Models of the structure of some rule-governed mathematical behaviors

Models of the structure of some rule-governed mathematical behaviors

CONTEMPORARY Models EDUCATIONAL PSYCHOLOGY 6, 237-249 (1981) of the Structure of Some Rule-Governed Mathematical Behaviors JOHN University R. BE...

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CONTEMPORARY

Models

EDUCATIONAL

PSYCHOLOGY

6, 237-249 (1981)

of the Structure of Some Rule-Governed Mathematical Behaviors JOHN University

R. BERCAN qf Arizona

This study investigated the extent to which various latent class models adequately described some elementary rule-governed mathematical behaviors. Two hundred and eighty-three children were given a fraction concepts test in which they identified fractional parts of sets of objects. Results supported the adoption of a set of three-class models including a mastery class, a nonmastery class, and a transitional class to describe the data.

Academic learning generally involves the acquisition of rules that permit learners to generalize acquired competencies across a variety of different tasks (Bandura, 1977; Gag&, 1977; Rosenthal & Zimmerman, 1978). Rule learning is particularly apparent in fields such as mathematics, which involve highly structured rule systems (Gag&, 1977). A substantial amount of research has been conducted on rule-governed behavior (Rosenthal & Zimmerman, 1978; Zimmerman & Rosenthal, 1974). Much of this work has focused on environmental variables, such as modeling, that influence rule acquisition (Zimmerman & Rosenthal, 1974). However, some attention has also been given to models describing the structure of rule-governed behaviors. Work on structure description has evolved from efforts to link concepts from behavioral psychology related to rulegoverned behavior to the idea of homogeneous item domains utilized in criterion-referenced assessment (Hively, Patterson, & Page, 1968). Models describing the structure of rule-governed behaviors advance explicit mathematical assumptions about structure that can be tested empirically. For example, Macready and Dayton (1977) have proposed two latent class models that conceptualize homogeneous item domains reflecting rule-governed behaviors. The Macready and Dayton models describe rule-governed behavior in terms of two categories: one representing rule mastery and the other representing nonmastery. The models hypothesize that when a set of tasks are within the same domain the task performance of individuals will reflect either the mastery or nonmastery categories. The models further assert that mastery does not imply perfect performance. Masters may make occasional errors. Likewise, nonmasters may inadvertently produce correct responses by such means as lucky guesses. Address reprint requests to Dr. John Bergan, Department University of Arizona, Tucson, AZ 85721.

of Educational

Psychology,

231 0361.476X/81/030237-13$02.00/O Copyright AU rights

0 1981 by Academic Press, Inc. of reproduction in any form reserved.

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JOHN R. BERGAN

Macready and Dayton’s first model assumes that within the mastery and nonmastery classes the probability of correct performance will be equal across tasks. Under model 2, the probability of attaining correct performance on one task is held to be independent on the probability of attaining correct performance on any other task for both masters and nonmasters. Bergan (1980) has suggested the use of a three-class model in describing the structure of rule-governed behaviors. This model was developed by Goodman (1975) in his work on response scaling. The three-class model includes mastery and nonmastery categories similar to those proposed by Macready and Dayton. However, people in the mastery category are assumed not to make mistakes and people in the nonmastery category are assumed not to make lucky guesses or other inadvertently correct responses. Inconsistent responders are assigned to the third class in the model. The model holds that the probability that individuals in the third class will respond correctly on one task will be independent of the probability that they will respond correctly on any other tasks. The third class in the three-class model allows for a transition stage between mastery and nonmastery. Individuals in the third class may be thought of as being transitional because the probability that they will perform correctly on any given task is greater than zero and because their performance is characterized by inconsistency suggesting that they have not yet acquired the role governing task performance. It could be the case that as Macready and Dayton suggest masters may make an occasional mistake and/or nonmasters may perform correctly now and then. Likewise, it could be that there are transitional learners who have not yet achieved rule mastery, but who are nonetheless beyond the stage of merely making lucky guesses. These possibilities suggest models incorporating some of the features of the Macready and Dayton models and some of the features of the three-class model. Combination type models may be constructed by assuming a latent class for transitional learners and by assuming probabilities of correct performance other than one and zero for masters and/or nonmasters. One model involving these assumptions might assert that the probability of correct performance for nonmasters would be equal across tasks and greater than zero, and that the probability of correct performance for masters would be independent across tasks. A second model could hold that the probability of correct performance would be equal across tasks for masters, zero across tasks for nonmasters and independent for transitional learners. A third model might assume that the probability of correct performance would be equal across tasks for both masters and nonmasters and independent across tasks for transitional learners. There is little information on the extent to which any of the models

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proposed to describe the structure of rule-governed behaviors provide an adequate description of such behaviors. Moreover, information comparing various models is lacking. The present study assesses the extent to which the models presented above afford an adequate description of the structure of some rule-governed elementary mathematics behaviors. The study compares various models to determine the occurrence of a transitional learning class and to ascertain the extent to which the performance of masters and nonmasters involves probabilities of correct responding other than one and zero. Mathematics was chosen for investigation because the rule systems representing mathematical operations are especially well-defined. A set of simple tasks was desired, which could be varied independently along a number of dimensions. The use of a number of different dimensions made it possible to assess the consistency with which various models afforded an adequate representation of different instances of rule-governed behavior. A series of tasks involving the identification of fractions was used to meet the requirements of simplicity and independent dimensions. The tasks varied in the values of the fraction numerators and denominators and in the shapes of the objects used to represent the fractions presented. METHOD Data for the study were obtained from performance on a fraction concepts test of children participating in an investigation by Bergan, Cancelli, and Karp (Note 1). The Bergan et al. research focused on a subset of items from the test reflecting potential prerequisite relationships among fraction concept tasks. The present analysis employed a subset of items reflecting potential homogeneous item domains.

Subjects One hundred and twenty-eight boys and 255 girls from 7 to 11years of age participated in the investigation. The children attended parochial schools in middle-class neighborhoods; 276 were Anglo, 100were Mexican-American, 6 were black and 1 was an American Indian.

Tasks All of the tasks used in the study required identification of some fractional part of a set of objects. The tasks varied in numerator value, denominator value, and object shape. Numerator values included 1 and 2. The denominators were 3 and 5. Four was omitted to avoid the problem of reducing a fraction to lowest terms. Some of the fraction identification problems involved sets of circles. Other problems employed sets made up of varying shapes including circles, triangles, squares, rectangles, and trapezoids. For instance, children were asked to identify one-third of a set of three objects which included a circle, a triangle, and a square. Eight different types of problems were derived from the variations described above. These included four problem types involving sets of circles and four problem types involving sets with varying shapes. The first problem type called for the identification of one-third of a set of three objects, the second involved one-fifth of five, the third two-thirds of three, and the fourth, two-fifths of five. Each problem was presented twice during the test yielding a total

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JOHN R. BERGAN

of 16 problems used in the study. Problems were arranged randomly. In each problem, the child was instructed to put an X on a fraction of the objects.

Procedures Testing was carried out in classroom settings. Prior to taking the test, the children were shown how to respond to the items without being informed of the correct answer. The test was not timed and all children managed to finish it. Testing was monitored by an experimenter and one assistant who answered children’s questions in those few cases in which there was difftculty comprehending the nature of the task.

RESULTS Data analysis was conducted on eight task sets each containing four fraction identification problems composed of two pairs of identical tasks. Identical tasks were included to make it possible to separate instances of inconsistency in responding to the same task from response inconsistency reflecting a lack of rule-governed behavior. The task sets were chosen to represent shape variations, fraction denominator variations, and a combination of shape and denominator variations. Numerators were held constant because other research (Bergan et al., Note 1) had shown that numerator variations could produce prerequisite relations among tasks rather than homogeneous task domains. Table 1 presents the observed frequencies for the possible response patterns for the various task sets examined in the study. Fractions with the prime sign indicate problems involving shape variations. Note the large number of extreme response patterns in the table. The cells associated with the extremes are the ones most clearly linked to mastery and nonmastery categories. All of the models examined in the present study are latent class models. These models assume that the association among a set of tasks can be explained by one or more latent (i.e., unobserved) variables each containing a number of latent (unobserved) classes (Goodman, 1974). Association among tasks is said to be explained when a x2 test supports the hypothesis represented by the model being examined. x2 tests for any given latent class model assess the extent to which maximum likelihood estimates of expected cell frequencies obtained under the model correspond to observed cell frequencies. When the correspondence between observed and expected frequencies is close, the value of x2 will be small and the model being tested can be described as affording an adequate fit for the data. Goodman (1974) has developed an iterative procedure for computing maximum likelihood estimates of expected cell frequencies for latent class models. Clogg (Note 2) has constructed a computer program which carries out the iterative process and computes appropriate x2 tests. Clogg’s program was used to assess model-data fit for the latent class models examined in the present study.

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STRUCTURE

TABLE 1 OBSERVED

FREQUENCIES

FOR ALL

TASK SETSO

Task sets

1 1 1 1 1 1 1122 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2211 2 2 2 2 2 2

1 1 1 2 2 1 1 1 2 2 1 1 2 2

1 2 1 2 1 2 1 2

1 2 2 1 2 2

178 5 5 2 6 2 0 9 8 2 2 4 2 2 2 54

157 7 6 1 6 3 1 3 4 1 6 4 5 3 8 68

171 4 6 5 6 2 1 12 11 3 6 4 2 6 7 37

163 10 6 4 8 4 I 6 10 1 2 2 2 1 4 59

164 4 12 10 4 4 1 8 2 3 1 10 1 2 1 56

161 13 3 9 5 1 1 14 12 3 6 3 5 2 5 40

159 8 13 10 5 1 1 10 5 3 5 3 3 2 2 53

159 9 5 13 7 0 1 13 14 3 2 5 3 4 5 40

” Fractions without the prime sign indicate sets of circles. Fractions with the prime sign indicate sets of varying shapes. The l’s under response patterns indicate a passing response, the 2’s a failing response.

Seven models were tested using Clogg’s program. To illustrate model characteristics, Table 2 shows for one item set latent class probabilities each of which indicates the probability that an individual will be in a given latent class and conditional response probabilities indicating the probability of correct task performance given that an individual is in a particular latent class. The item set used for Table 2 was the set involving one-third of a set of three circles and one-fifth of a set of five objects of varying shapes. Six of the seven models tested are shown in the table. The H’s indicate models, the 7~~‘slatent class probabilities, and the other 7~‘s conditional response probabilities. ~35 is the probability of a correct response under latent class t (t = 1, 2, or 3) for the first item involving one-third of three objects. & is the corresponding probability for the second one-third item. & is the probability of a correct performance for the first one-fifth item, and r% is the probability of a correct performance for the second one-fifth item. Note that model H, is not shown in the table. Model H, is the familiar independence model, which need not be presented in terms of latent class and conditional response probabilities. The fit of this model for any given data set is established by conducting the usual x2 test of independence among the cell frequencies in the contingency table containing the data under examination (see, for example, Hays, 1973). From the standpoint of

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JOHN R. BERGAN TABLE 2 LATENT

CLASS AND CONDITIONAL RESPONSE PROBABILITIES SELECTED MODELS FOR ONE TASK SET”

FOR

Conditional probabilities Model

37

67

TX n11

$7

.69 .31

.95 .I5

.95 .15

.95 .15

.95 .15

.70 .30

.95 .23

.97 .19

.91 .08

.94 .09

.53 .18 .29

1.00 .oo .69

1.00 .oo .68

1.00 .oo .44

1.00 .oo .52

.61 .17 .22

.98 .oo .63

.98 .oo .61

.98 .oo .31

.98 .oo .41

.52 .21 .27

1.00 .04 .74

1.00 .04 .76

1.00 .04 .50

1.00 .04 .59

.60 .19 .21

.98 .Ol .66

.98 .Ol .66

.98 .Ol .34

.98 .Ol .44

Latent Class Probabilities

u The item set used in the table is the one involving one-third of a set of three objects of the same shape and one-fifth of five objects of varying shapes.

latent structure analysis, the independence model can be viewed as a latent class model which assumes one latent class and independence among tasks within that class. As will be demonstrated below, the independence model serves as a useful benchmark against which other models may be judged, which is why it was included in the analysis. Model HP, the first model shown in the table, is the Macready and Dayton model that assumes equal response probabilities within mastery and nonmastery latent classes. Model H, is also a Macready and Dayton model. Note that under this model the conditional response probabilities vary across tasks. Model H, is the three-class model in which the probability of correct performance is assumed to be one for masters and zero for nonmasters. Note that the conditional response probabilities vary across tasks in the third latent class under model H,. This is the class representing transitional individuals. Under model Hq, the probability of correct performance on any one task is assumed to be independent of the probability of correct performance on any other task for individuals in the transitional latent class. Model H, assumes equal probabilities of correct performance across tasks for nonmasters. The third latent class in HZ is a

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transitional class identical in character to that assumed under Hq. Model H, is essentially the same type of model as H,. However, H, assumes that the probability of correct performance will be one across tasks for masters and that the probability of correct performance will be equal across tasks for nonmasters. Model H, asserts that the probability of correct performance will be equal across tasks for masters, equal across tasks for nonmasters, and that independence will prevail among tasks in the latent class representing transitional individuals. Table 3 presents likelihood-ratio x2 values testing model-data fit for the eight task sets examined in the study. Note that H* is included in the table rather than H,. The H* is used because in all cases except the one shown in Table 2 the attempt to estimate parameters under H, yielded estimates congruent with parameters associated with other more parsimonious models (i.e., models with more degrees of freedom). For example, in the case of the item set involving one-third and one-fifth of a set of circles, the effort to estimate parameters for H, resulted in estimates equivalent to those found under H,. When estimation of parameters under a given model eventuates in a result equivalent to a more parsimonious model, it is appropriate to use the more parsimonious model (Goodman, 1974). The H* indicates the variety of models produced by the attempt to estimate parameters under H,. Table 3 includes what Goodman (1974) has called hierarchical latent class models. Models H, and H3 are examples. These models are hierarchical in that H, implies H,. Model H, includes estimates of all of the parameters in H, plus the additional 6 parameters needed to establish conditional response probabilities under H,. Hierarchical models can be compared statistically. For instance, the x2 for H3 can be subtracted from the x2 for H,. The result gives x2 with 12 - 6 = 6 degrees of freedom. This x2 indicates the contribution to model-data fit derived from the removal of the restrictions on conditional response probabilities imposed under Hz. Hierarchical comparisons among the models tested indicate that H, and H, in general offer the best tit for the data. Indeed, there were only two cases in which these models did not fit the data adequately. An examination of the standardized residuals for observed and expected frequencies (Bishop, Fienberg, & Holland, 1975) was undertaken to get some clues as to the source of the two cases of lack of fit under H, and H,. The standardized residual for a given cell in a contingency table is given by (f, where fO is the observed frequency for the cell and fe is the fNEP2> expected cell frequency estimated under the model being tested. A large standardized residual indicates a cell in which the correspondence between expected and observed frequencies is low. Examination of the standardized residuals suggested that the source of the two cases of lack of fit for H, and H, was inconsistent responding to identical pairs of items.

215

z/3-2/J’ - 215’ ll3-‘ls 213-215 l/3-‘15’ 2/J- %

‘is-‘/s’

I/,-‘is’

Task sets

568.09 597.31 415.19 534.60 549.53 405.34 481.51 406.97

H,

II 11 11 11 11 II 11 11

df

c.01 <.ot ‘c.01 1.01 <.Ol <.Ol <.Ol <.Ol

p

16.96 29.46 16.62 20.42 51.84 44.66 47.30 42.08

Hz

12 12 12 12 12 12 12 12

df

<.25 <.Ol 1.25 1.10 <.Ol <.Ol 1.01 c.01

P 10.92 12.35 10.63 15.03 18.39 26.66 31.30 20.88

H, 6 6 6 6 6 6 6 6

df

x2 VALUES

TABLE3

<.lO 1.01 1.25 <.025 1.01 <.Ol c.01 c.01

P 9 9 9 9 9 9 9 9

22.86 14.58 23.35 16.31 14.30 36.84 14.50 41.59

<.Ol <.25 c.01 1.10 c.25 c.01 c.25 1.01

P

THROUGH

df

H,

H,

FOR MODELS

7.86 6.59 10.38 6.54 10.08 29.31 11.94 22.53

Hs

H,

8 8 8 8 8 8 8 8

df i.50 <.75 c.25 c.75 c.50 c.01 c.25 1.01

P 4.77 9.10 9.75 8.85 13.90 28.85 13.09 24.82

H,

8 8 8 8 8 8 8 8

df

1.90 c.50 c.50 <.50 c.10 <.Ol c.25 c.01

P

14.52 6.59 9.75 6.54 10.08 22.15 11.82 11.87

H,

8 8 8 8 8 8 8 8

df

c.25 1.10 c.50 c.50 1.25 c.01 <.I0 1.25

P

P z

w

F

z

5

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Table 4 presents latent class and conditional response probabilities for models H, and H, for all task sets examined in the study. The conditional response probabilities are of special interest. The probabilities estimated under H, tend to be congruent with an interpretation suggesting a mastery class composed of individuals who may make occasional errors, a nonmastery class comprised of individuals who never achieve a correct response, and a transitional class reflecting moderately accurate performance and independence across items. However, not all of the item sets produced conditional response probabilities congruent with this interpretation. More specifically, the conditional probabilities for the item set composed two-thirds of three circles and two-thirds of three objects of varying shapes imply a substantively different interpretation. For this item set, the first latent class is the transitional class. Conditional response probabilities are equal across tasks in this transition class. This state of affairs is contrary to what was presumed. The third latent class may be conceived of as the mastery class. The probability of a passing response is very high in this latent class. However, the conditional response probabilities are not equal across tasks as it was assumed they would be. The conditional response probabilities for H, contain some cases which conform to expectations. However, there are many cases in which the conditional probabilities estimated under H, suggest a different interpretation from that initially expected for this model. For models not conforming to expectations, items in the transitional class do not reflect independence. Rather, conditional response probabilities are equal across tasks. In the nonmastery class, response probabilities are not equal across tasks as it was thought they would be. Rather the items in the nonmastery category reflect independence. DISCUSSION

One major finding which emerged from the present study was the support which the data provided for a latent class reflecting a transition between nonmastery and mastery. Three-class models including a transitional class fit the data for seven of the eight item sets examined. None of the models tested afforded a fit for the eighth set. By contrast only three of the eight two-class models fit the data to an acceptable degree. Moreover, in the one case in which hierarchical comparison between a two-class model and a three-class model was possible, the three-class model improved significantly on the fit afforded by the two-class model. Examination of the conditional response probabilities for H, and H6 revealed that the characteristics of the transitional class are not consistent across task sets. In every case a model did exist in which the response probabilities within the transitional class were independent across items

TABLE LATENT

CLASS AND CONDITIONAL

4 FOR MODELS H, AND H,

RESPONSE PROBABILITIES

Conditional Task Sets ‘I3 -113’

Model

Latent 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Class Probabilities .71 .I5 .I4 .54 .21 .25 .62 .21 .17 .51 .20 .29 .25 .07 .68 .52 .26 .22 .69 .19 .12 .25 .46 .29 .63 .17 .20 .56 .21 .23 .67 .10 .23 .49 .23 .28 .61 .17 .22 .52 .21 .27 .68 .09 .23 .42 .31 .27 246

Probabilities

3:

7@

7q

.97 .oo .35 1.00 .80 .13 .97 .oo .25 1.00 .69 .02 .29 .oo .99 1.00 .16 .68 .96 .oo .51 1.00 .04 .81 .98 .oo .56 1.00 .02 .71 .96 .oo .47 1.00 .12 .68 .98 .oo .63 1.00 .04 .74 .95 .oo .46 1.00 .83 .29

.97 .oo .26 1.00 .80 .06 .97 .oo .33 1.00 .69 .04 .29 .oo .97 1.00 .16 .84 .96 .oo .33 1.00 .04 .80 .98 .oo .58 1.00 .02 .68 .96 .oo .41 1.00 .12 .83 .98 .ot-l .61 1.00 .04 .76 .95 .oo .39 1.00 .83 .20

.97 .oo .21 1.00 .80 .03 .97 .oo .33 1.00 .69 .03 .29 .oo .97 1.00 .16 .76 .96 .oo .30 1.00 .04 .82 .98 .oo .24 1.00 .02 .36 .96 .oo .25 1.00 .I2 .75 .98 .oo .31 1.00 .04 .50 .95 .oo .18 1.00 .83 .07

e .97 .oo .16 1.00 .80 .03 .97 .oo .5l 1.00 .69 10 .29 .oo .95 1.00 .16 .83 .96 .oo .32 1.00 .04 .75 .98 .oo .19 1.00 .02 .40 .96 .oo .31 1.00 .12 .66 .98 .oo .41 1.00 .04 .59 .95 .oo .I8 1.00 .83 .08

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as it was assumed they would be. However, in many cases alternative models which did not reflect transitional independence also fit the data to an acceptable extent. The findings suggest the plausibility of two kinds of transition states, one reflecting independence and the other indicating equal response probabilities across tasks. The two types of transition classes describe different, though not contradictory, ways of moving from nonmastery to mastery. The transition class reflecting independence across tasks suggests progress toward mastery in which learners master individual tasks independently and then achieve a mastery state by acquiring the rule-governing tasks in the domain under study. The transition class reflecting equal response probabilities across items indicates some degree of rule acquisition across tasks during the transition phase. Transition for this class implies rule acquisition coupled with moderate probability of accurate responding. It is worth noting that rule acquisition in the transition class occurred most often under H,. This model restricted the mastery state to individuals displaying perfect performance on all tasks. The results for H, support the view that when mastery is restricted to errorless performance, rule acquisition as evidenced by equal response probabilities across tasks tends to occur in the transition class. In contrast, the results for H, indicate that when mastery is conceptualized as including some performance errors, the transition class reflects independent responding across tasks. The support for a transition class revealed in the present investigation may have important implications for instruction. Different instructional procedures may be required to assist a learner to move from a transition state to a master state than to foster progress from a nonmastery state to a mastery state. Research is needed to investigate this possibility. A second significant finding revealed in the present study was that in order to achieve a model which fits the data to an acceptable degree, it tended to be necessary to make some allowance for response errors in either the mastery or the nonmastery class. The tasks used in the study were ones that might easily reward a well-conceived guess. Such inadvertent correct responses may be thought of as intrusion errors (Macready, Note 3). Although the items were not multiple choice, the response alternatives were limited. At the same time, there were ample opportunities for inadvertent mistakes on the part of children who had mastered the rule. Although the magnitude of error rates in the mastery and nonmastery classes tended to be small, performance errors in these two classes had a dramatic impact on model-data tit. In particular, the presence of mastery and nonmastery errors reduced the effectiveness of the Goodman threeclass model (HJ in affording an acceptable tit for the data. The findings for H, and HE indicate that the allocation of errors to

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mastery or nonmastery classes depends to some extent on one’s point of view about the mastery and nonmastery states. If one wishes to conceive of mastery as errorless performance, then errors that would have been allocated to the mastery class under a less demanding criterion are assigned instead to the transition class. Likewise if one wishes to conceive of nonmastery involving no correct responses, then what would have been nonmastery errors end up in the transition class. One might expect that the allocation of performance errors to mastery and nonmastery states would depend to some extent on task characteristics. For example, tasks in which the probability of correct guessing was very low might favor the allocation of errors to the mastery class. Research is needed to investigate factors affecting the allocation of performance errors to mastery and nonmastery conditions. Taken as a whole, the findings of this investigation support the feasibility of describing the structure of rule-governed behavior through the use of latent class models such as H, and H6. The findings suggest the immediate need to investigate rule structure across a variety of different kinds of tasks important to learning and development. In addition they open the way for the study of factors that might affect rule structure. For example, as indicated earlier, an abundance of research has investigated the effects of variables such as modeling on rule acquisition. Nothing is known about the effects of variables of this kind on the structure of rules. The use of latent class models makes it possible to study the influence of environmental variables such as modeling on rule structure. REFERENCES BANDURA, A. Social learning theory. Englewood Cliffs, N.J.: Prentice-Hall, 1977. BERGAN,J. R. The structural analysis of behavior: An alternative to the learning hierarchy model. Review of Educutional Research. 1980, 50, 625-646. BISHOP, Y. M. M., FIENBERG, S. E., & HOLLAND, P. W. Discrete multivariate analysis. Cambridge, Mass.: MIT Press, 1975. GAGNI?,R. M. Conditions of/earning (3rd ed.). New York: Holt, Rinehart & Winston, 1977. GOODMAN, L. A. A new model for scaling response patterns: An application of the quasiindependence concept. Journal of the American Statistical Association, 1975, 70, 755-768. GOODMAN,L. A. The analysis of systems of qualitative variables when some of the variables are unobservable. Part I-A modified latent structure approach. American Journal of Sociology, 1974, 79, 1179- 1259. HAYS, W. L. Statistics for the social sciences. New York: Holt, Rinehart & Winston, 1973. HIVELY, W., PATTERSON,H. L., & PAGE, S. H. A “universe-defined” system ofarithmetic achievement tests. Journal of Educational Measurement, 1968, 5, 275-290. MACREADY,G. B., & DAYTON, C. M. The use of probabilistic models in the assessment of mastery. Journal of Educational Stafistics, 1977, 2, 99- 120. ROSENTHAL, T. L., & ZIMMERMAN, B. J. Social learning and cognition. New York: Academic Press, 1978.

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ZIMMERMAN, B. J., & ROSENTHAL, T. L. Observational learning of rule-governing behavior by children. Psychological Bulletin, 1974, 81, 29-42.

REFERENCE

NOTES

1. BERGAN, J. R., CANCELLI, A. A., & KARP, C. Replacement and component rules in hierarchically ordered mathematics rule learning tasks. Unpublished manuscript. 2. CLOGG, C. C. Unrestricted and restricted maximum likelihood latent structure analysis: A manual for users (Working Paper No. 1977-09). Unpublished manuscript,

Pennsylvania State University, 1977. G. B. Personal communication, San Francisco, April, 1979.

3. MACREADY,