilves, mark I he fold as a hall: and then continue to do so, to build fourlhs and eighths.

j Children <.:reiltc tmits of units. web as a 'f()otstrip'

consisting traces of their feet glued to u roll of adding· machine tape. They measure in different-sized units

'l used a meter stick three : times, then there was a little left over. So, l lined it up from 0 and found 14 centimeters So, it's 3 meters, 14 ccntimetet·s in all.'

(e.g. 15 paces or 3 footstrips each of" which has five paces) and accurately relate the~e units. They also discuss how to deal with lefl:ovet· space, to count it as a who!e unil or as part of a unit.

Learning tr:ljectory fOr length measurements (adapred from Clem ems and Sarama, in press)

t1Jese types of tasks to prornote children's growth from the previous level. More com­plete learning trajectories provide multiple illustrations oft<1sks for eJch level (e.g. sec

Clements and Sanma, in press); however, keep in mind that these illustrations are simply examples- many approaches are possible and children's cultures and individual characteristics also need consideration.

Note the consistency between the standards for the grades preceding grc1dc 2. The kindergarten standards include comparing the lengths of two objects directly

335

Douglas H. Clements and julie Sarama

(by comparing them wlth each other). This is Figure 28.1's 'Length direct compare' level. The first grade Common Core standards include comparing the lengths of two objects indirectly by using a third object Figure 28. J 's 'Indirect length compare' level and the ability to 'Express the length of an object as a whole number of length units, by hying n1.ultiple copies of a shoner object (the length unit) end to end; understand that the length measurement of an object is tbe number of sam.c-size length units that span it with no gaps or overlaps', precisely, the 'End-to-end length measurer' level of figure 28.1. Finally, by the end of second grade, we reach the 'Length unit relater and repeater' ::md 'Length measurer' levels that ;~re consistent with the Grade 2 Comrnon Core goals previously described.

Teaching using learning trajectories

Learning trajectories' instructional tasks might offer a 'sketch' of a curriculum-8

sequence of activities. However, research suggests that they can and shotdd oiTer rnore. They should support teachers' use of formative assessment- the ongoing moni­toring of student learning to inform and guide instruction. Research indicates that formative assessment is an effective teaching strategy (Clarke, 2008; Clarke ct al.,

2002; National Mathematics Advisory Panel, 2008; Shepard, 2005). I .. Iowever, the strategy is useless for teachers unless they can accurately assess 'where students arc' in learning a mathematical topic and kno\V how to support them in learning the following level of thinking. The goal oflearning tiajcctories help define the math­ematical content that teachers have to teach and so have to understand well them­selves. The developmental progressions give teachers a tool to understand the levels of thinking at which their students are operating, along with the next level of think­ing that each student should learn. Then, matched instruction::tl tasks provide guidance as to the type of education;;] :1ctiviry to support thar le:1rni1:g :md hdp explain why tlwse activities would be particularly effective. Such knowledge helps teachers be more effective professionals. Next, we look at ways to put this all into practice.

The key to the usc of formative assessment is knowing what standards or goals one is trying to reach, where the students are starting, and how to help them move 6·orn there to the goal. Notice that these three formative assessrnent gucstions a.lign with the three components ofJearning trajectories, as shown in Figure 28.2.

Our second example is f"l·orn the learning trajectory of early counting-based addi­tion and subtraction. A study of textbooks in California showed the irnportance of teaching core concepts and rneaningful strategies for arithmetic, not simply 'facts' (Henry and Brown, 2008). The learning tr;~jectory, therefore, should include core knowledge, strategies, and skills.

Note that the following is one of two approaches to addition and subtraction involving counting-based strategies. A critical cornplemcnt to these is that of concep­tual subitizing and related visually and structurally based part-whole approaches (see examples in the patterns and structure curriculum, Mulligan et al., 2006, and chapter 6 in both; Sarama and Clements, 200<Jb). Both begin in the earliest years, not repre­sented here (sec references).

336

Developing young children's mathematical thinking and understanding

Fonnativc assessment questions Learning trajectories' components

l. Where arc you trying to go? The goal -Describes the mathematical concepts, stmeturcs, and skills

2. Where arc you now? The developmental progression-- Helps determine how the children arc thinking now and on the 'next step.'

3. How can you get there? The instructional activities- Provide tasks linked to each level of the developmental progression 1hat arc designed to engender the kind or thinking that will fOrm the next leveL Suggests feedback for specific errors.

1-1:~ure 28.2 Relationships bet\veen t:he majo1· questions of t(m·native assessment and the

components of a learning trajectory

1. The goal. Mathematically, whole-number addition can be viewed ;ls an extension of counting (National Research Council, 2009). The sum 7 + 5 is the whole number tlut results from counting up 5 numbers starting at 7; that is, 7, 8, 9, 10, 11, 12. As tedious as it would be to solve this way, the sum 194 + 74() is the number resulting from counting up 7 4fl numbers starting at 194.

As they 1nove through the learning trajectory, students learn to solve increasingly difficult problems. Some problems are more difficult simply beCl.use they involve larger numbers, of course. Unfortunately, such difficulties are often greater tlun they should be because too rnany curricula and teachers provide f~1r more practice on prob~ lcrns with smaller digits and negle-ct the larger single-djgit numbers (Ham:mn and Ashcraft, 1986). T eachcrs should ensure students receive more balanced experiences.

Beyond the size of the number, however, it is the type, or structure of the word problem that detennines its difficulty. Type depend~ on the situation and the unknowrt.

The situation cw be a 'Join' problem (have 2 apples, got 4 more) 01" 'Separate' prob­lem (had 2 apples, ate 1); a 'Part~part-whole' problern (3 are girls and 4 are boys- no action is suggested); or a 'Compare' problem Q"ohn has 4, Emily has 6). For each of these utegories, there are three quantities that play different roles in the problem, any one of which could be the unknown. In some cases, such as the unknown parts of 'Part-part-whole' problems, there is no real difference between the roles, so this does not affect the difficulty of the problem. In others, such as the 'result unknown', 'change unknown' (had 2, got sorne rnore, now has 6), or 'start unknown' (had sorne, got 4 more, now has 6) of 'join' problerns, the differences in difficulty are large. 'Result unknown' problems arc easy, 'change unknown' problems are moder­ately difficult, and 'start unknown' problems arc the most difficult. This is due in large part to the increasing difficulty children have in modeling, or 'act outing', each type. In summary, a main goal of this addition and subtraction learning tr:J.jectory is that children learn to solve arithmetic problems of dil:ferent types using counting strategies.

2 The developmental progression. A few selected levels for this component of learning trajectories are shown in the second column in Figure 28.3. Children

337

1-Ag~--Dcvciopmc·~-~;jp;~g,.(,SSi;~;;-r---~- ...... __ "Jn~-uctirH-;Ji~---~~----1 I . · ··· ·· · · ···· · · · · . .. I

. 'j I Jtmdp!oblcm> Chilmen ~;olvmg al, the dbove problem type~ I , Find Result 1 u~mg mampuln!Jve> or lhc•r Jmge•s to rcpres~n' obJects I

6--7

4---.'i

Find~ ~ums for joining (you ' had 3 apples and get 3

more, how many do you have in all?) and part-palt­whole (there are 6 g1rl~ and 5 boys on the playground, how m>my children were there in all?) problems hy direc1 modeling, counling-a/1, Wilh objects.

!low """'Y Jn all?' Coanrsnm21od, lhcncnwlt,out 3 blue, thon e<>unwaiiS

Solves take-away problems by separatmg with ob_iecls

rcmoming :J

I <JU.ll '- 1( 5 bells I "'"'"'''""'' "'"''' wlwl<• ''"known problen1s. tl1oy ongh1 solve

blue balk I low "'""Y mall?'

Dinosaur Shop 3 Customers at the shop asks students to combine fhcir two orders ~nd atld tlw contmns of two bo,.t·-s oi'!iJY dinosaurs (numbet· fbmes) and click a target numeral that represents the sum,

Off the Tree_ Student~ add two amounts of dots to identify their lola] number value. nnd then

I

I move forward a corresponding number I

of soac,cs on a game I boa;d, wh1ch is now

___ .... ___ ·1· .. ----~n~~~-:~. -~i_:r: _ ~~umerals ___ ··--- __ ··---- .... ____ . ·----· ·-- -··. --·--J : Counting; nra~cgics , 1 low Manv Now? I lave the children count objects as you I

I Finds sums for joming (you had 8 apples and get 3 more._.) amd part-part­whole (6 girls and 5 boys, .. ) problems with f\nger pMtems nnct!or by coumtng on

Crmming-up·to May "'Jvc mtssing addend (3 " - 7) or compa•e pltlbkm> by cnunting up; e g. oounl' '4, 5. (,, 7' whTJC ptllling Up f1ng""· #nd •hen cnu,ts '"

linger.<

place them in a box. Ask. 'How rnany are in the box j'

nnw?' Add one, repeating the questmn, then check the children's response~ by counting all the objects. Repeal, ~· checking occasionally. When children arc ready, sometimes add two, nnd eventually more, objects 1

Jom Resu/1 Unknown and l'arl-f'an-Whofe, Whole Unknown 'How much 1s 4and 3 rnorc'l'

Brif;htlden Students arc given~ numeral and 3

1\-amc with dots. They count on fi·om this numeral to identifY the total amount, :md then move forward a corresponding number of spaces

Eu.>y as Pic: Swdcllt,l add two numerals to !lnd 3 101111 number, and th~n move forward a corresponding number of spaces 011 ~ game board

nerivcr All type.\' ol' single-digit problems .

. ' Uses flexible Rlrategies and I>AMT See lex/_ derived combinations (e.g. ·7+7isl4,so7" 8is 15) to solve all typ<ls of problems. Can simultaneously think of 3 numb.ors witbtn :1 sum. and c~n move part of a number lo another, aware oi'lhe incrt,ase in one and the decrea~e in m10thcJ.

A.<b~l. '\Vil<~l'' 7 thmb:'1~8-> 1['1-IJ -->17-< 7)+!- t.1-' 1 ~

" '"~""-'"C 7 m"' 2 an<l 5. "lid 2 and~ lo moke ](). lhcn odd 5 more, 15

Tic- Tac- Total_ Draw a tic-lllc-toe board and write I he numbers l to 10. Players lake turn crossing out one oftbe numbcro a11d wri1ing il in the board. Whoever mnkes 15 first wios (Kamii, 1985)

]J Play cards, where Ace i~ worlh either I or II and 2 to I 0 are worth their values.

Deaicr g•vc.< <:Ycryune 2 curds, moludlng b~r.,cl[

On each mcmd, ~ach player, il'sum rs ks; than 21, can rcqac't <H1nlloc" card, m 'hold'

II" any n~w ea1d makes ihe -'<'111 nMc lhaFl 21, !h" playc1 is OHI Cunlinuo U1Hil everyone 'hold<'

The• rlay<·r whoso >um '" dooeollo 21 win;

VamMn<_ PI"Y lo II al firol

f'\i!.f-!rC 28.3 Partial learning trajectory for addition and subtraction (emphasizing counting st-rategies) (adapted ffom Clements and Sararna, in press)

Developing young children's mathematical thinking and understanding

develop increasingly sophisticated counting strategies to solve increasingly difficult problem types. For example, most initially use a counting-all procedure. At the 'Find result' level, given a prob1em of7 + 2, such children count out objects to f()ml a set of 7 items, then count out 2 more items, and finally count all those and say 'nine.' Children usc such counting methods to solve story situations if they understand tbe language in the story.

After children develop such methods, they eventually curtail them. Often indc·­pendent!y, children as young as 4 or 5 years invent 'counting on', solving the previ­ous problem by counting, 'Seven ... 8, 9. 9!' The elongated pronunciation may be substituting fi)r counting the initial set one-by-·one. It is as if they counted a set of7 items.

Children theu move to the COU11til~g-onjrom~lar;'?,er strategy, which is preferred by most children once they invent it. Problems such as 4 + 25, where the most 'count­ing on' work is saved by reversing the problem, often prompt children to start count­ing with the 25. Counting-on when increasing collections md the corresponding counting-back-from when decreasing collections are important numerical strategies fOr students to learn. However, they arc only beginning strategies. In the case where the amount of increase is unknown, children use counting-up-to to find the unknown :mwunt. lf 6 iterns arc increased so th:tt there are now 9 items, children may find the amount of increase by counting and keeping track of the number of counts, as in 'Siiiix 7, 8, 9. 3!' And if 9 items arc decreased so that 6 remain, children may count from 9 down to 6 to find the unknown decrease, as f()llows: 'Nine ... S, 7, 6. 3!' However, counting backwards, especially more than two or three counts, is dif .. ficult fOr most children unless they have consistent instruction. fnstcad, children might learn counrinJ<~up-to the total to solve a subtraction situ;Jtion. For example, 'I took away 6 rrom those 9, so 7, 8, 9 (raising a linger with each count)- that's 3 more left in the 9.' Students then learn to incorporate place value and other ideas.

3 The instructional tasks. As stated, instructional tasks are not the only way to guide children to achieve the levels of thinking embedded within the learning trajec­tories. I·--Iowever, those in the right-most column of Figure 28.3 are (simply) exam­ples of the type of instructional activity that helps promote thinking at the subsequent level. Thus, teachers implement, adapt, or use them as a template to gauge the appro­pliateness and expected effectiveness of other lessons, including those in published cunicula. Further, these tasks ne often useflll as problems for children to solve (via guided discovery), but teachers must make critical decisions concerning pedagogical strategies (Anthony and Walshaw, 2009), suclJ J.s whether tl1esc problems might be posed in a play context (e.g. Hirsh-Pasek and Golinkoff, 200S; Lee, 2010; Sarama and Clements, 2009a; van Oers, 2003, 2010), presented as small group activities (Clarke et al., 2002; Clements and Sarama, 2007, 2013; G1iffin, 2004; _Griffin et al., 2007), or other approaches. Some pr:inciples identified by researchers include (a) that social and cultural contexts that make sense to the children (Anthony and Walsluw, 2000; Perry et a!., 2008), (b) that they involve thoughtful and sensitive discourse (Anderson et a!., 2004; Anthony and Walshaw, 2009), (c) that a careful synthesis of child-centered and teacher-guided (intentional, sequential) experiences are included,

339

D<mglas II. Clements and Julie Samma

aJJd that connections het\veen home and school, 1nd cduc:ttors fiorn birth dlrough

the primary years, be strong :md continuous (Anthony and \\laJsba\v, :?OOY; National l~escarch Council, 2009).

In some c:Jses, there is evidence tlut cert1in aspects of the instructional tasks arc

especially cfft-ctivc. For example, ifstucknts need extra bdp in learning counting on

skills, there is thcor·y and empirical \vork that provides specific inst.n.JCtJOna] strategic~.

Afi:cr setting up the problem situation vvith objects (say, 5 + 3), the teacher guides

children to councct tbc numeral signif).'ing; the first addend to the objects in tbe first

set (Carruthers :Hld \Vorthington, 2006, describe the development of written syrn~

bols ;md dr:rwings beyond the scope of this chapter). Students then learn to recognize

tl1at tl1e bst object ofth:lt set is assigned the counting \OJOrd ('fivc'). Next, rhc teoKhn

helps the children understand that the first object in the second set "\vill ahvays be

assigned the next counting nmnhcr Students learn that they can start with the

'five' innncchatdy and count on. These underst:mdings and skills arc reinfi)]"ced "-Vith

additional problems aud a variety of specific, focused que~tions.

l.ksidcs c1rcfully addressing necessary ideas and subskills, this instruuional activity is

successfi.1l because it promotes psyclwh?J;iwl cunailnwnf (Clements ;md Burns, 2000;

K.rutetskii, 1976), an encapsulation process in \vbicb. one mental Activity' gradually

'stands in f(w' another mental activity. Children must learn that it is not necessary tn

enumerate each element of rhe first set. The teacher explains this, then demonstrates

naming the number of that set with an elongated mnnher \vord and a sweeping gestmT

of the hand before passing on to the second <Hkknd. El'konin and Davydov ( 1 iJ75) cbirn

that such abbreviated anions arc not eliminated but arc tr:msfencd to the position of

actions which arc considered as if they were cm~ed our and arc thus 'implicit'. A S\\iecp

ing lllnvcrnent t-,rivcs rise tn a 'mcmal plan' by which addition is pcrforrncd, because only

in this movement docs the child begin to view the [!;roup as a unit. The child becomes

aware of addition as distinct {i·om counting. This construction of counting on must be

based on physically present objects. Then, through introspection (considering the basis

of \)!·~e's own vnvs of ;,cri!lg), the o~jccr 5et is tr:msflwmc:d int:C! :! syrnbol.

A second cx:nnple of iustructionaJ. activities supported by specific research cvi~

dcnce, 6:nmd in the next level in 1-\gure 28.3, Deriver~+·/-, is the japanese approach

to developing tbc Break-Ap:{rt-to-Make--Tcn (Murata, 2004; note this is known as

'bridging through 1 0' in the UK and other countries, e.g. I--leirdsficld, 2005; BAMT.

see Murat<! and Fuson, 200(1). The BAMT strategy actually consists of a series of

instruction:ll activities involving several interrelated learning tr:~ectorics (see Murata

and Fuson, 2000; Sararna and Clements, :ZOO'Jb, for descriptions). Before lessons on

BAMT, children vvork on several rebtcd learning trajectories. They develop solid

knowledge of numerals and couming (i.e. move along the counting learning trajec~

tory). This includes the number stmcturc t{H teen numbers <ls H)+ another mm1her,

which is more straigbti{Jrward in Asian JanguJgcs :md Hindi than in English, Sp:mish,

and otllcr languages (' 1 J' is '1 U and 3'- teachers in the latter languages must be par~

ticularly attentive to this competence). They' learn to solve addition and subtraction

of numbers with totdls less than 10, often clmnking numbers into 5 (e.g.?.; ;Js S~plus-

3) :mel usiug visual rnodds. w·ith these levels of thinking established, children d.evdop

several levels of thinking within thl" Co1npositio11/ decomposition devdopmcnro:l

340

Developing young children's mathematical thinking and understanding

9+4 I

1

The line slants between the numbers, indicatlng that we need to find a pari11er for 9 to make 10.

9+4 1\

1 3

Four is separated into two partners, I and 3.

P(gure 28.4 Teaching BAMT

19\t;\ \j 3

10

The ring shows how the numbers combine to make 10.

Ten and three are shown to add to 13.

progression. For example, they work on 'break-apart partners' of numbers less than

or equal to 10. They solve addition and subtraction problems involving teen num­

bers using the lOs structure (10 + 5 = 15), and addition and subtraction witb three

;::dd.ends using Hh (t:_·.g. (\ + 4 + 7 ~-= 10-+ 7 = 17).

Teachers then introduce problems such as 9 + 4. They first elicit, value, and dis­

cuss child-ii'!Vented stmtcgJes and encourage children to use these strategies to solve a

variety of problems. Only then do they proceed to the use ofBA1\1T. They provide

supports to connect visual and symbolic representations of quantities. In the ex:nnple

9 + 4, they :~how 9 counters (or fingers) and 4 counters, then move 1 counter from

the group of 4 to make a group of 10. Next, they highlight the 3left in the group.

Then children are reminded that the 9 and 1 made 1 0. Next, children see 10 counters

and 3 counters and think 10-3. Last, representational drawings serve this role in a sequence such as shown in Figure 2R.4.

Conclusion

Young students can lean1 more mathenutics than many current prograrns provide.

Learning trajectories can help teachers support their students' learning of rnore pro­

fOund ideas in mathematics (space constraints have not ail owed explor.1tions of subi­

tizing, early number relationships, multiplication reasoning and so forth, sec Nunes

cr al., 2009). Current research in learning tn0ectories points the way toward more

effective and efficient, yet also more cre::ttive and enjoyable, mathematics.

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