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When do children trust counting as a basis for relative number judgments?
JOURNAL
OF tXPERIMENTAI.
CHILD
J’SYCHOLOGY
43,
32X-34.5
(
1987)
When Do Children Trust Counting as a Basis for Relative Number Judgments?
There inconsistent
are
to single know studied or the not
several explanations with their counting.
set\.
They
which number in conditions experimenter
ignore
count
may
of why Their
forget
the
children counting
results
make may
of counting
relative number judgment\ be unreliable or restricted when
is bigger. In four experiments children’\ which controlled for the&e explanations. counted made no difference. which implies
information
hecau\e
competent counters less than when conflicting length cues
they
mistrusted
their
6 years old judged conGhtentlq were present. Trusting counting
they
judge
or
not
judgment\ welt Whether the child that children did counting.
Very
few
with their counting over length to tell
whether two rows were the same preceded trusting counting to decide which row had more. The results are discussed in relation to children’~judgment\ uhen using matching and D. Klahr and J. 6. Wallace’\ t lY73). C‘o~!~iti\~c, P.tvc~l~ol~>,~v, 4. 301-327)
account
of number
development.
I’JX? Zc.idemK
Pre*\. IllL
Piaget (1952) reported children counting right but judging relative number wrong. He questioned the importance of counting for children’s number development. Recently counting has been much analyzed and children’s mastery of the procedure and recognition of principles implicit in it studied (Briars & Siegler, 1984; Fuson & Hall, 1983: Gelman & Gallistel. 1978; Gelman & Meek. 1983; Greeno, Riley, & Gelman, 1984; Wilkinson. 1984). Why children who know how to count do not use counting information in making relative number judgments has not been answered. There are several possible causes of counting inconsistent judgments. These are discussed, and then previous studies of the influence of counting on relative number judgment are reviewed. The first cause is lack of counting proficiency. Gelman and Gallistel (1978) distinguished three how-to-count principles: one-one. stable-order, The gratefully thanks Requests Institute
author
thank3
acknowledges Fitz Taylor for for reprints of Education.
the
htaff
and
children
in participating
school\.
In particular
the cooperation of Ms Rennell and MS Clery. The discussing the work and the editor and referee? for their should be sent to Richard Cowan. CDEP. University 34-27 Wohurn Square. London WCIH OAA. England.
the
authol
author alho commenta. of London
COUNTINGANDNUMBERJUDGMENTS
329
and cardinal. A child who does not know the cardinal principle, that the final number name uttered gives the cardinal-number value of the collection, cannot be expected to succeed in using counting to compare two collections. Gelman and Meek’s (1983) findings suggest this is an unlikely cause for 3-year-olds. The second cause is ignorance of how to adapt counting to determine cardinality of two or more simultaneously presented collections. Saxe (1977) found most 3-year-olds simply counted on when asked to compare the numbers of pigs and horses. Counting in the experience of preschoolers may be predominantly, if not exclusively, carried out with reference to a single collection of objects (Durkin. Shire, Riem, Crowther, & Rutter. 1986). Third, ignorance of relative magnitude. in unequal number comparisons, may be responsible. Schaeffer. Eggleston, and Scott (1974) and Siegler and Robinson (1982) found marked improvements in performance of digit magnitude comparisons between 3 and 6 years. However, in neither study did the numbers result from counting or refer to objects the child had seen. This may have made the task a “disembedded” one (Donaldson, 1978). Hughes (1981) found preschoolers’ performance of addition and subtraction problems improved considerably when the tasks were embedded by including reference to objects that had been shown. So children may judge that 6 cars is more than 5 cars before judging 6 to be more than 5. Furthermore. neither Schaeffer et al. (1974) nor Siegler and Robinson (1982) explicitly related knowledge of digit magnitudes to counting performance. A fourth explanation is memory failure. When asked to judge, children may have forgotten the number of items in one or both collections. All the above explanations of counting right and judging wrong are consistent with a view of number development which assigns a major role to counting. The first two specify developments in children’s counting that may be needed before it can improve their relative number judgments. The third points to knowledge implicit in the conventional count list, and the fourth explanation, memory failure. may result from either deficient knowledge of how to adapt counting or performance problems. In the former case children do not know that the result of counting the collections should be stored. in the latter they know it should but the interference caused by counting the second collection brings about the loss of the result of counting the first collection. In some of the previous judgment studies children have been found to err when none of the above causes seems likely (Fuson, Secada, & Hall, 1983; Michie, 1984a, 1984b; Saxe, 1977. 1979: Schonfield, 1986). Michie (1984a. 1984b) and Schonfeld (1986) only studied children who had passed a counting pretest, making lack of counting proficiency an unlikely explanation. However, Schonfeld (1986) does not report the
330
RICHARD
COWAN
details of his counting test, and it appears that Michie (1984b) did not test children’s counting up to the highest number, 8, used in the judgment task. Fuson et al. (1983) and Michie (1984a. 1984b) helped children count and reminded them of the numbers of items in the two collections if necessary immediately prior to asking them to judge. Memory failure and inability to adapt counting are thereby ruled out. Michie (1984a) asked children to judge rows of buttons on cards and collections of buttons counted into boxes. Most 4-year-olds judged right when the buttons went into boxes. By far the most errors were made on unequal number trials when the buttons were in rows on cards arranged so that length provided a conflicting cue to number. If ignorance of number magnitude was responsible then performance should have been similarly poor on unequal number trials in the boxes conditions. It was not. The influence of length cues on children’s number judgments has been frequently demonstrated. Brainerd ( 1979) studied children’s judgments of displays such as those illustrated in Fig. I. Children judged unequal number/equal length displays right more often than unequal number/ unequal length displays. Least often judged right were equal number/ unequal length displays. Indeed only IO children out of 200 from kindergarten through third grade judged all the displays right. Cowan (in press) found the same order of difficulty with display type. Adding guidelines to the displays, however, dramatically improved performance of 7-yearolds. In contrast, _F-year-olds showed some improvement when guideline:, were added but still typically judged wrong. Whether there are age-related differences in judgments after counting of these display types is one concern of the studies reported below. A second is whether the order of difficulty of display types can be replicated. Finally, to test whether children’s failure to use count information in judging is due to perceived unreliability of counting or lack of numerical understanding, judgments following the child’s counting are compared with judgments following the experimenter’s counting. Saxc ( 1977) found children who purposely miscounted to make their counting fit their lengthbased number judgment. The effects of greater confidence in counting when it is done by an adult may bc general improvements in judgment accuracy or reduced differences according to number version. Michie (1984b) reported performance on small (3 v\ 3. 3 vs 4) number versions was much more accurate than that on larger number versions (6 vs 6. 6 vs 8). Whether this reflected children’s greater confidence in counting smaller numbers or the availability of subitiLing could not be determined. If the effect of numerosity disappears when an adult counts, this would be strong evidence that it is confidence in counting that is responsible for it.
COUNTING Unequal
AND
number
Equal
NUMBER Unequal
length
Equal Unequal
number length
. .
. . . . . . . FIG.
number
Unequal length
.
331
JUDGMENTS
. . . . . . .
. . . . .
I.
Display
types
used by Brainerd
EXPERIMENT
(1973).
1
In previous studies either children’s counting accuracy was not assessed (Fuson et al. 1983) or judgment accuracy following accurate counting was not reported (Saxe 1977, 1979) or how children’s counting was assessed was not reported (Michie 1984a. 1984b; Schonfeld, 1986). Whether children make wrong judgments after counting right is not well established. In this experiment accurate counters from 3 to 5 years old are first identified. and then their judgments of relative number studied. When children are discouraged from counting they judge unequal number/equal length displays, shown in Fig. 1, right more often than unequal number/unequal length displays, and equal number/unequal length displays are least frequently judged right (Brainerd. 1973. 1979: Cowan. in press). Whether such an order of difficulty can be found for judgments following counting is tested. If children ignore count information because conflicting perceptual ones make them doubt their counting, then judgments will be more counting consistent following counts by a more competent counter, i.e.. an adult. If the superior accuracy ofjudgments of smaller number displays observed by Michie ( 1984b) reflects greater confidence in counting, then there may be less effect of numerosity following adult counts. Displays in which relative length does not conflict with relative number are included to check whether children’s errors on the length conflict displays, illustrated in Fig. I, are attributable to length-based responding. Method
There were 65 children (29 girls, 36 boys) who agreed to take part in the first phase. They all attended the same nursery school situated in a poor area of inner London. Their ages ranged from 3 to 5 years CT =
332
RICHARD
COWAN
4 years 1 month, SD = 6.4 months). The children were from a variety of ethnic groups but predominantly spoke English as their only language. Based on their performance in the first phase. 34 children. IO of whom were girls. were selected to take part in the second phase. They ranged in age from 3 years 2 months to 5 years (?? = 4 years 6 months, SD = 5.1 months). Matrrids
For the first phase there were 24 red cards consisting of 4 cards with single rows of each of the following numbers of dots on them: 3. 4. 8. 9, 15, and 16. The 4 cards of each numerosity differed with respect to the color of dots-on 3 of the cards the dots were yellow and on the other 2 cards the dots were blue. and in overall length of row. For the rows of 3 and 4 dots the two row lengths used were 54 and 102 mm. For the rows of 8 and 9 dots the two row lengths were 131 and 252 mm. For the rows of 15 and 16 dots the two row lengths were 737 and 466 mm. Within a row the dots were equally spaced. The dots were 7.5 mm in diameter. For the second phase there were 1X red cards with displays of rows of blue and yellow dots. There were 3 cards of each of the following six combinations of rows: a row of 3 yellow and a row of 3 blue dots. a row of 3 yellow and a row of 4 blue dots, a row of 4 yellow and a row of 3 blue dots. a row of 8 yellow and a row of 8 blue dots, a row of 8 yellow and a row of 9 blue dots, a row of 9 yellow and a row of X blue dots. The three displays for each row combination consisted of one in which the two rows were the same in length, one in which the row of yellow dots was longer, and one in which the row of blue dots was longer. The lengths of the rows in the equal length displays and the lengths of the shorter rows in Ihe other displays were the same as the shorter row lengths for that numerosity in the first-phase cards. The lengths of the longer rows were the same as the longer row lengths for that numerosity in the first-phase cards. The rows were 25 mm apart, from dot center to dot center, and the midpoints of the two rows were directly opposite each other.
First phuse. Children were tested individually but allowed to bring a companion of their own choice for reassurance. The room in which testing took place was familiar to the majority of children as a quiet place where they were read stories. Unaccustomed as the children were to both formal testing and the experimenter, efforts were made to put them at their ease, and they were allowed to interrupt the testing for a story break, or to abandon testing completely if they wished. The eight cards with rows of 3 and 4 dots were shuffled and presented
COUNTING
AND
NUMBER
JUDGMENTS
333
one by one with the child being asked to count the dots. If the child failed to count one or more of the cards correctly the session was terminated. Having correctly counted the first eight cards the child was then presented with the sets of cards with 8 or 9 dots on them. If a child miscounted a row the experimenter asked the child to count them again, carefully this time, with the experimenter moving his finger from dot to dot as the child counted. If the child made another error the session was terminated. If successful, with or without the assistance described above, the child then received the sets of cards with 15 and 16 dots on them. Again, if a child made an error assistance was provided. Notes were taken of idiosyncratic count lists. Second phnsc~. Children were tested alone in the same room that was used in the first phase. A conversation took place in which children agreed that they had the same number of arms as the experimenter, that they had the same number of fingers as the experimenter, and that they had a bigger number of fingers than legs. Displays of rows of 3 and 3 dots were shown and the children agreed that there were the same number of blue dots as yellow dots. both 2, despite the perceptible differences in length, and that a row of 3 dots had a bigger number of dots than rows of 2 dots that were the same length or longer. Following this conversation the main task was introduced. If the child had been assigned to the self-count group, the experimenter explained that he would be showing the child cards with rows of blue and yellow dots on them. First he would ask the child to count the blue dots and say how many there were. Then he would ask the child to count the yellow dots. Following this he would remind the child how many blue dots there were and how many yellow dots. Finally he would ask the child whether there were the same number of blue dots as yellow dots. If the child said there were not the same number, the experimenter would ask of which there were a bigger number. Then the 18 cards with displays of two rows of dots were shuffled and presented one by one to be counted and judged. If a child miscounted a row the experimenter requested a recount. Recounts were rare and invariably correct. Children in the experimenter count group were told that the experimenter would do the counting for them and that he was very good at counting. The procedure was otherwise identical to that for the self-count group. The experimenter did not miscount. The cards were presented so that the rows were vertical to the child’s line of vision. This was done to be comparable with previous studies (Cowan, 1984. in press) and to preclude the tendency of young children to choose the closer of two rows as having more.
334
RICHARD
COWAN
TABLE DISTRIBUTION
OF CHILDREN
I
ACCORDING TO FIRST-PHASE COUNTING IN EXPERIMENTS I. 1. ANI> 4
Highest
number
cards
PERFORMANCF
consistently
AND
counted
AN-
right
None item Experiment 3-year-olds 4-year-olds Experiment 5-year-olds Experiment h-year-olds
Level
(I
1 i
3
I
0
5
5
II
0
5
2
I
6
I4
c)
II
0
I
2
h
41
2 4
” With assistance. h Without assistance.
Results Children were classified according to the quality and consistency of their counting performance into four levels and two sublevels. To be considered as counting right on a particular trial the child had to both use the conventional English number words in order with no omissions and give the final number name uttered as the cardinal value of the set. Children were classed as Level 0 if they did not count right on every 3- and 4-item trial. Children were classed as Level I if they counted wrong on two or more of the 8- and 9-item trials. Children were classed as Level ?a if they needed assistance on the 8- and 9-item trials and counted wrong on two or more of the IS- and 16-item trials. They were classed as Level 2b if they needed no help on the X- and 9-item trials and counted wrong on two or more of the I_‘;- and Ih-item trials. Level 3a children were those who succeeded in counting right seven trials involving I5 and I6 items when assistance was provided. Level 3b children counted right on all the 15 and l6-item trials without assistance. The distribution of children across levels of counting according to age is shown in Table I. The association of counting level with age apparent in Table I was confirmed by a Spearman rank correlation test (r, -=z .5-t. p < .OOl) The children who took part in the second phase wcrc drawn from Levels 2 ’dnd 3 as follows: 6 from Level ?a. 10 from Level 2h. and all 8 Level 3 children.
COUNTING
AND
NUMBER
TABLE
MEAN
SCORES IN EXPERIMENTS
335
JUDGMENTS
2
1 AND 2 AS A FUNCTION AND NUMEROSITY
OF DISPI.AY TYPE, Suerr:cr
Display
GROUP.
type Equal
Unequal number/
Unequal number/
Unequal number/
number/
equal
unequal length
consistent length
unequal length
Item
length
Expt. I 3 vs 4. 3 vs 3 Self-count
I .8 (0.4)”
1.6 (0.X)
7.0
(0)
1.7 (0.X)
I.4
(0.9)
1.3 (1.0)
I.8
(0.6)
I .9 (0.3)
0.8
(0.9)
0.3
(0.5)
I .o (0.9)
0.7
l0.X)
1.6 (0.5) I.2 (0.9)
I.? (0.8) I .6 (0.7)
I .9 (0.3) I.7 (0.7)
3.0
(0)
2.0
(01
2.0 I.9
0.7
(0.9)
0.4
(0.7)
7.0 2.0
(0) (0,
Experimenter 8 vs 9. 8 vs x Self-count
count
Experimenter
count
count
I .9 (0.3) I .9 (0.3) I.5
Expt. 2 3 vs 4. 3 vs 3 Self-count Experimenter x vs 9, x vs 8 Self-count Experimenter
” Numbers
count
in parentheses
I.4 2. are
(0.7) (0.X)
standard
(0) (0.3)
I .4 (0.8) I.3
(0.X)
deviations.
For each right judgment the child was awarded one point. Judgments of equal number/consistent length displays were near ceiling level: 45 out of 48 were right. For the equal number/unequal length displays and the displays with unequal number and equal. unequal, or consistent length. scores were derived from combining the data from the different color versions. The means for these displays are presented in Table 2. A three-way analysis of variance with one between-subjects factor, counting agent, and two within-subjects factors numerosity and display type. was conducted. The only significant effects were main effects for numerosity, RI, 22) = 54.21, p < .OOl, and display type, Et3, 66) = 10.78, p < .OOl. The overall means for display types were as follows: 2.24 for unequal number/equal length, I .88 for unequal number/unequal length, 3.29 for unequal number/consistent length, and 3.35 for equal number/unequal length. Newman-Keuls tests confirmed all differences between means were significant, p < .Ol. except that between unequal number/consistent length and equal number/unequal length. The above analysis showed equal number/unequal length and unequal number/consistent length displays were best judged. However, in that
336
RICHARD
COWAN
analysis a child had to both say the rows were different in number and identify the more numerous row to be considered to judge an unequal number display right. Children’s errors on unequal number displays may be solely due to wrongly identifying the more numerous row. If that is true then no difference according to display type would be found when children’s judgments are analyzed according to whether displays were correctly judged to be the same or different in number. Children’s unequal number display judgments were recoded accordingly. Scores for equal number/unequal length displays were unchanged. A three-way analysis of variance was conducted with counting agent as a between-subjects factor and numerosity and display type as within-subjects factors. There were significant main effects for numerosity. F( I, 21) = 37.06. p --: .OOl, and display type, F(3, 66) = 1.84, p < .OS. The overall means for display types were as follows: 2.92 for unequal number/equal length. 3.63 fat unequal number/unequal length. 3~54 for unequal number/consistent length. and 3.25 for equal number/unequal length. There were no significant differences between means according to Newman-Keuls tests. When rows are the same length and judged to be the same number the judgment is length consistent. When rows differ in length and the longer row is judged to be more numerous. the judgment is length consistent. Children’s judgments on all displays were recoded accordingly to length consistency. and a two-way analysis of variance was conducted with counting agent as a between-subjects factor and numerosity as a withinsubjects factor. This revealed a significant main effect of numerosity. F( 1, 22) = 10.88, p < .OOl, and a significant interaction between counting agent and numerosity. Ft 1. 23) = 14.99. p i .OOl. The mean numbers of length consistent judgments according to counting agent and numerosity were as follows: self-count small numbers 3.50. self-count large numbers 5.58, experimenter count small numbers 4.17, experimenter count large numbers 4.00. Tests on simple main effects, recommended by Winer (1970), showed that on small number displays there was no significant difference between the groups, E’(1 , 73) = 1.6. but the self-count group made more length consistent judgments than the experimenter count group on large number displays. Ft 1, 21) = 8.99, p < .Ol. Also, whereas the self-count group made more length consistent judgments on larger number displays, F(1, 22) = 25.70, p c: .OOl, there was no effect of numerosity on the numbers of length consistent judgments made by the experimenter count group. RI, 21) = 0.17. As the children varied considerably in -judgment accuracy, age. and counting level, tests of association were carried out. According to Spearman rank correlation tests there were no significant associations between any of these variables: age vs judgment accuracy, I’, = .77. counting level vs judgment accuracy. Y, = .05. age vs counting level. r, = .35.
COUNTINGANDNUMBERJUDGMENTS
EXPERIMENT
337
2
In Experiment 1 accurate counters between 3 and 5 years old made wrong judgments after counting right. More errors were made on larger number displays than on smaller number displays and on length conflict displays than on length consistent displays. Within the range of counting performance studied there was no association between judgment accuracy and age or counting level. There were two unexpected findings. In contrast to previous studies children judged equal number displays better than unequal number displays when conflicting perceptual cues were present. Children’s difficulties with unequal number displays were mainly in determining the more numerous row. Even when this was taken into account, there was no suggestion that children were worse at judging equal number displays. Second, although who counted did not affect accuracy, it did affect frequency of length consistent judgments. When children counted themselves, they made more length consistent judgments on larger number displays. Whether these findings can be replicated with 5-year-olds with the same levels of counting performance is studied in Experiment 2. Method Subjects There were 44 children. 17 of whom were girls, who took part in the first phase. They all attended the same infant school situated in the same area as the school in Experiment 1. Their ages ranged from 5 years 1 month to 5 years I1 months (x = 5 years 7 months, SD = 3.2 months). Based on their performance in the first phase, 24 children, of whom 6 were girls, were selected to take part in the second phase. They ranged in age from 5 years I month to 5 years 11 months (x = 5 years 7 months, SD = 2.7 months). Materiuls
und Procedure
The same materials were used as in Experiment I. The procedures for the first and second phases were the same as those followed in Experiment 1, except that testing took place in a library area rather than in a small room. Results The Table level, phase 3a.
distribution of children according to counting I. There was a significant association between r, = .45, p < .Ol, The children selected to take comprised 5 from Level 2a, 11 from Level 2b,
level is shown in age and counting part in the second and 8 from Level
338
RICHARD COWAK
For each right judgment the child was awarded one point. No difference according to color version was observed so the data were combined. Means are presented in Table 3. No errors were made in judging the length consistent displays so these were not analyzed further. A threeway analysis of variance was conducted with counting agent as a betwecnsubjects factor and within-subject factors ofnumerosity and display type. unequal number/equal length. unequal number/unequal length. and equal number/unequal length. There were significant effects of numerosity. F( I, 23) = 54.46, p < .001, and display type, F(3. 44) = 19.62. [> <: .OOl. The interaction between numerosity and display type was also significant, E‘(?. 44) = 9.91. p < .OOl As can be seen in Table 2 there was little variation in scores on small number displays which were at ot near ceiling level. Newman-Keuls tests showed that performance on unequal number/unequal length displays was worse than on unequal number/equal length and equal number/unequal length displays (1’ c. .Ol), but the latter did not differ-. As for Experiment I, children.5 judgment5 of unequal number display\ were recoded according to whether the rows were judged to differ in number. For two reasons only unequal number/equal length display scores were analyzed further. First unequal number/equal length displays are the only unequal number type that have a conflicting length cue with respect to judging same or different in number. Second performance on unequal number/unequal length and unequal number/consistent length displays was errorless. A two-way analysis of variance was conducted with counting agent as a between-subjects factor and display type. unequal number/equal length and equal number/unequal length. ;IS a withinsubjects factor. The only significant effect was for display type. kI I. 22) = 3.78. p c .05. The means were 3.75 for unequal number/equal length and 3.33 for equal number/unequal length. Therefore children more often said that the rows in unequal number/equal length displays differed in number than that the rows in equal number/unequal length displays wcrc the same in number. Spearman rank correlation tests revealed no significant association between judgment accuracy and age. I’, ~7 .O9. or between -judgment accuracy and counting level. Y, = .?6. The latter does. however, approach significance. 11 c: . I. There was a marked association between counting level and age in the subset of S-year-olds who took part in the second phase, Y, = .62. p c .OI. In short, the results of the second phase of Experiment 2 are similar to those of Experiment I in that children made most wrong judgments when the displays were large number length conflict displays. in particular unequal number/unequal length displays. Also, who counted did not
COUNTING AND NUMBEK.IULXMENTS
339
and there was no association of judgment affect judgment accuracy. accuracy with age and no marked association with counting level. Unlike the children in Experiment 1. the j-year-olds in Experiment 7 did not judge equal number/unequal length displays right much more often than unequal number/equal length displays. and they did not make more length consistent judgments when they counted themselves. The explanation of the first difference is that Syear-olds rarely judged two rows to be the same in number ifcounting did not yield the same numbers. EXPERIMENT
3
Experiments I and 2 had rows of 8 or 9 dots as large number displays. and children were much more likely to judge them wrong than smaller number displays. Whether 5-year-old children make even more errors with even larger numbers still within their counting range is studied in Experiment 3. Method
Subjects were 12 children, 7 of whom were girls. They were drawn from the children who had participated in the first phase of Experiment 2 and consisted of I I Level ?b counters and I Level 3a counter. Their ages ranged from 5 years I month to 5 years I I months Cx = 5 years 8 months, SD = 3.4 months).
There were 18 red cards with displays of rows of blue and yellow dots. There were the 9 cards with rows of 3 and 4 dots as used in the second phases of Experiments I and _9 and 9 cards with rows of IS and I6 dots of rows and constructed on the same principles, i.e., same combinations same overall lengths as the corresponding number versions of the firstphase cards. The procedure was the same as for second-phase Experiment I. experimenter count group. Results For each right judgment the child scored one point. Data from the different color versions wet-c combined. There were no errors made in judging equal number, consistent length displays. In Table 3 the means for the other display types are presented. An analysis of variance was conducted with two within-subject factors. numerosity and display type. There were significant main effects of numerosity, Ft I. 1 I) = 27.96. 11 < .OOl. and display type. FC3. 3.3) = 5.81, p < .Ol. The interaction between display type and numerosity was also significant. F‘(3, 33) = 4.67. p i .Ol. The interaction results from the performance on all small
number displays being at or near ceiling level. A separate analysis ot scores on large number display types was therefore carried out. There was a significant effect of display type. E-(3, 33) = 5.65, \I i .Ol. NewmanKeuls tests showed unequal number/consistent length scores exceeded those on unequal number/equal length and unequal number/unequal length displays, I> < .Ol. and equal number/unequal length displays. 11 ,: .05. No other difference was significant. Children’s judgments of unequal number displays were recoded according to whether the rows were judged to differ in number. All judgments on small number displays and large number unequal number/unequal length and unequal number/consistent length displays were right. On large number unequal number/equal length displays the mean number of correct judgments was 1.33. As this is the same as the mean for large number equal number/unequal length displays no further analysis was conducted. There was no significant association between judgment accuracy and age, I’, = .I%. As all except one child was at the same level of counting no tests of association involving counting level were made. In this experiment, as before, children who could count judged wrong after the experimenter counted and made most of their errors on large number length conflict displays. There was less variation according to type of length conflict than in Experiments 1 and 2 and no evidence that displays with rows of 15 and I6 dots were significantly more likely to produce errors than those with rows of 8 and 9 dots.
COUNTINGANDNUMBERJUDGMENTS EXPERIMENT
341
4
In Experiments 1-3 most children who could count made one or more judgment errors after counting. The children so far studied were less than 6 years old. The counting and judgment performance of 6-year-olds is studied in Experiment 4. Method
In the first phase 50 children. including 2.5 girls, took part. They attended the same school as the children in Experiments 2 and 3. Their ages ranged from 6 years to 6 years 11 months (x = 6 years 5 months, SD = 3.2 months). In the second phase there were I2 girls and 12 boys, all Level 3b counters, whose ages ranged from 6 years to 6 years I I months Cx = 6 years 6 months, SD = 2.9 months).
For the first phase the same cards and procedure wet-e used as in Experiments I and 2. For the second phase the cards were those used in Experiment 3 and the same procedures followed as in Experiment 2. Results
The distribution of children across counting levels is shown in Table I. There was no significant association between age and counting level. I’\ = .19.
Scores were derived as before. No errors were made in judging number equal consistent length displays or any other small number displays. Means for the large number displays are presented in Table 3. As only three children made any judgment errors the scores were not analyzed further. GENERAL
DISCUSSION
Experiments 1-3 showed children made counting inconsistent judgments under conditions which made the four explanations discussed in the introduction implausible. Because only children who adequately counted the numerosities involved were tested, the errors cannot be attributed to counting incompetence. Neither memory failure nor ignorance of how to adapt counting seem likely explanations given the procedure. Having the experimenter count was supposed to increase countingbased judgments, because children would treat the experimenter’s counting as reliable. No improvement in judgments was observed but in Experiment I children between 3 and 5 years old made less length-based judgments
342
RICHARD C‘OWAN
when the experimenter counted. This was not expected. Although who counted did not affect accuracy in Experiments I, 2, and 4 it may if children are tested on numbers beyond their reliable counting range. In Experiment I equal number/unequal length displays were judged right more often than unequal number/equal length displays, which in turn were better judged than unequal number/unequal length displays. In Experiment 3 unequal number/unequal length displays were again least successfully judged but performance on unequal number/equal length and equal number/unequal length displays did not differ. It would be wrong to infer that children’s knowledge that same count number implies cardinal equivalence is more robust than their knowledge that different count numbers imply nonequivalence. This is because the difference between equal number/unequal length and unequal number/equal length displays disappeared in Experiment I and was revcrscd in Experiment 7 when children’s judgments were simply analyzed according to whether they correctly judged displays to bc different in number. What ma& unequal number displays difficult way using counting to identify the more numerous row. in particular when length contradicted counting, as in unequal number,/unequal length displays. In both Ilxpcrimcnts I and 2 unequal number/unequal length displays were Icss successfully .iutlged than unequal number/equal length display\. Distinguishing children’s USC of the relation between counts and cquibalcncc from their use of count information itbout relative magnitude seems important for understanding the major changes in children’s use of counting. Only 5 children in Experiments I-? judged all the display3 right when correct judgment of relative magnitude was required. Howcvcr. when judging unequal numbers displa\;h to differ is Ihe criterion fog correctness. the number of succedul children incrca\cs to 26. 6 in 3. Experiment I. I2 in Experiment 2. and X in Experiment Using length conflict displays makes rhc text of children‘s u\c’ of count\ demanding. and it i5 probable that in Gtuationa bvhere length neither conflicts with nor supports count information children may make more counting consistent judgmcnls. Using Icngth conflict dicrvation rulc4. by noting the Children Icarn to conscrvc. results of pretrnnsformation and pc’st-tn~nsformation subiti/ing. Estimating
COUNTINGANDNUMBERJUDGMENTS
343
is held to develop concurrently, but, in its early forms. it is the source of much inconsistency because it is based on unidimensional global perceptual attributes of sets such as length. To the three quantification operators discussed by Klahr and Wallace (1973, 1976) and Klahr (1984) could be added cardination or matching (Brainerd. 1973, 1979; Cowan. 1984, in press: Fuson et al., 1983). This is the process of determining relative number by pairing off each element in one set with one and only one element in another. Brainerd (1973, 1979) found matching rarely influenced the relative number judgments of children lower than third grade. This may be because young children need perceptual supports such as pieces of string or guidelines drawn between the elements (Fuson et al. 1983. Cowan, 1984, in press). Matching is unlike the other quantification operators, as it does not generate quantitative symbols for each set but directly determines relative magnitude. Unlike estimation it is concerned exclusively with discontinuous quantity. The developmental course of matching is not well-known except that it is present in 5ycarolds and trusted over length by 7-year-olds (Cowan, 1984. in press). Young children therefore have four methods of solving small number relative number judgment problems. Once number names are attached to the subitizing sublists, the same symbols for each set. and hence the same judgment, will be produced by subitizing and counting. If guidelines are provided, matching will depend on the display type. Small numbetproblems may thus be judged right because judgments based on estimating are outvoted rather than because they are recognized to be less accurate. In studies using large number displays. children usually have only one method other than estimating: counting here. matching in Cowan ( 1984. in press). The results in both situations attest to the limited nature of children’s learning. Children do not simply learn as a general principle to disregard information that conflicts with counting or matching. For matching, children most readily trust pairing over estimating when pairing indicates inequality and estimating indicates equality (unequal number/equal length). Next easiest is when pairing and estimating agree that the sets are unequal but disagree as to which is more (unequal number/unequal length). Finally. estimated inequality is ignored when pairing indicates equality (equal number/unequal length). The main difference bctwcen this order and that for counting is that children trust counting ovet estimating on equal number/unequal length displays before unequal number/unequal length displays. One explanation of the difference in order for counting and matching which fits the results is the difficulty children have in using unequal counts to determine which set has more. As Klahr (1984) and Fuson and Hall (1983) have recently pointed out. children do have to learn that same count number implies same numerosity. Furthermore they have to learn that different count numbers imply differences in numerosity. But they also have to Icarn to use unequal counts
344
RICHARD
COWAN
to determine relative number and this only becomes robust later. In this explanation developments in children’s counting knowledge have been identified as necessary for children to trust counting over estimating. An alternative explanation that emphasizes developments in estimating can be offered. Klahr and Wallace (1973) suggested that estimating develops from being unidimensional as a result of children trying to apply conservation rules to estimating products. If conservation rules have to bc acquired for estimating to develop and such development underlies the shift to counting consistent judgments, then the slowness of the shift is comprehensible. Several further predictions follow: conservation status rather than counting competence should predict judgment accuracy, simply reinforcing counting-based judgments should have little effect on judgments of conflict displays. and qualitative difference:, in estimating according to conservation status should be found. Finally, in discussing small number problems the view that subitizing combined with counting outweighed estimating was proposed. This could be tested. What happens when children can use both counting and matching on the same displays would also bc worth studying. REFERENCES
COUNTING
Klahr.
D. (Ed.
Klahr.
(1984).
Transition
1. Mechunisms D., & Wallace,
AND
NCIMBER
processes
of cognitive J. Ci. (1973).
in
345
JIIIXMENTS
quantitative
development.
dere/op,rtent. New The role of quantification
York:
In R.
Freeman. operators in the
J. Sternberg development
of conservation of quantity. C0g:niliL.r Psyholo~y, 4. 301-327. Klahr, D., & Wallace. J. G, ( 1976). Cop~itir~e developmrut: ,jt~ infi,r,ncrtir)lz-l)l.o(.(,.s.sil/!: ric,w,. Hillsdale, NJ: Erlbaum. Michie.
S. (1984a).
Number
P.syl~holog~. Michie.
S. (1984b).
Why
c!f Lkwhprnrr~/d Piaget.
J. (1952). G. B. (1977). 48, 151’-1520.
Saxe.
G. B. conservation.
Schonfeld. Early 912. Siegler. H. (Vol. Wilkinson,
B.,
preschoolers
Britid~
children.
are
reluctant
to count
Jo~rmrl
,,f’hlrcc,trril,/ftII
analysis
of notational
Developmental
relations
between
(1979).
Child
Dc~~~hprwtit,
Eggleston.
V.
1. S. (1986).
The
H.,
&
New York: C. (1984).
Genevan
Scott.
J. L.
notational
(19741.
and
Cattell-Horn rolution
counting
Number
Kegan
Paul.
~)r~,c,/op,~fc~rrr. and
number
development
conceptions aids.
M. (1981). The development P. Lipsitt (Eds.1, .4&111(~c.s Academic Children‘s
&
Child
in
Press. partial
RP~F.IV~D:
Ps~c/~~lo~~. 16. 78-64. B. J. ( 1970). Stcrfi.v/ic.tr/ prirzc,ip/e.s
of intelligence
n(,~,clcj/~r?lc’,rrrrl of numerical
itr child
knowledge
of the
compared:
~.~~c~lro/r~,~~.
22,
July
30.
1984:
REVISED:
January
;,I ~,.rl’r~i,,rc,,fttrl I?.
I987
X4-
understandings.
~lc~~~c~lopmet~t trncl hrlrrrric~~ cognitive
skill
of counting.
CoRnitir,cT Winer.
young
6, 357-379.
of numerical
R. S.. & Robinson. W. Reese & L.
Koutledge counting.
.I~uttxu/
50, 1X0-187.
Pyvcho/og~.
implementation
~riri.tlt
spontaneously,
London:
A developmental
Coynitirc
16). A.
in preschool
Psxddo~~. 2, 347-358. chi/d’s (.tm(,epfkm c$n~rnhr,r.
Tllr
Saxe.
Schaeffer. children.
understanding
54, 245-1.53.
dc.tixr~.
London:
McGraw-Hill.
In
OF tXPERIMENTAI.
CHILD
J’SYCHOLOGY
43,
32X-34.5
(
1987)
When Do Children Trust Counting as a Basis for Relative Number Judgments?
There inconsistent
are
to single know studied or the not
several explanations with their counting.
set\.
They
which number in conditions experimenter
ignore
count
may
of why Their
forget
the
children counting
results
make may
of counting
relative number judgment\ be unreliable or restricted when
is bigger. In four experiments children’\ which controlled for the&e explanations. counted made no difference. which implies
information
hecau\e
competent counters less than when conflicting length cues
they
mistrusted
their
6 years old judged conGhtentlq were present. Trusting counting
they
judge
or
not
judgment\ welt Whether the child that children did counting.
Very
few
with their counting over length to tell
whether two rows were the same preceded trusting counting to decide which row had more. The results are discussed in relation to children’~judgment\ uhen using matching and D. Klahr and J. 6. Wallace’\ t lY73). C‘o~!~iti\~c, P.tvc~l~ol~>,~v, 4. 301-327)
account
of number
development.
I’JX? Zc.idemK
Pre*\. IllL
Piaget (1952) reported children counting right but judging relative number wrong. He questioned the importance of counting for children’s number development. Recently counting has been much analyzed and children’s mastery of the procedure and recognition of principles implicit in it studied (Briars & Siegler, 1984; Fuson & Hall, 1983: Gelman & Gallistel. 1978; Gelman & Meek. 1983; Greeno, Riley, & Gelman, 1984; Wilkinson. 1984). Why children who know how to count do not use counting information in making relative number judgments has not been answered. There are several possible causes of counting inconsistent judgments. These are discussed, and then previous studies of the influence of counting on relative number judgment are reviewed. The first cause is lack of counting proficiency. Gelman and Gallistel (1978) distinguished three how-to-count principles: one-one. stable-order, The gratefully thanks Requests Institute
author
thank3
acknowledges Fitz Taylor for for reprints of Education.
the
htaff
and
children
in participating
school\.
In particular
the cooperation of Ms Rennell and MS Clery. The discussing the work and the editor and referee? for their should be sent to Richard Cowan. CDEP. University 34-27 Wohurn Square. London WCIH OAA. England.
the
authol
author alho commenta. of London
COUNTINGANDNUMBERJUDGMENTS
329
and cardinal. A child who does not know the cardinal principle, that the final number name uttered gives the cardinal-number value of the collection, cannot be expected to succeed in using counting to compare two collections. Gelman and Meek’s (1983) findings suggest this is an unlikely cause for 3-year-olds. The second cause is ignorance of how to adapt counting to determine cardinality of two or more simultaneously presented collections. Saxe (1977) found most 3-year-olds simply counted on when asked to compare the numbers of pigs and horses. Counting in the experience of preschoolers may be predominantly, if not exclusively, carried out with reference to a single collection of objects (Durkin. Shire, Riem, Crowther, & Rutter. 1986). Third, ignorance of relative magnitude. in unequal number comparisons, may be responsible. Schaeffer. Eggleston, and Scott (1974) and Siegler and Robinson (1982) found marked improvements in performance of digit magnitude comparisons between 3 and 6 years. However, in neither study did the numbers result from counting or refer to objects the child had seen. This may have made the task a “disembedded” one (Donaldson, 1978). Hughes (1981) found preschoolers’ performance of addition and subtraction problems improved considerably when the tasks were embedded by including reference to objects that had been shown. So children may judge that 6 cars is more than 5 cars before judging 6 to be more than 5. Furthermore. neither Schaeffer et al. (1974) nor Siegler and Robinson (1982) explicitly related knowledge of digit magnitudes to counting performance. A fourth explanation is memory failure. When asked to judge, children may have forgotten the number of items in one or both collections. All the above explanations of counting right and judging wrong are consistent with a view of number development which assigns a major role to counting. The first two specify developments in children’s counting that may be needed before it can improve their relative number judgments. The third points to knowledge implicit in the conventional count list, and the fourth explanation, memory failure. may result from either deficient knowledge of how to adapt counting or performance problems. In the former case children do not know that the result of counting the collections should be stored. in the latter they know it should but the interference caused by counting the second collection brings about the loss of the result of counting the first collection. In some of the previous judgment studies children have been found to err when none of the above causes seems likely (Fuson, Secada, & Hall, 1983; Michie, 1984a, 1984b; Saxe, 1977. 1979: Schonfield, 1986). Michie (1984a. 1984b) and Schonfeld (1986) only studied children who had passed a counting pretest, making lack of counting proficiency an unlikely explanation. However, Schonfeld (1986) does not report the
330
RICHARD
COWAN
details of his counting test, and it appears that Michie (1984b) did not test children’s counting up to the highest number, 8, used in the judgment task. Fuson et al. (1983) and Michie (1984a. 1984b) helped children count and reminded them of the numbers of items in the two collections if necessary immediately prior to asking them to judge. Memory failure and inability to adapt counting are thereby ruled out. Michie (1984a) asked children to judge rows of buttons on cards and collections of buttons counted into boxes. Most 4-year-olds judged right when the buttons went into boxes. By far the most errors were made on unequal number trials when the buttons were in rows on cards arranged so that length provided a conflicting cue to number. If ignorance of number magnitude was responsible then performance should have been similarly poor on unequal number trials in the boxes conditions. It was not. The influence of length cues on children’s number judgments has been frequently demonstrated. Brainerd ( 1979) studied children’s judgments of displays such as those illustrated in Fig. I. Children judged unequal number/equal length displays right more often than unequal number/ unequal length displays. Least often judged right were equal number/ unequal length displays. Indeed only IO children out of 200 from kindergarten through third grade judged all the displays right. Cowan (in press) found the same order of difficulty with display type. Adding guidelines to the displays, however, dramatically improved performance of 7-yearolds. In contrast, _F-year-olds showed some improvement when guideline:, were added but still typically judged wrong. Whether there are age-related differences in judgments after counting of these display types is one concern of the studies reported below. A second is whether the order of difficulty of display types can be replicated. Finally, to test whether children’s failure to use count information in judging is due to perceived unreliability of counting or lack of numerical understanding, judgments following the child’s counting are compared with judgments following the experimenter’s counting. Saxc ( 1977) found children who purposely miscounted to make their counting fit their lengthbased number judgment. The effects of greater confidence in counting when it is done by an adult may bc general improvements in judgment accuracy or reduced differences according to number version. Michie (1984b) reported performance on small (3 v\ 3. 3 vs 4) number versions was much more accurate than that on larger number versions (6 vs 6. 6 vs 8). Whether this reflected children’s greater confidence in counting smaller numbers or the availability of subitiLing could not be determined. If the effect of numerosity disappears when an adult counts, this would be strong evidence that it is confidence in counting that is responsible for it.
COUNTING Unequal
AND
number
Equal
NUMBER Unequal
length
Equal Unequal
number length
. .
. . . . . . . FIG.
number
Unequal length
.
331
JUDGMENTS
. . . . . . .
. . . . .
I.
Display
types
used by Brainerd
EXPERIMENT
(1973).
1
In previous studies either children’s counting accuracy was not assessed (Fuson et al. 1983) or judgment accuracy following accurate counting was not reported (Saxe 1977, 1979) or how children’s counting was assessed was not reported (Michie 1984a. 1984b; Schonfeld, 1986). Whether children make wrong judgments after counting right is not well established. In this experiment accurate counters from 3 to 5 years old are first identified. and then their judgments of relative number studied. When children are discouraged from counting they judge unequal number/equal length displays, shown in Fig. 1, right more often than unequal number/unequal length displays, and equal number/unequal length displays are least frequently judged right (Brainerd. 1973. 1979: Cowan. in press). Whether such an order of difficulty can be found for judgments following counting is tested. If children ignore count information because conflicting perceptual ones make them doubt their counting, then judgments will be more counting consistent following counts by a more competent counter, i.e.. an adult. If the superior accuracy ofjudgments of smaller number displays observed by Michie ( 1984b) reflects greater confidence in counting, then there may be less effect of numerosity following adult counts. Displays in which relative length does not conflict with relative number are included to check whether children’s errors on the length conflict displays, illustrated in Fig. I, are attributable to length-based responding. Method
There were 65 children (29 girls, 36 boys) who agreed to take part in the first phase. They all attended the same nursery school situated in a poor area of inner London. Their ages ranged from 3 to 5 years CT =
332
RICHARD
COWAN
4 years 1 month, SD = 6.4 months). The children were from a variety of ethnic groups but predominantly spoke English as their only language. Based on their performance in the first phase. 34 children. IO of whom were girls. were selected to take part in the second phase. They ranged in age from 3 years 2 months to 5 years (?? = 4 years 6 months, SD = 5.1 months). Matrrids
For the first phase there were 24 red cards consisting of 4 cards with single rows of each of the following numbers of dots on them: 3. 4. 8. 9, 15, and 16. The 4 cards of each numerosity differed with respect to the color of dots-on 3 of the cards the dots were yellow and on the other 2 cards the dots were blue. and in overall length of row. For the rows of 3 and 4 dots the two row lengths used were 54 and 102 mm. For the rows of 8 and 9 dots the two row lengths were 131 and 252 mm. For the rows of 15 and 16 dots the two row lengths were 737 and 466 mm. Within a row the dots were equally spaced. The dots were 7.5 mm in diameter. For the second phase there were 1X red cards with displays of rows of blue and yellow dots. There were 3 cards of each of the following six combinations of rows: a row of 3 yellow and a row of 3 blue dots. a row of 3 yellow and a row of 4 blue dots, a row of 4 yellow and a row of 3 blue dots. a row of 8 yellow and a row of 8 blue dots, a row of 8 yellow and a row of 9 blue dots, a row of 9 yellow and a row of X blue dots. The three displays for each row combination consisted of one in which the two rows were the same in length, one in which the row of yellow dots was longer, and one in which the row of blue dots was longer. The lengths of the rows in the equal length displays and the lengths of the shorter rows in Ihe other displays were the same as the shorter row lengths for that numerosity in the first-phase cards. The lengths of the longer rows were the same as the longer row lengths for that numerosity in the first-phase cards. The rows were 25 mm apart, from dot center to dot center, and the midpoints of the two rows were directly opposite each other.
First phuse. Children were tested individually but allowed to bring a companion of their own choice for reassurance. The room in which testing took place was familiar to the majority of children as a quiet place where they were read stories. Unaccustomed as the children were to both formal testing and the experimenter, efforts were made to put them at their ease, and they were allowed to interrupt the testing for a story break, or to abandon testing completely if they wished. The eight cards with rows of 3 and 4 dots were shuffled and presented
COUNTING
AND
NUMBER
JUDGMENTS
333
one by one with the child being asked to count the dots. If the child failed to count one or more of the cards correctly the session was terminated. Having correctly counted the first eight cards the child was then presented with the sets of cards with 8 or 9 dots on them. If a child miscounted a row the experimenter asked the child to count them again, carefully this time, with the experimenter moving his finger from dot to dot as the child counted. If the child made another error the session was terminated. If successful, with or without the assistance described above, the child then received the sets of cards with 15 and 16 dots on them. Again, if a child made an error assistance was provided. Notes were taken of idiosyncratic count lists. Second phnsc~. Children were tested alone in the same room that was used in the first phase. A conversation took place in which children agreed that they had the same number of arms as the experimenter, that they had the same number of fingers as the experimenter, and that they had a bigger number of fingers than legs. Displays of rows of 3 and 3 dots were shown and the children agreed that there were the same number of blue dots as yellow dots. both 2, despite the perceptible differences in length, and that a row of 3 dots had a bigger number of dots than rows of 2 dots that were the same length or longer. Following this conversation the main task was introduced. If the child had been assigned to the self-count group, the experimenter explained that he would be showing the child cards with rows of blue and yellow dots on them. First he would ask the child to count the blue dots and say how many there were. Then he would ask the child to count the yellow dots. Following this he would remind the child how many blue dots there were and how many yellow dots. Finally he would ask the child whether there were the same number of blue dots as yellow dots. If the child said there were not the same number, the experimenter would ask of which there were a bigger number. Then the 18 cards with displays of two rows of dots were shuffled and presented one by one to be counted and judged. If a child miscounted a row the experimenter requested a recount. Recounts were rare and invariably correct. Children in the experimenter count group were told that the experimenter would do the counting for them and that he was very good at counting. The procedure was otherwise identical to that for the self-count group. The experimenter did not miscount. The cards were presented so that the rows were vertical to the child’s line of vision. This was done to be comparable with previous studies (Cowan, 1984. in press) and to preclude the tendency of young children to choose the closer of two rows as having more.
334
RICHARD
COWAN
TABLE DISTRIBUTION
OF CHILDREN
I
ACCORDING TO FIRST-PHASE COUNTING IN EXPERIMENTS I. 1. ANI> 4
Highest
number
cards
PERFORMANCF
consistently
AND
counted
AN-
right
None item Experiment 3-year-olds 4-year-olds Experiment 5-year-olds Experiment h-year-olds
Level
(I
1 i
3
I
0
5
5
II
0
5
2
I
6
I4
c)
II
0
I
2
h
41
2 4
” With assistance. h Without assistance.
Results Children were classified according to the quality and consistency of their counting performance into four levels and two sublevels. To be considered as counting right on a particular trial the child had to both use the conventional English number words in order with no omissions and give the final number name uttered as the cardinal value of the set. Children were classed as Level 0 if they did not count right on every 3- and 4-item trial. Children were classed as Level I if they counted wrong on two or more of the 8- and 9-item trials. Children were classed as Level ?a if they needed assistance on the 8- and 9-item trials and counted wrong on two or more of the IS- and 16-item trials. They were classed as Level 2b if they needed no help on the X- and 9-item trials and counted wrong on two or more of the I_‘;- and Ih-item trials. Level 3a children were those who succeeded in counting right seven trials involving I5 and I6 items when assistance was provided. Level 3b children counted right on all the 15 and l6-item trials without assistance. The distribution of children across levels of counting according to age is shown in Table I. The association of counting level with age apparent in Table I was confirmed by a Spearman rank correlation test (r, -=z .5-t. p < .OOl) The children who took part in the second phase wcrc drawn from Levels 2 ’dnd 3 as follows: 6 from Level ?a. 10 from Level 2h. and all 8 Level 3 children.
COUNTING
AND
NUMBER
TABLE
MEAN
SCORES IN EXPERIMENTS
335
JUDGMENTS
2
1 AND 2 AS A FUNCTION AND NUMEROSITY
OF DISPI.AY TYPE, Suerr:cr
Display
GROUP.
type Equal
Unequal number/
Unequal number/
Unequal number/
number/
equal
unequal length
consistent length
unequal length
Item
length
Expt. I 3 vs 4. 3 vs 3 Self-count
I .8 (0.4)”
1.6 (0.X)
7.0
(0)
1.7 (0.X)
I.4
(0.9)
1.3 (1.0)
I.8
(0.6)
I .9 (0.3)
0.8
(0.9)
0.3
(0.5)
I .o (0.9)
0.7
l0.X)
1.6 (0.5) I.2 (0.9)
I.? (0.8) I .6 (0.7)
I .9 (0.3) I.7 (0.7)
3.0
(0)
2.0
(01
2.0 I.9
0.7
(0.9)
0.4
(0.7)
7.0 2.0
(0) (0,
Experimenter 8 vs 9. 8 vs x Self-count
count
Experimenter
count
count
I .9 (0.3) I .9 (0.3) I.5
Expt. 2 3 vs 4. 3 vs 3 Self-count Experimenter x vs 9, x vs 8 Self-count Experimenter
” Numbers
count
in parentheses
I.4 2. are
(0.7) (0.X)
standard
(0) (0.3)
I .4 (0.8) I.3
(0.X)
deviations.
For each right judgment the child was awarded one point. Judgments of equal number/consistent length displays were near ceiling level: 45 out of 48 were right. For the equal number/unequal length displays and the displays with unequal number and equal. unequal, or consistent length. scores were derived from combining the data from the different color versions. The means for these displays are presented in Table 2. A three-way analysis of variance with one between-subjects factor, counting agent, and two within-subjects factors numerosity and display type. was conducted. The only significant effects were main effects for numerosity, RI, 22) = 54.21, p < .OOl, and display type, Et3, 66) = 10.78, p < .OOl. The overall means for display types were as follows: 2.24 for unequal number/equal length, I .88 for unequal number/unequal length, 3.29 for unequal number/consistent length, and 3.35 for equal number/unequal length. Newman-Keuls tests confirmed all differences between means were significant, p < .Ol. except that between unequal number/consistent length and equal number/unequal length. The above analysis showed equal number/unequal length and unequal number/consistent length displays were best judged. However, in that
336
RICHARD
COWAN
analysis a child had to both say the rows were different in number and identify the more numerous row to be considered to judge an unequal number display right. Children’s errors on unequal number displays may be solely due to wrongly identifying the more numerous row. If that is true then no difference according to display type would be found when children’s judgments are analyzed according to whether displays were correctly judged to be the same or different in number. Children’s unequal number display judgments were recoded accordingly. Scores for equal number/unequal length displays were unchanged. A three-way analysis of variance was conducted with counting agent as a between-subjects factor and numerosity and display type as within-subjects factors. There were significant main effects for numerosity. F( I, 21) = 37.06. p --: .OOl, and display type, F(3, 66) = 1.84, p < .OS. The overall means for display types were as follows: 2.92 for unequal number/equal length. 3.63 fat unequal number/unequal length. 3~54 for unequal number/consistent length. and 3.25 for equal number/unequal length. There were no significant differences between means according to Newman-Keuls tests. When rows are the same length and judged to be the same number the judgment is length consistent. When rows differ in length and the longer row is judged to be more numerous. the judgment is length consistent. Children’s judgments on all displays were recoded accordingly to length consistency. and a two-way analysis of variance was conducted with counting agent as a between-subjects factor and numerosity as a withinsubjects factor. This revealed a significant main effect of numerosity. F( 1, 22) = 10.88, p < .OOl, and a significant interaction between counting agent and numerosity. Ft 1. 23) = 14.99. p i .OOl. The mean numbers of length consistent judgments according to counting agent and numerosity were as follows: self-count small numbers 3.50. self-count large numbers 5.58, experimenter count small numbers 4.17, experimenter count large numbers 4.00. Tests on simple main effects, recommended by Winer (1970), showed that on small number displays there was no significant difference between the groups, E’(1 , 73) = 1.6. but the self-count group made more length consistent judgments than the experimenter count group on large number displays. Ft 1, 21) = 8.99, p < .Ol. Also, whereas the self-count group made more length consistent judgments on larger number displays, F(1, 22) = 25.70, p c: .OOl, there was no effect of numerosity on the numbers of length consistent judgments made by the experimenter count group. RI, 21) = 0.17. As the children varied considerably in -judgment accuracy, age. and counting level, tests of association were carried out. According to Spearman rank correlation tests there were no significant associations between any of these variables: age vs judgment accuracy, I’, = .77. counting level vs judgment accuracy. Y, = .05. age vs counting level. r, = .35.
COUNTINGANDNUMBERJUDGMENTS
EXPERIMENT
337
2
In Experiment 1 accurate counters between 3 and 5 years old made wrong judgments after counting right. More errors were made on larger number displays than on smaller number displays and on length conflict displays than on length consistent displays. Within the range of counting performance studied there was no association between judgment accuracy and age or counting level. There were two unexpected findings. In contrast to previous studies children judged equal number displays better than unequal number displays when conflicting perceptual cues were present. Children’s difficulties with unequal number displays were mainly in determining the more numerous row. Even when this was taken into account, there was no suggestion that children were worse at judging equal number displays. Second, although who counted did not affect accuracy, it did affect frequency of length consistent judgments. When children counted themselves, they made more length consistent judgments on larger number displays. Whether these findings can be replicated with 5-year-olds with the same levels of counting performance is studied in Experiment 2. Method Subjects There were 44 children. 17 of whom were girls, who took part in the first phase. They all attended the same infant school situated in the same area as the school in Experiment 1. Their ages ranged from 5 years 1 month to 5 years I1 months (x = 5 years 7 months, SD = 3.2 months). Based on their performance in the first phase, 24 children, of whom 6 were girls, were selected to take part in the second phase. They ranged in age from 5 years I month to 5 years 11 months (x = 5 years 7 months, SD = 2.7 months). Materiuls
und Procedure
The same materials were used as in Experiment I. The procedures for the first and second phases were the same as those followed in Experiment 1, except that testing took place in a library area rather than in a small room. Results The Table level, phase 3a.
distribution of children according to counting I. There was a significant association between r, = .45, p < .Ol, The children selected to take comprised 5 from Level 2a, 11 from Level 2b,
level is shown in age and counting part in the second and 8 from Level
338
RICHARD COWAK
For each right judgment the child was awarded one point. No difference according to color version was observed so the data were combined. Means are presented in Table 3. No errors were made in judging the length consistent displays so these were not analyzed further. A threeway analysis of variance was conducted with counting agent as a betwecnsubjects factor and within-subject factors ofnumerosity and display type. unequal number/equal length. unequal number/unequal length. and equal number/unequal length. There were significant effects of numerosity. F( I, 23) = 54.46, p < .001, and display type, F(3. 44) = 19.62. [> <: .OOl. The interaction between numerosity and display type was also significant, E‘(?. 44) = 9.91. p < .OOl As can be seen in Table 2 there was little variation in scores on small number displays which were at ot near ceiling level. Newman-Keuls tests showed that performance on unequal number/unequal length displays was worse than on unequal number/equal length and equal number/unequal length displays (1’ c. .Ol), but the latter did not differ-. As for Experiment I, children.5 judgment5 of unequal number display\ were recoded according to whether the rows were judged to differ in number. For two reasons only unequal number/equal length display scores were analyzed further. First unequal number/equal length displays are the only unequal number type that have a conflicting length cue with respect to judging same or different in number. Second performance on unequal number/unequal length and unequal number/consistent length displays was errorless. A two-way analysis of variance was conducted with counting agent as a between-subjects factor and display type. unequal number/equal length and equal number/unequal length. ;IS a withinsubjects factor. The only significant effect was for display type. kI I. 22) = 3.78. p c .05. The means were 3.75 for unequal number/equal length and 3.33 for equal number/unequal length. Therefore children more often said that the rows in unequal number/equal length displays differed in number than that the rows in equal number/unequal length displays wcrc the same in number. Spearman rank correlation tests revealed no significant association between judgment accuracy and age. I’, ~7 .O9. or between -judgment accuracy and counting level. Y, = .?6. The latter does. however, approach significance. 11 c: . I. There was a marked association between counting level and age in the subset of S-year-olds who took part in the second phase, Y, = .62. p c .OI. In short, the results of the second phase of Experiment 2 are similar to those of Experiment I in that children made most wrong judgments when the displays were large number length conflict displays. in particular unequal number/unequal length displays. Also, who counted did not
COUNTING AND NUMBEK.IULXMENTS
339
and there was no association of judgment affect judgment accuracy. accuracy with age and no marked association with counting level. Unlike the children in Experiment 1. the j-year-olds in Experiment 7 did not judge equal number/unequal length displays right much more often than unequal number/equal length displays. and they did not make more length consistent judgments when they counted themselves. The explanation of the first difference is that Syear-olds rarely judged two rows to be the same in number ifcounting did not yield the same numbers. EXPERIMENT
3
Experiments I and 2 had rows of 8 or 9 dots as large number displays. and children were much more likely to judge them wrong than smaller number displays. Whether 5-year-old children make even more errors with even larger numbers still within their counting range is studied in Experiment 3. Method
Subjects were 12 children, 7 of whom were girls. They were drawn from the children who had participated in the first phase of Experiment 2 and consisted of I I Level ?b counters and I Level 3a counter. Their ages ranged from 5 years I month to 5 years I I months Cx = 5 years 8 months, SD = 3.4 months).
There were 18 red cards with displays of rows of blue and yellow dots. There were the 9 cards with rows of 3 and 4 dots as used in the second phases of Experiments I and _9 and 9 cards with rows of IS and I6 dots of rows and constructed on the same principles, i.e., same combinations same overall lengths as the corresponding number versions of the firstphase cards. The procedure was the same as for second-phase Experiment I. experimenter count group. Results For each right judgment the child scored one point. Data from the different color versions wet-c combined. There were no errors made in judging equal number, consistent length displays. In Table 3 the means for the other display types are presented. An analysis of variance was conducted with two within-subject factors. numerosity and display type. There were significant main effects of numerosity, Ft I. 1 I) = 27.96. 11 < .OOl. and display type. FC3. 3.3) = 5.81, p < .Ol. The interaction between display type and numerosity was also significant. F‘(3, 33) = 4.67. p i .Ol. The interaction results from the performance on all small
number displays being at or near ceiling level. A separate analysis ot scores on large number display types was therefore carried out. There was a significant effect of display type. E-(3, 33) = 5.65, \I i .Ol. NewmanKeuls tests showed unequal number/consistent length scores exceeded those on unequal number/equal length and unequal number/unequal length displays, I> < .Ol. and equal number/unequal length displays. 11 ,: .05. No other difference was significant. Children’s judgments of unequal number displays were recoded according to whether the rows were judged to differ in number. All judgments on small number displays and large number unequal number/unequal length and unequal number/consistent length displays were right. On large number unequal number/equal length displays the mean number of correct judgments was 1.33. As this is the same as the mean for large number equal number/unequal length displays no further analysis was conducted. There was no significant association between judgment accuracy and age, I’, = .I%. As all except one child was at the same level of counting no tests of association involving counting level were made. In this experiment, as before, children who could count judged wrong after the experimenter counted and made most of their errors on large number length conflict displays. There was less variation according to type of length conflict than in Experiments 1 and 2 and no evidence that displays with rows of 15 and I6 dots were significantly more likely to produce errors than those with rows of 8 and 9 dots.
COUNTINGANDNUMBERJUDGMENTS EXPERIMENT
341
4
In Experiments 1-3 most children who could count made one or more judgment errors after counting. The children so far studied were less than 6 years old. The counting and judgment performance of 6-year-olds is studied in Experiment 4. Method
In the first phase 50 children. including 2.5 girls, took part. They attended the same school as the children in Experiments 2 and 3. Their ages ranged from 6 years to 6 years 11 months (x = 6 years 5 months, SD = 3.2 months). In the second phase there were I2 girls and 12 boys, all Level 3b counters, whose ages ranged from 6 years to 6 years I I months Cx = 6 years 6 months, SD = 2.9 months).
For the first phase the same cards and procedure wet-e used as in Experiments I and 2. For the second phase the cards were those used in Experiment 3 and the same procedures followed as in Experiment 2. Results
The distribution of children across counting levels is shown in Table I. There was no significant association between age and counting level. I’\ = .19.
Scores were derived as before. No errors were made in judging number equal consistent length displays or any other small number displays. Means for the large number displays are presented in Table 3. As only three children made any judgment errors the scores were not analyzed further. GENERAL
DISCUSSION
Experiments 1-3 showed children made counting inconsistent judgments under conditions which made the four explanations discussed in the introduction implausible. Because only children who adequately counted the numerosities involved were tested, the errors cannot be attributed to counting incompetence. Neither memory failure nor ignorance of how to adapt counting seem likely explanations given the procedure. Having the experimenter count was supposed to increase countingbased judgments, because children would treat the experimenter’s counting as reliable. No improvement in judgments was observed but in Experiment I children between 3 and 5 years old made less length-based judgments
342
RICHARD C‘OWAN
when the experimenter counted. This was not expected. Although who counted did not affect accuracy in Experiments I, 2, and 4 it may if children are tested on numbers beyond their reliable counting range. In Experiment I equal number/unequal length displays were judged right more often than unequal number/equal length displays, which in turn were better judged than unequal number/unequal length displays. In Experiment 3 unequal number/unequal length displays were again least successfully judged but performance on unequal number/equal length and equal number/unequal length displays did not differ. It would be wrong to infer that children’s knowledge that same count number implies cardinal equivalence is more robust than their knowledge that different count numbers imply nonequivalence. This is because the difference between equal number/unequal length and unequal number/equal length displays disappeared in Experiment I and was revcrscd in Experiment 7 when children’s judgments were simply analyzed according to whether they correctly judged displays to bc different in number. What ma& unequal number displays difficult way using counting to identify the more numerous row. in particular when length contradicted counting, as in unequal number,/unequal length displays. In both Ilxpcrimcnts I and 2 unequal number/unequal length displays were Icss successfully .iutlged than unequal number/equal length display\. Distinguishing children’s USC of the relation between counts and cquibalcncc from their use of count information itbout relative magnitude seems important for understanding the major changes in children’s use of counting. Only 5 children in Experiments I-? judged all the display3 right when correct judgment of relative magnitude was required. Howcvcr. when judging unequal numbers displa\;h to differ is Ihe criterion fog correctness. the number of succedul children incrca\cs to 26. 6 in 3. Experiment I. I2 in Experiment 2. and X in Experiment Using length conflict displays makes rhc text of children‘s u\c’ of count\ demanding. and it i5 probable that in Gtuationa bvhere length neither conflicts with nor supports count information children may make more counting consistent judgmcnls. Using Icngth conflict di
COUNTINGANDNUMBERJUDGMENTS
343
is held to develop concurrently, but, in its early forms. it is the source of much inconsistency because it is based on unidimensional global perceptual attributes of sets such as length. To the three quantification operators discussed by Klahr and Wallace (1973, 1976) and Klahr (1984) could be added cardination or matching (Brainerd. 1973, 1979; Cowan. 1984, in press: Fuson et al., 1983). This is the process of determining relative number by pairing off each element in one set with one and only one element in another. Brainerd (1973, 1979) found matching rarely influenced the relative number judgments of children lower than third grade. This may be because young children need perceptual supports such as pieces of string or guidelines drawn between the elements (Fuson et al. 1983. Cowan, 1984, in press). Matching is unlike the other quantification operators, as it does not generate quantitative symbols for each set but directly determines relative magnitude. Unlike estimation it is concerned exclusively with discontinuous quantity. The developmental course of matching is not well-known except that it is present in 5ycarolds and trusted over length by 7-year-olds (Cowan, 1984. in press). Young children therefore have four methods of solving small number relative number judgment problems. Once number names are attached to the subitizing sublists, the same symbols for each set. and hence the same judgment, will be produced by subitizing and counting. If guidelines are provided, matching will depend on the display type. Small numbetproblems may thus be judged right because judgments based on estimating are outvoted rather than because they are recognized to be less accurate. In studies using large number displays. children usually have only one method other than estimating: counting here. matching in Cowan ( 1984. in press). The results in both situations attest to the limited nature of children’s learning. Children do not simply learn as a general principle to disregard information that conflicts with counting or matching. For matching, children most readily trust pairing over estimating when pairing indicates inequality and estimating indicates equality (unequal number/equal length). Next easiest is when pairing and estimating agree that the sets are unequal but disagree as to which is more (unequal number/unequal length). Finally. estimated inequality is ignored when pairing indicates equality (equal number/unequal length). The main difference bctwcen this order and that for counting is that children trust counting ovet estimating on equal number/unequal length displays before unequal number/unequal length displays. One explanation of the difference in order for counting and matching which fits the results is the difficulty children have in using unequal counts to determine which set has more. As Klahr (1984) and Fuson and Hall (1983) have recently pointed out. children do have to learn that same count number implies same numerosity. Furthermore they have to learn that different count numbers imply differences in numerosity. But they also have to Icarn to use unequal counts
344
RICHARD
COWAN
to determine relative number and this only becomes robust later. In this explanation developments in children’s counting knowledge have been identified as necessary for children to trust counting over estimating. An alternative explanation that emphasizes developments in estimating can be offered. Klahr and Wallace (1973) suggested that estimating develops from being unidimensional as a result of children trying to apply conservation rules to estimating products. If conservation rules have to bc acquired for estimating to develop and such development underlies the shift to counting consistent judgments, then the slowness of the shift is comprehensible. Several further predictions follow: conservation status rather than counting competence should predict judgment accuracy, simply reinforcing counting-based judgments should have little effect on judgments of conflict displays. and qualitative difference:, in estimating according to conservation status should be found. Finally, in discussing small number problems the view that subitizing combined with counting outweighed estimating was proposed. This could be tested. What happens when children can use both counting and matching on the same displays would also bc worth studying. REFERENCES
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