Construction and Representation of Orderings in Memory

CONSTRUCTION AND REPRESENTATION OF ORDERINGS IN MEMORY Kirk H . Smith and Barbee T. Mynatt BOWLING GREEN STATE UNIVERSITY BOWLING GREEN,

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I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . , 11. Review of Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Overview of the Experiments .. IV. Experiment 1: Retrieval from enngs ............................. A.

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B. Results and Discussion

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C. Conclusions and Implications . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 2: The Role of Determinacy in Constructing Partial Orderi A. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Experiment 4: Diverging and Converging Nodes . . . . . . . . A. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Experiment 5 : The Role of the Schema.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Conclusions from Experiments on Presentation Orders. . . . . . . . . . . . . . . . . . . IX . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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126 127 130 131 133 134 138 138 140 141 143 145 146 147 149 149 150

Introduction

Implicit in much of the recent work in cognitive psychology on the acquisition, retention, and retrieval of information on ordered relationships is the assumption that the linear order is a “good figure” (De Soto, 1960; Henley, Horsfall & De Soto, 1969). In general, when people are confronted with a set of asymmetric, transitive relations such as “A is greater than B,” there is a strong tendency to represent it in a single, complete ordering. Experiments by Barclay (1973), De Soto (1960), THE PSYCHOLOGY OF LEARNING AND MOTIVATION. VOL 16

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Potts (1972), and Trabasso, Riley, and Wilson (1975) provide a variety of demonstrations of the strength of this tendency. Unfortunately, the world is not a simple, well-ordered place. Much of our knowledge cannot fit neatly into the complete ordering schema. Important examples of partial orderings include family trees and causal relations. The present article is concerned with the conditions that facilitate construction of appropriate mental representations of partially ordered information. We begin with an examination of several domains of knowledge that are typically portrayed as networks of partially ordered concepts or events. A review of the experimental literature on networks and partial orderings follows. We then describe five experiments on partial orderings that have not been reported previously. These studies make two contributions. First, they point out the limitations of previously published studies and suggest the need for exploration of a greater variety of partial orderings. Second, the experiments were designed to determine whether the acquisition of partial orderings can be understood in the same terms as the acquisition of linear orderings . Our investigations of presentation order strongly suggest that the same theory is satisfactory for both partial and complete orderings. What common kinds of information form networks of partially ordered objects or events? One instance that is very familiar to cognitive psychologists is the hierarchy (see Fig. 1A). Networks of this type have been used to represent the grammatical relationships among the words in a sentence (Chomsky, 1957; Johnson, 1968), as well as a person’s knowledge about the semantic relationships among words or concepts (Collins & Quillian, 1969). Two influential theories of semantic memory combine both types of information in complex networks (Anderson & Bower, 1973; Norman, Rumelhart, & the LNR Research Group, 1975). These theoretical networks are relatively complex in that several different kinds of relationships between words (or lexical entries) are represented. Another example of a frequently encountered hierarchy of partially ordered entities is an organizational chart or chain of command. A second type of information that can form a partial ordering is a network of causal relations. Suppose, for example, that B, J, H, K, F, and D are events in a narrative. Several relations among two or more events are possible. In the simplest, B might be the sole cause of J. More complex possibilities are that J and K jointly cause H or that H is the cause of both F and D. Such causal networks are sometimes represented in graphs like those of Fig. 1. (The examples above describe the situation at the top of Fig. 1B.) Historical relationships are often portrayed in this way. A family tree represents a special case of a history described in causal terms. The flow of material through a manufacturing process often

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Fig. I , Examples of recently investigated networks of partially ordered entities from the following studies: (A) Nelson and Smith (1972), (B) Hayes-Roth and Hayes-Roth (1973, and (C) Moeser and Tarrant (1977).

can be usefully represented as a partial ordering, and the planning of complex development projects is frequently characterized this way (as in a “PERT Chart,” Moder & Phillips, 1964). Underlying all the examples is an implicit time line. However, to the extent that the relationships are partially ordered, time need not be fully specified. In Fig. lC, the precise time at which events H and M occur is unspecified. What is important is that both H and M precede N. For a variety of reasons, information about a network of relationships is often acquired sequentially. We learn about a family tree by listening to relatives talk about the individuals who comprise it. We read about causal relations one at a time. History is most often recounted serially. But even when we are not constrained by the serial nature of language as a medium of communication, experience enforces sequential acquisition on us. Our unguided experience with the interactions of a group of people necessarily follows a time line, even though what we eventually come to understand about a group may be most accurately reflected in a sociogram or organizational chart (cf. De Soto, 1960). Perhaps the most striking example of a network of relationships that is not completely ordered but must be translated into a serial representation is found in a computer program. For some purposes, a program must be understood as a set of complex logical relationships among the operations the computer can execute. However, the program must also be realized as a single, rigorously ordered series of symbolic statements. Writing a program involves a translation from the first kind of representation (in the programmer’s head) to the second. Often of more practical importance is the translation in the other direction, as when a program with logical errors is debugged or when someone other than the original programmer

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tries to correct or modify a program. (Sometimes even the original programmer has this problem after the passage of time.) The preceding examples make it clear that people need to be able to understand networks of partially ordered entities. The question is how this is done. The writings of De Soto (1960; as well as Henley et al., 1969) seem to imply that even when sophisticated people give the relationships their most thoughtful consideration, they cannot handle certain kinds of networks. Yet some of our examples indicate that such networks cannot be impossible to understand and remember. The purpose of this article is to explore certain variables that affect people’s ability to understand and remember networks of partially ordered entities, and to show that they are the same variables that affect the apprehension of complete orderings. 11. Review of Previous Research

As indicated above, the hypothesis that the linear order is a “good figure” comes from the work of De Soto. The most relevant finding for present purposes comes from an experiment in which college students had to learn 12 relationships among four people (De Soto, 1960). The task was to report for each ordered pair of names whether or not the first influenced the second. Subjects required approximately nine repetitions, or trials, to learn all 12 relations when the latter formed a complete or linear ordering. Roughly 12 trials were required when the ordering was not complete, that is, when it formed a partial ordering and resembled a hierarchy or organization chart. A thorough treatment of De Soto’s work is beyond the scope of this article; however, several comments are needed to place our work in the proper perspective. From the beginning De Soto was concerned with how certain social relations are perceived. Thus, the 1960 experiment also contained groups of subjects that learned a set of statements identical to those described above, except that “influences” was replaced by “likes. These groups had equal difficulty with complete and partial orderings, but performed better when the relationship formed a transitive symmetric structure. De Soto concluded that “influence” is understood to be an asymmetric relation, whereas “like” is symmetric. It should be noted that no logical inconsistency is involved in a situation in which A1 likes Bill, but Bill does not like Al, whereas A1 cannot be both older and younger than Bill, nor both the father and the son of Bill. The present study was concerned almost exclusively with the latter type of relationships. Thus, ”

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our work should be interpreted as an exploration of the role of completeness in understanding asymmetric, transitive structures. Two other aspects of De Soto’s work deserve comment. First, we are not concerned with the observation that a set of items with two or more conflicting orderings (e.g., on two different dimensions) produces cognitive strain (De Soto, 1961). This “predilection for single orderings” is easily confused with people’s tendency to reduce a partially ordered set of items (on a single dimension) to one (incorrect) ordering. Second, the present work is not intended to be a definitive treatment of how people understand and remember systems of causal relations or the logical structure of computer programs. We recognize that intransitivities play an important role in these instances. (In fact, the loop, one of the most important structures in computer programs, corresponds formally to what Henley et al., 1969 call a cycle.) However, many of the issues raised here, especially the methodological ones, must be faced in future studies of how people deal with even more complex networks than the ones considered here. The experiments considered so far have all used a paired associate learning procedure. A paper by Nelson and Smith (1972), which raises a number of important questions about partial orderings, explored a graphical form of presentation. Graphs are popular devices for facilitating comprehension of networks. Indeed, the word “network” is applied to partial orderings by an analogy between the graphs that are used and things like fishing nets. Examples are called flow charts, family trees, PERT charts, and sociograms. The value of a graph lies in its accurate and economical representation of the important aspects of a partial ordering. For example, Fig. 1B expresses not only the determinate relationships between K and H and J and H, represented by lines, but also the indeterminate relationship between K and J. (Each line in Fig. 1 represents an asymmetric, transitive relationship between two symbols. If the relationship is ‘‘greater than, ” then the symbol with the higher location on the graph is the larger.) Nelson and Smith examined learning and retention of the 34 determinate relations represented in Fig. 1A. The relations were presented either as a set of 34 associations between letters-that is, C M, G + M, B + M, . . . D +-K - o r as a diagram like the one in the figure. Tests required the subjects to make checkmarks in a 14 X 14 matrix for which rows indicated first letters of the associations (letters lower in the hierarchy in Fig. 1A) and columns indicated second letters (or letters higher in the hierarchy). The letters heading the rows and columns were assigned randomly from trial to trial. Thus, Nelson and Smith’s subjects had to learn how to translate one representation (a set of pairs or a graph) into

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another (a matrix). There is no special reason to assume that the matrix representation is unique or cognitively simpler than a set of pairs or a graph. Although all possible combinations are available in the matrix, the subject has to identify only the ones presented in the association condition and does not have to discriminate between logically incorrect combinations and those that are indeterminate. Nelson and Smith found that subjects learned the 34 pairings in fewer trials, retained more pairs, and required fewer trials to relearn them when they were presented graphically than when only the pairs were presented. The difference was particularly striking in the number of errors made in learning. These results indicate that the information conveyed by a graph enhances in some way a person’s knowledge about a set of partially ordered relationships. Of particular importance is the finding that learning was unaffected when the left-to-right ordering of the branches in the graphs was rearranged from trial to trial. Groups receiving graphs that changed in this way performed as well as groups that received identical graphs throughout learning. Apparently, the subjects were able to discriminate the essential features of the graphs they were shown from their nonessential aspects. Graphs of the kind shown in Fig. 1 contain a number of details extraneous to the information they represent. The order of the nodes from left to right in the drawing is irrelevant, and the length of the lines carries no meaning. The conventions for drawing graphs of this kind permit ordinal, and sometimes interval, information to be expressed under some circumstances-as when time is represented in a graph illustrating the political history of western Europe in the eighteenth century. The latter example also makes it clear that graphs representing networks are abbreviated and impoverished in important ways. The nodes or boxes and even the lines or arrows are presented in a symbolic shorthand; one must usually read an accompanying text to get the complete story. Although Nelson and Smith demonstrated that college students are able to master the information in a partial ordering (at least temporarily for the purpose of completing a laboratory experiment), a series of recently published papers seems to argue that the knowledge is qualitatively different from that acquired when a linear ordering is learned. This conclusion is based on the finding that the “distance effect,” universally found with linear orderings, has not been obtained with partial orderings. Briefly, the distance effect refers to the fact that in judging which of two items rank higher on a linear ordering, people tend to be faster and more accurate as the number of intervening things on the scale increases. In a typical demonstration of this effect, subjects first learn a completely ordered set of relations, A > B, B > C, C > D, D > E, E > F. They are

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then asked either to judge whether various test items of the same form are true or false or to pick out which of two things rank higher. Reaction times tend to be shorter and error rates lower in verifying B > E than B > C, C > D, and D > E. Both Potts (1972) and Trabasso et al. (1975) found the distance effect even when relationships of greater distance (e.g., B > E) were not presented until testing. In contrast, when Hayes-Roth and Hayes-Roth (1975) tested subjects who had learned the 11 relationships represented by Fig. IB, they obtained a reverse distance effect. That is, adjacent relations, involving two letters connected by a single line (e.g., B > J in Fig. 1B) were judged more quickly than remote relations involving letters connected by two or more lines (e.g., H > P). This is, of course, just the opposite of what had been found with linear orderings. Hayes-Roth and Hayes-Roth described their experiment within the context of the semantic memory literature, in which the typical material is composed of sentences about class inclusion (e.g., “Canaries are birds”), and reaction time increases with distance rather than decreases. They went on to demonstrate that repeated testing on remote relationships could change the observed effects of distance, a result that clearly has methodological implications for the verificationtime procedure used to study semantic relations. This latter aspect of their paper has been largely ignored, and subsequent work has focused on the failure to find the appropriate ‘‘distance effect”-remote relations faster than adjacent ones-in partial orderings. Moeser and Tarrant (1977) pointed out that the procedure used by Hayes-Roth and Hayes-Roth (1975) encourages subjects to learn the individual relations between letters as isolated units in memory, rather than to integrate them into a network. Moeser and Tarrant argued that people do not spontaneously integrate information into holistic representations except under special circumstances, and that none of these conditions had occurred in the Hayes-Roth and Hayes-Roth experiment. Moeser and Tarrant therefore changed a number of aspects of the learning situation. They pointed out that in the Hayes-Roth and Hayes-Roth experiment, the relationships were presented as abstract inequalities involving meaningless letter pairs. Arguing that integration is more likely to occur with concrete and familiar material, Moeser and Tarrant used sentences that related a set of male names in terms of age (e.g., “Hugh is older than Bob”). They also required subjects to learn specific ages for some of the names. And in one condition, they showed subjects a network representation (along the lines of Fig. 1C) and encouraged subjects to store information in this format. They argued that these changes should lead to integrated storage and the usual distance effect observed with linear orders. In fact, judgment times for adjacents and remotes were equal.

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It is not entirely clear how this last result should be interpreted. On the one hand, the finding of equally long times for adjacent and remote relationships can still be interpreted as evidence that partial orderings are not represented and accessed in the same way as linear orderings, even when special precautions are taken to ensure that all the information has been properly stored and integrated. On the other hand, both studies of complex, partial orderings (Hayes-Roth & Hayes-Roth, 1975; Moeser & Tarrant, 1977) differed from research on linear orders in several important ways. These differences might explain why the pattern of reaction times was different. One early hypothesis was that the experiments on partial orderings had used much larger structures. The number of elements in the partial orderings was 12, compared to at most 6 in Potts (1974) and Trabasso et al. (1975). Indirect evidence now suggests that the number of elements in the ordering is probably not responsible for the difference in results. Another line of investigation has looked more closely at the procedures used in this research. The results here are less clearcut; however, we argue in a subsequent section that these results tell us as much about the complex relationship between the distance effect and mental organization as they do about the difference between partial and complete orderings. There are two lines of evidence that the number of elements in an ordering cannot explain the difference between results from experiments on partial orderings and linear orderings. First, Pliske and Smith (1979) and Woocher, Glass, and Holyoak (1978) have reported distance effects using linear orderings of 12 and 16 terms, respectively. Second, Warner and Griggs (1980) had subjects learn a seven-term partial ordering under a variety of conditions designed to ensure that the information was correctly represented. In spite of these efforts, no distance effect was observed. Thus, it appears that the distance effect reflects the organization in memory of information from linear, but not partial, orderings. Unfortunately, no studies have directly compared distance effects in complete and partial orderings using the same procedures and testing the same relationships. The one study that has compared complete and partial orderings of the same size (14 elements in Moeser, 1979) failed to obtain a distance effect for both linear orderings and partial orderings. However, the procedure of this study was quite different from any discussed so far. Procedural variations seem to account for a good deal of the confusion in the literature on partial orderings. First, investigators have used a variety of methods to present the relationships that make up an ordering. Both Hayes-Roth and Hayes-Roth (1975) and Moeser and Tarrant (1977) used elaborate training sequences made up of many exposures to the

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relationships. By contrast, Pliske and Smith (1979) and Woocher et al. (1978) gave people a list to learn before coming to the laboratory and tested the success of this procedure by requiring each subject to recite the list in order. Second, the partial order studies have analyzed the time required to judge whether an assertion, such as “Carl is older than Mike,” is true or false (sentence verification procedure). Many of the linear ordering studies have used a procedure in which the subject is shown two terms (e.g., “Carl Mike”) side by side on a display screen and required to press a response key under the older of the two (twochoice procedure). Polich and Potts (1977) compared the two procedures and found that the verification procedure produces interactions between the presence of an end-anchor (highest or lowest ranked element in the ordering) and whether the sentence is true or false. Not only were such interactions absent in the two-choice procedure but Polich and Potts also reported that the overall variability of the response times was significantly less with this procedure. The importance of these procedural variations is dramatically illustrated by the two experiments that constitute a master’s thesis by Pliske (1978). In the first, unpublished experiment, subjects learned a 12-term linear order and were tested on series of adjacent relationships and a selected subset of the possible remote relationships. The method was designed to follow as closely as possible the one used by Moeser and Tarrant (1977), but without any special training on how to represent the ordering. Response times were highly variable; and the effect of distance, although evident and statistically significant, was not nearly as straightforward and compelling as that obtained in the second, published experiment (Pliske & Smith, 1979), in which subjects studied the list on their own and were tested with the two-choice procedure. Such variations in method may be more important than has been previously realized. For example, we cannot rule out the possibility that the distance effect is to some extent a reflection of the acquisition and testing procedures used in these larger sturctures. Two recent papers by Griggs and his students (Griggs, Keen, & Warner, 1980; Warner & Griggs, 1980) explored the procedural variations already discussed, along with several others. In no case were distance effects obtained for partial orderings. Elaborate preliminary instructions about the nature of partial orderings and their representation in graphical form did not lead to distance effects. Warner and Griggs (1980) found that without exposure to a graphical representation of the information, less than 60%of their subjects’ responses were consistent with the correct seven-term partial ordering. Only when subjects were required to draw

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the correct graph from memory on two consecutive trials and were then trained on the adjecent comparisons to two consecutive correct trials, did they respond correctly to remote comparisons on the test. Even this rigorous program did not lead to a distance effect, although this approach can be criticized because the extensive training on adjacent comparisons may have facilitated responses to them. Warner and Griggs’s third experiment comes the closest to matching the procedures of the earlier studies with large linear orderings. Following preliminary instructions on the nature of partial orderings and their graphical representation, subjects were given a graph of either a 7- or 12item partial order and told to memorize it for a second experimental session. At the beginning of the second session, the subject had to draw the structure both before and after familiarization with the test procedure. Finally, testing made use of a modification of the two-choice procedure similar to that used by Pliske and Smith and Woocher et al. The modification involved the addition of a third button to be used when a pair of items were indeterminate, that is, not ordered by the information given to the subject. In spite of the procedural similarities, Warner and Griggs found a reverse distance effect for the 12-item partial order used by Moeser and Tarrant. (It should be noted that 2 of the 20 subjects in this condition of Warner and Griggs’ experiment were unable to learn the correct structure.) In summary, 6 years of research suggests that the information in partial orderings is more difficult to memorize than similar completely ordered information. However, after receiving special instructions about the nature of incomplete ordering, accompanied by graphs and practice in using them, most college students are able to judge whether or not a given pair of elements is ordered and to draw correct inferences about the relationship between pairs of elements that are ordered but not specifically presented. The only evidence that the partial orderings are represented in an inherently different fashion is the failure to find a distance effect for judgments requiring inferences. The correct interpretation of this difference depends on how the distance effect is interpreted for complete sets of elements (cf. Potts, Banks, Kosslyn, Moyer, Riley, & Smith, 1978). In any case, these conclusions are based on investigations of a remarkably small sample of different partial orderings. No arguments have been offered to support the claim than the sample is representative. If our survey is complete, exactly four partial orderings have been examined, the two 12-item orderings shown in Fig. 1B and lC, a 7-item ordering studied by Griggs and his students, and a 14-item ordering that Moeser (1979) compared with a complete ordering.

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111. Overview of the Experiments As part of a research project dealing with the construction of linear orders (Foos, Smith, Sabol, & Mynatt, 1976; Mynatt & Smith, 1977; Smith & Foos, 1975), we became interested in partial orderings or networks because they seemed to be a rich domain into which we could extend our theory of constructive processes (see Foos et al. and Smith’s section of Potts et d.).Our work has focused on the construction of four-, five-, and six-element linear orderings. With such small sets of relationships, the construction of a branch or node from two relations (e.g., A > B, A > C) did not appear to be fundamentally different from the construction of a linear ordering (e.g., from A > B, B > C). Indeed, an early study in our laboratory (Smith & Mynatt, 1975) indicated that four- and five-term partial orderings were no more difficult to construct than similar-sized linear orderings. These preliminary observations were in sharp contrast to the previously published studies we have reviewed. In what follows, the first experiment we report was an investigation of the distance effect in retrieving information from partial orderings. One condition of the experiment was essentially a replication of the experiments by Moeser and Tarrant (1977) and Warner and Griggs (1980). It differed from the previous studies mainly in the additional procedures included prior to testing in order to guarantee that subjects understood the indeterminacy of partial orderings and had learned the specific adjacent relationships. A second condition tested another structure with the same number of elements (1 2) but a configuration similar to the hierarchical network studied by Nelson and Smith (1972). The second experiment reported below was designed to explore the diversity of structural configurations possible in 12-element partial orderings. A less elaborate testing procedure was used, and the focus was on whether subjects could answer questions and draw accurate diagrams on the basis of a set of sentences describing a partial ordering. The sentences were continuously in view in order to eliminate the effects of memory storage and retrieval. Results of the first two experiments were interpreted as evidence that, at least for college students, partial orderings can be learned and the resulting knowledge is not fundamentally different from what is learned in a linear ordering. The last three experiments, using four- and five-element orderings, were concerned with the process of construction. How are the relationships in individual sentences combined to form mental networks? The first of these experiments contrasted the process of extending a linear ordering with the process of building a node or branch (e.g., the structure

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involving J, R , and D, at the top of Fig. 2). The second experiment investigated the construction of different types of nodes; the third was concerned with the effects of context on constructive processes. Different contexts-in this case the sentence frames used to express relationshipswere expected to elicit more or less appropriate representational schemas from the subjects’ permanent memory. Throughout these last three experiments, the Foos et al. theory of constructive processes was extended and modified to apply to networks or partial orderings.

IV.

Experiment 1: Retrieval from Partial Orderings

None of the recent studies of partial orderings has found a distance effect, in which responses to remote relationships are faster than to adjacent ones. Various attempts (Hayes-Roth & Hayes-Roth, 1975; Moeser & Tarrant, 1977; Warner & Griggs, 1980) suggest that this failure cannot be attributed to differences in training procedures, response measurement procedures, or to the number of elements in the structure. However, only four configurations have been investigated. Each of these four structures seem arbitrarily complex and unlike anything a college student might have encountered previously. The conclusion that distance effects cannot be obtained in any partial ordering is obviously premature. A more familiar and intuitively simpler configuration of relationships that form a partial ordering is a hierarchy or family tree. The present experiment compared retrieval time for information in the hierarchical network shown in Fig. 2 with comparable performance for the irregular network used by Moeser and Tarrant. The two structures have the same number of elements (12). The elements were one-syllable given names from Battig and Montague’s (1969) norms, and the relations between them were described as age relations. A.

METHOD

The subjects were 22 undergraduate students at Bowling Green State University. Their participation partially fulfilled a course requirement for introductory psychology. The 11 students in the “hierarchy” condition worked with the relations graphed in Fig. 2. The 11 students in the “irregular” condition worked with the relations that form the irregular partial ordering (see Fig. 1C) studied by Moeser and Tarrant, and Warner and Griggs. The training phase was based to some extent on the procedures used by Nelson and Smith. It had several steps and attempted to expose the

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Fig. 2. The hierarchical network that subjects in the hierarchy condition of Experiment 1 learned.

subjects to the various properties of the order in a thorough but relatively unstructured way. The subjects were first given a sheet containing 11 sentences describing the age relation between adjacent elements from the order and were asked to draw a diagram representing the information presented in the sentences. The experimenter checked these drawings and discussed any inaccuracies with the subjects. The subjects were then asked to verify four other diagrams representing the same information in four somewhat different ways. These diagrams had lines that were longer or shorter or of varying length and had different arrangements of the branches. Two of the diagrams had minor errors in them, and two had no errors. The purpose of this phase was to allow the subjects to see that a variety of diagrams could be accurate. Again the subjects’ responses were immediately scored and discussed. The subjects then read 26 sentences describing possible age relations between the names and decided whether each sentence was true, false, or indeterminate based on the information presented in the original sentences. Feedback was also given to the subjects on these decisions. The next training task required the subjects to fill in a 12 X 12 matrix which had the 12 names printed along the left and top borders. Subjects were instructed to place a checkmark in every cell in which the name from the row was older than the name from the (intersecting) column. These responses were corrected and discussed. Up to this point, the subjects had the original sentences and all other materials available for reference. However, the subjects were told at the outset that eventual memorization of the relationships would be necessary. At this point, the subjects were asked to study any or all of the materials as long as they wished, until they felt they knew the material completely. They were then given one final test in which they placed check marks on another arrangement of the 12 X 12 matrix. Following the training phase, subjects retrieved information from memory about the relative age of pairs of names in the partial ordering. Pairs of names appeared on an Owens-Illinois Digi-Vu screen, and response times were recorded as subjects pressed one of two marked keys

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on a keyboard under the name of the older person or pressed the space bar to indicate an indeterminate relation. Presentation of the name pairs and response records were under the control of a Nova 1220 computer. Each block of trials consisted of 78 name-pair presentations. The composition of the trials depended on the condition. For the hierarchical structure, on each block of trials all 26 possible determinate relations were presented, including 11 adjacent relations, 9 remote relations with a step size of 1, and 6 remote relations with a step size of 2. (Step size is defined by the number of elements between the two test items.) Each of these was presented once in a left-to-right order on the screen, and again in a reversed order. A subset of 26 of the 80 possible indeterminate relations were also included in each block. To equate the hierarchy condition with the irregular condition as much as possible, subjects in the latter condition were likewise tested on 26 determinate relations presented in both forward and reverse orders and 26 indeterminate relations. However, these relations do not exhaust all the possible relations in either category. Of the determinate relations, 7 were adjacent pairs, 5 had a step size of 1, 5 a step size of 2 , 4 a step size of 3, and 3 a step size of 4, and 2 a step size of 5. Each subject was required to complete three blocks of trials in which their error rate did not exceed 5%. B.

RESULTSAND DISCUSSION

The mean number of trial blocks required to meet the criterion was 3.7 blocks for the hierarchy group and 4.4 for the irregular group. The difference was not significant, although it was in the direction of our original hypothesis that the irregular structure is more difficult to master than the hierarchy. Response times on error trials were replaced with the mean of response times for correct trials of the same type. The response times from the irregular condition parallel other reported results: There was no evidence of a distance effect like that found for linear orders. In fact, adjacent relations, with a mean response time of 2.80 sec, produced significantly faster responses than the nonadjacent relations, with a mean of 3.11 sec, F(1, 10) = 14.7, MSe = 2.16. However, response times to pairs from the hierarchical structure showed distance effects: Response time decreased as step size increased (see Table I). Analyses showed that response time to the adjacent pairs, with a mean of 3.51 sec, was slower than to the remote pairs, with a mean of 2.31 sec, F ( 1 , 10) = 72.6, MS, = 8.25, and that 1-step pairs, with a mean of 2.91 sec, were slower than 2-step pairs, with a mean of 1.41 sec, F(1, 10) = 69.5, MS, = 7.74. A more detailed analysis of the data for the irregular condition failed to

I25

Orderings in Memory

TABLE I MEANRESPONSETIME(SEC)To DETERMINATE PAIRSAS A FUNCTION OF STEPSIZEFOR THE HIERARCHY CONDITION Step size

0

1

2

1.49 3.52 3.14

1.41

N

1.34 3.40 3.60 4.50 3.66 4.06

Meanb (with J): Mean (without J):

3.51 3.99

2.91 3.63

1.41 -

Older term in paira J R

D P

G

“The actual elements used in the structure were one-syllable first names. For convenience, only the first letter of the name is used. Refer to Fig. 2 for the placement of each term in the structure. bThe last two rows present weighted means. Note that certain names have more than one relation of a certain type, so that the means given do not always equal the means of the corresponding column of table values.

reveal any obvious trends or patterns. However, the data from the hierarchy condition suggest that the obtained distance effect consisted of two components. First, any pair of names containing J, the element at the top of the hierarchy, led to faster responding than pairs not containing J. As can be seen in Table I, this was true for both adjacent pairs, F(1, 10) = 57.7, MSe = 13.09, and for pairs of step size 1, F(1, 10) = 40.0, MS, = 14.09. (All pairs of step size 2 involve J.) The fact that J is the correct response to any pair containing it appears to confer on it a special status. Subjects can store this specific information with the term in memory and use it in making a rapid, categorical decision in much the same way that Pliske and Smith’s (1979) subjects used the gender of names in a linear order to make rapid decisions when all names of one gender preceded names of the other gender in the ordering. However, even when pairs containing J were removed from the analysis, there remained a distance effect of the kind typical for linear orderings. The last row of Table I presents the means for adjacent and remote (step size 1) pairs that did not include J. The difference was significant when tested with a comparison that was not orthogonal to those given earlier, F(1, 10) = 24.7, MSe = 1.43. A second component of the distance effect shown in Table I is a significant increase in response times to adjacent pairs from the second

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Kirk H.Smith and Barbee T. Mynatt

level of the hierarchy (R and D) compared to the third level (P, G, and N), F( 1, 10) = 7.69, MS, = 8.01. (This planned comparison is orthogonal to all but the last one presented above.) This pattern of results suggests a search process operating like the spread of activation (cf. Collins & Loftus, 1975) from the top of the hierarchy (J) and directed downward. If such a search process is assumed to terminate only when both members of pair have been located (as would be necessary to classify a pair as indeterminate), then most of the results in Table I fall into place. For example, responses to pairs of step size 1 containing R and D (3.52 and 3.74 sec, respectively) took longer than the adjacent pairs containing these letters (3.40 and 3.60 sec, respectivley). The apparent distance effect is the result of averaging response times for adjacent pairs at different levels of the hierarchy, (i.e., pairs involving P, G, and N). The results in Table I can also be seen to display an effect similar to that of a propositional fan (Anderson, 1976). Response times were shorter to pairs containing R, with only one subordinate, than to D, with two. And response times to pairs containing G, N, and P, with one, two, and three subordinates, respectively, increased as expected. (The mean response time to indeterminate pairs, 4.35 sec, was significantly longer than the mean to determinate pairs, 2.82 sec, t(l0) = 8.21, SE, = .375; however, we could find no easily interpretable trends in the times for indeterminates.) C. CONCLUSIONS AND IMPLICATIONS

The pattern of results from the hierarchical structure displayed a traditional distance effect (when analyzed in the usual way), whereas for the irregular structure the pattern was reversed. What conclusions should be drawn from this outcome? Following the logic of other studies of partial orderings published within the last 8 years, we might conclude that the hierarchy was represented and stored like a linear ordering, but the irregular structure was not. It is interesting to speculate on the direction research on partial ordering might have taken if Hayes-Roth and HayesRoth (1975) had chosen to investigate a hierarchical structure. We contend that the search for a set of conditions that lead to a distance effect with partial orders has been misdirected. The presence of a distance effect can be the result of averaging the response time data for adjacents and remotes (or even for rank-ordered distances) in ways that obscure the effects of very different variables and processes. Our data for the hierarchical structure illustrate this kind of confusion quite clearly. One hypothesis that is consistent with all the data discussed so far is a two-process model such as Pliske and Smith (1979) have suggested to explain retrieval times for linear orderings. The general form of this

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model contains two components, a rapid decision process based on specific categorical information about the elements of an ordering and a slower systematic serial search process that moves from element to element along the learned connections among them. For linear orderings, an example of categorical information is the gender of names used in an ordering. The parallel in the hierarchy would be responses to pairs of names containing the topmost element (J). An example of serial search processes in linear orderings is the proposal that subjects search the ordering from the ends inward (cf. also Woocher et a/., 1978). The parallel for a hierarchy is the spreading activation notion discussed above. The irregular structure may have been learned in the same way as the hierarchy (or a linear ordering, for that matter), and the retrieval processes may have been basically the same. The difference is that irregular structures do not have a small number of elements that can be uniquely categorized by their structural properties; and sequences of the serial search (or the pathways of spreading activation) may be more idiosyncratic from subject to subject, or even from trial to trial for the same subject. In effect, irregular structures such as Figs. 1B and 1C may be interpreted as situations in which only the underlying search processes are manifest in the data because the effects of other retrieval processes, especially rapid categorical decisions, have been randomized. However, if this interpretation is correct, investigations of partial orderings should have used many more irregular structures. It also follows from this interpretation that patterns of retrieval time reflect a great deal more than the incorporation of a set of relations into an integrated memory representation. Because our goal was understanding how people construct and represent partial orderings, we drew an important lesson from our first experiment. The time taken to retrieve comparative information from an ordering is influenced by many factors other than the process of understanding and representing the information as an integrated whole. A more appropriate methodology is needed to investigate how people combine relationships and construct integrated representations of partial orderings. There is no doubt that the process is more difficult for partial orderings than for linear orderings, but it can be done. The question is how.

V.

Experiment 2: The Role of Determinacy in Constructing Partial Orderings

In reviewing research on partial orderings, we noted that a very limited number of structures have been intensively investigated. Whatever conclusions have been reached cannot really be generalized to "partial order-

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ings,” but must be confined to these few specific structures. The obvious remedy is a systematic exploration of the domain; however, the number of possible configurations of partial orderings is surprisingly large and diverse. A feeling for this diversity can be gotten from Fig. 3, which illustrates five possible partial orderings of 12 elements. The only configuration in Fig. 3 that has been investigated previously is Fig. 3B, the ordering devised by Moeser and Tarrant (1977). At present there is no way of guaranteeing that the structures shown are representative of partial orderings, even with the restriction that exactly 12 elements be ordered. The partial orderings in Fig. 3 were chosen according to several principles derived from our intuitions about the possible sources of difficulty people have in understanding and remembering the relations in such structures. Figure 3B was included as a reference point, since it has been investigated not only by Moeser and Tarrant, who introduced it, but also by Warner and Griggs (1980) and by us (reported in Section IV). It has been clearly established that with appropriate background and procedures of presentation, college students can learn the relationships involved in this ordering and can draw the correct inferences about remote relationships (even though the speed of their performance may not correspond to that of subjects who are retrieving information from a linear ordering). The question raised here is whether people have more or less difficulty with the other structures. If so, we wanted to isolate 1he reasons for these difficulties. A

B

C

H

I

H

L

I\

F

I

I

F

I

G

D

E

47\7\/D ,7 A l

J

J

E

G

G

D

\ I..

I F I

\ /

H

B

K

G

I

L

D

/A\ E

/A\ F

G

H

A\ I

J

Fig. 3. The partial orderings investigated in Experiment 2.

K

L

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The fundamental difference between partial and linear orderings is that the former leave the relationship between some pairs of elements indeterminate. One obvious way in which partial orderings can differ is in the extent of this indeterminacy. For example, 12 elements have 66 possible pairwise relationships. A linear ordering specifies 11 of these, which in turn determine all of the remaining 55 relationships. Thus, a linear ordering is a complete ordering; there are no indeterminate relations. By comparison, Moeser and Tarrant’s partial ordering (Fig. 3B) leaves 25 of the 55 potentially determinable relations indeterminate. The remaining four structures in Fig. 3 were chosen for study in part on the basis of their level of indeterminacy. Figure 3A is in some ways a much simpler, more orderly configuration than Fig. 3B, which has received so much attention; yet Fig. 3A has the same degree of indeterminacy. Specifying 11 of the 66 possible relationships between pairs of elements leaves 25 of the remaining 55 relationships indeterminate. By contrast, Fig. 3C seems intuitively very similar to Fig. 3B, but in fact has more indeterminate relations-32 out of 55, instead of 25. Figure 3E, which is a hierarchy similar to that investigated in Section IV, has 46 indeterminate relationships, a still higher level of indeterminacy. (The hierarchy in Fig. 2 has 45 indeterminate relations.) Figure 3D was constructed to have almost the same number of indeterminate relations (45) as Fig. 3E, but to look very different, at least superficially. The five structures selected for study can be seen to possess levels of indeterminacy similar to either Moeser and Tarrant’s irregular structure or our hierarchy, with the exception of Fig. 3C, which falls in between. In fact, Fig. 3C has a little more than half indeterminate relations (58%), although it “looks” a lot more like Fig. 3B than it does like Figs. 3A, 3D, or 3E. The only difference is that in Fig. 3B, H is greater than B, whereas in Fig. 3C, H is less than A. One further dimension distinguishes the five structures. Figures 3A, 3B, and 3C all have one long linear ordering of 7 elements, ABCDEFG, whereas in Figs. 3D and 3E, the longest chain is 3 elements long. Although the differences outlined above may not correspond to the relevant cognitive dimensions of such configurations, they seem to reflect the diversity that exists within the domain of 12-element structures. In order to find out whether these structures differ in difficulty, we gave subjects a set of 11 statements describing adjacent relationships among the 12 elements and tested whether they could draw a diagram representing the partial ordering and answer questions about the nonadjacent relations implied by the ordering. We selected a paper-and-pencil version of the task in which all 11 specified relations were available for

Kirk H. Smith and Barbee T. Mynatt

I30

inspection during testing and subjects could work at their own pace. Our purpose was to find out whether people could understand partial orderings of this size and complexity, independent of the demands made on memory to retain the 11 relationships. The present experiment was designed to determine whether there are aspects of partial orderings for which people do not have a readily accessible schema. A more global source of difficulty may be the fact that comparisons of age per se, especially among several persons or objects, are most easily made in terms of numerical values rather than rankings. Thus, people may assume that a set of age comparisons form a complete ordering. Another familiar relationship, ‘‘parent of, ” implies relative age but only within broad limits. A set of sentences such as “Mary is the mother of Ted” and “Sam is the father of Rita” might be expected to lead to better understanding of a partial ordering. Although the effect of changing the sentence frame seems minimal, De Soto (1960) found that the same structure differed in difficulty depending on the sentence frame used. In the present experiment, the partial orderings were presented as either parent-of or older-than sentences, although the test sentences were age comparisons (older than) in all cases. A.

METHOD

The subjects were 200 undergraduate students drawn from the same source and in the same manner as for the first experiment. Five groups of 40 students were assigned to work with each of the five partial orderings. Within each group, half of the subjects drew the diagram first, then answered questions; the other half answered questions first. The subjects in the diagram-first conditions, when given the instructions for answering questions, were told that they could refer back to their diagrams. The subjects were run in small groups of 2 to 8. Verbal instruction asked them to follow a set of printed instructions and to ignore whatever a neighbor might be doing, because each person had different materials. Each subject received a three-page booklet containing materials and instructions. For both tasks, 11 sentences appeared at the top of the page. For half the students, the sentence had the form “Beth is older than Dave”; for the other half, the sentence had the form “Beth is the mother of Dave.’’ The 11 sentences were the set of adjacent relations necessary to describe one of the 12-element networks shown in Fig. 3. The six male and six female names used as elements were again chosen from Battig and Montague’s (1969) norms. The order of the sentences on the page was randomly determined. For the diagram-drawing task, the instructions asked the subject to draw a tree or diagram which would accurately represent the age relation-

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ships among the people described by the sentences. They were told to use arrows to connect the names, with the head of the arrow pointing toward the younger person. Space for the drawing was provided at the bottom of the sheet. For the question-answering task, 33 statements were listed below the 11 sentences defining the order, and the subject was told to read each statement and decide whether it was true, false, or indeterminate based on the information in the 11 sentences at the top. Three columns of blanks were printed next to the statements with the headings, “True,” “False,” and “Can’t Tell,” allowing the subject to put a check in the appropriate column to indicate an opinion about each statement. The types of statements used with each of the five partial orderings were randomly chosen from among the 132 possible statements in proportion to the occurrence of each type. For example, for Fig. 3E there are 20 possible true statements, 20 possible false statements, and 92 indeterminate relations. Of the 33 statements tested, approximately 68% (23) were indeterminate, 15% (5) were true, and 15% (5) were false. The indeterminate statements were randomly selected from all possible indeterminates. The indeterminate statements were randomly selected with the restriction that the proportions of adjacent and remotes were matched to their proportion of occurrence in the set of all possible determinates. B.

RESULTSAND DISCUSSION

The number of subjects in each condition who drew completely correct diagrams on the basis of 11 relationships is shown in Table 11. In general, subjects tended to be more successful when the sentences described the TABLE I1 NUMBEROF SUBJECTSWHOSEDIAGRAMS WERECOMPLETELY CORRECT IN EXPERIMENT 2 Structure Condition * Diagram first Older Parent Questions first Older Parent

A

B

C

D

E

9 10

7 9

6 9

6

7

8 9b

5 8

8 6

4

8 8

9 96

*A total of 10 subjects were tested in each condition

8

bDue to an error, two subjects, one in each of the indicated conditions, were not requested to draw a diagram.

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Kirk H. Smith and Barbee T. Mynatt

parent-child relationship than when the same elements were related in age, 83% vs 70%, respectively; x2(1) = 5.28, p < .05. The differences among the five structures were not significant; x2(4) = 7.75, p < .lo. There was also no indication that answering a set of 33 questions about the implications of the 11 sentences had any impact on success in drawing a diagram, or vice versa. Of the 200 subjects, 8 drew linear orderings and 4 drew two separate orderings. These errors were not confined to any one condition, however. The errors made in answering questions were compiled separately for determinate and indeterminate relations and converted to percentages, which are shown in Table 111. These values were submitted to an analysis of variance in which type of question, determinate or indeterminate, was a repeated measure. The .01 level of significance was used as a criterion for discussing any comparison. The findings are presented here in order, beginning with the effect that accounted for the greatest proportion of total variance attributable to experimental manipulations. Subjects made significantly more errors on indeterminate than on determinate relations, F( 1, 180) = 107.1, MSe = 324.6, and this difference accounted for 37% of the variance. The effect was consistent across all conditions of the experiment. The five structures differed significantly in difficulty, F(4, 180) = 16.2, MSe = 306.8, accounting for another 21% of the variance. However, examination of Table I11 suggests that the effect of structure was different for indeterminate and determinate relations. The interaction was significant, F(4, 180) = 10.5, MSe = 324.6, and accounted for 15% of the variance. The percentages in Table I11 make clear that our original hypothesis that difficulty is affected by the amount of indeterminacy was incorrect. Subjects made the greatest number of errors on Fig. 3C, in which the level of indeterminancy is intermediate between Figs. 3A and 3B (with TABLE I11 PERCENTAGE OF ERRORS ON QUESTIONS ABOUT RELATIONSHIPS IN FIVE STRUCTURES TESTEDIN EXPERIMENT 2" Figure 3 structure Type of relationship

A

B

C

D

E

Overall

Determinate Indeterminate Overall

7 25 16

17 27 22

12 51 31

21 16

5

8 13 11

10 29

OPercentages based on different numbers of questions for different structures of Fig. 3.

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relatively few indeterminate relations) and Figs. 3D and 3E (with many). For Fig. 3C, questions about indeterminate relationships seemed to pose the greatest difficulty. The only subjects who failed to get any indeterminate questions correct were, with a single exception, in this condition (five subjects out of six). Figure 3D was more difficult than Fig. 3E because the indeterminate relations led to more errors, but Moeser and Tarrant’s structure (Fig. 3B) was more difficult than Fig. 3A because the determinate relationships produced more errors. Long-chain structures (Figs. 3A, 3B, and 3C) did not emerge as strikingly more difficult than short-chain structures (Figs. 3D, 3E). Structures with certain elements connected to many other elements (especially Fig. 3E and, to a lesser extent, Fig. 3D) were not different from structures with little multiple connectedness (e.g., Fig. 3A). (The average number of elements connected to a given element-a measure of the fan effect, Anderson, 1976-is the same for all the Fig. 3 structures.) The effect of the sentence frame on the question-answering data was consistent with the subject’s success in drawing correct diagrams. Fewer errors were made on questions about partial orderings based on “parent” sentences than on “older” sentences, F ( l , 180) = 13.5, MS, = 306.8. This significant difference, which accounted for 4.4% of the total variance, was primarily due to the reduced numbers of errors to questions about indeterminate relations when “parent” sentences were used. This interaction was also significant, F( 1, 180) = 9.4, MS, = 324.6, accounting for 3.3% of the variance. Finally, the only effect of procedure was on determinate questions. Fewer errors were made on determinates when subjects had first drawn a diagram of the ordering (6%) than when they answered questions first ( 14%), whereas errors on indeterminates were unaffected (29 and 28% error rates for diagram-first and questions-first conditions, respectively). This interaction accounted for only 2.5% of the variance but was significant, F ( l , 180) = 7.3, MS, = 324.6. One interpretation of this result is that a diagram does not aid the subjects’ understanding of indeterminate relations, although it helps in making inferences about remote relationships. However, subjects received no feedback on the diagrams they made. Thus, their errors on questions after drawing a diagram may have been a reflection of their initial misunderstandings in composing a diagram. C . CONCLUSIONS

The results of this experiment show that partial orderings can differ considerably in difficulty. By difficulty we refer to the problems people have in understanding the implications of a set of relationships that does

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not assign each element a unique rank. The results for both diagram drawing and question answering support the contentions of earlier investigators that the necessary schema for representing partial orderings is not very salient. Our subjects generally did better on both diagrams and questions when the sentences described parent-child relations than when they simply stated that one individual was older than another. We assume that parent-child relationships awaken a schema of age ordering that admits indeterminancies. That subjects in the parent-child condition did better on questions about the relative age of members of a family and that the principal gain in performance was in answering indeterminate questions are especially compelling evidence for the importance of the schema. These results also indicate that our subjects, who were college students, did not lack the appropriate schema altogether. Rather, they seemed to be much less likely to use it with pure age relations. Instead of a predilection for linear orderings in all situations, people appear to gravitate toward the linear ordering in situations where both (a) complete ranking is possible and (b) interval measurement makes sense. Although our results indicate that much of the difficulty with partial orderings has to do with indeterminate relations, no conclusion about the importance of various structural variables seems possible. In particular, the amount of indeterminacy (the proportion of indeterminate relationships among the unspecified ones) in a partial ordering does not affect its difficulty in any simple, monotonic way. (The only hypothesis consistent with our data is that difficulty increases as the ratio of indeterminate to determinate relations approaches unity. However, even this idea is limited; structures with roughly the same ratios of indeterminancy showed differences in accuracy of question answering.) One important lesson about structural variables may be gleaned from our results. There is no reason to suppose that a haphazardly selected configuration of partially ordered relationships is representative of such structures. It is important to stop generalizing about “partial orderings” and begin a more systematic search for the processes people use in integrating and understanding this kind of information. VI.

Experiment 3: Node Construction

Presentation order is another factor that has been suggested as a possible explanation for the difficulty people have in correctly integrating the components of a partial ordering. Moeser (1979), for example, compared three groups of subjects (Experiment 3), one which learned a partial ordering and two which learned a linear ordering. One of the latter groups

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was exposed to the sentences describing relationships (specifically, “Todd is older than Herb”) using what Moeser labeled a “match” order, that is, A > B, B > C, C > D, . . . , M > N. The other linear order group received the sentences in a “nonmatch” order so that the ninth sentence introduced two names not previously mentioned (a nonmatch situation). Three more sentences were given before a sentence referred again to a name in the seventh sentence, and not until the twelfth sentence was there sufficient information to complete a 13-element ordering. All subjects received instructions on how to represent partial and linear orderings with diagrams and how to answer questions about them correctly. Moeser found that subjects in the linear match condition did better than those in the other two conditions, which did not differ. On the basis of this finding, she concluded that some of the difficulty in integrating and understanding a set of partially ordered relationships is due to the fact that partial orderings must always be presented in nonmatch orders. In spite of its plausibility, Moeser’s argument does not really explain either her results or the difficulty of partial orderings. The terms “match” and “nonmatch” are taken from a theory of linear order construction proposed by Foos et al. (1976). Detailed application of this theory to the presentation orders Moeser used reveals that they are not of comparable difficulty. In the Foos et al. theory, match orders require the subject to add new elements to the ordering by one of two processes, both of which involve locating a single element in common between the ordering previously constructed and the new relationship. (This common element is the “match.”) The M1 process detects a match at the end of a previously constructed ordering. For example, given B > C > D, the relation D > E is added by the M1 process. The M2 process detects a match at the beginning of a previously constructed ordering, so that given B > C > D, the relation A > B is added by the M2 process, which has been shown to be slightly more difficult than the M1 process. A nonmatch order in the Foos et al. theory requires at least one nonmatch situation, for example, when A 3 B > C > D is established and E > F is to be added eventually (but must be held temporarily separate in memory). There is no reason why a partial ordering has to be presented in a nonmatch order. For example, in Moeser’s 16element ordering, once the central %element linear ordering has been presented, the remaining comparisons may be added by processes comparable to M1 and M2. For example, given the previously constructed chain B > C > D > E, branches can be added by relations such as C > Q (parallel to M1) and X > D (parallel to M2). While Foos et al. did not consider such nodeconstruction processes, the process of locating a common element (the “match”) would be the same. Whether a node is more difficult to con-

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Kirk H. Smith and Barbee T. Mynatt

struct than a line is an empirical question, which the present experiment considered. Moeser did, in fact, introduce a nonmatch situation in presenting the partial ordering. However, the nonmatch was not introduced at precisely the same point in the presentation order for partial and linear orderings. For the partial ordering, the first true nonmatch occurred on the eleventh comparison and was immediately followed by a resolution (or “doublematch,” D1, process). A recent study by Foos and Sabol (1981) has reported that as the number of comparisons between the nonmatch and its resolution increases, performance on nonmatch orders declines. Whereas the nonmatch was immediately resolved in the partial ordering (no intervening comparisons), three comparisons intervened in Moeser’s linear nonmatch condition. The Foos and Sabol data were based on shorter, simpler orderings (strings of six letters to be constructed from five letter pairs), so the results may not be strictly comparable. However, the point is that Moeser’s three conditions confound a number of variables known to affect the difficulty of constructing linear orderings. The linear match condition will obviously be the easiest, not only because a match order was used but also because only the M1 process was required. The linear nonmatch condition has a substantial delay following the nonmatch, making it more difficult than the partial ordering; but the latter contains match processes that result in nodes, the difficulty of which is unknown. The difference between MI and M2 processes suggests that other match processes are unlikely to be as easy as M1. A good deal of the confusion about the difficulty of different presentation orders is probably due to the fact that subjects cannot master a 14element ordering of any kind in one exposure. Beginning with the second repetition, the classifications of Foos et al. do not apply, strictly speaking. The subjects have heard all the elements once, so no comparison can pose a nonmatch of the kind envisioned by the theory. Moreover, a question such as the relative difficulty of adding a node is best posed in designs similar to those used in unraveling the effects of presentation order in constructing linear orderings. Therefore, in the remaining experiments, we limited our attention to orderings with only a few elements and explored a wide range of presentation orders. We have not been exhaustive, however, as were Foos et al. The reason is that partial orderings create a much more extensive range of possibilities, even with orderings of only 5 elements. In the first experiment, we compared the difficulty of constructing either a linear order or one of the three partial orders shown in Fig. 4. The presentation orders are given in Table IV along with an analysis of the construction processes specified by the Foos et al. theory. All presen-

Orderings in Memory

A

B

C

A

A

A

/ \

B

I

I37

I I

I

E

c

c

D

D

I

I

B

'E D

E

Fig. 4. The three partial orderings constructed by subjects in Experiment 3. They differ in the location of the node or branch relative to the four-term linear ordering ABCD.

tation orders consist of three sentences that can be combined to form a four-term linear ordering. Half the presentation orders in Table IV accomplished this with process M1 (orders 1, 2, 5 , 7, and 9) and the other half with process M2 (orders 3 , 4 , 6, 8, and 10). The last sentence in each presentation order either completed a five-term linear ordering or resulted in one of the networks. Completing a linear ordering required either process M1 (orders 1 and 3) or process M2 (orders 2 and 4). In the TABLE IV MEANPROPORTION OF CORRECT TRIALSAS A FUNCTION OF PRESENTATION 3 ORDERIN EXPERIMENT Constructive processb involved Presentation order"

After second sentence

1. AB,BC,CD,DE 2. BC,CD,DE,AB 3. CD,BC,AB.DE 4. DE,CD,BC,AB

MI M1 M2 M2

5 . AB,BC,CD,AE

M1 M2 MI M2 MI M2

CD,BC,AB,AE AB,BC,CD,BE CD,BC,AB,BE AB,BC,CD,CE 10. CD,BC,AB,CE

6. I. 8. 9.

After third sentence Linear orderings M1 MI M2 M2 Networks MI M2 M1 M2 MI M2

After fourth Sentence

Mean proportion correct

MI M2 M1 M2

.62 .43 .35 .42

Nd Nd Nd Nd Nd Nd

.42 .45 .33 .29 .51 .28

<"Forconvenience, all presentation orders describe the linear ordering ABCDE or one of the three networks shown in Fig. 2. bConstructive processes are identified by the abbreviations of FOOSe r a / . (1976).

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remaining 6 presentation orders, the last sentence required the formation of a “node” or branching point in the mental representation. The process of adding a branch or creating a node was labeled process Nd. A.

METHOD

A total of 24 sets of four sentences were tape-recorded and presented to subjects at a rate of one sentence every 6 sec followed by the spoken signal “Recall.” After a pause of approximately 25 sec for the subject to write down the constructed array, another set of sentences began. One block of 12 sets of sentences used the form “A is the father of B,” and a second block used sentences of the form ‘‘A is older than B ,’’ where A and B were chosen from the following set of five names: Bob, John, Fred, Paul, Tom. Within each block, half of the sets described a five-term linear array and half described one of the three types of five-term networks shown in Fig. 4. Presentation orders 2 and 3 occurred twice in each block, and the remaining eight orders occurred once. The subjects were 36 undergraduate students recruited in the same way as for the first two experiments. They were tested in small groups varying in size from 2 to 7. The instructions read to the subjects included examples of arrays and practice trials with feedback. Half of the subjects received the block of trials with father-of sentences first, the half received the block with older-than sentences first. Two different randomizations of each set of trials were recorded on magnetic tape, and each group of subjects received a different combination of conditions and randomizations. Responses were written on sheets containing two unequal columns of eight empty rectangles with connecting lines. The subjects were told that the boxes represented up to five generations, with the possibility of one or two persons belonging to each generation. B.

RESULTSAND DISCUSSION

Each trial was scored on the basis of whether all five names were correctly placed on the response sheet. The proportion of completely correct orderings constructed was analyzed in an analysis of variance using the .05 level of significance unless otherwise indicated. The relational term used in describing the lines and networks had no effect. Subjects’ performance was also not influenced by the order in which they were tested on the two relational terms, and none of the interactions involving these effects of sentence content was significant. The mean proportion of linear orderings correctly constructed (.a) was not significantly greater than the mean proportion of partial orderings

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(.38), although it cannot be concluded with certainty that there is no difference, F(1, 35) = 2.91, MS, = .23, p < .lo. Type of structure did not interact with any other factors in the design. The failure to find significant semantic or structural effects may be due to the task and, in particular, to the response sheets used. For example, older-than sentences describing networks might not be particularly difficult to understand and process in the context of a diagram that specifically implies the possibility of a node or branch. The evidence does not rule out the conclusion that even with considerable procedural support, networks may be more difficult to construct than lines. An alternative interpretation of this outcome in terms of the constructive processes involved is given below. The mean proportion of correct trials for the 10 orders of presentation is given in Table IV. These data were analyzed using a set of orthogonal planned comparisons plus several additional nonorthogonal comparisons. Overall, trials requiring mainly process M1 (orders 1, 2, 5, 7, and 9) resulted in more correct responses than trials requiring mainly process M2 (orders 3, 4, 6, 8, and 10). The mean proportion correct for the former was .46 and for the latter .37, F(1, 35) = 12.87, MS, = .174. This difference is completely consistent with earlier findings by Foos et al. (1976) and Mynatt and Smith (1977). However, detailed analysis of the 4 presentation orders describing a linear ordering suggests that difficulty did not increase in a perfectly linear fashion with number of M2 processes. For example, order 4, with three M2 processes, was easier than order 3, with only two. Similar discrepancies in orders 5 through 10 suggest that the sequence in which the processes are executed may be important, although neither Foos et al. nor Foos and Sabol reported any decrement in performance when the matching elements occur in relations separated in the presentation order by several other comparisons; for example, B > C, C > D, D > E, A > B was no more difficult than B > C, A > B, C > D, D > E. However, these studies did not examine partial orderings. The results suggest that working from the top to the bottom of a line and then forming a node at the bottom (order 9) was one of the easiest presentation orders, whereas working in a reverse direction and forming the node at the bottom (order 10) was the most difficult. This pattern appeared to be reversed for orders 5 and 6. The most difficult sequence was one in which the node had to be formed in the middle of a previously constructed linear ordering (orders 7 and 8 compared to orders 5, 6, 9, and lo), F( 1, 35) = 7.21, MS, = .135. One possible explanation for this pattern of results is based on the assumption that process Nd involves a search through previously stored elements in the representation to locate

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one that matches (a basic component of the match processes, M1 and M2). In such a search process, a middle element might be less salient than one on the end. However, on this line of reasoning, the salience of the “middle” term C in order 9 would have to be explained by the recency of its occurrence in the preceding sentence. When the results for both linear and partial orderings are considered together, we conclude that process Nd is no more difficult than process M2; but, like M2, Nd is more difficult than MI. The mean proportion correct for orders involving MI on the last sentence (orders 1 and 3) was .49 as compared to .42 for orders involving M2 on the last sentence (orders 2 and 4) and .38 for orders involving Nd on this sentence. Nonorthogonal comparisons support this conclusion. A comparison of all orders involving MI on the last sentence to those involving M2 or Nd was significant, F ( 1 , 35) = 6.69, MS, = .250. A comparison of orders involving M2 on the last sentence to those involving Nd was not significant, F ( l , 35) = 1.48, MS, = .174. The preceding account makes clear that process Nd, which is a fundamental process in integrating a partial ordering, is basically a match process in the terminology of Foos et al. (1976). That is, after a pair of elements, AB, has been stored, the processing of a new relationship, AC, requires that the subject find the two occurrences of the element A in order to form a node. Foos et al. argued that process M2 involves an additional rearrangement of the terms that process M1 does not. Similarly, process Nd may involve a mental rearrangement of comparable difficulty. The implication is that partial orderings may be more difficult to construct than linear orderings because all the possible orders of presentation require some degree of mental rearrangement, whereas linear orderings can be presented in an optimal order requiring few, if any, such rearrangements.

VII.

Experiment 4: Diverging and Converging Nodes

The third experiment, described in Section VI, was deliberately designed to avoid the full range of complexity that is possible with networks. Two important omissions are addressed in Experiment 4. First, one of the largest and most reliable effects found in the work on constructing linear orders has been the difference between match and nonmatch orders (see Foos et al., 1976; Mynatt & Smith, 1977; Smith & Foos, 1975). However, the orders of presentation in the third experiment were what Foos et al. designated “match” orders. In the fourth experiment, a limited number of nonmatch orders were introduced in order to explore the generality of this important effect.

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The second omission in Experiment 3 has to do with the type of network structures considered. In all cases, they had nodes that opened downward in a diverging manner. For example, a father, A, could be described as having two sons, B and C. However, it is also possible to construct nodes in which two terms converge on a third term. For example, the information that A and B are the parents of C could be represented in a converging node. These two types of nodes convey somewhat different information, and the process of forming a converging node, which we shall now refer to as process Nc, may be psychologically different from the process of forming a diverging node (process Nd) investigated in the third experiment. In the present experiment, all possible orders of presentation were incorporated into a design investigating the four network structures shown in Fig. 5. As can be seen, all of the networks involve four elements and contain one node. The Fig. 5 structures have been classified as either convergent or divergent, and as symmetrical or asymmetrical. It is important to keep in mind that these labels refer to the “shape” or configuration of the network. In what follows, we discuss the relationship between shape and the order of presentation and argue that the effects of the latter on constructive processes are more important than the shape of the resulting network. A.

METHOD

A total of 48 sets of three sentences were tape-recorded and presented to subjects at a rate of one sentence every 6 sec followed by the signal “Recall.” After a pause of approximately 24 sec for the subject to write down the constructed array, another set of sentences began. The sets of Divergent

Convergent

A

Symmetrical

I

A\

/”\

C

/B C

I

D

D

A Asymmetrical

B

C

I

D

I

B\

/“

D

Fig. 5 . The four types of four-term partial orderings studied in Experiments 4 and 5 . Divergent and convergent orderings differ in the orientation of the node (down and up, respectively). “Symmetrical” and “asymmetrical” refer to the configuration of the entire network.

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sentences were presented in two blocks of 24. Within each block, half of the sentences described divergent networks and half described convergent networks. All possible presentation orders were used; these are shown in Table V. Within each block, the 24 sets of sentences were randomly ordered. Half of the subjects received one randomization first, and half received the other randomization first. The sentences were of the form “A is the mother of C” or “B is the father of C,” where elements were chosen from the following set of names: Ann, Kate, Sue, Bob, Mike, Jim. After elements had been randomly assigned to arrays, the correct relation, “mother of” or “father TABLE V MEANPROPORTION OF CORRECTLY CONSTRUCTED NETWORKS AS A FUNCTION OF PRESENTATIONORDERSIN EXPERIMENTS 4 AND 5 Constructive processes involved0

Mean proportion correct

~

Presentation order I . AC,BC,CD 2. AC,CD,BC 3. CD,AC,BC

4. AB,BD,CD 5 . BD,AB,CD 6 . BD,CD,AB 7 . CD,BD,AB 8 . AB,CD,BD 9. CD,AB,BD 10. BC,BD,AB 1 1 . AB,BC,BD 12. BC,AB,BD 13. AC,CD,AB 14. CD,AC,AB 15. AB,AC,CD 16. AC,AB,CD 17. AB,CD,AC 18. CD,AB,AC

After second sentence

After third sentence

Experiment 4

Convergent, symmetrical networksb Nc M1 - 1 MI Nc M2 Nc Convergent, asymmetrical networks MI Nc M2 Nc M2 - 2 Nc Nc M2 - 2 N Nc N Nc Divergent, symmetrical networks M2 - I Nd MI Nd M2 Nd Divergent, asymmetrical networks M1 Nd M2 Nd Nd MI - 2 Nd MI - 2 N Nd N Nd

Experiment 5

.77 .75 .76

.84

.87 .70 .72 .77 ,123

.82 .69 .78 .76 .68 .69

.85 .77 .88

.85 .81 .79

.87 .75 .95

.91

.43

.92

.65 .63

.85 .79

.78 .85 .78 .74 .69

Osee the text for a complete discussion. bFor symmetrical networks, each order of presentation consisted of two different presentation orders that cannot be logically distinguished, that is, order 1 could be either AC,BC,CD or BC,AC,CD. Both equivalent orders were presented to subjects, and the mean appears in the table.

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of,” was added and alterations were made in cases where two names of the same sex converged on a term. The subjects were 30 students from the same source as in previous experiments. The subjects were tested in small groups varying in size from 2 to 6. They were read a set of instructions which included examples of arrays and were given practice trials with feedback. The answer sheets used by the subjects contained six rectangular boxes arranged in two columns of three for each array. The subjects were instructed to write the names in any space they wished (although they were encouraged verbally and through examples to use an oldest-to-youngest order) and to connect the names by arrows pointing from parents to children. An answer was scored as correct only if all four names were recalled and correctly connected by arrows. B.

RESULTSAND DISCUSSION

The data from the fourth experiment were analyzed in the same manner as those of the third. The mean proportion of correctly constructed family trees is shown in Table V. Overall, the mean proportion of correctly constructed divergent structures (.8 1) and was significantly greater than the mean proportion of convergent structures (.69), F(1, 29) = 30.14, MS, = .195. Because the principal difference between the two types of structures is which of the two constructive processes, Nd or Nc, was required, it appears that converging nodes are more difficult to construct than diverging nodes. Although this conclusion seems intuitively correct, the explanation is by no means obvious. A converging node in a family tree constrains the entries, because the two parents cannot be of the same sex, whereas diverging nodes do not have this constraint. This additional constraint might appear to aid, rather than hinder, the subject in constructing a converging node. Perhaps the best explanation of the difference is that because family trees have a definite direction descending from older to younger generations, converging nodes in some sense extend “upwards” (or in the opposite direction). Process Nc appears to involve a mental rearrangement process similar to that involved in process M2. Experiment 5 was designed to deal, at least indirectly, with this issue. Overall, the two asymmetrical structures with a mean proportion correct of .70 were significantly more difficult to construct than the symmetrical ones with a mean of .80, F(1, 29) = 29.09, MS, = .104. However, a careful consideration of the cognitive processes involved in constructing the two types of networks reveals a confounding. The center columns of Table V present an analysis of the constructive processes

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required in each order of presentation. Of interest at the moment is the presence of process N for the second sentence in orders 8, 9 , 17, and 18. These orders result in a nonmatch situation following the presentation of the second sentence, according to Foos et al. (1976). That is, the first two sentences in these orders introduce all four elements of the network, but there are no matches between duplicate elements. The subject is forced to maintain two separate relationships and await the information in the third sentence. Foos et al. argued that the nonmatch situation increases the difficulty of construction. Note that a nonmatch situation can only occur in the asymmetrical structures of this experiment. A planned comparison revealed that orders requiring process N (orders 8, 9, 17, 18) were significantly more difficult than the other orders of presentation for asymmetrical structures (orders 4-7 and 13- 16), F( 1, 29) = 110.51, MS, = .169. We would, therefore, argue that the “symmetry” of constructed networks does not make them easier to construct. Rather, asymmetrical networks may appear to be more difficult when nonmatch orders of presentation are included in the design. The difference between match and nonmatch orders of presentation also displayed an interaction with the type of node; the difference between match and nonmatch orders was greater for convergent than for divergent structures, F(1, 29) = 13.40, MS, = .151. The foregoing interaction appeared to be the result of the unique difficulty of order 9. The difference between orders 8 and 9 was significandy greater .than the difference between 17 and 18, F(1, 29) = 6.43, MS, = .127. Although previous research with linear ordering has found that presentation orders like orders 9 and 18 are more difficult than orders resembling orders 8 and 17, the explanations offered appear to have no bearing on the extreme difficulty of order 9. This issue is further discussed in Section VIII. The unusual difficulty of order 9 probably also accounted for a significant interaction between symmetrical-asymmetrical and convergent- divergent networks, F(1, 29) = 5.94, MS, = .187. Within all match orders, a comparison of orders requiring process M1 to construct a line with those requiring process M2 showed the expected difference favoring M1, F(1, 29) = 6.32, MS, = .149. This result, of course, replicated previous findings with linear orders. The remaining match processes shown in Table V involve adding the last element of the network to a previously constructed node. This situation can be analyzed into two factors. The first factor is locus of the attachment point: The new element can be added to the top or the bottom of the three-element branching construction. This distinction is identical to that between processes MI and M2, and in Table V, these processes retain the designations “MI” and “M2.” However, a qualifying numeral has been added

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to indicate the second factor, which is whether one or two points of attachment are available. To see more clearly what is at stake here, consider order 1 in Table V. A converging node is first formed from A > C and B > C. The third sentence, C > D, then requires attachment of D to the bottom of the network (paralleling process Ml), but just one point of attachment is available at the bottom, namely, C. Now, consider orders 15 and 16. From A > B and A > C, the subject can form a diverging node. Again, the third sentence requires that the fourth element in the network, D, be attached to the bottom (paralleling process M1 again); however, two possible points of attachment, B and C, are available. The potentially different processes are tentatively labeled processes M 1-I and M1-2. However, a contrast comparing orders involving processes M1-1 and M2-1 with those involving processes M1-2 and M2-2 was not significant, F(1, 29) < 1; the difference very slightly favored the latter orders, with two available points of attachment. A second contrast comparing orders involving processes M1- 1 and M1-2 with orders involving processes M2-1 and M2-2 also failed to reach significance F(1, 29) = 2.75, MS, = .128, although attaching an element to the bottom (MI) was slightly easier (mean proportion .85) than attaching an element to the top (M2) of a tree (.82). (The interaction of these two comparisons is a contrast of a subset of orders describing converging and diverging orders. This contrast, which is not orthogonal to similar ones described above, was significant, as would be expected.) The preceding indicates that converging and diverging networks differ subtly in ways other than the requirements of processes Nc and Nd. However, the relevant analyses suggested that the confounded variables account for little of the variance in the data collected in Experiment 4. The failure to find a significant difference between processes M 1 and M2 for partial orderings is somewhat disappointing. However, the difference between processes M1 and M2 for linear orders has proven statistically elusive and always accounts for a small fraction of the variance that is due to orders of presentation.

VIII. Experiment 5: The Role of the Schema The two preceding experiments make clear that college students can readily learn simple network structures when the component relations are presented in the proper context (i.e., as a family tree) and in an order that does not make too great a demand on memory. Both experiments demonstrate the importance of presentation order and its effect on successful

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construction. Moreover, the constructive processes involved in constructing partial orderings appear to have much in common with those used in linear order construction. However, the family tree schema is not only well known but also has a conventional orientation. That is, most people find it natural to conceive of a family tree as a network with a top (usually headed by a significant ancestor) and a bottom (representing the most recent generation). Very infrequently, the network is conceptualized as a system of roots with the ancestors at the bottom. Whatever the specific representation, there is a specific, conventional format for organizing a family tree. In the last experiment, we explored the process of constructing a network with a much vaguer format and set of conventions for representation. The fifth experiment used an adaptation of a task used by Hayes ( 1 965) to study problem-solving strategies. During one phase of his procedure, Hayes had subjects learn configurations of code names forming “spy rings” in which not all spies could talk to each other. The subjects learned the configurations by studying a list of the pairs of names making up the spy ring. In the present experiment, a similar cover story was used to present the same four-term convergent and divergent structures used in the fourth experiment. Numbers were used instead of code names, and each spy could pass messages in only one direction. The stricture that spies could pass messages in only one direction was added to simulate the directional nature of the relations “older than,” “father of,” and “mother of.” The relations are directional in the sense that correct recall required that one term be marked or tagged as the older (vs the younger) person or source (vs the receiver) of messages. The term “direction” is here intended to be distinct from orientation. Orientation applies to the complete memory representation, whereas direction applies to the relation between any two terms. Thus, by changing the context in which the relations were presented in the last experiment, we attempted to influence the subject’s selection of a representational schema while maintaining the directional character of individual relations. A.

METHOD

The method of the fifth experiment was the same as that of the fourth with the following exceptions: The four elements of the network to be presented on each trial were randomly selected from the set of 10 digits. The subjects were told that the numbers stood for spies and that their task was to figure out spy networks from the three pairs of digits which would be read. The digits were read simply as pairs, for example, “3, 8,” with no sentence frame. The instructions indicated that the first-named spy

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could pass messages to the second-named spy, but messages could not go in the reverse direction. When subjects heard the verbal signal “Recall,” they were to draw the networks on an answer sheet containing two columns of three small circles for each trial. The numbers were to be placed in the circles and arrows drawn between the circles indicating in which direction spies could pass messages (see Fig. 6). Each of the 34 subjects received a total of 48 networks presented at a rate of 4 sec per relation. (Pairs can be read at a faster rate than sentences and still be comfortably understood.) B . RESULTSAND DISCUSSION

The mean proportion of trials on which the correct spy network was produced was analyzed in the same way as in Section VII,B, and the values for each presentation order are shown in the last column of Table V. In the interest of clarity, the outcome of the analysis will be presented in the same order as Section VI1,B. In contrast to Experiment 4, the difference between convergent and divergent networks (processes Nc and Nd, respectively) was not significant, F(1, 33) = 1.08, MS, = .183. The mean proportions were .78 and .80 for convergent and divergent structures, respectively. This result could be due to the fact that the schematic framework of the spy ring differs from a family tree in lacking a well-defined orientation. No convention requires one spy to be in a certain place relative to another, so converging nodes and diverging nodes can be represented in a nearly equivalent fashion, as illustrated in Fig. 6. This finding is especially striking when one considers that the individual relations, as presented in this study, had a directional component. That is, subjects were told that “AB” meant that A could talk to B , but not vice versa. Within asymmetrical structures, where nonmatch orders are possible, Converging

Diverging

Fig. 6. Two examples of the response format used in Experiment 5 . For each trial, the subject was provided with a set of six circles in the configuration shown and was to fill them in using arrows and the elements of the ordering (which were actually digits for the subjects). For purposes of illustrating the form of the responses, the two symmetrical networks in Fig. 5 have been represented.

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these orders proved more difficult than match orders, F(1, 33) = 10.96,

MS, = .163. The mean proportion correct on nonmatch orders was .70,

in contrast to .80 on match orders. Although this result is similar to what has been found in previous work on linear orders as well as in the fourth experiment, the effect was not so large in the present experiment, nor was there an interaction with convergent and divergent structures, F < 1. Moreover, order 9 was not strikingly more difficult than the other orders. These results are consistent with the notion that a spy network imposes no particular orientation on the representation. The reasoning involved can be seen best with examples. First, consider the nonmatch situation in the family tree setting. The relationships “A is the father of B” and “C is the father of D” in that order might suggest that A and C are on different levels of the tree, with A above (or older than) C, whereas “C is the father of D” followed by “A is the father of B” should, according to the explanation, suggest that C is above A. Subjects might be expected to order the two relationships tentatively by level using the order of mention. This would be a correct assumption if A is the father of C, but not if D is the father of A. In the latter case, some mental rearrangement would be required. (The rearrangement is part of the process D2 described by Foos et af., 1976). When “mother of” is permitted in the sentences, it becomes possible for A and C to be on the same level, but “A is the father of B” and “C is the mother of D” together seem to imply more complex relationships, for example, half-brothers and half-sisters. Now consider the same relationship in a spy network. “A talks to B” and “C talks to D” leave no implications about any third relationship. A could talk to C or C to A. Logically, there is even the possibility that the next relationship might be “C talks to B” or “A talks to D,” although such networks never occurred in the experiment. The point is that the spy network schema involves many fewer presuppositions about possible configurations and probably encourages subjects to remain uncommitted in the nonmatch situation. This may also explain why the difference between match and nonmatch orders is not as large in Experiment 5. The only remaining comparison of interest which was significant in Experiment 5 contrasted match orders requiring process M1 (mean proportion correct .86) with those requiring process M2 (.68), F( 1, 34) = 5.75, MS, = .180. This difference may seem inconsistent with the argument about the lack of orientation of spy networks. However, in all cases, the presentation orders compared begin with the construction of a threespy line, that is, A talks to B, B talks to C, or B talks to C, A talks to B. Because the construction of a linear ordering is involved, one might expect processes M1 and M2 to play a role. Note that the contrast did not

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involve presentation orders in which the fourth element was added to a previously constructed node. No significant differences among these orders were obtained. C.

CONCLUSIONS FROM EXPERIMENTS ON PRESENTATION ORDER

The results of the last three experiments suggest that at least some kinds of networks are no more difficult to construct than linear orderings under the right circumstances. Moreover, the difficulty does not appear to be due solely to the type of structure. Rather, the context and form in which relationships are presented is quite important, as well as the order in which the component relations are presented. For the latter variable, the results quite strongly indicate that what Foos et al. labeled nonmatch presentation orders increase the difficulty of constructing a network, just as they do in constructing a linear ordering. In other respects, constructing a node, the structure that makes partial orderings more complex than linear orderings, turns out to be much like combining two relations to form a line. When people listen to sentences and attempt to respresent the information they contain, a large number of contextual factors including the choice of words in the sentences determines the type of representation ultimately formed. With most previous studies of constructive processes, both instructions and the investigator’s semantic choices generally activated a single familiar schema of representation-a line or a ranking. Once this schema has been invoked, it is probably very difficult to represent indeterminacies correctly. We would argue that this has happened in previously published network studies. Subjects were expected to represent partial orderings (or networks) on the basis of relational information that seemed to be fully ordered. In our experiments, we have shown that this presupposition can be readily broken and that college students have an appropriate way of representing partially ordered relations. The last experiment demonstrates that at least two such representational schemas differ subtly in their presuppositions.

IX.

Summary

Our program of research began from two points: a model of the constructive processes used in integrating the information in a linear ordering and a review of the recent literature on partial orderings. The first experiment was designed to clarify an implicit assumption in recent studies of partial orderings. We believe the focus on retrieval times has been mis-

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Kirk H. Smith and Barbee T. Mynatt

directed so far as understanding why people have difficulty mastering these structures. Retrieval times probably reveal more about search processes in partial orderings than about their representation. The second experiment we described also implied a criticism of recent studies. The use of partial orderings with a large number of elements and arbitrary structural characteristics may have obscured the similarities between partial and linear orderings when an integrated representation of them is being constructed. These two experiments set the stage for the last three, which verified the importance of activating an appropriate representational schema for the subject and demonstrated that many constructive processes are either the same as, or quite similar to, those described in earlier models of how linear orderings are constructed. ACKNOWLEDGMENTS The research described here was supported by grants BNS 75-19313 and BNS 77-16860 from the National Science Foundation to Bowling Green State University with the authors as coprincipal investigators. We would also like to acknowledge the many helpful discussions of this work we have had with Richard A. Griggs, Shannon D. Moeser, and Rebecca M. Pliske, although our views differ sharply with theirs in various ways. Finally, we would like to thank Mrs. Waltraud Vogel for her effort and patience in typing the manuscript of this contribution.

REFERENCES Anderson, J. R. Language, memory, and thought. Hillsdale, New Jersey: Erlbaum, 1976 Anderson, J. R., & Bower, G. H. Human associative memory. New York: Holt, 1973. Barclay, J. R. The role of comprehension in remembering sentences, Cognirive Psychology, 1973,4, 229-254. Battig, W. F., & Montague, W. E. Category norms for verbal items in 56 categories: A replication and extension of the Connecticut category norms. Journal of Experimental Psychology Monograph, 1969, 80, No. 3, Pt. 2. Chomsky, N. Syntactic structures. The Hague: Mouton, 1957. Collins, A. M . , & Loftus, E. F. A spreading-activation theory of semantic processing. Psychological Review, 1975, 82, 407-428. Collins, A. M., & Quillian, M. R. Retrieval time from semantic memory. Journal of Verbal Learning and Verbal Behavior, 1969, 8, 240-247. De Soto, C . B. Learning a social structure. Journal of Abnormal and Social psycho lug.^, 1960, 60, 417-421. De Soto, C. B. The predilection for single orderings. Journal of Abnormal and Social Psychology, 1961, 62, 16-23. Foos, P. W., & Sabol, M. A. The role of memory in the construction of linear orderings. Memory and Cognition, 1981, 9, 311-377. Foos, P. W., Smith, K. H., Sabol, M. A., & Mynatt, B. T. Constructive processes in simple linear order problems. Journal of Experimental Psychology: Human Learning and Memory. 1976, 2 , 759-766.

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