1986 Understanding of Number Concepts in Low

8/14/2019 1986 Understanding of Number Concepts in Low

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B . D E N V I R A N D M . B R O W NU N D E R S T A N D I N G O F N U M B E R C O N C E P T S I N L O W

A T T A I N I N G 7 - 9 Y E A R O L D S : P A R T I . D E V E L O P M E N TO F D E S C R I P T I V E F R A M E W O R K A N D D I A G N O S T I C

I N S T R U M E N TABS TRA CT. T hree studies were carried out into the development of num ber concepts inpupils aged 7 to 9 y ears who w ere considered to be 'low attainers' in mathem atics. This paperreports the search for a descriptive framework, the development of a diagnostic assessmentinstrument and a longitudinal study. A subsequent paper (Denvir and Brown, 1986) reportstwo teaching studies. Support was found for the two m ain hyp otheses namely that:(i) A framework can be identified which describes the orders in w hich children ac qu irenum ber concepts(ii) Th is framework can be used to develop a diagnostic assessment instrument w hich willprovide a description of pupils ' understanding of num ber.The aspects of num ber w hich we re considered were counting, addition, subtraction andplace value. The numb er skills which children had grasped were inferred from solutions andstrategies offered by them to questions pose d in a series of interviews.

B A C K G R O U N DT h e r a t i o n a l e f o r th e p r e s e n t s t u d y d e v e l o p e d a s a r e s u lt o f b o t h a u t h o r s 'i n v o l v e m e n t in t h e S c h o o l s C o u n c i l p r o j e c t ' L o w A t t a in e r s i n M a t h e m a t i c s5 - 1 6 ' ( D e n v i r e t a l . , 1 9 82 ). V i si ts t o a la r g e n u m b e r o f sc h o o l s m a n i f e s t e da n e e d f o r d i a g n o s t i c a s s e s s m e n t l in k e d t o p r e s c r i p ti v e t e a ch i n g . T h i s n e e dw a s s u b s e q u e n t l y s t r o n g l y s u p p o r t e d b y B e n n e t t ' s f i n d i n g s ( B e n n e t t e t a l . ,1 98 4) t h a t t e a c h e r s w e r e fr e q u e n t l y u n s u c c e ss f u l in m a t c h i n g n u m b e r t a s k st o t h e c o n c e p t u a l s t a g e s r e v e a l e d b y 6 a n d 7 y e a r o l d s . C h i e f l y i n t h e l a s td e c a d e c o n s i d e r a b l e r e s e a r c h h a s b e e n c a r r i e d o u t , n o t a b l y i n t h e U . S . ( e .g .,G e l m a n a n d G a l l i s te l , 1 97 8; S c h a e f f e r e t a l . , 1 97 4; S t e f f e a n d J o h n s o n , 1 97 1;C a r p e n t e r a n d M o s e r , 1 97 9) b u t a l s o i n F r a n c e ( C o m i t i, 19 81 ; D e s c o u d r e s ,1 92 1; V e r g n a u d , 1 9 82 ) t h e U . K . ( H u g h e s , 1 98 1; M a t t h e w s , 1 9 83 ) a n d I s r a e l( N e s h e r , 1 9 8 2) i n to t h e d e v e l o p m e n t o f n u m b e r u n d e r s t a n d i n g i n 3 - 1 1 y e a ro l d s , e s p e c i a l l y i n c o u n t i n g , a d d i t i o n a n d s u b t r a c t i o n . S i n c e 1 98 1, w h e n t h i sp r e s e n t s t u d y b e g a n , a c o n s i d e r a b l e l i t e r a t u r e h a s b e e n p u b l i s h e d a n d t h i sn o w i n c o r p o r a t e s s i gn i fi c an t w o r k i n p l a c e v a l u e ( B r o w n , 1 981 ; B e d n a r za n d J a n v i e r , 1 98 2; S t e f fe , 1 98 3; R e s n i c k , 1 9 8 3 ) a n d a c l e a r m o v e t o w a r d sp r o p o s i n g t h e o r i e s w h i c h w i l l e x p l a i n h o w u n d e r s t a n d i n g d e v e l o p s( R e s n i c k , 1 9 8 3 ; R i l e y e t a l . , 1983) .

T h e w o r k d e s c r ib e d i n t h is p a p e r b u i l d s o n m a n y o f t h e e a r li e r f in d i ng s .C a r p e n t e r a n d M o s e r ( 1 98 2 ) d e sc r i b e t h r e e m a i n t y p e s o f s tr a t e g y f o rs o lv i ng s im p l e a d d i ti o n p r o b l e m s ; ' C o u n t A l l ' , ' C o u n t O n ' a n d ' R e c a l l ' a n dt h e f ir st t w o o f t h e se a r e r e g a r d e d b y t h e m ( C a r p e n t e r a n d M o s e r , 1 98 3) a n do t h e r a u t h o r i t i e s ( e . g . , F u s o n e t a l . , 1982; Steffe e t a L , 1983) a s cha rac t e r i s t i c

Educa t iona l S tud ies in M athe ma t ics 17 (1986) 15-36.9 1986 by D. Reide l Pub l i sh ing Company.


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16 B. DENVIR AND M. BROWNof distinct stages of development of number understanding. Consequentlyit should be possible to make inferences about a child's understanding ofnumber from observing that child's repertoire of strategies. In part icularSteffe et al. (1983) and Fuson et al. (1982) regard the ability to counton as crucial to an understanding of number. Steffe regards the childwho is unable to make 'deliberate extensions', i.e., to count on in additionwith the intention of keeping a record of the increment, as 'numericallypre-operational'. Riley et al. (1983), building on the work of Steffe et al.(1971), Carpenter and Moser (1979, 1982) and Nesher (1982), explain thedevelopment of pupils' ability to solve different semantic categories ofaddition and subtraction word problems in terms of their 'part-part-wholeschema'. Resnick (1983) also uses this schema to explain children's develop-ment of place value understanding. The child is thought to progress throughthree stages in the ability to represent numbers with base ten blocks. Atstage one two digit numbers are seen as a whole comprised of two parts,such that one par t is all of the tens and the other part is the remaining units.By stage two the child appreciates that there are other non-canonicalequivalent representations in which there are more than ten units. AtResnick's stage three the child can map from transactions with blocks to thewritten algorithm and vice versa, giving semantic meaning to carry andborrow marks.

AIMSThe general intention of this study was to shed light on the learning ofnumber concepts by children aged seven to nine years who were considered tobe low attainers in ma them atics in a way that w ould help teachers to provideeffective learning experiences.The aims were to:

(i) find a framework for describing low attainers' acquisition of numberconcepts;

(ii) develop a diagnostic instrument for assessing children's understandingof number; and

(iii) design, carry out and evaluate a remedial teaching programme.It was proposed that whilst the orders of acquisition of many skills would

be independent, the hypothesised framework would contain 'hierarchicalstrands', within which the acquisition of some 'easy' skills would be aprerequisite condition for the acquisition of more difficult skills. These'strands' would be the means of ascribing developmental stages or levels ofnumber understanding to pupils.


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NUMBER CONCEPTS OF LOW ATTAINERS 17METHOD

All the data in the assessment part of the study were gathered from individualinterviews with 7-9 year olds in which items were presented orally and inwhich frequent use was made of practical materials. The work was carriedout in three stages:

(i) Responses made in the pilot study helped to identify which skills itwas appropriate to assess.

(ii) The assessed skills were extended and defined more precisely duringthe main assessment study and the items to assess the skills were developedand refined. Pupils taking part in the main assessment study also partici-pated in a longitudinal study.

(iii) Finally the Diagnostic Assessment interview was trialled on a widersample of pupils. Organisation of the samples is shown in Table I.

Throughout the Pilot and Main Assessment Studies predictions weremade about what hierarchical dependencies there might be between acqui-sitions of different number concepts. These were based on observation ofpupils' behaviour, reflection on their responses, logical analyses of themathematics and evidence from research literature. From the DiagnosticAssessment Interview results with a wider sample of pupils, relationshipsbetween performances on every pair of skills was investigated numericallyusing item-item Loevinger coefficients (Loevinger, 1947). The DiagnosticAssessment Instrument was also used to examine changes in performanceof a group of seven pupils over a'period of two years.

RESULTS: THE ASSESSMENT INTERVIEWThe interviews aimed to elicit each child's repertoire of strategies for dealingwith number. The aspects which were considered included:(i) Strategies for adding and subtracting small numbers in 'sums' andword problems.

(ii) Commutativi ty of addition.(iii) Enumerating grouped collections.(iv) Strategies for adding larger numbers: place value.(v) Piagetian tasks: conservation of number and class inclusion.

These aspects will be discussed separately below.Strategies fo r Adding and Subtracting Small NumbersSimilar responses were observed in this study to those found by Carpenterand Moser (1979, 1982, 1983) and they are classified according to Carpenter


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NUM BE R CONCE PT S OF L OW AT T AINE RS 19and Moser ' s t e rmino logy . On a few occas ions , ch i ld ren s imply ' knew' theanswer to a ' sum' (Reca l l ed Fac t ) . In th i s example , however , Pa l uses' De r i v ed Fac t ' :

BD: Sarah has some sweet s. She has 5 to f fees and 6 f ru i t gums . H owmany sweet s has she go t a l toge ther?

Pal : Eleven.BD: E l even . Ho w d i d y o u d o t h a t?Pa l: E as y . Ad d o n - y o u ad d o n - I ad d ed o n f iv e. I t o o k . . , l ik e

I too k o f f one . O ff s ix , added bo th f ives then pu t bac k a o ne an dadded i t up . I t ' s e leven.

San u s e s ' Co u n t i n g o n ' :BD: E igh t add seven?San: Fi f teen.B D : H o w d i d y o u d o t h at ?San: Co u n t ed fo rward s .BD: Co u n t ed fo rward s . W h a t n u mb er s d i d y o u s ay?San: I said n ine, ten , e leven, twelve, th i r teen, fou rteen , f i f teen.

' Co u n t i n g u p ' is g ro u p ed t o g e t h e r wi th ' Co u n t i n g O n ' an d ' Co u n t i n g Back ' .In th i s example Br i 'Coun t s Up ' ; the answer i s the increment needed tocoun t f rom the smal le r to the l a rger .

BD: Al ice has seven fe l t t ipped pens , G ra ha m has e leven . H ow m an ym o r e h a s G r a h a m g o t ?

Bri: (Using f ingers) Eight , n ine, ten , e leven ( looks at f ingers) . . .F o u r .

an d C l o co u n t s b ack :BD: At a b i r thd ay par ty there a re e l even peop le a l toge ther . Som e are

ch i l d r en an d s o me a r e g ro wn u p s . T h e re a r e fo u r g ro wn u p s ,h o w m an y o f th em a re chi ld r en ?

C lo : E r . . . E l e v e n . . . F o u r g r o w n u p s ( pa u se ). T h e r e ' s o n l y s ev e nchi ldren.

BD: Sev en ch i ld r en . H o w d i d y o u d o t h a t?C l o: O h . . . E r . . . I h ad e lev en. An d I h ad fo u r . I wen t e lev en , t en,

nine, e ight for the g row n ups and I bel ieved seven, s ix , f ive, four ,th ree , two , one to be the ch i ld ren .In 'Coun t ing Al l ' (wi th model s ) the ch i ld uses phys ica l ob jec t s such as

f ingers o r coun ters to model the p rob lem, e .g . :


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2 0 B. D E N V I R A N D M . B R O W N

B D :

D o n :

J o h n h a d 2 5 c o n k e r s , h e g i v e s e l e v e n t o hi s b r o t h e r , h o w m a n yh a s h e g o t l e f t ?( C o u n t s 2 5 s t o n e s , c o u n t s e l e v e n a n d r e m o v e s t h e m , r e c o u n t sr e m a i n i n g g r o u p ) F o u r t e e n . H e ' s g o t f o u r t e e n l e f t .

TABLE II(a)Frequency of strategies for carrying out additions and subtractions used by each childduring Pilot Study

Clo Don Bri Pal LoyDerived Fa ct 0 0 14 1 0Recalled Fa ct 4 2 2 5 ICou nting On/Back 16 0 5 11 4Counting All 9 16 4 5 9(with models)

A f u r t h e r d i s t i n c t i o n a p p e a r e d i n t h e main s t u d y b e t w e e n ' c o u n t i n g a l l 'a n d c o u n t i n g f r o m o n e ' . I n r e s p o n s e t o '5 + 3 ' ? C h . t y p i c a l l y c o u n t e d a l l ,a s k i n g f o r c u b e s a n d c o u n t i n g o u t f i rs t f iv e, t h e n t h r e e , t h e n c o m b i n i n g t h et w o c o l l e c ti o n s a n d c o u n t i n g f r o m o n e . O n t h e o t h e r h a n d :

P h : F i v e a d d t h r e e ? ( S t a r e s w i t h f i x e d g a z e s t r a i g h t a h e a d ) . 1, 2 , 3 ,4 , 5 . . . 6 , 7 , 8 . Eig ht .

TABLE II(b)Frequency of strategies for carrying out ad ditions and subtractions used by each childduring Main Study

Ch Je Pe Ph Dn Jy ThRecalled Fa ct 2 2 1 8 4 4 11Counting On /Bac k/Up 0 0 7 13 7 23 20Counting from One 0 0 1 2 3 0 0Count Al l (direct 12 30 24 7 19 6 1physical modelling)In the main study, no use was made of 'Derived Fact'.

T h e t h r e e c h i l d r e n ( D o n , C h , J e ) w h o d i d n o t u s e a c o u n t i n g o n s t r a t e g yu s e d c o u n t a l l w i t h m o d e l s a l m o s t e x c l u s iv e l y : t h e i r r e s p o n s e s t o d i s c u s s i o n so f th e c o u n t a ll s t r a t e g y c o n f i r m e d t h e e vi d e n ce t h a t c o u n t i n g o n w a s n o ti n t h e i r r e p e r t o i r e .

I n J e ' s r e s p o n s e t o a c o m b i n e p r o b l e m :


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NUMBER CONCEPTS OF LOW ATTAINER S 21

BD: Ga ry has some red marbles and some green marbles . He has 9red marbles and 6 green marbles. H ow m an y marbles has he got?Je: W ha t was i t again?

Je: (Co unt ing f ingers) 1, 2, 3, 4, 5, 6, 7, 8, 9 red a n d . . . H ow m an ygreen?

BD: (Repeats ques t ion)Je: M m m . . . can I u se your f ingers?Be: (Spreading f ingers for coun ting) Use which you wan t.Je: ( 'Pu ts aw ay ' her nine an d co un ts BD 's) 1, 2, 3, 4, 5, 6. (No w

'spre ads ' her nine again - co un tin g each one) 1, 2, 3, 4, 5, 6, 7,8, 9, (lets her nine 'go' and points to BD's) 10, 11, 12, 13, 14, 15.

BD: Fif teen. An d yo u used m y fingers as well as yours .BD: Cou ld yo u have done i t wi th jus t you r f ingers?Je: No.BD: W hy no t?Je: I ain ' t got enou gh.BD: Suppose i t was 9 red and 7 green . Could you work that out wi th

your f ingers?Je: (Puts up 9 f ingers , again cou ntin g each one). Nine. An d wh atwas it?BD: 9 red and 7 greenJe: (Puts away 9 f ingers and counts 7 of her own, then looks a t

them) Can I use yours?These responses conf i rm work by o ther authors (S tef fe et al., 1983;

Fus on, 1982; Ca rpen ter a nd Mose r, 1982) in which a transit io n is observedbetween 'co unt a l l' and ' coun t on ' and th is is expla ined theoret ica l ly (S teffeet al., 1983). N o such clear-cut tran sition occu rs from eithe r of these strategiesto the use o f recalled facts . Responses given by these low attain ing 7-9 yearolds were s imilar to responses given by younger children in the Carpenterand Moser (1982) and Riley et al. (1983).

Comm utat iv i ty o f Addi t ionThere were three methods of determining whether chi ldren perceived thatthe add i t ion o f numbers was com muta t ive . No te was t aken o f whether, ina series of writ ten sums, they rem ark ed on pairs of sums such as 3 + 6 an d6 + 3; their s trategy for solving sums in which the smaller adde nd ca mefirs t (e.g., 3 -t- 18) was no ted; an d the t ime children too k to respo nd to sumsof the form 1 + n an d n + 1 was noted. These t imes are show n in Table III .


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2 2 B. D E N V I R A N D M . B R O W NTABLE I I I

Com mutat ivity of addition for sums o f the form n + 1, 1 + n. Time (inseconds) taken to give answers to sums of the type n + 1, 1 + n (0 < n < 10)in the Main Study

Je Pe Ph Dn Jy Th2 + I 2 1 1 2 2 13 + 1 3 1 1 1 I I4 + 1 8 1 1 2 1 15 + 1 8 1 1 1 1 16 + 1 9 1 1 1 1 l7 + 1 10 1 1 1 1 I8 + 1 8 1 I 1 1 19 + 1 9 1 1 1 1 11 + 2 2 6 1 3 3 11 + 3 3 10 1 ! 1 11 + 4 8 9 1 1 1 11 + 5 6 8 1 5 5 11 + 6 11 9 1 1 1 11 + 7 10 12 1 1 1 11 + 8 10 5 1 1 1 11 + 9 7 14 1 2 2 1

Je solved nearly all the sums by coun ting all, but Pe, had two distinct strategiesf o r n + l a n d l + n.N.B. These sums were all given in one interview, but no t in the tabulated order.B D : ( s h o w s 4 + 1 )P e : ( L a u g h s ) F i v e ( O n e s e c o n d t a k e n t o a n s w e r) .B D : A n d h o w d i d y o u d o t h at ?P e : I t ' s e a s y . I t w a s v e r y e a s y . I j u s t d o n e i t.B D : O k a y . ( S h o w s 8 + 1 ).P e : N i n e . ( O n e s e c o n d t o a n s w e r ) .B D : A n d w h a t a b o u t t h a t o n e ?P e : J u s t d o n e i t.B D : H o w d i d y o u d o i t?P e : E i g h t a d d o n e i s n i n e .B D : O k a y . ( s h o w s 1 + 7 ).P e : S e v e n . . . s e v e n . . . o n e a d d s e v e n . . . o n e a d d s e v e n . . . I

d u n n o . ( T e n s ec o n d s to " I d u n n o " ) .B D : H o w c o u l d y o u w o r k i t o u t ?P e : I d u n n o r e a l l y . . . I c o u l d d o i t o n m y f i n ge r s .

T h i s s u g g es t s a s t r a t e g y f o r d e t e r m i n i n g w h e t h e r c h i l d r e n w h o u s e ac o u n t i n g o n s t r a t e g y f o r a d d i t i o n a p p r e c i a t e a n d c a n u s e t h e f a c t t h a ta d d i t i o n is c o m m u t a t i v e .


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N U M B E R C O N C E P T S O F L O W A T T A I N E R S 23Counting Grouped CollectionsCh i l d ren were a s k ed t o s ay ' h o w man y ' w e re in g ro u p ed co l lec t io n s u n d e rvar ious cond i t ions ; g roups m igh t con ta in 2 , 5 o r 10 ind iv idua l it ems ; theremay o r may no t be ungrouped s ing le i t ems inc luded in the co l l ec t ion ; andind iv idua l i tems in the g roup ed co l l ec t ion m ay o r ma y n o t be v i s ib le a t thet i me o f t h e en u mera t i o n .

In one i t em four o paq ue bags , each con ta in ing t en sweets were opene d intu rn and the ch i ld encouraged to examine the con ten t s . Hav ing es tab l i shedtha t e ach bag , in it s tu rn con ta ined t en sweet s, the four bags a nd th ree loosesweet s were d i sp layed and the ch i ld asked 'How many sweet s? ' .

Fr om the seven ch i ld ren in the ma in s tu dy sample , there were s ix subs tan-t ia l ly d i fferent responses :1 . Coun t a l l as one (CA1)

Ch and Je bo th gave the same response :Je: (Co unts , poin t ing to each bag and each sweet in turn) 1 , 2 , 3 , 4 ,

5, 6, 7. Seven.2 . Guesses (G)

Pe~B D :Pe:B D :Pe:BD :Pc:BD :Pe:

T w e n t y .T wen t y . Ho w d i d y o u d ec i d e t h a t ?I jus t guessed.Do you th ink your guess i s r igh t?N o . . . dun no rea lly .Ho w co u l d y o u f i n d o u t i f i t was r i g h t?Tip ou t the sweet s and coun t them.Yes , y o u co u l d t i p t h em o u t . Co u l d y o u d o i t an y o t h e r way ?N o . . . I d o n ' t th i n k so . (P ro ceed s t o t ip o u t s weet s f ro m eachbag , pu t them toge ther in one co l l ec t ion , then coun t in ones ) .

3. Co u n t i n o n es - Gu es s e s n u m b er in each b ag C1 / G)D n: (Gazes at first bag) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (transfers gaze to

next) 11, 12, 13, 14, 15, 16, 17, 18, 19 (next) 20, 21, 22, 23, 24,25, 26, 27, 28 (next) 29, 30, 31, 32, 33, 34, 35, 36, 37 (looks atloose sweets) 38, 39, 40 . . . . for ty .

Dur ing th i s t ime Dn d id no t appear to use her f ingers and d id no t seemto th ink , when ques t ioned , tha t she had used them. I t does seem l ike ly ,


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24 B . D E N V I R A N D M . B R O W Nhowever, that she was matching her count to some mental image of tenobjects. In Fuson's terms Dn was 'counting entities' and in Steffe's she wasa 'counter of figural items'.4. Counts

Jy:in ones (C1)How many? (She looks at a bag, puts up ten fingers and counts,touching each finger in turn on her lower lip) 1, 2, 3, 4, 5, 6, 7,8, 9, 10 (and repeats, transferring her gaze to each bag in turn)1 1 . . . 20, 2 1 . . . 30, 3 1 . . . 40 (then points to each loose sweet)41, 42, 43.

. 'Place Value' strategy, incorrectly co-ordinated (PVc)Th: Dunno.BD: Could you work it out?Th: I could count in tens. (Point to each of the four bags in turn) 10,

20, 30, 40. (Points to each of the loose sweets in turn) 50, 51, 52.6. 'Place Value' strategy (PV)

Ph: For ty three.BD: Forty three. How did you decide it was 43?Ph: (Points to each bag, then each sweet in turn) 10, 20, 30, 40, 41,

42, 43.Th's error could easily have been passed off as a 'careless slip'. But

evidence from her behaviour as well as from other children interviewed atother stages of the study suggests tha t children who make this type of errordo so frequently.Table IV summarises children's strategies and changes in strategies forenumerating collections grouped in tens and ones during the Main Study.

Results support Resnick's (1983) notion that children find it easier tocount when only one 'denomination', i.e., only ls or 10s or 100s has to becounted than when 2 or 3 denominations are present. The results suggestthat several skills are needed for an 'economical' solution:

(i) knowledge of the number word sequence for tens;(ii) appreciation of the structure of the grouped collection;

(iii) the ability to stop the 'tens number word sequence' and begin the'ones number word sequence';

(iv) ability to co-ordinate the end of the groups o f ten and the beginningof the individual items with the change in the number word sequence.


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N U M B E R C O N C E P T S O F L OW A T T A I N E R STABLE IV

Strategies for enumerating collections grouped in tens in the Main Study

25

Date Ob je ct s Individual Tens only or Strategiesused by:used items visible? tens and ones?Y/N 10s/10s & ls Ch Je Pe Ph Dn Jy ThJan (1) Str aw s Y 10s & lsJan (1) Di en esbase Y 10s & lsten blocksFeb (1) Sw eets N 10s & lsMarch (1) Sweets N 10s & lsMarch (1) Straws Y 10s & lsMarch (2) Straws Y 10sMarch (2) Sweets N 10s & ls

C1 C1 C1 PV c C1 PVc C1C A 1P V C1PVc PVc CA10 ClCA1 CA1 G PV C1 C1 PVcC1/G C1/G G PV CI CA10 CA5C1 C1 C 1P V C1 CA10 PVC1 C1 C1 PV C1 PV PVC1 C1 Cl PV PV PV PV

C1 = counts in ones.CA1/CA5/CA10 = counts each 'package' as 1(5, 10).G = Guesses.C1 /G = Counts in one s, guessing how many count words to say for each 'package'.PVc = 'Place Value' strategy incorrectly co-ordinated.PV = 'Place Value' strategy.T h u s , c h i l d r e n m a y p a s s t h r o u g h a s e q u e n c e o f s t a g e s i n u s in g a p l a c e

v a lu e s t r a t e g y t o e n u m e r a t e a c o l l e c t i o n o f g r o u p e d i t e m s , n a m e ly :( 1) C o u n t i n g e a c h ' p a c k a g e ' a s o n e , r e g ar d le s s o f n u m e r o s i t y o f

p a c k a g e s ( - - . i n c o r r e c t a n s w e r ) .( 2) Co u n t in g e a c h i t e m a s o n e ( ~ c o r r e c t a n s w e r ).(3 ) A t t e m p t a t p l a c e v a l u e s t r a t e g y b y c o u n t i n g g r o u p i n g n u m b e r

f o r e a c h p a c k a g e a n d o n e f o r e a c h u n p a c k e d i te m , b u t l a c k i n gc o - o r d i n a t i o n ( ~ i n c o r r e c t a n s w er ) .

(3b ) Success fu l p lace va lue s t r a tegy .

Strategy for Adding Larger Numbers: Place ValueD u r in g t h e p i l ot , f o r o r a l l y p r e s e n t e d a d d i t i o n a n d s u b t r a c t i o n s u m s w i thla rger num be rs ( e .g ., 18 + 24 ), ch i ld ren usu a l ly r eso r te d to ' c ou n t a l l ',a l t h o u g h a f ew ' c o u n t e d o n ' f o r a dd i t i o n. N o c h il d s h o w e d a n a p p r e c i a t i o no f t h e p l a c e v a lu e s t r u c tu r e o f n u m b e r s , b y f o r e x a m p le , a d d in g t e n s a n du n i t s s e p a r a t e ly . I t i s, o f c o u r s e , l i k e ly t h a t c h i l d r e n w o u l d h a v e r e s p o n d e dd i f fe r e n t ly i f t h e ' s u m s ' h a d b e e n p r e s e n t e d i n t h e c o n v e n t io n a l ' v e r t i c a la lg o r i t h m ic ' f o r m a t . I n t h e m a in s tu d y c h i l d r e n w e r e i n s t e a d p r e s e n t e d w i thth e m o s t s im p le o r a l q u e s t i o n s i n w h ic h a ' p l a c e v a lu e s t r a t e g y ' f o r a d d i t i o nmi gh t be observed , e .g ., ' 20 + 4 ' . I t was u sua l ly very easy to tel l, f ro m the


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2 6 B . D E N V I R A N D M . BRO W Nt im e t a k e n ( c f. Re s nik , 1 98 3) w h e th e r t h e r e h a d b e e n a n ' a u to m a t i c ' r e s p o n s e' 2 4 ' o r t h e c h i l d h a d c o u n te d o n : ' 2 1 , 2 2 , 2 3 , 2 4 ' . F u r th e r e v id e n c e w a sp r o v id e d b y th e c h i l d ' s d e s c r ip t i o n o f t h e s t ra t e g y . F o r e x a m p le f o r 4 0 + 9 ,K a r r e p l i e d

'40 + 9? (Put s up 9 f ingers) 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 'a n d h a d c l e a r ly c o u n t e d o n , b u t C o r m u s t h a v e u s e d a ' p l a c e v a lu e s t r a t e g y 'w h e n h e t o o k u n d e r o n e s e c o n d to r e p ly ' 4 9 ' a n d s a id i n d e s c r ib in g h i ss t r a tegy :

' W e l l I h a d 4 0 . A n d I h a d 9 , a n d i t ' s 4 9 'T a b l e V s h o w s th e it e m s a s k e d a t d i f fe r e n t s ta g e s o f t h e m a i n s tu d y a n d

the s t r a teg ies u sed by each ch i ld .TABLE V

Strategies used in M ain A ssessment Study for adding two digit numbersDate Item Strategy for:

Je Pe Ph Dn Jy ThJan (1) 38 + 7 DM DM DM CO CO COFeb (1) 20 + 4 DM CO CO CO CO COFeb (1) 40 + 9 DM CO CO CO CO COFeb (1) 10 + 6 DM CO CO CO CO PVFeb (1) 20 + 6, 30 + 6, 40 + 6, 50 + 6 DM CO PV CO CO PVFeb (2) 5 + 20 DM DM PV CF1 CO PVFeb (2) 6 + 30 DM DM PV DM CO PVFeb (2) 27 + 10 DM DM DM DM CO CODM = direct modelling with physical objects - countCF1 = counts from one (without physical objects).CO = counting on.PV = place value strategy.

all.

T h e r e s u l t s s u g g e s t t h a t a d d in g t e n t o a 2 - d ig i t n u m b e r i s m o r e d i f f i c u l tt h a n a d d in g u n i t s t o a d e c a d e n u m b e r . T a b l e V , l ik e T a b le s I I , I I I , a n dI V s u g g e s t t h a t p u p i l s ' r e s p o n s e s t o t h e s e f o u r c a t e g o r i e s o f q u e s t i o n sw e r e h ig h ly c o n si s t e n t : o n n o n e o f t h e m d id a n y c h i l d u s e q u i t e d i f fe r e n ts t r a t e g i e s e a c h t im e . F u r th e r m o r e t h e c h a n g e s w h ic h d id . o c c u r b e tw e e nin t e r vi e w s s h o w n in T a b le s I V a n d V a r e t o w a r d s a m o r e s o p h i s t i c a t e ds t r a tegy , ind ica t ing lea rn ing .

T h e s e r e s u l t s a r e g e n e r a l l y i n a g r e e m e n t w i th Re s n i c k ' s ( 1 9 8 3 ) m o d e l o fn u m b e r d e v e l o p m e n t i n w h i c h s he se es a n e l a b o r a t i o n o f t h e c o n c e p t o fc a r d in a l i ty a n d t h e p a r t - p a r t - w h o le s c h e m a a c c o u n t in g f o r c h i l d r e n ' s s o lu t i o n


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N U M B E R C O N C E P T S O F L O W A T T A I N E R S 27

to word problems and also for the stages through which chi ldren progressin their grasp o f place value concepts. A new m ode l of chi ldren's e laborationof the part-part-whole schema for cardinal number is proposed here andshown in Table VI . This i s an extens ion and modi f i cat ion of Resn ick ' smodel. It does not represent a 'natural' progression since it relates tochi ldren within an educational system and within a culture that heavi lyemphasises certain aspects of mathematics. At a general cultural level , thestandard num ber wor d sequence and canon ical form are emphasised; withinthe educa tional system th-ese are usual ly also heav i ly empha sised, bu t otheraspects ma y also be emp hasised and this is l ikely to affect no t only the rate ,but poss ib ly the order of e laborat ion .It w as noted abo ve that chi ldren rarely count b ack for ' take aw ay' but anappreciat ion of this strategy and an awareness of the inverse nature ofaddi t ion and subtract ion appeared to be important aspects in mentalcom putat ion , e .g . , for 43 - 28

M iw : 4 3 ta ke 2 8 . . . fo rt y ta ke 2 0 . . . 8 . 2 0 . . . 2 0 . . . 3 f ro m 8 . . .i s f i v e . . . 5 , 19 , 18 , 17 , 16 , 1 5 . . . f if teen .

S imi lar observat ions were ma de by F uso n (Fu son e t a l . , 1982).T A B L E V I

E la b o r a t io n o f c a r d in a l i ty o f n u m b e rS t a g e o f e la b o r a t io n E x a m p le o f n u m b e ro f p a r t - p a r t- w h o le b e h a v io u rsc h e m a

A d d i t io n a l c o n c e p tf r o m earlier stage

N o n e C a n s ta te h o w m a n y i n a s im p l ecol lec t ion

1. (a ) C a n c o u n t o n t o a d d(b) Can so lve 'compare and

'miss ing addend'2 . ( a ) C a n c o u n t g r o u p e d c o l l ec t io n

(b) Can add tens and unit s ,mental ly , when there i s noregrouping

3 . C a n a d d , m e n t a l ly w i t hregrouping

4 . C a n t a k e a w a y , m e n t a l ly , w i t hregrouping5. M ental ly can so lve shar ing

p r o b le m s

T h e w h o le i s t h e su m o fthe parts - neither mo renor lessT h e p a r t s m a y h a v edifferent sizes (tens ando n e s )

The different sizes arerelated to each other(I0 I = 1 10): alln u m b e r s m u s t b e incanonical formNumbers can be expressedin non-canonical formNumbers can be expresseda s p r o d u c ts o f o t h e rn u m b e r s


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28 B. DEN VIR AND M. BROWNPiagetian TasksT h i s co n s is t en cy o f r e sp o n s e was n o t ap p a r en t i n ev e ry ty p e o f q u es ti o n .Per fo rmance on the P iage t i an t asks (number conserva t ion , c l as s inc lus ion)was mu ch mo re v a r i ab l e a s i t was o n an o t h e r i t em s i mi l a r t o t h e n u mb erconse rva t ion t ask b u t spec i fi cal ly des igned fo r th i s s tudy . Ch i ld ren wereasked to co m par e two co l l ec t ions o f s imi la r bu t d i s t ingu i shab le ob jec t swhich w ere g roup ed d i f fe ren t ly (F igure 1 ).

collectionof bluestors ~-X ~'~ ~

Fig. 1. Illustration of an item in which pupil is asked to compare two collectionswhich aregrouped differently.Th e ch i ld was asked : 'Are there mo re b lue s t a rs, mo re ye l low s ta rs o r the

same n um be r o f b lue and ye l low? ' Severa l s tra t eg ies were observed : on ly thefirst one is successful:( i) Co u n t i n g each co l lec t i on o n e b y o n e an d co m p ar i n g (C1 )( ii) Co mp ar in g the num ber o f ' sing le ' i t ems in each co l l ec t ion and

ignor ing the g ro upe d i t ems (CS) , e .g . , i n second i t em (Tab le VII )says 4 x 4 + 2 > 3 x 6 + 1 'becau se 2 there and only 1there ' .

( ii i) Co mp ar in g the to ta l num ber o f ' packages ' in each co l l ec tion ,i r respec tive o f s i ze, i .e ., com par i ng the nu m ber o f g roup s p lust h e n u m b er o f si ng le s b u t i g n o r i n g t h e n u m b er o f i tems p e rgr ou p (C N P) , e.g., in the first i tem 3 x 4 + 2 > 2 x 7 + 1because 5 there and on ly 3 there ' .

( iv ) Co m p ar i n g t h e n u m b er o f i t ems p e r g ro u p b u t i g n o r in g t h en u m b er o f g ro u p s an d t h e n u mb er o f s in gl e i tems (CG N) , e .g .,in first i tem says 2 7 + 1 > 3 x 4 + 2 'bec au se 7 is m or et h an 4 ' .

As Tab le VII show s on ly th ree o f the seven ch i ld ren showe d a cons i s t en tresponse . Fu r th erm ore whi l s t two assessments a re insuf fi c ien t fo r d rawingm any con clus ions , there appears to b e no cons i s t en t t rend in the changes o fs t rategy.


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N U M B E R C O N C E P T S O F L O W A T T A I N E R STABLE VII

Strategies for comparing two collections which are grouped differently

29

Date Item Ch Je Pe Ph Dn Jy ThFeb.

March (1)

Comp ares2 7 + l (bl ue CGN CGN CI C! CNP C1 CSstars) with 3 x 4 + 2(yellow stars) in divid ualstars visibleCompares 4 x 4 + 2 (blue CNP CNP CG N C1 C1 C1 CSstars) with 3 x 6 + 1(yellow stars) in divid ualstars visible

I n t h e f in a l v e r s i o n o f t h is t a s k t h e c o m p a r i s o n w a s b e t w e e n d i f f e r e n tc o l o u r e d s w e et s g r o u p e d t o g e t h e r in c a r d b o a r d b o x e s s o t h a t , a l th o u g he a c h i n d i v i d u a l s w e e t h a d p r e v i o u s l y b e e n e x a m i n e d b y t h e c h il d , a t t h e t im et h a t t h e c o m p a r i s o n w a s m a d e e a c h i n d i v i d u a l s w e et w a s n o t d i r e c t l y v i si b le( F i g u r e 2 ) .

Fig. 2.

orange sweets:5 in each box,3 ' loose ' ones

2 in each box

Fin al version o f item for com parison of two collections grou ped differently.

R E S U L T S : T H E D I A G N O S T I C A S S E S S M E N T I N T E R V I E W

I n a l l , 4 7 s k i l ls w e r e a s s e s s e d f o r e a c h o f t h e 4 1 p u p i l s i n t e r v i e w e d i n t h eD . A . I . T h e s e s k i ll s w e r e s h o w n i n T a b l e V I I I ( a ) .T h e r e w e r e t w o m a j o r o u t c o m e s o f t h e a n a ly s is o f c h i l d re n ' s p e r f o r m a n c e s

i n t h e D . A . I . b a c k e d u p b y t h e q u a n t i t a t i v e d a t a c o l le c t e d i n t h e a s s e s s m e n ts t u d i e s :

( a) L e v e ls o f P e r f o r m a n c e( b) T h e D e s c r ip t i ve F r a m e w o r k

Levels o f PerformancesW h e n e a c h o f th e s k i ll s w a s o r d e r e d a c c o r d i n g t o f a c il it y a n d e a c h o f th ep u p i l s o r d e r e d b y o v e r a l l r a w s c o r e i t w a s p o s s i b l e t o g r o u p t h e s k i ll s i n t o


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30 B. DENVIR AND M. BROWNTABLE VIII(a)

Skills assessed in D.A.I. in order o f difficulty (hardest first). The categories of word problemsis that given by Riley e t a l . (1983)

3.6.47.4.45.7.5.2.20.33.

15.46.12.8.24.25.34.9.

10.1.17.26.13.16.19.21.II .14.22.18.23.40.27.35.38.39.44.28.29.30.32.37.31.41.36.42.43.

Mentally carries out two digit 'take away' with regroupingUses multiplication facts to solve a 'shar ing' word problemPerceives 'compare (more) difference unknown' word problem as subtractionModels two digit 'take away' with regrouping using 'base ten' apparatusAppreciates concept of class inclusion, without any hint or helpMentally carries out two digit take away without regroupingUses multiplication facts to solve a 'lots of' word problemMentally carries out two digit addition with regroupingUses counting back/up/down strategy for 'take away'Bundles objects to make a new group of ten in order to facilitate enumeration of acollection which is partly grouped in 10s and lsUses repeated addition or repeated subtraction for a 'sharing' word problem

Partly appreciates concept of class inclusionUses derived facts for additionMentally carries out two digit addition without regroupingCan repeat the number sequence for counting in 10s from a non-decade two digitnumberCan repeat the number sequence for counting backwards in 10s from a non-decadetwo digit numberMakes quantitative comparison between two collections which are groupeddifferently'Knows answer' when taking ten away from a 2-digit number

'Knows answer' when adding ten to a 2-digit numberModels two digit addition with regrouping using 'base ten' apparatusKnows number bonds (not just the 'doubles')Interpolates between decade numbers on a number lineModels two digit 'take away' without regrouping using 'base ten' apparatusUses repeated addition for a 'lots of word problemSolves 'compare (more) difference unknown' word problemCounts in 2s and ls to enumerate a collection grouped in 2s and ls'Knows answer' when adding units to a decade numberModels two digit addition without regrouping using 'base ten' appara tusCounts in 5s and ls to enumera te a collection grouped in 5s and lsSolves 'compare (more) compared set unknown' word problemCounts in 10s and ls to enumerate a collection grouped in 10s and lsKnows numbers backwards f rom 20Orders a selection of non-sequential two-digit numeralsAppreciates structure o f grouped collectionsSolves sharing problems by direct physical modellingSolves 'lots of' problem by direct physical modellingAppreciates conservation of numberAppreciates commutativity of addition for sums of the form 1 + nUses a counting-on strategy for additionReads a selection of non-sequential two-digit numeralsRepeats numbers in correc t sequence for count ing in 2s, 5s and 10sUses counting on strategy when provokedRepeats numbers in correct sequence to 99Knows numbers backwards from 10Compares collections and states whether equalCan say numbers in correct sequence to 20, can solve addition and take away bydirect physical modellingMakes 1 : 1 correspondence


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N U M B E R C O N C E P T S O F L O W A T T A I N E R S 31TAB LE VIII(b)

Performance of wider sample in Diagnostic Assessment Interview classified according tolevel of performance

~upl l HameHFER A or BquotienL

Ski l l Ho .TABLg y l l l ( s )

)6

, i.~i!::%.::>..:>>::!:!.i "': ",".~i!~i~ii.!i~i!i~ii233

i I U '.'.., :..::.v,..... :.)':'..'.;:'.,7:Z.2'.v:7::)v:z,: :r ":-::.)', 4 ' : ' + i " . % ~:?i:2"~!:i':i:i.::i:i:):i:i~:i.~'!:i:i':):.!:i:?.!:i~r :i':Z:'i:',~ ...:::: , ~:,.~ :. " >.:.?:.::?:::ii:U.:::..::::;,~.."?;~.::,!::i?::::::?.:.:,i:.:::::::::::::::::::::::::::::::::::::::::::::::::::::'.'...'..:..ii!:::.:.


18/22


19/22

N U M B E R C O N C E P TS O F L O W A T T A I N E R S 33

~ '

(|

Fig. 3-Descriptive framework, n indicates skill n in Table VIII(a), ~ indicates skitl mpre-requisite for skill n, an d ( ~ Q indicates strong connection between skill i and skill .

u s ed an d t h e t ech n iq u es ad o p t ed ap p ea r ed t o y i e ld a co h e r en t an d s en s ib l ed es c r ip t i o n o f ch i l d r en 's u n d e r s t an d in g , b u t n ev e r th e le s s th e m e t h o d h ass o m e im p o r t an t l im i t a t i o n s .

The ch i ld ' s under s tand ing i s in fer r ed f rom the s t r a teg ies which tha t ch i ldis o b s e r v ed t o u s e. i n f ac t t h e chi ld m a y h av e t h e u n d e r s t an d in g b u t s im p lychoo se to use o the r s t r a teg ies a l l the t ime. Fu r th erm ore there is a mot i va t ionaspec t . I t i s l ike ly tha t some ch i ld ren on ly p roduce the i r mos t soph is t ica tedr eas o n in g wh en t h ey a r e hig h ly m o t iv a t ed t o s u cceed . A l th o u g h b o th t h es el imi ta t ions were r ecogn ised f rom the ou tse t and in te rv iewing s t r a teg iesdes igned as f a r as poss ib le to ove rcom e the d i f ficul ti es, never the less in som ecases the r esu l ts ma y p rov id e an und eres t i ma te o f pup i l s ' ab i l it i es .

The Longitudinal StudyT h e D . A. I . was u s ed t o ex am in e ch an g es i n p e r f o r m an ce o f sev en p u p i l si n t e rv i ewed ap p r o x im a te ly s ix m o n th ly o v e r a p e r i o d o f two y ea rs . T h e


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34 B. DEN VIR AN D M. BROWN

Level OfPerformonce nAssessmentInterview

LEVEL OF PERFORMANCE WITH TIME

,.w.p h . f "f !h _

3 8 13 20Time in months(a )

Row Score nAssessmentInterview

RAW SCORE WITH TIME

~ xP h

Dn ~ C h

Ch0 3 8 13 ZO

Time in months(b)

Fig. 4. Ch ange n performance with time for pupils in the longitudinal study.

t iming of these interviews is show n in Table IX. Figures 4(a) and 4(b) sho wtheir progress in terms of changes in raw score (i .e. , total number of skillssuccessfully performed) and in level of performance, respectively. Severalpoints emerged from an examination of these results :

(i) All the children made progress at nearly every stage of the study, butthis progress was, in most cases, very slow; so slow, in fact, that a lesssensit ive instrument might not have registered any change for some of thepupils .


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N U M B E R C O N C E P T S O F LO W A T T A I N E R S 35(ii) Whils t no two pupils acquire skil ls in the same order , the match

between each pupi l ' s per formance a t each in terv iew and the h ierarchicalf ram ewor k was extremely good - only 3 ski lls were a t any t ime acquired'out of order ' .

(iii) Each pupi ls ' per formanc e in re la t ion to the f ram ewo rk was used asthe basis for two remedial teaching s tudies which are described in a forth-coming paper (Denvir and Brown, 1986) .

S U M M A R YI f one assumes that there is a developm ent aspect to chi ldren ' s learn ing ofnumber, useful prescriptive teaching aris ing from diagnostic assessmentneeds to take accoun t o f three d i f ferent aspects of learn ing:

( i) the orders in which children learn, i .e . , a framework describingacquis it ion;

( i i) where each individual child is within the framework;( i i i) how the individual progresses from one skil l to another , i .e . , how

individuals learn.This paper has descr ibed , wi th in a l imi ted range of per formance for a

smal l aspect of the mathe mat ic s curr iculum ho w the f i r st two aspects havebeen considered. The third aspect was also dealt with during the researchstud y and will be reported in a subsequ ent pap er (D envir and Brown , 1986).

R E F E R E N C E SBednarz, N. and Janvier, B.: 1982, 'The understanding of numeration ', Educational Studies in

Mathematics 13, 33 57.Bennett, N., Desforges, C., Cockburn, A., and Wilkinson, B.: 1984, The Quality of Pupil

Learning Experiences, Erlbaum, New Jersey.Brown, M.: 198 I, 'Levels of understanding of numbe r operation , place value and decimals in

secondary school children', unpublished Doctoral Dissertation, University of London,Chelsea College.Carpenter, T. P. and Moser, J. M.: 1979, 'An investigation of the learning of addition and

subtraction', Theoretical Paper No. 79, Madison, Wisconsin Research and DevelopmentCenter for Individual Schooling.Carpenter, T. P. and Moser, J. M.: 1982, 'The dev elopment o f addition and subtraction

proble m solving skills', in T. P. Carpenter, J. M. Mos er and T. A. Rom berg (eds.),Addition and Subtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 9-24.

Carpenter, T. P. and Mo ser, J. M .: 1983, 'Acquisition of addition an d subtraction concep ts',in R. Lesh and M. Landau (eds.), Acquisition o f Ma them atics Concepts and Processes,Academic Press, New York, pp. 7-44.

Comiti, C.: 1981, ~ premieres acquisitions de la notio n de nom bre par l 'enfant', EducationalStudies in Mathematics 11, 301-318.


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3 6 B. D E N V I R A N D M . B R O W NDenvir, B. and Brown, M.: 1986, 'Understanding of number concepts in low attaining 7-9

year olds, pa rt II: The teaching studies', Educationa l Studies in Mathematics 17 (2) (in press).Denvir, B., Stolz, C., and B rown, M.: 1982, 'Low attainers in m athematics 5-16: Policies an dpractices in schools', Schools Council Working Paper 72, Methuen Educational, London.Descoudres, A.: 1921, 'Le developm ent de l'enfant de deux a sept ans', Delachaux et Niestle,

C.A., Paris.Fuson, K. C.: 1982, 'Analysis of the counting on procedure ', in T. P. Carpen ter, J. M . M oser

and T. A. Romberg (ed.), Addition and Subtraction: A Cognitive Perspective, Elbaum, NewJersey, pp. 67-81.

Fuson, K. C., Richards, J. and Briars, D. J.: 1982, 'The acquisition and elaboration of thenumb er word sequence', in C . J. Brainerd (ed.), Children's Logical and MathematicalCognition, Springer Verlag, New York, pp. 33-91.

Gelm an, R. and Gallistel, C. R.: 1978, The Child's Understanding of Number, HarvardUniversity Press, Cambridge, Mass.Hughes, M.: 1981, 'Ca n preschoo l children add and subtract?, Educational Psychology 1,207-219.

Loevinger, J.: 1947, 'A systematic approach to the con struction an d eva luation o f tests o fability', Psychological M onographs 61, No. 4. American Psychological Association.

Matthews, J.: 1983, 'A subtraction experiment with six and seven year old children ',Educational Studies in Mathematics 14, 139-154.

Nesher, P.: 1982, 'Levels of description in the analysis of addition and subtraction wordproblem s', in T. P. Carpenter, J. M. Mo ser and T. A. Rom berg (eds.), Addition andSubtraction: A Cognitive Perspective, Erlbaum, New Jersey, pp. 25-38.

N.F.E.R.: 1969-80, "Basic Mathem atics Tests', National Foundation for EducationalResearch. NFER-Nelson, London.Schaeffer, B., Eggleston, V. H ., and Scott, J. L.: 1974, 'Nu mb er develo pmen t in young children',Cognitive Psychology 6, 357-379.Steffe, L. P.: 1983, 'Children's algorithms as schemes', Educationa l Studies in Mathematics 14,109-125.Steffe, L. P., von Glaserfeld, E., Richards J., and Cobb, P.: 1983, Children's Counting Types:Philosophy, T heo ry and Application, Praeger, New York.

Steffe, L. P. and J ohnson , D. C.: 1971, 'Problem solving performa nce of first-grade children',Journal fo r Research in Mathematics Education 2, 50-64.Resnick, L. B.: 1983, 'A development theory of number understanding', in H. P. Ginsberg(ed.), The Development o f Mathematical Thinking, Academ ic Press, New Y ork, pp. 10% 151.

Riley, M. S., Greeno, J. G., and Heller, J. I.: 1983, 'Dev elop men t of children's ability inarithmetic', in H. P. G insburg (ed.), The Development o f Mathematical Thinking, AcademicPress, New York, pp. 153-196.Vergnaud, G.: 1982, 'A classification of cognitive tasks and operations of thought involved inaddition and subtraction problems', in T. P. Carpenter, J. M. Moser, and T. A. Romberg(eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum , Ne w Jersey, pp. 35-59.

S h e l l M a t h e m a t i c s E d u c a t i o n U n it ,C e n t r e f o r E d u c a t i o n a l S t u d ie s ,K i n g s C o l l e g e ( K . Q . C . ) ,U n i v e r si t y o f L o n d o n ,5 5 2 K i n g s R o a d ,Che l sea ,L o n d on S W I O O U A,E n g l a n d

1986 Understanding of Number Concepts in Low

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