Learning and utilization of conjunctive and disjunctive classification rules: A developmental study

JOURNAL

OF EXPERIMENTAL

Learning

and

Classification

CHILD

4, 217-231 (1966)

PSYCHOL433Y

Utilization

of Conjunctive

Rules:

A Developmental

Disjunctive Study1p2

L. KING

WILLIAM University

and

of

Colorado3

Children of 6, 9, and 12 years of age, as well as adults, learned both conjunctive and disjunctive classification rules under instructions denoting the relevant attributes in advance. In addition, they were asked to identify which of the two rules was employed in a subsequent problem. The conjunctive rule was more difficult than the disjunctive at every age. Moreover, subjects of every age showed transfer to subsequent problems employing the same rule. Although rule learning ability differed for children of different ages, rule identification did not. Sources of rule difficulty were tentatively identified.

In many studies of conceptual behavior S learns to classify a series of stimuli into two groups. In one (the positive class), all stimuli are examples of the concept, and in the other (the negative class), no stimuli are examples of the concept. Moreover, there is usually a principle enabling those who possess it to classify all stimuli correctly. These principles, or concepts, have been shown to consist of two components (Haygood and Bourne, 1965). One designates the specific stimulus characteristics, or relevant attributes, embodying the concept, and the other designates the type of the concept or rule. For the concept, red and/or square, the relevant attributes are redness and squareness, and t,he rule is an inclusive disjunction. A thorough discussion of the different types of bidimensional connectives or rules may be found in Haygood and Bourne’s report (1965). These investigators demonstrated that Ss knowing the rule in advance but not the relevant attributes (attribut’e identi‘This research was supported in part by Grant MH-08315-01, from the National Institute of Mental Health, U. S. Public Health Service, and in part by Grant APAfrom the National Research Council of Canada. ‘I would like to thank Lyle E. Bourne, Jr., for his invaluable assistance in planning and executing this research. I would also like to thank Mrs. Ruth Raimy of the Central School Administration of the Boulder Valley School District; Mr. T. N. Hovde, Principal, Crestview Elementary School in Boulder; Mr. M. V. Chase, Superintendent, St. Vrain School in Longmont; and Mr. J. 0. Pope, Principal, Central School in Longmont for making available the children needed for this study. ‘Now at Dalhousie University, Halifax, Nova Scotia. 217

fication) made mo~~c errors than & knowing the, rclcvant, attributes 111 advance, but not the rule (rule learning). Most significantly, they showed that Ss knowing neit’her of these component,s had a much more’ difficult task (complete learning) than those knowing one or the other component. The purpose of the present investigation was to explore the development of rule learning in hope of elucidating the underlying processes. Haygood and Bourne (1965) suggest that rule learning depends on two processes, encoding the stimuli into a small number of mediators and learning to assign these mediators to the appropriate response class. The mediators are presumed to correspond to the four possible combinations of the two relevant stimuli, i.e., both attributes present (PP) , one present and the other absent (PA or AP), and both attributes absent (AA). It is not asserted that encoding is complete before assignment begins. Two characteristics of their data suggest that these four at,tribute contingencies (PP, PA, AP, and AA) become the effective stimuli for S. First, treating each attribute contingency as a separat.e problem shows presolution error rate to be approximately at a chance (50%) level. Second, observing just one of each contingency enabled some experienced Ss to identify the rule being employed. This second finding suggests that some Ss had developed an implicit understanding that any bidimensional rule can be specified by the way t.he four attribute contingencies are assigned to the two response classes. A conjunctive rule assigns the PP contingency to the positive class and the remaining three to the negative class. In contrast, the inclusive disjunction assigns the AA contingency to t.he negative class and the remaining three to the positive class. The relative difficulty of these rules will be investigated in the present study. The disjunctive rule has bren found to be more difficult than the conjunctive in attribute identification t,asks (e.g., Bruner, Goodnow, and &4ust,in. 1956; Conant and Trabasso, 1964; ancl Haygoocl and Bourne, 1965), in rule-learning tasks (Haygood and Bourne, 1965). and in complete learning tasks (e.g., Hunt and Kreuter, 1961; Neisser and Weene, 1962; Wells, 1963; and Haygood and Bourne, 1965). Wells (1963) suggested relative rule difficulty might be due to familiarity and might be climinished for young children having little differential experience with the two rules. The present study was designed to explore this possibility. Another possibility is that disjunction is more difficult than conjunction because important, attribute contingencies are more difficult to encode under the former. According to this rationale, one would not expect a decrease in relative rule difficulty since the process of encoding the stimuli (into the four attribute contingencies) ought to increase in difficulty for both rules as age decreased.

LEARNING

CLASSIFICATION

RULES

219

The task of distinguishing which of two previously learned rules is being employed is called rule identification. This process, too, will be investigated. After learning both the conjunctive and disjunctive rules one needs to observe a single stimulus displaying only one relevant attribute (PA or AP) ; if it is positive, the rule must be a disjunction; if it is negative, a conjunction. Formally, t,his task is one of logical exclusion? a type of problem children do not solve until they are seven or eight (Burt, 1919; Piaget, 1953). This conclusion is based on verbally administered tests. It may not hold, however, for children (as young as six) receiving extensive training with the information t’hey will need to integrate. METHOD

Materials The stimuli were 81 $&inch thick, wooden blocks displaying every combination of four tertiary stimulus dimensions. The dimensions and their values were: color (red, blue, and green), size (small, medium, and large), shape (triangular, square, and octagonal), and number of white dots (one, two, and three). Twelve cardboard cards, each depicting (verbally and pictorially) one of the twelve attributes (levels) served as reminders of the relevant, attributes in the rule-learning problems. Subjects Groups of four boys and four girls were chosen from a kindergarten class (mean age of 6 years and 4 months), from a third grade (mean age of 9 years and 1 month), from a sixth grade (mean age 12 years). Four men and four women attending the University of Colorado also served as subjects. Somewhat different information was available for each group from the school records: for the kindergarten children the Nctropolitan Readiness Test,s (the mean percentile rank was 76). For the 9-year-olds the Lorge-Thorndike Intelligence Test (the mean IQ was 127) and the Metropolitan Achievement Tests (mean percentile rank was 87). For the 12-year-olds the California Test of Mental Maturity (the mean IQ was 118) and the Iowa Every-Pupil Test of Basic Skills (the mean percentile rank was 71). Procedure The sequence and purpose of each experimental task is described below. Stimulus description. In order to establish that S and E agreed regarding the descript,ion of the stimuli, every S described nine stimuli chosen to collectively exhibit every attribute three times. If S attended to

“irrelevant” rected him.

aspects

of t,he stimuli,

e.g., t,exturc

of the surface,

E cor-

Unidimension~al Attribute Identijkation. The purpose of this task was primarily to familiarize S with a sorting problem and, secondarily, to obtain information about attribute identification. The X was informed t’hat his task was to “tell” the positive blocks from the negative ones and that he could “tell” whether the block was positive or negative by looking at it. A single attribute (e.g., all red ones) designated the positive class. The words “good” and “bad” were used in place of positive and negative with the younger children to facilitate understanding. The X was then given a “hint” informing him which stimulus dimension was relevant to solution. The three attribute cards depicting the relevant dimension (e.g., red, blue, and green) were placed in view and X was told these cards were to remind him t,o attend to the relevant dimension (e.g., color). The E corrected S after each response and allowed him 45 trials during which to begin the criterion run of ten consecutive correct responses. If S committed errors after trial 45, he was told the correct answer and given ten trials to utilize this information; no S’s failed to do so within the 10 trials allowed. Rule learning. Half of the Xs received three conjunctive rule learning problems first and three disjunctive problems second, while the other half received the problems in reverse order. The instructions stressed that two attributes were important for sorting the stimuli into the two groups. These attributes were displayed on cards throughout the experiment. On each problem the subject was given two positive and two negative examples which remained in view throughout the problem. As in attribute identification, E placed each stimulus before S who categorized them at his own rate ; receiving feedback after each response. The first stimulus for any problem was always a positive instance. The criterion for solution was ten consecut,ive correct responses. Following solution, S was asked t.o explain how he dist’inguished t,he positive st,imuli from the negative ones. The stimulus sequences were arranged so that each of the four attribute contingencies (PP, PA, AP, and AA) appeared at least once every ten trials. Moreover, no more than four consecutive stimuli were all positive or negative, and an equal number of positive and negative stimuli appeared. Aside from these restrictions the stimulus sequences were generated from a table of random numbers. If S failed to begin his criterion run within 100 trials, E explained the solution to him and continued for an additional 50 trials. All Ss solved all problems within 150 trials. After the first three problems were completed, S was informed that the next problem would be somewhat different from the last one.

LEARNING

CLASSIFICATION

RULES

221

Rule identification. After the last rule learning problem S was asked to describe the two different rules. The S was encouraged to refer to the relevant attributes displayed, as in rule learning, when describing the two rules. If he could not explain the two rules, he was informally tutored and the problem was not begun until he could correctly distinguish them. The 5 was told to discover which rule was being employed. The E emphasized that S need only to respond as he had done in the past. Otherwise the procedure was identical to the other rule-learning problems. Half the Ss received a conjunctive problem and half a disjunctive problem. The experimental design. The major design for the rule learning problems was a 2 X 2 X 4 repeated measures factorial plan. The between-S variables were order of administration for the two rules (conjunction first, disjunction first), sex (male, female), and age (six, nine, twelve and college students). In addition, each cell was fractionated into two blocks, three conjunctive problems and three disjunctive problems; making rule difficulty a within S comparison. Two variables, the particular sequence of positive and negative instances, and the order of the relevant attribute pairs for each problem were deliberately confounded with sex. There were two sequences of randomly determined positive and negative instances; and these were counter-balanced so that the females received one order and the males the other. Likewise, there were two orders of the six pairs of relevant attributes (one for each problem), which were composed of all combinations of color and shape. The females received one order and males the reverse one. This confounding was deemed necessary because of the large number of cells which would otherwise have been generated had these variables been made orthogonal to all the others. RESULTS

and transfer in mLle le~arning. An analysis of variance was performed on a square root transformation of the data rather than the raw data because the distributions were skewed (Walker and Lev, 1953, pp. 463). The analysis is summarized in Table 1. Performance improved with increasing age up to, at least, the 12-yearold level, and disjunctive rules were more difficult than the conjunctive rules at every age level as shown in Fig. 1. Furthermore, all age groups showed transfer from earlier to later problems based on the same rule as shown in Fig. 2. The results, thus far examined, indicate that some children as young as 6 years of age can learn abstract conjunctive and disjunctive classification rules. This conclusion must be tempered, however, in light of the Rule dificulty

TABLE ~MMARY

OF AKALYSIS

1 FOR ERRORS"

OF VARIANCE

TX ~~UI.E LEARXING

[performed on (n + l)l’z] Variables

Mean square

Age (A) il

::

Il.

1632 .347!) :3 .062( ) I ,107n .0146 .OOYO .3595 .4800 ‘) 6493 s, 1250 I. 1149

Order of rules (0) Sex (S) x 0

AXS oxs AXOXS wgps Rules (R) Problems/Rules R X A RX0 P/R x A Residual

(P/R 1

I

:; :: 1 :< 16 1 4 3 1 12 1.39

1.1919 1 .4682 .7440

23 .2500 .7200 6.3700 2.4300

1.9050 .0166 .7480 12.9108 10.8713 1.4917 1.5948 1.9644 -

a A similar analysis performed on trials t,o criterion yielded subst,antially results.

<.Ol NS < .o.i NS NS NS NS < .Ol <.Ol NS NS NS the same

fact that four of the 6-year-olds did not solve a disjunction and one did not solve a conjunction within the 100 trials allowed. These Ss had to be verbally tutored by E. Nevertheless, they understood the solution once it was attained, as indicated by the fact t’hat they all showed very good transfer to following problems. By the third problem (of the type requiring tutoring), three of the five children made no errors, one made one error, and one made two errors. It might be noted that E tutored the v=Dlsjunctlve problems A-Conjunctive problems

6

9

12

A

AGE

FIG. 1. Mean iunction.

errors summed across all three problems

for conjunction

and dis-

LEARNING

CLAVSIFICATIUX

223

RlXES

child only in the specific solution, not a general one. The analysis of variance indicates that there is no significant change in relative rule difficulty at any age (the rule X age interaction was not significant). However, this may reflect the fact that the number of errors made by 6-year-olds on disjunctive problems was limited by the artifical ceiling imposed on the number of trials. While the number of “nonsolvers” suggests that, the disjunctive rule is disproportionately more difficult for the 6-year-olds, a final statement must await more extensive data. .--. o-----o ---e---e

1 DIS.J”r&“E3 PROBLEMS

~ONJ”2NCTd PROBLEMS P’Ic.

2. Mean

errors

6yr. olds 9 yr. olds 12yr. olds adults

according

to rule,

problem,

and

age

Rule identification. It is possible to distinguish between a disjunctive and a conjunctive rule with no more than one error. The S must guess at the first AP or PA contingency which is presented-if it is a positive instance he can deduce that the rule is a disjunctive one; if it is a negative instance, that the rule is a conjunctive one. Therefore, if S made more than one error he was classified as failing the rule identification phase of the experiment. According to this criterion, six of the 32 Ss failed to identify the rule being employed; three failed on a conjunctive rule and three on a disjunctive rule. While the data are too few to draw

3

2

1

0 Number of errors. h Proportion.

Problem PP PA : AP AA Problem PP PA AP AA Problem PP PA AP AA

1 3 0 0

2 8 1 0

20 24 38 6

25 .75 .oo .oo

.18 .73 .09 no

.23 .27 .43 .07

0 2 2 0

0 1 0 n

2 5 7 0

P

no .50 .50 .oo

.oo 1 00 .oo .oo

.14 .36 .50 .oo

N

P*

0 0 1 1

0 1 0 0

0 3 2 0

N

P

IN

.oo .oo .50 .50

.oo 1.00 .oo no

.oo .60 .40 .oo

12 yr.-olds

CONTINGENCY

Conjunction

Na

ATTRIBUTE

9 yr.-olds

EACH

6 yr.-olds

FOR

~-.____

ERRORS

RULE

0 1 1 0

0 4 0 0

0 3 2 0

N

.oo .60 .40 .oo

P

.OO .50 .50 .oo

.oo 1.00 00 .oo

Adult

LEARNING

TABLE

2

6 3 5 16

2 30 5 25

5 15 60 55

N

P

.20 .lO .17 .53

.03 .48 .08 .40

.04 .ll .44 .41

6 yr.-olds

ACCORDING

0 6 4 0

1 11 0 0

0 10 17 19

N

AM:.

.oo .60 .40 .oo

.08 .92 .oo .oo

.oo .22 .3T .41

P

Disjunction !I yr.-olds

TO RULES,

PROBLEMS

0 2 1 0

0 3 0 1

1 :; 10 8

N

.oo .67 .33 .nn

.oo .7r, on .“iI<

05 .14 .45 .36

1’

12 yr.-olds

.~ND

,oo 50

.on .%I

.oo .80

3 .50 0 no ______~

3

0

1

0 4 0

.41 18 11

.34

.oo .“5

I’

0 s

x

Adult ~___-

w g 2

7 2: r

F

<

LEARNING

CLASSIFICATION

RULES

225

any definite conclusions, they suggest that, excepting adults, there is little difference between Ss of 6, 9, and 12 years of age with regard to this ability since two children failed at each of these ages. Five of the six children who failed the test for rule identification did not perform in an optimal manner on the last rule learning problem of the type they failed to identify, suggesting that rule learning was not, yet complet,e for these 8s. An optimal manner was defined as making no more than one error on each type of attribute contingency. It was felt that if the S had learned a rule he would not exceed this limit. Attribute contingerxies as stimuli. The proportion of errors for each contingency in each rule may be found in Table 2. The distribution of errors in this table seems to result from several factors. First, the relative frequency of occurrence for the contingencies differ for each rule. In the conjunctive rule, 50% of the stimuli were PP’s and 50% were divided equally among the other three contingencies (16.5% for each). In the disjunctive rule 50% of the stimuli were AA’s and 50% were divided equally among the other three contingencies (16.5% for each). This difference in the proportion of contingencies for each rule was caused by the necessity of presenting an equal number of positive and negative instances. At no age do the data conform to the distribution expected from the relative frequency of occurrence of each contingency. Moreover, as age or experience increases, the obtained frequencies of errors depart more and more from the relative frequency expectation. More of the errors occur on the PA and AP contingencies, and fewer on the PP or AA contingencies in either rule. Excluding the performance of the 6-year-olds in the disjunctive rule, only two of the 31 errors made on the third problem concerned PP or AA (for both rules combined). Acquisition: continuous or discontinuous? In order to evaluate the manner in which acquisition took place each contingency was treated as a separate problem. The number of errors for each attribute contingency was divided by the number of opportunities for making an error on that contingency yielding the proportion of presolution errors. (The number of opportunities is equal to the number of times the attribute contingency is presented before the last error for that conting,ency.) Table 3 shows the data for each age, rule, and contingency separately and combined. A proportion of errors/opportunities of less than .50 (the chance value) might be taken as evidence for a continuous acquisition process. (This is the usual interpretation but not the only one possible.) In general, the data conform to an interpretation of discontinuous acquisition in which the subject changes from either a chance level (indicated by a proportion of .50) or a below chance level (indicated by a proportion greater than

226

\\'lI,I,l.\\l

I..

Kl.Xls

.50) of responding to loo:/, correct, responding for any single attribute contingency. The italicized numbers in Table 3 appear to be exceptions t,o t,his rule. These exceptions oww on the PP and AA contingencies for the 6- and 9-year-olds and for t.he ,4,4 contingency for adults (in diujunction). The speculation offered to account, for these discrepancies is that some Ss treated the AA contingency initially as a number of specific

&OI'ORTION OF ERRORS RELATIVE ERRORS FOR EACH ATTRIBUTE

TABLE 3 TO THE x;CMBER OF OPPORTUXITIES COXTIXGEWY IX RULE LEARNISP

Conjunction &e

b-j

A

9

I”

Adult

Att.

Cont. PP PA AP AA Total PP PA AP AA Total PP PA AP AA Total PP PA AP AA Total

Tot,al (1 IX = Number for t,he contingency

of errors. before

Disjunction

E

0

E/O

E

0

23 36 41 6 106 2 10 10 0 22 0 9 4 1 14 0 10 3 0 13 155

99 47 43 15 204 R 13 10 0 32 0 9 4 1 14 0 10 3 0 13 263

23 .77 95 40 ..i2 22 ii 1.00 0 on .69 0.00 1.00 1.00 1.00 1.00 0.00 1.00 1.00 0.00 1.00 .59

13 48 74 96 231 1 30 24 19 74 1 s 14 17 40 0 16 22 17 55 400

37 97 112 291 53; 3 35 32 70 140 1 ‘1 19 36 65 0 25 28 59 112 x54

0 = Opportunity last error on that

Total E/O SC Ii”, 66 .z3 43 33 186 .i5 .27 53 1.00 .89 .74 .47 .62 0. no .64 i9 :29 .49

.4i

FOR

C + D

I!:

0

E/O

36 54 115 112 33i 3 40 34 19 96 1 17 1s 1S 54 0 26 25 17 68 555

136 144 155 306 741 12 48 42 in 172 1 1s 23 37 i9 0 35 31 59 125 1117

.2 6 -58 .74 .37 45 25 .8X .Sl ‘27 .56 1.00 .94 .78 .49 .68 0.00 74 :s1 .29 .54 .50

for errors-i.e., number of presentations contingency. X = 28 for each cell.

stimulus patterns, not as a single abstract stimulus. Each of the specific subpatterns might be learned in an insightful fashion and yet collectively yield a gradual acquisition curve for the contingency. In a pilot study one 12-year-old spontaneously reported that she had learned to categorize the AA instance by memorizing several subpatterns of attributes rather than a single abstract description of ‘[not A nor B.” Unidimensional hypotheses in B-year-olds. An unanticipated result was the asymmetry in the number of errors for the AP and the PA con-

LEARNIh-;G

CI~ASSIFIC.\TIOS

RI-12-Y

227

tingencies as depicted in Table 2. The effect is especially pronounced and based on a large number of errors for the younger Ss in the first disjunctive problem (and to a lesser extent on the second disjunctive and conjunctive problems). There seem to be two causes for this phenomenon. The first reflects the fact that, by chance, PA occurred before -4P in probIem two ; and S’s forced to guess would frequently err on the first contingency but not on the other. The second cause is attributed (post h’oc) to the tendency of the 6-year-olds to entert.ain a undimensional hypothesis, e.g., “all blue ones are positive” when the correct rule was, e.g., “all blue and/or square ones are positive.” The undimensionnl hypothesis based on one relevant attribute is correct for the PP, PA (or AP but not both), and AA contingencies in either rule. Such an hypothesis would produce asymmetry between the AP and PA contingencies. The unidimensional S would call one contingency positive when it was negative in conjunction, and one comingency negat,ive when it was positive in disjunction. Since employin g a unidimensional hypothesis is an individual characteristic, the data from each probIem for each R were categorized into one of two groups. In one, the number of errors on the PA and AP contingencies were approximately equal, and in the other t,hey were not equal. The criterion for inequality was a ratio of errors of at least 2:l. Of course, only those problems in which at least two errors were made on PA4 and/or AP could be included. For the 6-year-olds there were 17 cases of inequality and five cases of equality, while for the older Ss there were but nine cases of inequality and 19 cases of equality. These results are consistent with the suggestion that unidimensional hypotheses were employed by the 6-pear-olds hut probably not by older 8s. 17erbal statement of the yule. After reaching criterion on each problem of each rule, the S was asked to explain how he knew the positive (good) blocks from the negative (bad) ones. His explanations were categorized by the writer and independently by another judge, into three mutually exclusive and exhaustive categories of correct, indeterminate, and incorrect. An explanation was considered correct if it contained sufficient, information to specify either the positive or negative class; it was considered incorrect. if it would lead to incorrect classifications, and all other instances were considered indeterminate. The two judges agreed on better than 90% of the 224 cases and, of the 23 discrepancies, only two involved a disparity of more than one category. The assignment. of the conflicting cases was made by discussing the criteria and reaching a joint agreement. Since the 6-year-alds were discrepant from the other age groups and since performance was substantially the same for both rules, only the data for the combined rules for the 6-year-olds and the older

Ss are reported. For the 6-year-olds the frequency of the correct, incorrect, and indeterminate explanations was 21, 14, and 13, respectively, and for the combined older groups they are 128 correct, 10 incorrect, and 6 indeterminate. Moreover, inspection of the verbalizations suggested t’hat the 6-year-olcls did not spontaneously improve the quality of their verbal explanations with increasing experience. These findings demonstrate that Ss (especially 6-year-olds) could solve the problems and shoxv transfer without being able to correctly explain the solmion. The “sex” effect. The significant effect for “sex” cannot he interpreted as indicating that females perform better than males since sex was deliberat.ely confounded with other variables. Data from a separate group of Ss, four at each age level, in which sex was not confounded, yield numerically similar results. The average number of errors on two problems (either of a conjunctive or disjunctive nature) was 5.68 for males and 1.87 for females. This mean difference is not statistically significant, but the data are inadequat’e to make a reasonable test. Unfortunately. the school year ended before more Ss could be tested. Attribute identifimtion. The mean errors for the 6-, 9-, and 12-yearolds and adults were 16.88, 3.38, 19.63, and 0.75, respectively. The differences among these means were significant F = 24.04, df = 3,24, p < .Ol. There was also a significant age X sex interaction F = 4.43. df = 3,24, p < .05 which appeared to be due to the fact that the 6-yearold boys made approximately twice as many errors as the B-year-old girls, while the reverse was true of 12-year-old girls who made approximately twice as many errors as the 12-year-old boys. It is surprising that the 12-year-olds made more errors than the 9-year-olds. Perhaps these Ss were seeking more difficult solutions than younger ones. DISCUSSION

Disjunctive rules were more difficult than conjunctive ones for all age levels. Furthermore, there was no t.endency for relative rule difficulty to decrease with decreasing age. The fact that four 6-year-olds failed a disjunctive problem, but only one failed a conjunctive problem, suggests the opposite-that with decreasing age disjunctive rules become more difficult than conjunctive rules. Rule utilization appeared largely independent of initial diffculty since groups differed widely in their ability to learn the rules but not in their ability to identify them. Children requiring assistance in order to learn a rule showed ,transfer to subsequent problems. This result suggests,the existence of three stages in rule learning. In the first, children are not able to discover the rule nor to profit from verbal tutoring. In

LEARNING

CLASSIFICATION

RULES

229

the second they are not able to discover the rule by induction but can learn to utilize it with the aid of verbal tutoring. In the third, they can discover the rule and utilize it without verbal tutoring. Haygood and Bourne (1965) suggest that bidimensional rule learning is dependent on two processes: encoding the stimuli into mediators corresponding to the four attribute contingencies and learning the assignment of each contingency to the appropriate response class. They pointed out that both processes would probably occur simult,aneously. The data from this investigation appear to be consistent with such an interpretation. The distribution of errors and the sudden acquisition for each contingency support the idea that they are or become the effective stimuli for 8. At the very least they demonstrate the utility of this approach for finding regularity in the data, Moreover, one would predict from this model that once S had encoded t,he stimuli for a rule he would learn subsequent problems with no more than one (“information”) error for each contingency. A sufficient (though not necessary) indicator for encoding was defined as a pattern of errors in which not more than one error was committed for each attribute contingency. Except for two Ss, any S showing this pattern for one problem showed it also on subsequent problems based on the same rule. The fact that 8s did not transfer to different rules suggests that Ss will not usuahy form a completely abstract code from experience with only one problem. The disjunctive rule was more difficult than the conjunctive largely because the AA contingency was more difficult in disjunction than its counterpart (in information)PP in conjunction. This result is analogous to the finding from attribute identification studies that information transmitted by negative instances is not as well utilized as information transmitted by positive instances (Smoke, 1932; Hovland and Weiss, 1952; Bruner ‘et ‘al., 1956; and Conant and Trabasso, 1964). In the present study the difficulty associated with the AA cont.ingency might result from Xs beginning the task as if it were a complete learning task, i.e., they do not begin by using the information contained in the rulelearning instructions. If this were the case, they would make more errors on the AA contingency since it is a negative instance and Ss are prone to use a strategy (Bruner et al., 1956) in which they look for attributes common to positive instances. Another interpretation, which might be saying the same thing, is that the AA contingency was more difficult to encode. Both interpretations refer to the same principle-that the absence of stimulus attributes is more difficult to ut,ilize than their presence. The PP and the AA contingences differ with respect to the number of

230

\VILLIAM

I,.

I
unique stimulus patterns they exhibit. This follo\\.s from the fact, that the absence of a pair of relevant attributes map be concretely expressed by the presence of many alternative attribute pairs. For this experiment, not redness and not squareness could be expressed by any of 36 different patterns while redness and squareness could be exprcsecd by only nine different patterns. Of course, Se understanding the instructions ought not to attend to any irrelevant attributes, however, this assumption would seem unwarrented. If S attends t’o irrelevant attributes at all, he would be at a greater disadvantage in disjunction as compared to conjunction since there are a greater variety of negative AA instances in disjunction than t.here are PP instances in conjunction. The rules are equivalent in the amount of information to be learned only when S can use the information contained in negative instances as efficient’ly as t’he information contained in positive instances. The four Ss who failed to discover the disjunctive rule for themselves appeared to be perseverating with a unidimensional hypothesis. The ratio for AP to PA errors was 3:1, 6:1, 8:1, and 12:0 for t’hese Ss. The single S who failed to solve a conjunction made approximately an equal number of errors on these contingencies. It would appear that for some reason the unidimensional hypotheses are more difficult to infirm with disjunctive rules as compared to conjunctive ones. This might result from the greater difficulty of finding an alternative hypothesis in a disjunction rule as compared to a conjunctive one. In a conjunctive rule the fact that there were only nine patterns for the positive class (all containing the two relevant attributes) would facilitate adoption of the correct hidimensional hypot.heses, especially if S attends to attributes common to positive instances. The 6-year-olds differed in yet another respect from the older SP. although they were able to solve all problems (albiet with assistance), they had a larger proportion of incorrect and indeterminat.e explanations than the older Ss. A correct verbalization was not neccsenly for solution or transfer, suggesting that direct verbal mediation was not necessary. In general, the results of this study support Haygood and Bourne’s analysis of rule learning and offers no support to the idea that. relativcl rule difficulty diminishes with decreasing age. They suggest, instead, that it increases. Rule difficulty seems to depend on the ease of encoding the important attribute contingencies for each rule. It is likely that SP attend to irrelevant attributes even though the instructions denote the relevant attributes for them. Thus, encoding would include learning t,o attend to the relevant stimuli as well as learning to consider them in all combinations (the attribute contingencies-). At all ages studied, rule learning may be desrribed as mediated hypothesis testing behavior,.

LEARNING

CLASSIFICATION

itUL&

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