Dot counting by brain damaged subjects

BRAIN AND COGNITION 17, 116-137 (1991)

Dot Counting by Brain Damaged Subjects

XAVIER SERON,* GERARD DELOCHE,? ISABELLE FERRAND,~ JACQUES-ANDRE! CORNET,* MICHEL FREDERIX,$ AND TH$R~SE HIRSBRUNNER~

*Unite de Neuropsychologie Experimentale de I’Adulte (NECO), Voie du Roman Pays, 20, B-1348 Louvain-la-Neuve, Belgium; TSalle Racine, Service du Professor Pierrot-

Desseilligny, H6pital de la Salpetiere, 47 Bd de I’Hbpital, Paris Cedex 13, France; $Centre Neurologique William Lennox, Allee de Clerlande,

1340 Ottignies, Belgium; and BHdpital Psychiatrique Cantonal, CH-2018 Perreux, Switzerland

In this study, we examine how brain damaged adults (aphasics, right-brain lesioned subjects (RBD), and demented subjects) perform a basic education skill: determining the cardinality of different sets of objects (dots). The RBD subjects encountered more difficulty with the spatial correspondence components of the task (correct pointing to the dots), while the aphasics experienced more difficulty with the verbal components (the production of the correct number-word sequence). The deficit evidenced by the demented patients was less systematic. However, qualitative analyses of patients’ behavior suggested an organization that tended to minimize the impact of their cognitive deficits on the object-counting task, and an analysis of their counting indicates that the basic counting principles proposed by Gelman (1982) and Fuson (1988) may be preserved. 8 1~1 Academic

Press. Inc.

In this study, we analyze how brain damaged adults (aphasic subjects, right-brain lesioned subjects, and demented subjects) perform a basic skill: the establishment of cardinality of a set of simultaneously presented objects (dots) under normal viewing conditions and without time pressure. In such a task, it has been established that normals use different procedures depending on the size of the array. For small arrays, up to three or four elements, the numerosity could be established by means of subitizing; normals use a predictable set of procedures, that is, the rapid visual estimation of a number of items (Atkinson, Campbell, & Francis, 1976; Burgess & Barlow, 1983; Ginsburg, 1978; Kaufman, Lord, Reese, & Volkman, 1949; Mandler & Shebo, 1982). For large arrays counting

Address reprint requests to X. Seron, UCL-NECO, 20 Voie du Roman Pays, Louvain- la-Neuve, B-1348 Belgium.

116

0278-2626191 $3.00 Copyright 8 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

DOT COUNTING 117

procedures are required. In children, counting is generally accomplished by means of a one-to-one correspondence between each object in the array and the number-word sequence. In adults, counting objects may be done with various and mixed strategies, visual or manual, each stimulus at a time or tbrough iterative countings of small subitized collections of two or three dots (Beckwith & Restle, 1966; Van Oeffelen & Vos, 1982).

Given its relevance for concepts of number and cardinality and its impact on the mastery of basic arithmetical skills (such as addition and subtrac- tion), the development of counting abilities has been extensively studied in children from psycholinguistic as well as conceptual perspectives (Fuson, 1982, 1988; Gelman & Gallistel, 1978; Gelman & Meek, 1983, 1986). Several authors have stressed that counting of objects requires, in addition to the emission of the number-word sequence, the successive and precise assignment of words to items. According to Gelman (1982) the counting of objects implies five different constitutive principles: (1) the one-to-one principle (every item in a set must be assigned a unique tag), (2) the stable number-word order principle (the number words must be emitted in the correct conventional sequence), (3) the cardinality principle (the last counted word refers to the numerosity of the set), (4) abstraction (counting applied to heterogeneous items), and (5) the object-irrelevant order principle (the order in which objects are counted does not change the cardinality of the set). Gelman (1986) has suggested that these five principles function as “the initial competence children bring to the task of acquiring skill at counting.” Nevertheless, there is still considerable debate among developmental psychologists (Briars & Siegler, 1984; Gel- man & Meek, 1986; Greeno, Riley, & Gelman, 1984; Wilkinson, 1984) about what constitutes the conceptual prerequisites for counting (conceptual competence) and what develops through progressive exercise adjustments (procedural and utilizational competencies). One significant aspect of this theoretical debate is that it has led several authors (see Fuson, 1988, in particular) to analyze in detail the spatial and temporal structure of the behavior that children produce in various counting tasks as well as the influence of physical (size, spatial arrangements) and contextual factors (time pressure) on children’s counting strategies. From these analyses it is thought that because words are organized temporally and objects are located spatially, Gelman’s one-to-one principle requires an inter- mediary act to connect spatial object arrangements with the temporal word string. This mediation is accomplished by a manual or visual pointing act. The one-to-one principle can be operationalized by two one-to-one correspondence rules (Fuson, 1988): one, intrinsic to the subject’s own activity, consisting of the temporal adjustment between the emission of the individual number word and the pointing act (the word-pointing correspondence); the other corresponding to a one-to-one spatial correspondence between pointing and the object location in the array (pointing-

118 SERON ET AL.

object correspondence). Counting thus requires double coordination: a temporal method between words and pointing, and a spatial method between pointing and the elements in the array.

Although it is one of the most basic numerical and arithmetical skills, the evaluation of numerosity of a set of objects by subitizing and/or counting procedures has not as yet been studied extensively in neuropsychology. There is, however, a neuropsychological analysis of oral and written (arabic and alphabetic) counting in three ratios (by ones, twos, and fives) by aphasics but in situations in which no entities were actually counted (Seron & Deloche, 1987). The results of the Seron and Deloche (1987) study indicated a strong influence of the notational system on distribution and number of errors, pointing to the specificity of how some procedures underly the production of number sequences. However, the counting ratios seemed less relevant for explaining variations in the performance of aphasics. The counting of objects or dots has only been examined in the context of lateral visual hemifield accuracy in dot enumeration with adults and with children of 5, 9, and 11 years of age (Kimura, 1966; McGlone and Davidson, 1973). Nevertheless, the complex structure of the processes involved in estimating the numerosity of an array of items points to the possibility that very different selective disorders could result in the breakdown of evaluating the number of concrete objects. In 1967, Warrington and James compared the performance of right- and left-hemisphere damaged subjects with that of control subjects on two tachistoscopic tasks: a number estimation task of three to seven dots and a number-letter detection task in three visual field conditions that were compared to a dot-counting task (arrays of grouped dots and dashes in linear and loop arrangements). On the whole, the performance on the dot-counting task was not related to lesion laterality, but, in the right- brain damaged group, there was a strong association between number estimation and counting. Warrington and James (1967) interpreted these right-hemisphere deficits as visuospatial agnosia in which the subject cannot integrate spatial information in a visuomotor task (counting of scat- tered items), but they offered no description of the visuospatial components of the task.

Given the lack of research on subitizing and counting strategies in brain damaged subjects, the present study is necessarily only exploratory. Never- theless, on the basis of what is known about temporal and functional aspects of the processes in object-counting tasks, one may predict that aphasic subjects with speech output errors may have difficulty producing the correct sequence of number words because of the inherent linguistic characteristics of aphasia. In this case, it would be important to determine if aphasic subjects are able to produce numbers orally while pointing to dots; it could be that, as observed in children, counting behavior itself is intact, whereas its coordination with the pointing activity is too demanding

DOT COUNTING 119

of available attentional resources. Right-brain damaged subjects, on the other hand, might have problems with the spatial aspects of the task, that is, the point-object correspondence rule and the object-irrelevant order principle. In regard to subitizing, the empirical data are so sparse in neuropsychology that we can propose no a priori hypothesis of what problems brain damaged subjects might encounter in this respect. Finally, comparison of the performance of aphasics and right-brain damaged subjects with that of a group of demented subjects is of interest since, in cases of aphasia and right-brain lesions, one may conservatively presup- pose that the conceptual principles underlying counting are not themselves disturbed but that, given verbal or spatial disorders, their actualization could be at fault. In cases of severe dementia, however, one may expect that the subjects would not be able to use basic verbal and pointing skills in a purposeful way even if they were partially preserved. This could indicate a true alteration of the conceptual principles underlying quantity estimation and may indicate a general deficit in action coordination.

METHODS

Subjects There were four groups of subjects in this study; 20 control subjects (11 males, mean

age: 50.9, range 39-77), 20 aphasic subjects (15 males, mean age: 52.6, range 26-75) 20 right-brain damaged subjects (15 males, mean age 46, range 18-66) and 7 demented subjects (3 males, mean age: 72.4, range 58-80).

For the aphasics, the etiology was vascular in 17 cases and tumoral in 3 cases. The aphasic types were established according to the classic criterion (Goodglass & Kaplan, 1972): six Broca’s, five Wernicke’s, three conduction, three anomia, two global, and one mixed.

Right-brain damage (RBD) was due to a cerebrovascular accident in the majority of the subjects (17) tumors in two cases, and trauma in one case. Eight subjects presented clinical signs of neglect, and six had visual-field defects.

The demented subjects were diagnosed as presenting Senile Alzheimer’s Disease on the basis of clinical and neuropsychological criteria (DSM-IIIR, 1987). All demented subjects presented severe memory deficits, some were disoriented and confused, and all were an- osognosic of their cognitive disorders with the entry symptoms of more than 1 year in duration. In four cases, the EEG was in the normal range; for two subjects, computerized tomography showed a moderate diffuse atrophy. The subjects were not clinically depressed and had no history of a cerebral vascular accident (CVA). Nevertheless, in the absence of complete discriminatory neuropsychological screening, we decided to consider this group as composed of demented subjects with no further specification of dementia type of severity.

Procedures

Stimuli. The material consisted of 12 white stimulus cards (15 x 10 cm) with a variant number of green dots (diameter: 9 mm) displayed at intervals of approximately 6 mm in curved lines of different forms (see Fig. 1). The 12 cards were presented, respectively, 3, 16, 30, 19, 25, 1, 14, 20, 7, 28, 17, and 12 dots, and were shown to each subject in that order. The cards with 1 and 3 dots permit subtizing on the whole stimulus pattern.

120 SERON ET AL.

The 16 dots stimuli card The 17 dots stimuli card

The 19 dots stimuli card The 25 dots stimuli card

The 30 dots stimuli card

FIG. 1. Examples of card stimuli.

The dot-counting task. The cards were presented to aphasic and right- brain damaged subjects in the hemifield ipsilateral to the brain lesion under a free vision condition. Cards were also presented free vision but centered in the visual field to the control and demented subjects. For each card the subject was asked to write down the number of dots in digit form. There were no time constraints. Each subject thus performed 12 dot-number quantifications. Subjects were not constrained in the way they performed the task (e.g., they were allowed to touch the cards if they wished), the only requirement being the production of the written digit response.

The ord counting task. The subjects were asked to orally count in series by ones, twos, and fives, with no objects or cards present. After specifying the sequence type, the examiner indicated orally the first number name (1, 2, or 5) and stopped the subject’s production at 30.

DOT COUNTING 121

TABLE 1 ERFORS OF CARDINALITY

Groups

Number Number of of

subjects responses

Number of

errors

Percentages of

errors

Controls 20 240 5 2% Aphasics 20 240 32 13% RBD 20 240 31 13% DS I 84 29 34%

Scoring Methods

The dot-counting task. All the counting sessions were videotaped. Any response that differed from the correct cardinality was scored as an error. We attempted to identify the counting method used by subjects on each item, irrespective of whether the result was correct or erroneous.

The oral counting task. Any event that departed from the correct conventional sequence was classified according to a scoring system that has already been presented in detail elsewhere (Seron & Deloche, 1987). The main categories of events recorded were local errors, which involved only one item but did not disrupt the entire sequence, and global errors, which changed the general pattern of the verbal sequence.

RESULTS

The Dot-Counting Task

Global analysis. As indicated in Table 1, the control subjects made a negligible number of cardinality errors (2%). The aphasics and RBD subjects, made 32 (13%) and 31 (317) o errors, respectively, while the demented subjects made 29 errors (34%). Post-hoc comparisons indicated that the controls made significantly fewer errors than any other group (Mann-Whitney tests; controls/aphasics: lJ = 110, p < .Ol; controls/RBD: U = 115, p < .025; controls/demented, U = 17.5, p < .OOl), that the error scores of the aphasics and the RBD subjects were not significantly different (U = 196) and that the demented subjects made significantly more errors than the aphasics (U = 41.7, p < .05) or the RBD subjects (U = 39, p < .05). There were subjects in the brain damaged group who made no errors; eight aphasics, eight RBD subjects, and one demented subject.

A positive correlation between the stimuli set size and the percentage of errors was observed in the RBD subjects (Spearman rank-order correlation (p = 0.89, p < .Ol), in the aphasics (p = 0.79, p < .Ol), and less markedly, in the demented subjects (p = 0.61, p < .05). The influence of the stimuli set size on subject performance is illustrated in Fig. 2.

122 SERON ET AL.

lOO-

90 -

60 -

70 -

60-

50 -

40 -

30 -

+-OoNTRoLs + APHASICS 4Fm

+ SDAT

0 2 4 6 6 1012141616202224262630

number of dots

FIG. 2. Proportion of correct responses as a function of problem size.

We also examined error magnitude, that is, the difference between the subjects’ responses and the cardinality of the sets. On the five errors made by the controls, the error differences were 2 1 or 2 in two cases each, and -3 in one case. Error differences of rt 1 represented 42, 47, and 48% of the total number of errors for the RBD, aphasic, and demented groups, respectively; the percentages rose to 81, 75, and 70% for error differences equal or less than 2 2 for the same groups. Concerning the direction of errors (underestimation versus overestimation), underestimation accounted for 55% in the RBD group, 44% in the aphasic group, 40% in the demented group, and three of the five errors made by the control subjects.

Error Analjwis

Scoring procedures. The errors and types of counting behavior were analyzed according to a system of analysis which differentiates a counting task into separate components, providing some hypotheses on the locus of the deficit.’

To locate the origin of the subjects’ difficulties, four behavioral categories were distinguished:

(1) The dot-referral category includes: -visual control (VC): Simply looking at the dot pattern.

’ The data have also been submitted to an analysis developed by Fuson (1988) with very young children which permits a distinction between time and spatial correspondence errors. However, such an analysis can only be performed on counting behavior with both explicit pointing and overt accompanying behavior. Only 53 of our 106 errors occurred in a context that allowed Fuson’s analysis. Of these 53 errors, 25 were correspondence errors, including 8 spatial errors by RBD patients and 8 errors by 1 demented subject.

DOT COUNTING 123

-manual control: pointing or marking behavior. -continuous manual pointing (CM): pointing to every dot one after

the other. -segmented manual pointing (SMx): pointing to dots located at regular

intervals of units (pointing to the second, fourth, sixth, etc. positions were scored SM2).

-position maintenance (PM): temporary maintenance of one finger on a dot while performing another spatial activity with other fingers.

-graphic marks (GMr): writing on or between dots (bars, digits, etc.). -try again (TA): stopping, counting resumes from the initial position

of another point, but only the response of the last trial is recorded. (2) The accompanying behavior category includes: -oral counting (OCx): production of a number-word sequence in units

of x. Oral counting was either continuous in counting by ones (OCl) or segmented in counting by steps (e.g., counting by twos: OC2).

-idiosyncrasies (Id): repetitive emission of the same sounds such as uh . . . uh . . . uh . . .

-finger counting (Fg): using the fingers to count dots or groups of dots.

-graphic marks (GMa): subject writes marks (such as bars) on the response sheet along with or after his pointing behavior.

-oral counting to a particular number (OPx): subject counts the dots to a certain number and then restarts from 1. At the end of counting, he adds or multiplies the times the counting was performed (e.g., for the item with 30 dots, the subject may count from 1 to 10 three times and then add together 3 times 10 or multiply 10 by 3; noted as OPlO).

(3) The calculation (C) category includes those instances when a subject computed (added or multiplied), for example, two subparts of dots already counted (e.g., “6 and 10, 16”).

(4) Transcoding (T) category includes when a subject produces a cardinality in another modality or notational system before writing the response in the digit code.

Behavioral Results

The dot-referral category. For the analysis of the subjects’ reference to the dot array, we disregarded arrays of one and three dots. In fact, both of these stimulus cards were evaluated by subitizing and were thus visually processed. Table 2 presents the dominant style of referral acts used by the subjects. A referral act was scored dominant for one subject if it was used on at least 7 stimuli sets out of 10 (70%).

The percentage of subjects using a predominant dot-referral act was constant across the control (85%), aphasic (SO%), and RBD subjects (85%), but only 57% of demented subjects used a dominant referral style. The most frequent behavior was manual pointing, while visual control

124 SERON ET AL.

TABLE 2 PERCENTAGE OF DOMINANT DOT REFERRING BEHAVIORS

Groups

Controls (n = 20) Aphasics (n = 20) RBD (n = 20) DS (n = 7)

Visual control

25% 10% 15% -

Continuous Segmented manual manual

20% 40% 50% 15% 60% 10%

42.9% 14.3%

Graphic marks Total

- 85% 5 80%

- 85% - 57%

(beyond the limits of subitizing) was infrequently used as a dominant referral behavior, Among the controls, more subjects used segmented manual control than continuous manual pointing while the reverse occurred in the brain damaged groups. However, some aphasics used segmented manual pointing by fives and by tens. When demented subjects presented a regular referral behavior, it was most often continuous manual pointing with little regular or well-organized referral behavior.

A comparison of the four groups on the number of stimuli processed using visual control evidenced no significant differences (Kruskal-Wallis, df = 3; H = 6.16). Moreover, no significant differences were observed when the groups were compared on the percentage of stimulus cards that were referred to by continuous manual pointing (Kruskal-Wallis, df = 3; H = 6.52, n.s.).

Another spatial aspect of manual control was considered in terms of the left or right location of the subject’s departure point. However, since the shapes of the curved lines varied (some had their two ends in the same left or right part of the stimulus cards), the analysis was limited to only five stimulus cards (stimulus size: 7, 14, 19, 20, and 30) which clearly presented different left and right extremities. For each group, the number of right versus left departures per subject was compared for each of the five stimuli. A Wilcoxon matched rank test for related samples indicated that the departure point was more frequently chosen on the left side by the controls and the aphasics (controls: n = 16, T = 12, p < .005; aphasics: IZ = 13, T = 8.5, p < .005), while there was no predominant departure point for the RBD subjects (n = 18, T = 48.5, n.s.) or demented subjects (n = 7, T = 5.5, n.s.). Indeed, six RBD subjects and one demented subject presented a clear tendency to begin their manual referral behavior at the right end. Comparison of the groups using the index of laterality as the dependant variable did not reveal any significant differences between the groups (Index = Left entries minus Right entries divided by Total entries; Kruskal-Wallis: df = 3; H = 4.5 n.s.).

The accompanying behavior category. A particular accompanying behavior was predominant for a subject when it appeared on at least 7 of

DOT COUNTING 125

TABLE 3 PERCENTAGE OF DOMINANT ACCOMPANYING BEHAVIORS

Groups Silent Oral

continuous Oral

segmented Graphic marks OPlO Total

Controls Aphasics RBD DS

75% 5% 15% - 95% 30% 35% 10% 5% 5% 85% 50% 35% 10% 95% - 43% 28.6% 71.6%

the 10 items. Here, too, the stimulus cards with one and three dots were excluded from the analysis. The percentages of subjects by groups presenting dominant accompanying behaviors are shown in Table 3.

If silence as an accompanying behavior is included, 95% of both controls and the RBD subjects, 85% of the aphasics, and 72% of the demented subjects presented a dominant accompanying behavior. Controls were more regularly silent than brain damaged subjects. When oral counting behavior occurred in brain damaged subjects, it was preferentially counting by ones whereas only one control showed oral counting as a dominant accompanying behavior.

Comparison of the groups on the number of silently processed items indicated significant differences (Kruskal-Wallis, df = 3, H = 15.5 p < .Ol). Two-by-two group comparisons indicated that the controls were significantly more silent than all the brain damaged groups (Mann-Whit- ney: controls/aphasics U = 85, p < .Ol; controls/RBD U = 137.5, p < .05; and controls/demented U = 10.5, p < .OOl). However, the RBD subjects were more silent than the demented subjects (U = 102, p < .025). The four subject groups were also compared according to the type of accompanying oral behavior, e.g., continuous or by steps. The comparison was made for each subject with the percentage of continuous counting calculated on the stimuli that provoked oral counting as the dependent variable. The group comparisons indicated a small but significant difference (Kruskal-Wallis: df = 3; H = 6.66, p < .05): the comparison of the groups two-by-two indicated only that the controls produced less continuous oral counting than did the RBD subjects (Mann-Whitney: U = 14.5, p < .025).

Calculation category. Calculation as a dominant category occurred in only five aphasics. Calculation occurred infrequently for the controls and demented subjects and never in the RBD subjects. In fact, calculation could not be identified in the controls at all when they performed the task visually and silently. Calculation could be detected only when the subjects stopped their count (and the related pointing act) before the end of the dot array and evaluated visually (perhaps by subitizing) the not

126 SERON ET AL.

TABLE 4 PERCENTAGE OF DOMINANT TRAN.WODING AND CALCULATION BEHAVIORS

Subjects Transcoding Calculation


20% - 55% 25% 55% - 57% -

yet counted dots and then explicitly added the number to the last number word of the sequence. Detection of calculation was thus dependent on the presence of manual pointing and explicit oral counting. Nevertheless, calculation was also inferred when some aphasic subjects used manual segmented pointing by fives that consisted of first counting three dots then two (or two, one and two dots) and taking a position maintenance at the fifth dot. However, given the small number of calculation behaviors, no statistical analyses can be made.

Trunscoding category. A group comparison of the number of items that elicited transcoding behavior indicated that the groups differed significantly (Kruskal-Wallis: H = 18.60, p < .OOl). Two-by-two group comparisons indicated that the controls showed less transcoding behavior than the three brain damaged groups (Mann-Whitney controls/aphasics: U = 88.5, p < .Ol; controls/RBD: U = 114.5, p < .Ol; controls/demented: U = 9.5, p < .OOl). The aphasics and the RBD subjects did not differ, while the demented subjects produced more transcoding behavior than did the aphasics (Mann-Whitney: U = 28, p < .Ol) and the RBD subjects (Mann-Whitney: U = 21, p < .Ol).

Most likely the brain damaged subjects presented more dominant transcoding behaviors than did the controls because their dominant accompanying behavior was accomplished by means of explicit oral counting (especially by ones), frequently speaking the last number word before writing it down (Table 4).

Association between Referral and Accompanying Behavior

Correct assignment of the cardinality value to a collection of objects requires correct association between the accompanying and the referral behavior. Any discrepancy of the progression ratios used in one of these two behavior categories, as for example, counting by twos but pointing by ones, should result in an erroneous cardinality estimation.

Comparisons of referral and accompanying behavior indicated globally no aberrant associations (see Table 5). Of the counting sequences produced by the controls, 79% were counted silently. The remaining sequences revealed quasi-perfect parallelism between counting and referral

DOT COUNTING 127

TABLE 5 PERCENTAGE OF C&WRRENCES BETWEEN REFERRING AND ACCOMPANYING BEHAVIORS

Groups Silent

counting

Counting by one’s Counting by X and continuous and segmented

pointing pointing by X Others

cooccurrences


79.4% 5% 15.5% - 34.5% 35.5% 7% 23% 52.5% 34.5% 9.5% 3.5% 7.14% 47.1% 27.1% 18.6%

ratios: in all groups continuous manual pointing was accompanied with counting by ones, and segmented manual pointing was accompanied by counting in ratios that paralleled the pointing segmentations. When the task was executed under visual referral control, the accompanying behavior was diverse in each group. The demented subjects, however, never used visual control.

Considered globally, these results indicated that the association principles between accompanying and referral behaviors were generally respected by the four groups.

Error Analysis According to the Taxonomic Behavioral Categories

The distribution of errors according to the four categories (see Table 6) varied according to the four groups.*

The control subjects produced one error in visual control, three omission errors, and one calculation error. The aphasic subjects produced mostly accompanying behavior (8.6%) and calculation errors (18.6%) in which the accompanying behavior errors were mainly local omissions of a word in oral counting by ones. The RBD group produced primarily dot-referral errors. These referral errors were of different types, e.g., omissions of one or several dots located at the left end of the curve, local omissions of one or several dots in continuous manual counting, or in segmented manual pointing between two dots and, finally, pointing to and counting a nonexisting dot. Those dot-referral errors occurred frequently at the change of curvature in a given line. Although less numerous, accompanying behavioral errors of the RBD subjects were evident: some consisted of stopping oral counting before the end of a complete continuous pointing when the subject was pointing to dots located at the left end of the array. Except for 1 single calculation error, the demented group produced an

* Note that the total number of errors according to our descriptive categories would be greater than the total number of erroneous cardinality responses. A subject may have produced several procedural errors on the same stimuli set which produced only one cardinality error.

TABL

E 6

DIS

TRIB

UTI

ON

OF

ERR

OR

S

Gro

ups

Ref

errin

g Ac

com

pany

ing

erro

rs

erro

rs

Cal

cula

tion

erro

rs

Tran

scod

ing

erro

rs

SYN

CR

O

erro

rs

?

Con

trols

(n

= 2

0)

3114

7 (2

.04%

) -

- - -

- l/2

40

(0.4

%)

Apha

sics

l/1

81

11/1

28

8143

21

140

2112

4 81

240

(n =

20)

(.5

5%)

(8.6

%)

(18.

6%)

(1.4

%)

(1.6

%)

(3.3

%)

RBD

19

1159

61

92

- 41

90

5/24

0 (n

= 2

0)

(11.

95%

) (6

.5%

) -

- (4

.4%

) (2

.1%

)

DS

11/6

8 7/

60

l/84

1317

4 7/

57

W34

(n

= 7

) (1

6.17

%)

(11.

7%)

(1.2

%)

(17.

6%)

(12.

3%)

(7.1

%)

DOT COUNTING 129

equal proportion of all error types. However, of the 45 errors made by the demented subjects, 20 were committed by the same subject. Their dot-referral errors were omissions, multiple points on the same dot, a point between dots, and a general disorganization of manual pointing by twos. The accompanying behavioral errors were local number-word omissions in counting by twos, perseverative counting from one stimulus card to the next, and changing from counting forward by twos to backward counting by ones. The transcoding errors for the demented subjects were either no-response or erroneous including transcoding into the alphabetic code, an analogic response (the subject writes down the dots instead of an Arabic number), and 1 error writing the oral counting sequence instead of just the last cardinal number (violation of the cardinality principle). Three demented subjects produced occasional synchronization errors between pointing and counting that consisted of counting in one set and pointing in another (local violation of the one-to-one principle).

Oral Counting Task

Since oral counting is one of the main components of an object-counting task, it was included to investigate whether the performance on oral counting sequences produced in a noncardinality context would predict subject performance.

The errors produced on the three oral sequences (by ones, twos, and fives) were pooled for the aphasics and the RBD subjects. The differences between the two groups were evident: aphasics produced a total of 94 errors whereas RBD subjects produced only 2 errors (Mann-Whitney: z = 5.99 p < .Ol). There were no significant differences by the group as a function of counting sequences (Friedman: x2 = 2.039 n.s.).

Only six subjects in the demented group were able to participate in the oral counting tasks. Five subjects counted correctly by three’s, three subjects counted by twos and two subjects by fives.

Comparison of Dot-Counting with Oral Counting

The RBD subjects made no errors in the oral counting task, and when they used oral counting in the dot-counting task they also made no errors.

For the aphasics, total error scores for the two tasks using Spearman rank correlations were nonsignificant (p = .08 n.s.). Such a lack of correlation is not surprising, given the presence of the many sources of error in the object-counting task (i.e., transcoding, pointing, nonverbal accompanying behaviors, etc.). A second correlation was run on a subgroup of aphasic subjects who presented oral counting as the dominant accompanying behavior taking into account only their counting errors in the object-counting task. This analysis also produced nonsignificant correlations (p = .19). Since the above correlations were based on the error scores from pooled oral counting tasks, a third correlation was computed

130 SERON ET AL.

on counting errors in the object-counting task and the counting errors made in the same set in the oral counting task. Nevertheless, the correlation still did not reach significance (p = - .19 n.s.).

On the basis of these results, it is reasonable to conclude that for the aphasic group, there is no direct relationship between the number of counting errors in dot-counting tasks and that in oral counting tasks. However, not all of the aphasic subjects used oral accompanying behavior in the dot-counting task. Therefore, a Mann-Whitney U test was computed to determine if the aphasics who counted the objects silently vs. the aphasics who used oral counting accompanying behavior differed in efficiency on the oral counting task. Results indicated that subjects who made the most errors on the oral counting task counted the objects silently (U = 17, p = .Ol). I n summary, when aphasic subjects counted orally in the two tasks, there was no correlation between counting scores (even when the analysis focused on the same counting sequences), but they generally committed fewer errors in the oral counting task (error range 0 to 2). Subjects who committed many errors in the oral counting task generally counted silently on the dot-counting task.

For the demented subjects, the correlation between the scores in oral counting and in the comparable oral sequences in dot-counting was also nonsignificant (p = .06). This was likely due to the demented group using counting by ones in the dot-counting task in which they made few errors.

DISCUSSION

In this study we have analyzed a basic numerical skill, the counting of objects (dots) by normal adults, aphasics, right-brain damaged, and demented subjects. Analysis on a group level demonstrated that all of the brain damaged subjects committed more cardinality errors than the controls but that there was considerable intersubject heterogeneity in performances within each group. As a whole, counting of dots does not seem to be dramatically altered in the aphasic and the RBD groups, the total number of errors being under 15% for both groups. For the demented group, the deficit is more marked, the error score being 34.5%. The magnitude of the individual errors was small, however, as the three groups (aphasics, RBD, and demented) made less than 30% of their errors at a distance of ?2 from the correct cardinality. Our first conclusion, therefore, is that counting of objects is a basic numerical skill that seems relatively resistant to cerebral damage.

Our analysis of four main activities (object referral, accompanying behaviors, transcoding, and calculation) revealed a few differences in strategies among the groups. The control subjects used mainly visual control or segmented manual pointing, proceeded silently, and rarely responded by transcoding oral to Arabic digit forms. The RBD, aphasic, and demented groups preferentially used continuous manual pointing and as-

DOT COUNTING 131

sociated oral overt counting. They also showed more explicit calculation and transcoding behavior than did the controls. The demented subjects transcoded more often than the aphasic and RBD subjects. Finally, the controls and the aphasics more often began their pointing activity from the left end of the arrays; the RBD and the demented subjects had no dominant entry points. A plausible interpretation of these results might be that there is a hierarchy of complexity in the behavioral organization adopted by the subjects which resembles normal performance during the acquisition of counting skills. Normally, in cases of visual inspection of an array without manually pointing to the material, the fixation and ocular saccades cannot take advantage of the external cues offered by finger pointing, that is, the progression in the array and the partition between counted and to-be-counted objects depend on the mentality “marked” objects. Such mental processing is attention consuming for immediate spatial memory (Fuson, 1988). By contrast, in continuous manual pointing, visual inspection is triggered by simple moves of the finger from one dot to the next, and each finger position on the array realizes and maintains the partition between the counted and the to-be-counted dots. Continuous manual pointing is probably a less attention demanding task since each point is named in the simplest and first-acquired number word sequence. For segmented pointing, the finger must move from one point to another not adjacent in the array. It is thus necessary to recall visually the departure point, to check (by subitizing) the intermediate points to be skipped, and to locate the arrival point. Such a strategy requires a memory algorithm different from those used in the above strategies (Fuson, 1988). This final strategy may also be more attention demanding since oral counting by twos, threes, or fives is less automated than counting by ones, and the segmentations require calculation (repetitive addition of segments or final multiplication of the segments). Possibly, some of the brain damaged subjects used the most secure and easiest method to perform the task since the three pathological groups preferentially used continuous finger pointing with counting by ones. This method is less demanding than other strategies, such as visual referral control or segmented pointing. Such externalization of counting procedures has been observed by Fuson (1988) with high school students when they were confronted with difficult counting tasks (large disorganized arrays). The demented subjects also used continuous manual pointing, but it should be recalled that three of these subjects presented no regular or well-organized referral behavior and they had severe difficulties in oral counting by twos and fives. The control subjects, for their part, distributed their strategies in a variety of ways; either they used visual inspection or counted by different ratios, having no particular difficulty in segmenting the array by twos, threes, or fours or in counting and calculating the segments.

It is difficult to identify the exact nature of the difficulties that the three

132 SERON ET AL.

brain damaged groups encountered in performing the tasks or to determine the reasons why a subject selected one behavioral strategy over another. For the RBD subjects, one possibility may be that they chose continuous pointing with counting by ones because of the inherent spatial difficulties in the pointing component of the task, whereas the aphasics might have used counting by ones because it is an overlearned verbal sequence and thus more easy for them to perform than counting in larger segments. In the former, it is the spatial pointing component that would determine the selection of the behavioral strategy, while in the latter it is the verbal component that might play the leading role.

Those results, of course, are based on group analyses and there was likely great intraindividual heterogeneity within each group. A detailed analysis of single case performance might highlight the nature of the difficulties and also make evident the strategies a given patient uses to handle a counting task. We briefly illustrate that point with the following case.

Individual Analysis

PM, a 79-year-old demented woman produced 10 cardinality errors on the dot-counting task. There were no dominant referral or accompanying behaviors evident since she did not establish a preferred way of counting or pointing on the stimuli array. Her counting behavior was examined in detail to determine the manner in which her particular strategies differed from the norm. We observed that when the stimulus array changed in orientation, her counting behavior was accompanied by a simultaneous change in pointing. In fact, when the dots were in line, PM used segmented manual pointing by twos or threes, but when confronted with a change in line orientation, she adopted a continuous counting procedure of one- by-one, returning to segmented pointing and counting only after the line orientation returned to a straight line. In this case, the presence in the same dot array of different counting and pointing procedures was not an indication of unstructured disturbed behavior but could rather be interpreted as an adaptive strategy which facilitated the discrimination between counted and to-be-counted dots in difficult spatial contexts.

LG, another demented patient, also showed a spatial adaptation to the task. She fragmented the curved arrays in simpler subparts of quasi-linear vertical (parts 1,3,4; see Fig. 3) or horizontal (part 2) arrays. This strategy allowed her to make an adjustment to changes in line orientation, but she had to be able to separate the dots already counted from those still to be counted. Such a requirement is especially critical at the borders of the subparts and could, as in LG’s case, produce errors by counting a dot again after it had already been considered as pertaining to two different subparts.

Parallel to these adaptations for variant spatial characteristics of dot

DOT COUNTING 133

(1) one a a fiveteen ( 15)

(2) two l l fourteen (14) l sixteen (16) (3) three

l a a thirteen (13) a seventeen ( 17)

(4) four

(5) five 0 a twclv,(12) l eighteen (18)

(6) six l l clcven (1 l* nineteen ( 19)

(7) seven 0 l 0 l twenty (20)

eiht nine ten (8) (9) (10)

FIG. 3. Counting and referring acts of patient LG on the 19-dot stimulus card.

arrays, patients are able to adjust their counting behavior according to their language difficulties: For example, GA, a 57-year-old Wernicke’s aphasic, presented a regular adaptation in counting and pointing ratios; he began to point and count the dots by threes and then revised and counted by ones. The patient’s revisions seemed independent of the spatial characteristics of the dot array but occurred at regular moments in the oral counting sequence probably due to the fact that counting by threes was sufficiently automated only at the beginning of the number-word string. Another strategy used by other aphasic subjects that may be the result of linguistic difficulties is a particular segmented counting consisting of segmentation by fives, e.g., adding subsegments of three and two (3 + 2) or of two, one, and two (2 + 1 + 2) dots. These subjects used calculation (adding or multiplying the subsegments) instead of oral counting. Such a strategy has the advantage of segmenting the dot lines into groups of dots within the perceptual limits of subitizing (Beckwith and Restle, 1966; Van Oeffelen and Vos, 1982, 1984). A strategy such as this may appear overly complicated but was in fact an adaptive strategy al- lowing more accuracy than the classic counting by ones, which would not have worked given the specific and severe verbal deficits in oral counting of these subjects.

CONCLUSION

The object-counting task we used may be considered a problem-solving situation for some of our brain damaged subjects; they were confronted with patterns of dots arranged in various orientations and asked to evaluate the cardinality of the arrays. There are several ways to determine cardinality: (1) dots could be referred to one after the other or in a succession of small segments; (2) the referral act itself could be manual, graphic, or

134 SERON ET AL.

visual; and (3) the accompanying behavior could be implicit (silent) or explicit (aloud) with counting or calculation. It was predicted that brain damaged subjects, given their specific cognitive deficits, would experience difficulties during particular components of the task. As we hypothesized, the RBD subjects demonstrated more difficulties with the spatial components of the task while the aphasics experienced more difficulties with the verbal components. However, these deficits appeared only as a function of the strategies adopted to complete the actual counting task. This study indicates that brain damaged subjects regularly presented a behavior organization that tended to minimize the impact of their cognitive deficits on the object-counting task. Brain damaged subjects use, more frequently than controls, the simplest, least resource consuming and the most prim- itive (from a developmental point of view) strategy: continuous dot pointing associated with oral counting by ones. Some aphasics, who were among the least efficient in oral counting, tended to count the dots in silence. Some demented subjects and aphasic subjects who presented apparently disorganized behavior initially used referral and counting strategies that bypassed their cognitive deficits. This may be interpreted as a successful adaptation according to the task requirements. We have briefly illustrated some of these adaptations in subjects as segmental changes at critical spatial points, fragmentation of a curved line into linear fragments, aban- donment of complex counting behavior when a limitation is reached in the verbal sequence, and a return to simple calculation procedures when complex counting is no longer efficient. These results indicate that the brain damaged subjects had a clear understanding of the task objective and they were at least partially able to structure the task into components as a function of the task requirements as well as their residual capabilities. This points to preservation of the basic principles of counting (Gelman, 1982).

First, the cardinality principle was always respected, that is, the brain damaged subjects understood that they had to evaluate the cardinality of a set of dots and had to produce (in the written modality) only one number representing the cardinality of the set. Even the demented subjects who were unable to write the correct digit responses tried in each case to write on paper only one number element corresponding to the oral answer. Furthermore, in no instance did brain damaged subjects produce written responses not belonging to the number lexicon.3

For most of the subjects, the one-to-one principle also seemed well preserved. There were, of course, some one-to-one violations but, in

3 The only violation of the cardinality principle was made by a demented subject who, instead of writing the last number word he had produced, wrote the entire word sequence he had produced in the digital code. This occurred on the three-dot card, for which the subject wrote down the digits corresponding to his oral counting.

DOT COUNTING 135

comparison with the total number of errors, they were not numerous and seemed reasonable in terms of performance variables, since dots were more often omitted at complex spatial places in the array or, in cases of neglect, in the contralateral field. One may thus reasonably hypothesize that, given the cognitive deficits, some situational factors render the actualization of the correspondence principles difficult for some subjects. These violations, since they were local and linked to situational variables, point to instrumental difficulties altering the correct counting performance rather than to an alteration of the knowledge of the correct principles.

The same reasoning applies to the word-order principle. Of course, some subjects made local errors in counting (omission or repetition of a word in the sequence) but none regularly uttered scrambled number-word sequences or, for example, verbal sequences that systematically decreased instead of increased. Here, too, the difficulties were not due to a lack of knowledge, but rather the existence of performances difficulties.

The two additive object-indifference and the order-indifference principles of Gelman (1982) were not directly testable given our material and in- structions. However, if the brain damaged subjects we studied were not affected in terms of their knowledge basis underlying counting, our results point to the importance of considering counting performances as a function of the different components of a task as sensorimotor, verbal, or numerical. It is important here to keep in mind the exploratory nature of this study, since results from each group are only indicative of that group of patients given the small size of our groups, the heterogeneity of patients, and the intrinsic complexity of the task of counting. At the group level, our study has provided a first sample of the kind of difficulties which may be observed in brain damaged patients. Future research focussing on single cases should be designed to probe more deeply into the relationship between the cognitive neuropsychological deficits and the different components of the counting task. It would also be interesting to analyze counting performance with an emphasis on visuoperceptive constraints, e.g., the proximity and number of dots, spatial arrangements (disorganized or especially difficult arrays such as a circular arrangement of dots), as well as the visual frequenty of items. It may be that these variables have an influence on the strategies used by normal adults in object-counting tasks (Atkinson et al., 1976, Van Oeffelen & Vos, 1982, 1984). It may then be possible to identify the functional parts of the processes that are impaired by cerebral lesions and to determine more precisely if the deficits are specific to the principles of counting or are of a more general nature, such as visual short-term memory difficulties in pattern identification. It might also be useful to place more constraints on the task so that the specific adjustments to different quantities can be evaluated. Finally, lon- gitudinal studies of counting by demented subjects promise to be important for improving our understanding of the decline of cognitive performance.

136 SERON ET AL.

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Dot counting by brain damaged subjects