results on repeated countings of the same set. First children get 10;
then they carefully count again and get 11. Not only is their
counting unreliable; they also see nothing wrong in it! There is no
contradiction in counting ten fingers one time, and eleven the next*.
(ppl2, Ginsburg 1977)
This c o n t r a d i c t i o n was also report ed by Siegel ( 1983) . Chi l dr en ob t ai n i ng d i f f e r e n t t o t a l s as a r e s u l t of counting a set of o b j ect s may be the r e s u l t of counting accuracy, counting s t r a t e g y or t-he choice of obj ect s to be counted. Counting accuracy and counting s t r a t eg y although c o r r e l a t e d were p a r t i a l l y independent from each other; counting accuracy could break down as a r e s u l t of sensor i - mot or
i m p a i r m e n t wnereas c o u n t i n g s t r a t e g y was n o t so v u i n e r a o x e l o
impairments (Saxe 1 9 7 7 , 1 979) .
Chi l dren r e l i e d h e a v i l y on counting techniques in order to solve
problems (Hughes 1986). He report ed on J u l i e t t e seeming to place more r e l i a n c e on her f a u l t y counting procedure than on her i n i t i a l , and
c o r r e c t r ep r e s e n t a t i o n of b r i c k s with f i n g e r s . In the same study he found t h a t Andrew knew his number word sequence t horoughly from one to ten and could s t a t e c o r r e c t l y the t o t a l of two q u a n t i t i e s of b r i cks placed in a box. When he checked by a c t u a l l y counting the b r i c k s he miscounted through i n c o r r e c t matching of word and touch.
Procedural s t r a t e g i e s in counting have been much researched ( S c h a e f f e r , Eggleston and Scot t 1974; Ginsburg 1977; Gelman and
G a l l i s t e l 1978; Wagner and Walters 1982; S t e f f e , et al 1983; Fuson and Hal l 1983; Baroody and P r i ce 1983; McConkey and McEvoy 1986; Hughes 1986). McConkey and McEvoy (1986) studied how c h i l dr en with severe l ea r n i n g d i f f i c u l t i e s learned to count. They considered the basic number s k i l l s to be:
a) The a b i l i t y to r o t e count from 1 to 20 wi thout omissions. b) The a b i l i t y to recognise the numerals 0- 9.
c) The a b i l i t y to count out a q u a n t i t y of o bj ect s from 1 to 20. They considered these t hr ee s k i l l s as being independent with
competence in one not meaning a b i l i t y in another. They i d e n t i f i e d the f o l l o w i ng steps to be i nvolved in counting a pre- det ermi ned set of obj ect s:
a) I d e n t i f y the items making up the set .
c)
Uive eacn item in cne sei one, ana a n i y o ne , numuer name.d) Remember the ob j e c t s which had been counted and those which remai ned.
e) Real i se t h a t the l a s t number named was the t o t a l f or the set .
They o u t l i n e d four d i f f e r e n t mental checks t h a t had to be made f or each obj ect as a c h i l d counted: "What number am I at now?"; "What is the next number in the sequence?"; "Have I counted t h i s ob j e c t or not?"; "Are t her e any more obj ect s to be counted?". They discussed some of the d i f f i c u l t i e s a c h i l d and t eacher might have in deci di ng where a procedural counting e r r o r had occurred, quoting two examples of a c h i l d making an e r r o r of execution in counting an a r r ay of four obj ect s:
X X X X . . . 1 , 2 , 3 , 5
Wrong even though the r u l e of only touching each obj ect once was kept.
X X X X . . . 1 , 2 , 3 ,4 ,5
Wrong, even though the r u l e of number words was c o r r e c t , one o bj ect was touched t wi ce; one- t o- one correspondence word and touch were c o r r e c t .
They observed t h a t the c h i l d was u n l i k e l y to r e a l i s e t h a t an e r r o r had been made f o r d i f f e r e n t reasons and may as a consequence have r e s or t e d to guessing. Here l ay the problem f o r t eachers because t h e r e was no easy way of ex p l a i n i ng the e r r o r s . Correct counting demanded p e r f e c t r u l e keeping on a l l p o i n t s . I f a student could not process
In counting f or c a r d i n a l i t y procedural accuracy was ot v i t a l
importance, the l a r g e r the set to be counted the more l i k e l y e r r o r s were to a r i s e ( S c h a e f f e r , Eggleston and Scott 1974; Gelman and G a l l i s t e l 1978). Accurate counting with smal l er sets was, in p a r t , due to s u b i t i z i n g (Ginsburg 1977; Fuson and H a l l , 1983). Counting could i n v ol v e "sequence" (moving from item to item when d i r e c t i o n was i mportant) and " c oo r di n a t i on " (moving from item to item when d i r e c t i o n was not i mpor tant ) (Von G l a s e r s f e l d , S t e f f e and Richards 1983). In c o - o r d i n a t i n g countable u n i t s to number words i t did not matter whether the word or the u n i t came f i r s t , what did matter was i f they became out of step (Von G l a s e r s f e l d , S t e f f e and Richards 1983).
Hughes (1986) conducting his "br i cks in the box task" observed
c h i l d r e n r e l y i n g on a d i r e c t vi sual image of br i c k s "hidden" in a box. They would tap at d i f f e r e n t places on the closed l i d of the box
w h i l s t count ing. Gelman and G a l l i s t e l (1978) noted t h a t v a r i a t i o n in colour or item type had l i t t l e , i f any, e f f e c t on counting accuracy. Fuson and Hall (1983) described the e x t e r n a l behaviour of c h i l d r e n w h i l s t they were counting by matching successive sequence words to
items in a well def i ned set . Objects not f i x e d were p h y s i c a l l y moved from the uncounted to the counted and p a i r i n g of countable to the sequence word was o f t en observed with a poi nt i n g a c t i o n .
S t e f f e et al (1983) were concerned with the i n t e r n a l r e p r e s e n t a t i o n s t h a t were i nvolved in count ing. They considered counting to be the production of a counting word and of a "counting u n i t item" (a mental c o n s t r u c t i o n ) . In percept ual counting the counter produced u n i t items from the concrete m a t e r i a l s present.
beiman ana b a i i i s t e i u v / b ) o u m n e a t i v e " p r i n c i p l e s ' n e c e s s a r y t o t
accurat e counting: one-one p r i n c i p l e ; s t a b l e order p r i n c i p l e ; c a r di na l p r i n c i p l e ; a b s t r a c t i o n p r i n c i p l e ; order i r r e l e v a n c e p r i n c i p l e . They ■found l i t t l e evidence of c h i l d r en at t empt i ng to give the same
"numberlog" to a p a r t i c u l a r item during r e - c o u n t s . - -Contrary to t h i s view Wagner and Walters (1982) found p r e - s c h o o l e r s tended to use a l i s t exhaustion scheme. When a set was l ess than a c h i l d ' s known count sequence then double tagging would occur to use a l l the count sequence. When the set was g r e a t e r than the count sequence then terms would be made up in order to exhaust a l l the set items. They also argued t h a t t h e i r study suggested t h a t Gelman and G a l l i s t e l ' s (1978) evidence f o r a s t a b l e - o r d e r p r i n c i p l e was weak. Baroody and Pr i ce (1983) and Baroody and Ginsburg ( 1 9 8 4 ) -found l i t t l e evidence of the l i s t exhaustion scheme in t h e i r resear ch. They found c h i l d r e n tended to use s t a b l e nonconventional sequences across the counting t asks. Fuson, Richards and B r i a r s (1982) and Baroody and Ginsburg (1984) found c h i l d r e n counted in t hr e e p o r t i o n s : an i n i t i a l convent i onal s t a b l e p o r t i o n ; a s t a b l e nonconventional p o r t i o n ; a f i n a l nonst abl e "spew" p o r t i o n . Some c h i l d r e n moved s t r a i g h t from the i n i t i a l
convent ional s t a b l e p o r t i on to the nonstable "spew" p o r t i on (Baroody and Ginburg 1984). R e p e t i t i o n of terms during the s t a b l e
nonconventional por t i on seemed to be i n c o n s i s t e n t with the s t a b l e - o r d e r p r i n c i p l e and with a "uniqueness scheme" which
demonstrated the understanding f or the need to generate a sequence of d i s t i n c t terms (Baroody and P r i ce 1983; Baroody and Ginsburg 1984) . However, when count i ng, c h i l d r en who appr eci at ed the s t a b l e - o r d e r p r i n c i p l e would avoid r epeat i n g standard or non-standard terms t h a t they remembered using p r e v i o u s l y . Consequently a spew or repeated term per se was not i n c o n s i s t e n t with a s t a b l e - o r d e r p r i n c i p l e
I D d f u u u y d 11 U P i I l d U U I y .
HMI (1979) report ed t h a t r e s t r i c t i o n to one type o-f apparatus could r e s t r i c t the c h i l d ' s understanding of number. Bologna (1982) st at ed the d i f f i c u l t y of teachers being able to recognise the st r en gt hs and weaknesses of the vari ous types of apparatus in developing a c h i l d ' s understanding of number.
Chi l dren were able to use several basic counting s t a t e g i e s in sol vi ng a d d i t i on and s u b t r a c t i o n problems bef ore they r ecei ved formal
i n s t r u c t i o n (Carpenter and Moser 1982) . In sol vi ng a d d i t i o n problems they i d e n t i f i e d t hr ee types of s o l u t i o n s t r a t e g i e s : d i r e c t modelling with f i n g e r s , or physi cal objects? using the counting sequence; using r e c a l l e d number f a c t s . In t he counting s t r a t e g i e s t h r e e d i s t i n c t methods were observed:
a) "Counting a l l wi t hout models" - beginning the counting sequence with one and conti nui ng u n t i l the answer was reached, wi t hout used physical r e p r e s e n t a t i o n s .
b) "Counting on from f i r s t " - counting on from the f i r s t addend. c) " C o u n t i n g - o n - f ro m- l a r g e r " - counting on from the l a r g e r addend.
In sol vi ng s u b t r a c t i o n problems using counting s t r a t e g i e s Carpenter and Moser (1982) observed "Counting down from" which was a backwards counting sequence from the l a r g e r number and "Counting up from g i v e n " , which was counting on from the smal l er number to reach the l a r g e r . They found c h i l d r e n were able to solve c e r t a i n types of s t o r y problems using counting s t r a t e g i e s although the s o l u t i o n s u s u a l l y used only mani pul at i ves and forward counting.
u ni i o r e n naa a i t t i c u u y i n c o u m i n g an T r u t n n u m u e r s u u n e r uncm une