@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Arts, Faculty of"@en, "Psychology, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Pachev, Gueorgui Stefanov"@en ; dcterms:issued "2009-02-18T22:41:59Z"@en, "1996"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The development of children's reasoning has often been associated with the attainment of strategies, higher-order concepts, or a broader knowledge-base. Several researchers have argued that these achievements are parallelled and influenced by a child's capacity to process a fixed amount of information in the context of a reasoning task. It is assumed that children's processing resources increase with age and that this increase allows children to acquire the more complex forms of reasoning, advanced strategies, etc. Insufficient resources, on the other hand, prevent children from performing at a higher level. The present study had two goals: (1) to provide evidence for the increase of resources with age, and (2) to explore the effects asociated with the limited-capacity of the processes involved in reasoning. Two dual-task paradigms were combined in the pursuit of these goals. One paradigm allowed for identifying the level of a task, at which performance was capacity-limited. Age-groups with different amount of resources were expected to exhibit capacity-limited performance at different task-levels. The second approach, based on introducing additional processing load, allowed for comparing the effects of charging the Capacity-limits of different processes involved in reasoning. Eighty-six children from three age-groups were given a matrices-completion task at four levels of difficulty. The task was performed either alone or concurrently with a second task. The secondary task was administered in the beginning or in the middle of some trials, thus disrupting processes at the initial stage and processes at the executive part of the solution. Capacity-limited performance was detected at the third and fourth level of the task for the first two age-groups, respectively. There was an indication that the oldest subjects would exhibit capacity-limited performance at levels beyond the fourth one. Reasoning performance deteriorated when the secondary task disrupted the operation of the executive processes. The additional load introduced in the beginning of a trial, however, did not affect the level of reasoning performance or resulted in improvement in certain conditions. The results were interpreted as supporting the hypothesis that the capacity factor operates as a necessary but not a sufficient condition for the development of reasoning."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/4773?expand=metadata"@en ; dcterms:extent "6108234 bytes"@en ; dc:format "application/pdf"@en ; skos:note "CAPACITY CONSTRAINTS ON REASONING: D E V E L O P M E N T A L ASPECTS by GUEORGUI STEFANOV PACHEV B. A., University of Sofia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Psychology) We accept this thesis as conforming to-the-sequired standard T H E UNIVERSITY OF BRITISH COLUMBIA March 1996 © Gueorgui Stefanov Pachev, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date PrpML Jjl . 6 DE-6 (2/88) Abstract The development of children's reasoning has often been associated with the attainment of strategies, higher-order concepts, or a broader knowledge-base. Several researchers have argued that these achievements are parallelled and influenced by a child's capacity to process a fixed amount of information in the context of a reasoning task. It is assumed that children's processing resources increase with age and that this increase allows children to acquire the more complex forms of reasoning, advanced strategies, etc. Insufficient resources, on the other hand, prevent children from performing at a higher level. The present study had two goals: (1) to provide evidence for the increase of resources with age, and (2) to explore the effects asociated with the limited-capacity of the processes involved in reasoning. Two dual-task paradigms were combined in the pursuit of these goals. One paradigm allowed for identifying the level of a task, at which performance was capacity-limited. Age-groups with different amount of resources were expected to exhibit capacity-limited performance at different task-levels. The second approach, based on introducing additional processing load, allowed for comparing the effects of charging the Capacity-limits of different processes involved in reasoning. Eighty-six children from three age-groups were given a matrices-completion task at four levels of difficulty. The task was performed either alone or concurrently with a second task. The secondary task was administered in the beginning or in the middle of some trials, thus disrupting processes at the initial stage and processes at the executive part of the solution. Ill Capacity-limited performance was detected at the third and fourth level of the task for the first two age-groups, respectively. There was an indication that the oldest subjects would exhibit capacity-limited performance at levels beyond the fourth one. Reasoning performance deteriorated when the secondary task disrupted the operation of the executive processes. The additional load introduced in the beginning of a trial, however, did not affect the level of reasoning performance or resulted in improvement in certain conditions. The results were interpreted as supporting the hypothesis that the capacity factor operates as a necessary but not a sufficient condition for the development of reasoning. IV Table of Contents Abstract ii List of Tables' vi List of Figures viii Chapter 1: Introduction 1 • The Problem 1 The Concept of Capacity: Information-Processing Perspective 5 Chapter 2: Review 23 Neo-Piagetian Models of the Relation between Reasoning and Capacity 23 The Relation between Reasoning and Capacity in Neo-Piagetian theory: Evidence and Problems 33 Chapter 3: Hypotheses and Method 51 Objectives and Hypotheses 51 Method 55 Participants 55 Equipment 56 Tasks .57 Measures and Design 60 Procedure .71 V I Chapter 4: Results 74 Association between Concurrent Performance on the Secondary Task and Primary Task Performance 74 Group 1 76 Group 2 . 9 1 Group 3 101 Effects of Distraction on Reasoning Performance 108 Group 1 . 109 Group 2 116 Group 3 120 Chapter 5: Discussion 123 Age-Differences in Capacity 123 Effects of Capacity Limits on Reasoning Performance 129 References 134 List of Tables 1. Summary of the empirical evidence 50 2. Trials for the one-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses 66 3. Trials for the two-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses 67 4. Trials for the three-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses 68 5. Trials for the four-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses 69 6. Capacity-limited performance tests for Group 1: perception condition 77 7. Capacity-limited performance tests for Group 1: memory condition 78 8. Group 1: Means and standard deviations for the two measures of primary task performance (ho-distraction trials) across task conditions and levels of task difficulty 84 9. Group 1: Means and standard deviations for the secondary task performance measures 86 10. Capacity-limited performance tests for Group 2: perception condition 92 11. Capacity-limited performance tests for Group 2: memory condition 93 12. Group 2: Means and standard deviations for the two measures of primary task ^ performance (no-distraction trials) across task conditions and levels of task i I difficulty 96 13. Group 2: Means and standard deviations for the secondary task performance measures 14. Group 3: Means and standard deviations for the two measures of primary task performance (no-distraction trials) across task conditions and levels of task difficulty 15. Group 3: Means and standard deviations for the secondary task performance measures List of Figures 1. Hypothetical performance-resource function 2. Sharing of resources between two concurrent tasks 3. Unobservable parameters, measures, and their interrelations according to Hunt and Lansman's (1982) model 4. An example of a trial for the \"perception\" condition 5. An example of a trial for the \"memory\" condition 6. Group 1: Secondary task performance by task condition, level of difficulty, and distraction condition. 7. Group 2: Secondary task performance by task condition, level of difficulty, and distraction condition. 8. Group 3: Secondary task performance by task condition, level of difficulty, and distraction condition. 9. Hypothetical data pattern 10. Group 1: Primary task performance (percentage correct) by performance measure, task condition, level of difficulty, and distraction condition. 11. Group 2: Primary task performance (percentage correct) by performance measure, task condition, level of difficulty, and distraction condition. i 12. Group 3: Primary task performance (percentage correct) by performance measure, task condition, level of difficulty, and distraction condition. ' . 1 Chapter 1 Introduction The Problem The purpose of this work is to explore the question of how children's performance on a reasoning task at different ages is influenced by limits on their processing capacity. In brief, it is hypothesized that: (1) there is an age-related increase in the quantitative characteristics of different processes involved in the solution of a reasoning problem, and (2) reasoning performance is constrained by the capacity characteristics of these processes in that a lack of sufficient resources would prevent individuals from performing at a higher level. If these hypotheses are confirmed, the results will add to the description of a developmental factor that parallels and influences the attainment of more effective strategies, higher order concepts, or more complex forms of reasoning with development. This, in turn, can help the explanation of a number of developmental phenomena. To illustrate, an analogy can be made with adults' performance on a six-term transitive reasoning problem. Adults will usually fail unless there is some way of visualizing the elements and their relations. This failure will not be due to an inability in making transitive inferences; it is the number of elements or the number of intermittent steps to be carried out that make this task difficult. By analogy, one may suppose that a five-year-old experiences similar difficulties when confronted with a three-term transitive task. The child may be able to carry out the necessary comparisons between the elements of each pair but the number of required comparisons may exceed the child's processing resources. 2 This explanation seems obvious and consistent with the common-sense. Its conceptualization and empirical test, however, have proved to be difficult. First, capacity is not the only factor that determines performance. Several common findings from the area of reasoning development suggest that there are important qualitative differences in the way younger and older subjects approach and solve a reasoning problem. For example, young subjects often exhibit systematic patterns of mistakes; older subjects (i.e., subjects with presumably sufficient processing resources) sometimes fail to solve a reasoning problem correctly; the influence on the solution process of domain-specific knowledge and experience with the type of task is also well documented. Thus, one source of difficulty is associated with the need to distinguish between the effects of capacity limitations and the effects of other factors on performance. Second, a number of unresolved and controversial questions arise with respect to the notion of capacity: what is capacity; how is it quantified; how is it measured? In the context of the example above, one may ask whether the inferior performance of younger subjects on the three-term task and older subjects on the six-term task is due to inability to apprehend all necessary components of the task, to a deficient memory for the premises of the task, or to a failure to coordinate all premises into a logical inference. Several attempts to answer these and similar questions have been made. In the cognitive developmental literature these attempts are most often associated with the so called \"neo-Piagetian\" tradition. The authors from this tradition have either proposed alternatives to Piagetian models of development by including additional developmental factors, or have tried to extend Piaget's theory to areas not covered by empirical research from a Piagetian perspective. These efforts of the neo-Piagetian theorists are aimed at providing a more accurate and detailed account of children's cognitive development. The assumption that an increase in individuals' capacity underlies age-related regularities in cognitive development has been accepted as central by several theorists (Case, 1985; Halford, 1982, 1993; Pascual-Leone, 1970, 1984). On the basis of this assumption they have addressed important issues and offered explanations for a number of developmental phenomena. For example, Pascual-Leone and Sparkman (1980) have argued for the advantages of Pascual-Leone's theory in explaining the transitions between Piagetian stages, the phenomenon of horizontal decalage, and the effects of a task's information-processing load on performance. Horizontal decalage is one of the phenomena considered in Chapman's (1987) structural-functional model. By assuming maturational capacity constraints on cognitive development, Case (1985) offered an explanation for a number of facts: the failure of training studies to produce stage advancement in some children; the considerable cross-task parallels in intellectual development; the similarities in the rate in which cognitive and physical development decelerate; the relatively universal character of cognitive development up to the ages of 16 -18. To account for these problems and phenomena, neo-Piagetian theorists have proposed several lines of evidence for the role of capacity as a necessary but not a sufficient condition for cognitive development. The general goal of the present work is consistent with this line of research. The proposed study, however, differs from the \"mainstream\" research in two important aspects. 4 First, although the general hypothesis of neo-Piagetian theories that capacity acts as a necessary but not sufficient condition for the development of reasoning is preserved, the approach to the test of the hypothesis is new. So far, the main body of empirical evidence comes from correlational studies, or studies where performance on tasks with established capacity demands has been predicted by subjects' performance on tasks designed to measure capacity directly. Although the latter approach is more reliable than the former, both types of evidence rely heavily on task analytical procedures and \"direct\" measures of capacity with questionable validity. An exception is provided by several studies by G. Halford and his colleagues (Halford, 1993; Halford, Maybery & Bain, 1986; Maybery, Bain & Halford, 1986). A dual-task procedure for detecting capacity-limited performance instead of direct measures of capacity have been used in these studies. This alternative approach is followed in the present work but it is applied to subjects from different age groups. It is argued that if there are age-related changes in the amount of processing resources, then capacity-limited performance will be detected at the easier versions of the task for younger subjects and at the more demanding task versions for older subjects. Second, it is assumed that reasoning, as a complex activity, is constrained by the capacity limits of different processes that comprise a particular task. So far, the studies have concentrated on a single process, according to the model of reasoning activity adopted by the particular author. The present study is aimed at exploring the capacity characteristics of several component processes in the context of a reasoning task and their relative influence on performance through the use of a dual-task methodology. 5 In what follows, the concept of capacity and capacity limitations as used in the adult cognitive literature is considered first. Two approaches to capacity, which the present work builds upon, are briefly reviewed and issues related to measurement of capacity are discussed. The next chapter reviews four contemporary developmental models of the relation between reasoning and capacity and summarizes the empirical evidence generated by these models in support of the neo-Piagetian hypothesis that capacity is a necessary but not a sufficient condition for the development of reasoning. The assumptions that underlie the present study, the particular hypotheses and the description of the method are outlined in Chapter 4. The final two chapters contain the results of the study and their discussion in terms of the stated objectives and hypotheses. The Concept of Capacity: Information-Processing Perspective The idea that there are quantitative limits to the human ability of processing information receives special attention within the information processing perspective, where the view of the cognitive system as a channel through which information flows and is transformed makes the description of the channel's limitations an important study task. Despite this importance, there is no commonly accepted approach to the conceptualization and study of capacity. Two approaches that are relevant to the goals of the present study will be described briefly below. The first one depicts capacity as a characteristic of a mechanism. A particular cognitive function is presented as carried out by a finite collection of elements, each performing a well defined operation on the information. The result is additive, failures of a particular element influence the overall result in a specific way, with the magnitude of the influence depending upon the role of the element in the mechanism. Explanation of performance is in terms of the qualities of the participating structures. Capacity, in this view, characterizes the elements (and the overall mechanism) in terms of the quantity of specific, concrete work that they perform. Thus, the capacity of a store is the number of items it can hold, the capacity of a filter is quantified as the number of items it lets through, the capacity of a process is characterized by the number of items that can be manipulated for a certain amount of time. This approach to capacity will be referred to here as \"specific\" capacity view. The important problem in approaching the relation between reasoning and capacity from this point of view is the conceptualization of the \"workspace\" of reasoning and the particular processes involved in the reasoning activity. Two models of the \"workspace\" have prevailed in the field: Atkinson and Shiffrin's (1968, 1971) short-term memory store model and Baddeley and Hitch's (1974) working memory model. R. Shiffrin (1976) reviewed the capacity limitations as revealed by studies in several areas of memory research and concluded that all of them (except for the results on masking) \"can be traced to a relatively small set of limitations in a single system: the active memory system, called short-term store (STS)\" (Shiffrin, 1976, p.213). The review is based on a version of a model proposed by Atkinson and Shiffrin (1968, 1971). According to the model the memory system consists of sensory input channels and two memory structures, an active but temporary structure called short-term store (STS) and a permanent repository called long-term store (LTS). The STS is considered to be the 7 activated subset of LTS. Sensory information enters the system and is encoded in a series of stages. The process of encoding is the activation of the inactive features contained in the LTS through their contact with the sensory \"input and, in fact, this is the process of constructing the temporary STS. The loss of information from STS is assumed to be equivalent to the reversion of a currently active feature to a stable inactive mode in LTS. The cause of such transformation is interference by an activity in STS that prevents the maintenance of information in an active state through; rehearsal or other control processes. Thus, the limitation with most far-reaching consequences is the rate of loss of information from STS. The same constraint on the quantity of information in active state applies to the stages after selection when certain information is designated for retention or for use in the processes of controlled search or decision making. For example, the evidence for automatic processing implies that with practice the process of search can be speeded up, processing accuracy can be increased, as can be the number of items selected in the search (see Schneider & Shiffrin, 1977; Shiffrin & Schneider, 1977). In addition, there are several ways to expand the limits of the controlled processing (e. g., through categorizing and unitizing, as discussed by Shiffrin, 1976). However, no matter how fast and accurate the search process and how many items identified in the search, the loss-rate limitation always stays and restricts the output. Therefore, the capacity characteristics of the participating processes will influence the overall capacity of the system only within the range set by the capacity for storage. In this case the short-term memory span would be a relatively accurate measure for the capacity of the system. This notion of capacity and the way of its measurement are relevant to several problems in memory research. The 8 question is whether it would be relevant to the idea that the short-term store is at the same time the workspace of reasoning. A. Baddeley and G. Hitch (1974) presented evidence against the view of the STS as the workspace of reasoning and proposed the concept of working memory as their alternative for a system that serves the complex cognitive tasks of language comprehension, learning and reasoning. The argument against the STS model can be summarized in two points. First, if the short-term store acted as a working memory necessary for the performance on a reasoning task, one would expect patients with a grossly defective short-term store to show many other cognitive problems. In fact, such patients often seem to encounter very few practical problems in coping with the information-processing demands of everyday life. Second, if the capacity of the system determined the number of items that can be held in an active state, then the maintenance of a memory load of a number of items while performing a reasoning task would impair the solution. The experimental test of this proposition yielded similar results for reasoning, comprehension and learning tasks: with the increase of concurrent load, performance declined but the degree of disruption was far less than predicted. The disruption affected mainly the time for performance, while the error rate remained more or less constant. These results indicate that memory load does interfere, implying some overlap of processing with the reasoning task, but even loading subjects' memory to capacity still leaves them able to reason accurately. In addition, the pattern of disruption by the concurrent task suggests that storage capacity is not the main capacity constraint on 9 reasoning. What is necessary is a more detailed view of the processes involved in performing a reasoning task and the capacity limits of these processes. In the model of working memory proposed by Baddeley and Hitch (1974) and elaborated in later works (for reviews, see Baddeley, 1983, 1986, 1992), storage and control processes are (at least theoretically) separated and viewed as different subsystems. In brief, it is hypothesized that the core of the system is a central executive responsible for coordinating the information from the subsidiary systems. The central executive is assumed to function like a limited capacity attention system capable of selecting and operating control processes and strategies. Unfortunately, it has proven the most difficult both to analyze and conceptualize. Thus, the research has concentrated on the subsidiary slave systems with the hope being the gradual whittling down of the functions that need to be assigned to the central processor (Baddeley, 1983, p.315). The articulatory loop is one of the stores explored in more detail. It stores speech coded information and makes use of a subvocal rehearsal system. The loop is considered to comprise two components — a phonological store that can hold acoustic or speech based information for 1 to 2 seconds, and an articulatory process, analogous to inner speech. It was established that memory span for words is inversely related to spoken duration of the words. Subjects can generally remember about as many words as they can say in about 2 seconds (the time for which the traces fade away in the passive phonological store). Thus, the model provides an explanation of the tendency for the digit span of children to increase with age: as children get older, they are able to rehearse faster (Hitch & Halliday, 1983). 10 The second system explored in some detail is the visuo-spatial scratch-pad. Its functions are considered to be maintenance and manipulation of visuo-spatial images. The scratch-pad seems to comprise, in analogy to the articulatory loop, a store linked with a rehearsal process — in this case the one used voluntarily to control eye movements. Hitch and Halliday (1983) report interesting findings of age related changes in the use of the two systems in memory tasks. Older children tend to use the articulatory loop to remember picture names as indicated by the phonemic similarity effect and the disruption by articulatory interference. Younger children's performance is not affected by either of these factors but is sensitive to visual similarity. In general, an advantage of the working memory model, as compared to the short-term store model discussed previously, seems to be the potential to separate analytically the different functional parts of the system and thus, to provide a more detailed account for the system's operation. However, the important task of providing a way for estimating the processing capacity involved in reasoning is not as yet solved. The potential of the working memory model in this direction seems to be associated with the analysis of the central executive, which still seems to be the \"area of residual ignorance\" (Baddeley, 1983, p.315). In summary, both models under this view represent'performance as strictly dependent upon the specific structures that participate in carrying out the solution of a reasoning problem. In order to estimate the impact of capacity constraints on reasoning performance one should first specify an explicit model of reasoning activity. The-model should specify the participating processes and a consistent mode of operation should be 11 assumed under all conditions. Two general empirical approaches can be applied on the basis of this model. One can correlate reasoning performance with measures of the capacity characteristics of the participating processes in an attempt to construct a model that explains in full performance on the reasoning task. The other approach consists in disrupting the operation of the separate processes by offering a specific concurrent task (e.g., additional memory load in order to disrupt the operation of the short-term store) in a dual-task situation. Both approaches have been used but the results indicate that the assumption of a consistent mode of operation does not always hold and that subjects are quite flexible in overcoming the disruption of the secondary task. The second approach to capacity, discussed below, is an attempt to account for this flexibility by including an additional factor that determines performance, namely, the amount of effort (or capacity) invested in the solution of a task. The approach is best exemplified by the resource theories of attention (e.g., Kahneman, 1973; Norman & Bobrow, 1975; Navon & Gopher, 1979). These theories treat capacity limits as limits in the amount of processing, that is, mental work, that can be devoted to a task. A basic assumption of this perspective is that there is a general limit on people's capacity to perform mental work. Thus, the inability to perform a task with excessive capacity demands or two tasks at once may not derive from a structural bottleneck at any particular stage of processing, but rather from a non-specific depletion of a limited pool of resources. Put figuratively, according to the former view the limits of capacity are imposed by the \"walls\" of the channel; according to the latter, it is the processing efficiency between these walls that further determines the permeability of the channel. Another assumption shared by capacity theories is that the level of performance depends upon the amount of resources allocated to the task. In Kahneman's (1973) theory, for example, the efficiency of processing within the limits imposed by structural constraints is controlled by four factors: (1) enduring dispositions, assumed to reflect the rules of involuntary attention; (2) momentary intentions; (3) evaluation of demands; (4) arousal level. Examples of situations, in which results are better when one is more concentrated, are numerous. Performance failures, from this point of view, can be due not only to structural factors like unavailability of appropriate strategies or lack of the necessary operations, but also to a depletion of individuals' resource pools. Figure 1 presents graphically the relation between resources and performance on a hypothetical task. The sector from the origin to point A shows that the quality of performance is directly dependent upon the amount of resources applied. Norman and Bobrow (1975) proposed the term \"resource-limited\" to designate this type of performance. The sector between points A and B depicts what is known as \"data-limited\" performance. That is, no matter how many resources are allocated to the solution of a problem, there is a point beyond which performance will not become any better. An example is the situation of listening to a radio when the signal is masked by static. At a certain point the signal-to-noise rate of the radio transmission is such that despite increased efforts invested in understanding the message, it does not become intelligible. 14 This second assumption has been further specified with respect to the dual-task situations. Capacity theories assume that when two activities are to be carried out concurrently, the available resources are divided between them. In cases where the joint demands of the two tasks exceed the available resource pool, performance on one or both is at lower levels, or, is capacity-limited. Such a situation is presented graphically by plotting performance on one task against performance on a second, concurrent task on Figure 2. The sectors AB and CD reflect data-limits on Task 1 and Task 2 respectively. The region from B to C depicts the capacity trade-off between the two tasks which in this hypothetical case results in deterioration of performance on both. This kind of plot is known as \"performance operating characteristic\" (POC) (Norman & Bobrow, 1975). Comparing the performance plots on Figure 1 and Figure 2, it should be noted that the theoretically assumed, unobservable resource variable on the abscissa in Figure 1 is substituted with an observable performance variable in Figure 2. That is, in a dual-task situation, the quality of performance on one task may serve as an index for the resource expenditure on the other task. Due to this possibility of indexing resources with performance on another task, the dual task paradigm has been accepted as the main method for estimating demands and resources. The rationale behind the application of this procedure requires additional assumptions and specific trade-off arrangements. In particular, the demands of the two tasks together should exceed the available resources. If the tasks were too easy then performance on both would be data-limited and no interference would be observed. 16 Second, the same amount of resources should be allocated to one of the tasks when it is performed alone and when it is performed concurrently with the second task. This is achieved by manipulating the instructions: one of the tasks is designated primary and subjects are instructed to give it priority in the dual-task situation. When these conditions are met, the quality of performance on the other (secondary) task will depend upon the spare resources, that is resources that remain unused after the necessary amount is allocated to the solution of the primary task. In cases where the secondary task is sensitive to resource variation, the quality of performance on it will allow inferences about the capacity demands of different primary tasks or about the relative amount of resources applied by different individuals to one and the same primary task. For example, performance on two tasks that have different capacity demands may be indistinguishable if both are within the capacity range of the subjects. The dual task situation allows for distinguishing them. When paired with one and the same secondary task, the task with fewer demands on resources will be accompanied by a better performance on the secondary task. For the purposes of the present work it is important to evaluate the extent to which the dual-task approach allows us to distinguish between individuals or groups of individuals with respect to their available capacity. Logic similar to that involved in distinguishing between capacity demands of tasks can be applied. Individuals with different resources may perform equally well on an easy task. In a dual-task situation, their performance on the secondary task will differ: the individuals with more available 17 capacity will perform better. Thus, secondary task performance can be used as an index of individuals' capacity. The problem is how to use this index in dealing with the question of the role of capacity in development. In testing the hypothesis that capacity is a necessary but not sufficient condition for the development of reasoning, the important issue is to show that the amount of available resources sets limits on the level of difficulty at which an individual can perform successfully. In terms of the dual-task approach, individuals with a worse performance on the secondary task in the concurrent-tasks condition should not succeed on the more difficult versions of the primary task. In addition, one should expect a strong association between secondary and hard primary tasks performance, if the difficulty levels of the primary task differed in terms of demands on resources only. Predicting performance on the basis of secondary task indices of capacity is central for Hunt and Lansman's (1982) formal model of the role of resources in determining individual performance. In brief, it is assumed that individuals differ in three general characteristics that determine performance. These are: (1) structural parameters pertaining to primary task performance; (2) structural parameters pertaining to secondary task performance; and (3) total resource capacity (Hunt and Lansman, 1982, pp. 218-219). Performance on the primary and secondary tasks in both the single and dual task condition is expressed as a function of various combinations of these unobservable variables. Information theory is then used to generate specific predictions concerning the relationships between performance measures. 18 The idea behind this approach is that if there is a causal relationship between a set of unobservable characteristics and each of a set of observable measures, then knowledge of one observable measure may provide information concerning another. The implementation of the idea in a particular model is depicted in Figure 3. The unobservable variables E 1 (structural parameter pertaining to primary task performance), E 2 (structural parameter pertaining to secondary task performance), and R (total resources), are connected to the observable variables by arrows, whose direction illustrates causation. Performance on the hard primary task is designated as a target variable (i.e., the measure that should be expressed in terms of the other observable variables). It is clear from Figure 3, that performance on the hard primary task depends on E 1 and R. Therefore, any measure that depends on one or both parameters will provide information concerning these parameters and, thus, improve the prediction of the target variable performance. In particular, the authors demonstrate that performance on the secondary task in the dual task condition provides information concerning performance on the difficult version of the primary task in that performance on both can be expressed as being determined by the same unobservable variables. Statistically, one should expect that a reliable linear relation exists between the two measures if the postulated causal relations are true. Two other measures that depend on the unobservable variables participating in the prediction and may influence the observed relation, are the performance on the easy primary task and performance on the secondary task, when both are performed alone. Easy primary, performed alone Secondary task, performed alone Secondary task, concurrent performance Hard primary, performed alone Figure 3. Unobservable parameters, measures and their interrelations according to Hunt and Lansman's (1982) model. 20 The authors propose that eventual shared variance between these two measures and the performance on the hard version of the primary task is partialled out in the statistical test Of the relation. In summary, the dual-task approach to measuring capacity provides an index of individual capacity (secondary task performance), which can be used for predicting performance on a reasoning task (Hunt and Lansman's \"easy-to-hard\" paradigm). A success in the statistical prediction of performance on the difficult primary task with performance on the secondary task in the dual-task condition would indicate that the observed differences in primary task performance are due to limited resources. Such a result bears direct relevance to the claim that capacity acts as a necessary but not sufficient condition for the development of reasoning. The second view of capacity, as exemplified by resource theories of attention, makes no specific architectural predictions, or more precisely, accepts certain architectural assumptions of the alternative approach. Explanation of performance, however, is based not only on the quantitative characteristics of the specified processes, but also in terms of the allocated capacity relative to task demands. That is, the approach allows for taking into account an additional, \"intensive\" aspect of performance. The purpose of measuring capacity under this view is to express the result in terms of the quantity of abstract work invested in it. As such, capacity can be used as a characteristic of a task (task demands), of performance (allocated capacity), of a structure or a process (processing capacity), or of an individual (available resources). Approaching the problem about the relation between reasoning and capacity from this point of view 21 avoids the need for an ad hoc identification of the processes or structures participating in a solution and the need for specifying the units into which capacity of the particular process or structure is quantified. Performance on one and the same secondary task, accompanying the different stages of the solution, can serve as an index of the capacity demands of the compartment processes. There is, definitely, an overlap of the phenomena that the two uses of the term capacity, described in this section, are intended to capture. It can be argued that these are, in fact, two ways of expressing and describing the capacity factor in performance. The first view describes capacity in terms of specific processing units. The second view depicts capacity in terms of abstract, \"pure\" quantity and allows for including in the explanation the \"intensive\" properties of performance. Thus, the two approaches could usefully complement each other. The concept of capacity, as used in the present work, bears upon both views described above and is an attempt at capitalizing on the advantages of each approach. First, similar to the \"specific\" approach to capacity, it is assumed that the term refers to a characteristic of a process, task or individual. Descriptions of capacity as \"mental energy\", \"mental space\", \"(resource) pool\", etc., can be accepted only as useful but limited metaphors. Second, similar to the \"non-specific\" capacity view, it is assumed that capacity characterizes the quantity of abstract mental work that an individual, applying certain processes, can handle at a time, or the quantity of mental work that is required for carrying out successfully a particular task. Finally, it is assumed that the overall success on a task depends on how much effort is invested in solving the problem. Thus, in a dual-22 task situation, although the two tasks involve different operations and processing units, the success on each will depend on how much effort the individual can allocate to them. r r 23 Chapter 2 Review Neo-Piagetian Models of the Relation between Reasoning and Capacity In the literature on cognitive development the study of capacity limitations is associated mainly with the work in what is known as the neo-Piagetian tradition. The goal of this research is to provide a more detailed and accurate account of cognitive development by adopting a more functionalist stance, often borrowing concepts and models from the information-processing approach and the problem solving research. At the same time, neo-Piagetians have more or less tried to preserve basic tenets from Piaget's structural theory of cognitive development. As stated in the previous chapter, several neo-Piagetian theorists have raised the hypothesis that capacity acts as a necessary but not sufficient condition for the development of reasoning. In brief, insufficient capacity would prevent performance at higher levels of reasoning. In contrast, the more capacity available, the more sophisticated form of reasoning would be exhibited, given that all other conditions are met. Different views, however, characterize the understanding of capacity in neo-Piagetian theories. A brief description of these views will precede the discussion of the evidence in support of the hypothesis. In Pascual-Leone's theory (1970, 1984, 1987), performance is considered to be determined by the dynamic interplay of two groups of factors. The first group comprises the basic units of the mental apparatus: operative schemes, figurative schemes, and executive schemes. The second includes as factors the most general architectural 24 constraints of the system that is studied. The M-operator (the capacity construct in Pascual-Leone's theory) denotes the nonspecific mental attentional energy or mental space. The function of the M-operator is to boost the mental schemes necessary for performance. Its action is monitored by the executive schemes dominant at the given moment. -Operationally, the power of the M-operator (M-power) is defined as the maximum number of schemes that can be activated in a single mental centration. The power of the M-operator, according to Pascual-Leone, increases endogenously with age. M-power is partitioned into two additive components: e — capacity used by the task executive to represent the problem goal and the initial strategy; and, k — the capacity for activation of additional schemata. The e-component is assumed to develop up to the second or third year of life and to remain constant afterwards, while the ^-component increases to late • adolescence. The rate of growth is constant and the range is from one additional scheme at age 3 ~ 4, to six or seven additional schemes at approximately the age of 16. These estimates were originally inferred from a logical analysis of Piagetian tasks, and later were empirically verified using a variety of tasks (see Pascual-Leone & Goodman, 1979, for references). Despite the parallels that can be drawn between this concept of M-capacity and the \"nonspecific\" capacity notion as described in the previous section, Pascual-Leone's approach to the empirical study of capacity has been quite different from the approach to capacity measurement under the \"nonspecific\" capacity view. This difference is due to the assumption that M-capacity can be quantified in terms of the number of 25 schemes that are applied to a solution of a problem. Thus, Pascual-Leone proposes an extensive and detailed procedure of task analysis for establishing the capacity demands of a task. Available capacity is estimated by means of tasks considered to be direct measures of individuals' capacity. These estimates of individuals' M-power and the assessment of task demands through task analysis have been used to provide evidence for the relation between reasoning and capacity. Task analysis has been most often identified as the weakness of the theory (Gelman & Baillargeon, 1983; a similar argument is leveled by Case, 1985). As mentioned above, Pascual-Leone has proposed a detailed and elaborate procedure for task analysis. Nevertheless, there is some arbitrariness in determining the units of analysis because the task analytical procedure is not derived from the theory in a rigorous manner (see also Chapman, 1987). Another problem is the prediction of performance with measures of M-power. The M-operator is not the only activating factor according to the theory. Other operators can serve as scheme-boosters. For example, the interactions of the operators for learning (L-, and C-operators) with the M-operator are considered to determine both the scheme that is formed and the speed of its formation (see deRibaupierre & Pascual-Leone, 1979). The direct influence on activation is the I-operator, which serves to inhibit the irrelevant to the current task structures (see Pascual-Leone 1987,1991). The joint action of all these factors produces a dynamic synthesis of schemes in the field of activation. However, since performance is a function of the joint action of several factors, why is it predicted with M-power measures? Averaging the performance level across several tasks 26 of the same demand has been proposed as a safeguard against such bias. This is supposed to reduce the variance caused by factors other than M-power in predicting performance on cognitive tasks. Nevertheless, there is no guarantee that the reduction is sufficient. The other three theories adopt an approach to the definition of capacity that is closer to the \"specific\" view of capacity. That is, specific assumptions are made regarding the processes and structures involved in the particular cognitive activity and capacity characterizes quantitatively the operation of these processes or structures. Executive Processing Space is the capacity construct in Case's (1985) theory, and is defined as the \"maximum number of independent schemes that a child can activate at any one time\" (p. 289). It is further subdivided into \"operating space\" (allocated to the activation of new schemes) and a \"short-term storage space\" (the proportion of the total processing space devoted to maintenance and retrieval). This subdivision does not imply two different capacities, each with their own limit, but a single capacity that can be allocated to the two functions. Case (1985) provided data from several studies concerning the nature of the growth of short-term storage space with age. It was hypothesized that the total processing space within each period remains constant, while the increase of storage reflects the decrease of operational space. In short, the greater the operational efficiency, the more space available for storage. Supporting evidence for this hypothesis is the high correlation between speed and span as revealed in a series of studies with each age group (see Case, 1985, pp. 354-365). In addition, a study of adults whose counting rate was artificially reduced to the level of six-year old children by counting in a nonsense 27 language, showed the same relation. The span performance of these adults was comparable to that of six-year olds. For the explanation of these changes in capacity, Case raises a tentative hypothesis for the possible physiological correlate of the changes in efficiency. In short, increased processing efficiency is related to the established fact of progressive myelinization of nerve fibres with development. The myelinization is related to the increased speed of linear transmission and reduced amount of lateral transmission. One advantage of Case's approach to capacity, as compared to that of Pascual-Leone, is the attempt at specifying in more detail the different processes involved in solving a problem and the quantitative limitations associated with them. Reasoning activity is described as involving two types of processes: the processes of storage and maintenance of information, and the processes of active organization and manipulation of this information for producing the solution. The assessment of capacity in this case, however, is more or less the assessment of the capacity of the short-term storage space, which is characteristic for a particular stage of development. In face of the evidence for the relative independence and different processing areas of the executive and storage processes (Baddeley & Hitch, 1974), and for the general increase of processing speed with age (Kail, 1988), such a treatment of capacity, at least when the question is about the capacity constraints on reasoning, seems insufficient. A more detailed account of the processes involved in reasoning is necessary as is an approach to measuring the capacity limits of these processes. Another important aspect, which is missing in Case's treatment of reasoning, has been pointed out by D. Kuhn (1983) in a critique of an earlier version of the theory. According to Kuhn, Case analyzes subjects' ability to execute the sequence of steps that lead to a success on a task. What is missing is the more developmentally challenging aspect of knowing that these are the appropriate strategies to apply, i.e., the process of active construction of the problem situation by the subject (see Kuhn, 1983, pp. 94 -99). The capacity constructs proposed in Chapman's (1987) model and in Halford's (1982, 1993) theory are both examples of the \"specific\" approach to capacity in that they characterize the limitations of particular processes. Both differ from Case's treatment of capacity in that they attempt to concentrate on the quantitative aspects of processes that are specific to reasoning, rather than on the capacity limitations of the overall system involved in a performance of a task. The conceptualization of these processes, however, is quite different in the two models. Chapman (1987) based his model on a constructivist approach to reasoning and, following Piaget, described the formal properties of the age-specific forms of reasoning as rooted in the interiorization of action and the coordination of mental operations in operatory structures. Three types of schemes are the functional units of the model: representational, procedural and operational schemes. \"Representational schemes\" refer to sensory, perceptual and cognitive representations of the permanent and simultaneous properties of comparable objects or classes. They provide the content that is coordinated in.the process of solving a task by the operational and procedural schemes. \"Procedural schemes\" are defined as \"transformations effected by the child successively in 29 time and in pursuit of a goal\" (Chapman, 1987). Finally, in defining \"operational schemes\", Chapman emphasized the importance of understanding them as internalized reversible actions. Reversibility here means that the operation is integrated in a structure with other operations that compensate the transformation it designates. In this case, the application of the operation is a simultaneous coordination of implications and does not involve temporal sequence. This simultaneous coordination marks deeper understanding and reaching the conclusion by necessity. The model is aimed at investigating the \"form of reasoning\", which is defined as the type of inferential relation uniting children's judgments (conclusions) with the explanations (premises) of those inferences. The operation of the model involves simultaneous coordination of the values provided by representational schemes in an \"inferential scheme\" (Chapman, 1987). In a later version of the model, this process is referred to as assigning a value to an operatory variable (operatory variables are defined as the \"aspects or dimensions of the task situation that the subject recognizes as potentially varying within the experiential context of the task\" (Chapman & Lindenberger, 1989). Regardless of the difference in terminology, a basic assumption of the model is that the structural act of assigning a value to an operatory variable corresponds to the functional consumption of a fixed amount of attentional capacity. This fixed amount of capacity is considered a \"unit\". In terms of the operation of the model then, the capacity requirements of a given form of reasoning will be equal to the number of operatory variables that are assigned values simultaneously in employing that form of reasoning in a particular task. The concept of capacity demands can be defined as the number of 30 representational schemes that must be coordinated for the solution of a given task. Defined this way, the model provides guidelines for the analysis of the capacity demands of the tasks, for the estimation of the capacity necessary for a particular form of reasoning, and for the clarification of the requirements towards the tasks used as independent measures of individuals'capacity. In task analysis, the estimation of task demands has usefully complemented the analysis of the structural aspects based on Piaget's operatory logic. The advantage of this approach is that it allows for analyzing the tasks in terms of their formal properties and for deriving the capacity demands of those tasks from the quantitative dimensions of those properties. Task analysis of typical Piagetian tasks indicated that demands increase regularly by stage (see Chapman, 1987, pp. 310-311). With respect to measuring capacity, Chapman did not propose new measurement tasks but used task analysis to demonstrate the relative validity of measures proposed by Pascual-Leone and Case. For example, as a result of the comparison between forward-and backward digit span as measures of capacity, the latter was evaluated as more relevant to the task of measuring capacity involved in reasoning because it entails the assimilation of items to a reversible scheme of temporal order, i.e., a scheme in which the forward order simultaneously implies its inverse. Halford's treatment of reasoning is based on a complexity metric derived from category theory (MacLane, 1972), which allows for a unified approach to the assessment of the complexity of a task and the structural complexity of the representations and structures used in a solution. The purpose of this is to anchor the complexity classification 31 of the tasks and the levels of reasoning on objective criteria. As Halford put if (Halford, 1982, p. 360), the fact that transitive reasoning has the structure of the binary operations characterizing Level 2 reasoning is a mathematical truth and not an intuitive judgment. The core of the metric is the process of structural mapping, which is defined as the rule for assigning elements of one structure to elements of another in such a way that any functions, relations or transformations between elements of the first structure correspond to functions, relations and transformations in the second structure (see Halford, 1993, p. 71). Four levels of task complexity are described, which differ in the number of elements and relations determining the problem space. According to Halford, the level at which a task will be approached depends upon the complexity level of the representations that the subject is capable of processing. The different levels of structure mapping require relevant means of representation, that is, concepts at the respective level of structural complexity. The complexity of concepts, according to Halford, is determined by their dimensionality, or by the number of independent units of information required to define a concept. The units themselves may have arbitrary informational size. Their number is related to the number of arguments in a predicate. Thus, one-dimensional concepts are predicates with one argument (e. g., category membership); two-dimensional concepts are defined as predicates with two arguments (binary relations and bivariate functions); etc. In brief, the notion of limited capacity in Halford's theory (1993) is associated with the limits in capacity for representing structure and is quantified in terms of the independent dimensions that can enter into a representational structure. Reasoning is 32 capacity limited in the sense that the structure mapping processes impose processing demands that depend on the dimensionality of the mapped structures. Thus, tasks that require concepts of high dimensionality to be mapped will impose high processing load that exceeds the capacity of young children. Halford's concept of capacity can be clearly classified as an example Of the \"specific\" approach to capacity. What distinguishes it from the other such concepts, is the attempt to consider the capacity limitations of processes that are intrinsically related to the activity of reasoning. Even in his earlier (1982) book, where short-term memory is assumed to be the \"workspace\" for reasoning, Halford emphasizes the \"... information-processing load imposed by the requirements of matching the symbol system to the environment system in a consistent way\" (p. 361) and takes no account of loads imposed in any other way. In the later (1993) book, an attempt is made at the explication of the concept of representational dimensionality on the basis of a computational model with connectionist architecture. In terms of the model, the independent dimensions are represented by separate vectors in the processing space. The tensor product of these vectors characterizes the concept that is applied. The argument is based on the established properties of neural networks that discriminability of items is proportional to the number of units used for the representation and the conduction speed of the neural computation. Thus, keeping the two factors constant, the increase in the number of items (vectors) entered into the computation will decrease their discriminability. Although still speculative, this hypothesis offers a way for accounting for the effects of capacity limitations and for eventual age changes in these limitations. In brief, it is suggested that 33 the number of dimensions that can be processed in parallel increases with age, possibly due to differentiation of representation. This would not increase overall capacity but will permit more complex structures to be processed and more complex concepts to be understood. It is clear that the overall amount of information is not limited, as far as the separate vectors that enter into a computation are of arbitrary informational size. The changes are in the number of such independent vectors that enter into any one , computation or decision. Also limited in this way are the orders of interaction and levels of structure that can be represented. The Relation between Reasoning and Capacity in Neo-Piagetian Theories: Evidence and Problems The theoretical models described above, although similar in several aspects, offer quite different interpretations of capacity and the influence of capacity limitations on reasoning. The different views should be kept in mind in evaluating the evidence because they determine to a great extent the processes upon which the research efforts of the authors were concentrated. The evidence provided by neo-Piagetian theorists for the relation between reasoning and capacity can be divided into three groups according to the approach taken for the empirical validation of this relation. The first group is comprised of evidence from correlational studies. Assuming that capacity has the role of a developmental constraint, it is natural to predict that there will be a close correspondence between levels of reasoning and levels of capacity. Two 34 objections have been leveled to this type of evidence. First, some critics have argued that the observed correspondences can be explained as an artifact of correlated age changes (Brainerd & Reyna, 1989; Howe & Rabinowitz, 1990). Second, several authors (e.g., Halford, 1993; Howe & Rabinowitz, 1990) have noted that the magnitude of the correlations does not exceed that typically found between many pairs of cognitive tasks and there is no method of determining whether the correlations are based on capacity. Both objections are valid, but only if the correlational evidence were the only evidence presented by neo-Piagetian theorists. Several studies in the frame of Pascual-Leone's theory include data about the association between performance on M-capacity measurement tasks and performance on cognitive tasks, the capacity demands of which have been assessed by means of task analysis. This relation has been established for a number of cognitive tasks and for different ages (for a review, see Chapman, 1981; see also Johnson & Pascual-Leone, 1989; Morra, Moiso & Scopesi, 1988; deRibaupierre & Pascual-Leone, 1979). However, this is not the only evidence and the reported correlational data are used as an initial stage in the studies. A similar argument can be made about several studies reported by Case (1985). He found, for example, that the average scores for_both cognitive tasks and capacity scores coincided closely with those predicted for each age (Chapters 6-11). It should be noted, however, that these chapters cover the initial \"descriptive stages\" of his project. Chapman (1987, 1990) argued that the mere detection of correspondences between capacity and cognitive development does not directly test the necessity-but-not-sufficiency relation between capacity and reasoning predicted by neo-Piagetian theories. '35 The former pattern of results might exist even if the latter relation did not occur. Chapman and Lindenberger (1989) proposed a direct test of the specific prediction on the basis of the statistical technique of prediction analysis (Hildebrand, Laing & Rosental, 1977). In brief, this is a technique which allows for testing predictions based on a logical relation between nominal or ordinal variables by partitioning the contingency table into \"permitted\" and \"non-permitted\" (error) cells. The comparison between the observations actually found in error cells and the expected errors (determined from the marginal totals) yields the test statistic (DEL) which is a measure of the extent to which the number of observed errors is less than expected by chance. For example, in testing the hypothesis that a certain level of cognitive development is necessary for a corresponding level of moral development one should expect the cell determined by performance on cognitive tasks under the specified level and performance on moral tasks at and over the respective level, to be empty. In Chapman and Lindenberger's study, a task analysis based on Piaget's operatory logic was used to determine the task demands of typical Piagetian tasks for class inclusion, transitivity, multiplication of classes, and multiplication of relations reasoning. A minimum of three units of capacity was found to be necessary for successful performance on such tasks. Two tasks, Backward Digit Span and Pascual-Leone's Figural Intersection Test, were used as measures of individuals' capacity. The results of 120 first, second, and third grade children on the measurement tasks were used to predict their performance on the reasoning tasks. More specifically, it was expected that the tasks would be solved by children with at least three units of capacity as determined from their performance on the measurement tasks. The prediction was confirmed for all tasks but the class inclusion one, which was solved by nearly all children suggesting that the particular version of>the class t ; inclusion task could be solved without operational reasoning. Another line of research, pursued in Chapman's laboratory, addressed the controversial question about the \"true\" ages at which particular cognitive competencies develop. This topic is characteristic of neo-Piagetian research. Several attempts had been made to demonstrate through task analysis that children could solve versions of Piagetian tasks at an earlier age, because those versions had lower capacity demands (e.g., Case, 1985, Chapter 11; Halford, 1987; Pascual-Leone & Smith, 1969). Pachev, McBride, Carpendale and Chapman (1993) applied the technique of prediction analysis and tested particular performance predictions in addition to ordering the versions of the tasks according to their demands through task analysis. The first i experiment compared the performance of forty eight children from three age groups (4 - 6, 7-8, 9 - 11) on two versions of a transitivity task. One version of the task was assessed as requiring the standard (according to Chapman's task analysis) for transitivity tasks: 3 units of capacity. The second version was designed'to permit a solution by means of a functional scheme that requires 2 units only. Subjects' capacity level was assessed by means of the Backward Digit Span and the Opposites Test. One of the predictions tested in the study is of interest here. It was expected that children who have only two or less attentional capacity \"units\" should be able to pass the task when a functional scheme is applicable, but not in cases when a functional solution is impossible. This predictions was confirmed. For both capacity measures and in both conditions of the transitivity task, the 37 number of children who passed the task with two or less capacity units was much less than the number expected to fall into this category solely by chance under the null hypothesis of independence. In addition, clear evidence for a tendency to infer weight as a function of size was obtained, but only for children who would otherwise have used a nonoperational form of inference. The second experiment compared performance on a standard and easier version of a class inclusion task. It was estimated that in order to solve the standard version of the class inclusion task the child must simultaneously attend to the supraordinate and subordinate classes. Therefore, three variables must be evaluated and the operation of class addition would require a minimum of three units. The decrease in the capacity demands of the easy version of the task was achieved through omitting the comparison of the subclasses by asking the subjects to compare them before the test question was posed. Sixty-eight children, divided evenly into two age groups (5 - 6, 7 - 8) were given two class inclusion tasks (differing in the materials used and the dimension by which the supraordinate class was labeled) and two capacity measures (Backward Digit Span and Opposites Test). Each class inclusion task was given under one of two conditions: the \"prior question\" condition (subjects were asked to compare the subclasses before the class inclusion question was posed) and, the \"no prior question\" condition (standard version). It was hypothesized that children with two or less \"units\" of capacity should be able to pass the class inclusion task in the \"prior question\" condition, but not in the \"no prior question\" condition. This prediction was confirmed when Backward Digit Span scores, 3 8 and the mean scores from the two capacity measures were used in the prediction. The Opposites Test (Case, 1985) failed to yield a significant prediction. DeRibaupierre and Pascual-Leone (1979) proposed a binomial test method for testing the necessity-but-not-sufficiency of capacity for the development of reasoning. The model resembles the prediction analysis method of Hildebrand, Laing and Rosental (1977) in that it is based on determining error (here called \"critical\") cells in the contingency table. The procedure afterward is different. Expected frequencies for each critical cell are computed from the marginal frequencies. These expected frequencies are added and divided by the total number of responses in the contingency table. The result is the expected probability p that a response falls by chance within the critical cells and q of falling in the noncritical cells, with N equal to the total frequency in the table and X equal to the total frequency in these critical cells. Using the binomial tables, one can find the probability, with which the obtained pattern is due to chance alone. In the study, deRibaupierre and PascualrLeone applied the binomial-test model to test the prediction for the formal-operational stage that subjects with M-power lower than e + 6 would not be able to perform above a certain level on formal-reasoning tasks. The M-power of the subjects (12 and 15 year olds) was assessed by means of the Figural Intersection Test and the Compound Stimuli Visual Information task. Both tests were proposed by Pascual-Leone (1970, 1978; Pascual-Leone & Smith, 1969) as measures of M-capacity as defined in his theory. The performance level on the cognitive tasks was determined by task-analysis of subjects' performance on five formal tasks: versions of Balance Task, Projection of Shadows, Pendulum, Flexibility of Rods (Inhelder & Piaget, 39 1958) and Control of Variables (Scardamalia, 1977). Several other tests were included in the test battery which controlled for the influence on performance of the other \"operators\" postulated in Pascual-Leone's theory (e. g., F- and L-operators were controlled by including Witkin's Embedded Figures Test and the Hidden Figures Test). In addition, the order of administering the tasks in four separate sessions, allowed the results on the first two sessions to be treated as \"pretest\" scores and the results from the second two sessions as \"posttest\" scores. The predictions of interest here were confirmed for all but one of the four scores determined by the combination of age group and type (pretest vs. posttest) of score: Group 12 pretest (p = 0.16), group 12 posttests (p = 0.02), group 15 pretests (p = 0.03), group 15 posttests (p = 0.05). The predictions based on the total scores were also significant: group 12 total (p = 0.01), group 15 total (p = 0.004), total pretest (p = 0.005), total posttest (p = 0.001). It should be noted that these results were achieved by using in the prediction the average score of subjects across tasks and the higher score from the results on the M-power measurement tasks. This type of approach was considered by the authors as consistent with the interpretation of the relation between reasoning and capacity in Pascual-LeOne's theory. Having in mind, however, that the problem of direct measurement of capacity is still a controversial issue, and the weaknesses of Pascual-Leone's task-analytical procedure mentioned in the previous section, the results should be regarded with caution. More recent studies in the framework of Pascual-Leone's theory (e. g., Johnson & Pascual-Leone, 1989; Morra, Moizo & Scopesi, 1988; Stewart & Pascual-Leone, 1992) 40 that deal with the relation between capacity and particular forms of reasoning usually approach the specific hypothesis by means of the binomial test. Stewart and Pascual-Leone (1992), for example, in their study of the relation between M-power and moral reasoning report a probability of p = 0.04 that the pattern they found is due to chance. This result was supported by a prediction analysis procedure (DEL = 0.527; Z(DEL) = 4.444; p < .001). The problems with measuring capacity, however, are evident in this study as well. This time the average of the scores of each subject on the two measures used (Figural Intersection Test and Compound Stimuli Visual Information test) was included in the statistical tests as a more reliable measure of mental capacity. The prediction analysis and the binomial-test approaches to testing the relation between reasoning and capacity provide more compelling evidence for the role of capacity as a developmental constraint when compared^ to the mere registration of associations between measures of capacity and measures of cognitive development. It should be noted, however, that each of the studies described above required certain \"adjustments\" (averaging across reasoning tasks, averaging across measurement tasks, use of highest results, etc.) for a successful prediction in some cases. This indicates one of the weaknesses of this approach: it relies on direct measures of capacity which have questionable validity. In addition, the approach relies on a undifferentiated view of capacity, and does not allow for a detailed study of the compartment processes of reasoning and their quantitative characteristics. This last task is usually addressed in studies with experimental or quasi-experimental designs. 41 One set of such studies, reported by Case (1985, pp. 331-348), deals more directly with the question of whether the size of the short-term storage space sets limits on the complexity of the executive control structures assembled. All studies use similar logic: if two groups with different short-term storage space are exposed to the opportunity to learn a new structure, only the group with sufficient short-term storage capacity will benefit from the training. This hypothesis has been confirmed for dimensional tasks. Of interest here is an experimental study with adults. The different span of the storage space has been experimentally induced by providing a different amount of training in counting in an artificial language. When exposed to an opportunity for learning a new structure, which involved the trained operation as a component, the group with induced higher short-term storage space benefited considerably more. The strength of the evidence from these studies should be evaluated keeping in mind the context of the theory that generated these results. As mentioned in the previous section, the important aspect of approaching the task and constructing it as a problem situation by the subject was not well addressed in this model. A particular algorithm for solving a task can be learned, a strategy for approaching the problem can be learned, as well, but the problem is whether the application of the algorithm or the strategy in this case reflects a developmental achievement. From this point of view, the regularities found in these studies are important but they pertain to the role of capacity as a constraint on learning rather than to capacity as a constraint on development. Halford's application of a mathematical scheme derived from category theory (MacLane, 1972) to the problem of task analysis has been positively evaluated as adding 42 rigor to the problem of task analysis. The application of the complexity metric to the problems of measurement of capacity and of deriving the task demands, however, has been criticized. More specifically, in his earlier works Halford (1982) had relied on the use of short-term memory span measures as measures of capacity. In his later works, however, Halford adopts an approach that avoids the measurement of capacity by independent means. Instead of independent measures of capacity, Halford applies methods for detecting capacity-limited performance (Norman and Bobrow, 1977), namely Hunt and Lansman's (1982) easy-to-hard paradigm for assessing individual differences in resources. Another feature of Halford's later works is the use of the dual-task paradigm for identifying the key processes involved in reasoning and describing their specific functions. It is this new approach to the problem of the relation between reasoning and capacity in the aspect of development that is of greatest interest for the present work. Two sets of studies on transitive and class inclusion reasoning using the dual-task methodology will be described briefly below. Consistent with the proposed understanding of capacity constraints as constraints on the dimensionality of representations, Halford predicted that the main difficulties experienced with transitive tasks would be associated with the integration of premises when the relation between nonadjacent premise components is to be judged. Maybery, Bain and Halford (1986) provide evidence for this in a study using a dual task approach with adults. In their study, subjects were presented with successive displays of the premises and the target relation and had to indicate whether the target was consistent with the premises by pressing different buttons for consistent or inconsistent target relations. 43 The task had two conditions. In one condition, involving premise integration, the target demanded the establishment of the relation between non-adjacent elements. For example, \" J is above T\", \"N is below T\" (premises); \"N is above T' (target). In the second, control condition, the target display involved establishing the relation between elements from one of the premises ( e. g., premises: \"R is above G\" and \"L is below S\"; target: \" \"L is above S\"). Vocal reaction to a tone has been used as a secondary task, with probe reaction time as an index for interference. The tone was administered with each premise and target as well as before and after the presentation of the primary task stimuli. Premise integration was expected to occur with the presentation of the second premise, resulting in longest reaction time to the probe at this phase. The results supported the prediction: reaction time was increased for the probe, which accompanied the presentation of the second premise and in the experimental condition only. Other factors, like processing of negatives or increased problem solution time had no effect on this pattern of results. In addition, the pattern was not changed by the decrease of the solution time over trials. Halford, Maybery and Bain (1986) report two experiments using the dual-task approach to transitive reasoning with children. The first experiment is an attempt at replication of Baddeley and Hitch's (1974) findings that memory load interfered with reasoning only for a near the limit load and for more difficult problems. Eighteen children, (5 - 6-year-olds) were given two- and three-term transitive problems as primary tasks and had to perform them alone or concurrently with either articulation of a word repeated several times or with a short-term retention and rehearsal of two color pairs. Pilot experiments showed that passive retention did not interfere with reasoning. Both the 44 active rehearsal and the articulation condition (the latter to a lesser degree) increased the time and decreased the accuracy of the solution. These effects were larger for the more difficult three-term task. These results, and results like those reported by Maybery et al. (.1986) demonstrate the possibility for applying the dual task-approach to studies with children. The problem with this version of the dual-task paradigm is that the observed interference might be due to either structural or output effects. The second experiment of Halford et al, (1986) provides evidence for the capacity nature of the interference. In this experiment the two- and three-term transitive tasks were used as the easy and hard primary task respectively, and the secondary task was remembering and rehearsing two color pairs. Subjects were 36 children within the age range 3;4 — 5;9. The easy primary task and the secondary task were performed both alone and together, while the hard primary task was performed alone only. The hypothesis for capacity limited performance was assessed by using Hunt and Lansman's (1982) paradigm. The results confirmed the capacity nature of the interference as evidenced by the successful prediction of performance on the three-term transitive task (criterion) by both accuracy and latency measures of performance on the secondary task, performed concurrently with the easy primary task. This approach has been extended to class inclusion reasoning. Halford proposed that class inclusion inferences were made by mapping the problem into a pragmatic-reasoning schema (i.e., an induced from experience familiar analogue of the inclusion relation). The class inclusion concept, according to Halford's task analytic scheme, is a ternary relation because it involves relations between a superordinate and two subsets. At 45 least part of the difficulties experienced by children, it was hypothesized, could be explained with the system mapping required by the task. Halford (1993) described two studies.(Leitch, 1989; Halford & Leitch, 1989) in which the structure-mapping hypothesis has been tested on the basis of tasks having the logical structure of class inclusion. These tasks preserve the inclusion hierarchy but avoid difficulties caused by the unusual linguistic form of the inclusion question. The minimum case of an inclusion hierarchy would consist of two elements that have at least one attribute in common and at least one on which they differ. Thus, in the studies described by Halford (1993), children had to choose a pair of toys, which shared a certain dimension but differed in another, from a series of pairs that were of three kinds: inclusive, identical (no different attribute) and disjoint (no common attribute). In one of the studies, children aged 3 ~ 6 years received four such problems that required mapping the set of objects into a schema consisting of one common attribute and two distinct attributes. Children less than 5 years old showed performance at a chance level, which supported the hypothesis that the difficulties of young children with the inclusion schema might be at least partly due to the complexity of the mapping. The second study tested the capacity nature of these difficulties using the easy-to-hard paradigm. Subjects were children aged 3 ~ 8. The hard primary task required the recognition of a pair of stimuli that formed a minimal inclusion hierarchy (similar to the pairs from the first study). The easy task required children to recognize whether two stimuli were the same or different along only one dimension. Probe reaction time to a tone was used as an index of performance on the secondary task. Performance on the easy primary task, performed jointly with the secondary task, predicted performance on the inclusion isomorph task, with performance on the two predictors performed separately partialled out. That is, positive evidence of capacity-limited performance was obtained. Through the use of the dual-task methodology Halford has addressed two of the questions that are of interest in the present work. First, the method of selective distraction has been applied (Study 1 in: Halford, Maybery and Bain, 1986) to the problem for the nature of the capacity constraints associated with reasoning. Additional capacity load had been introduced through the secondary tasks for rehearsal and articulation. The results confirmed the expectations that active processing (rather than passive retention) is more disrupting. Second, the easy-to-hard paradigm was applied to provide evidence for the capacity character of the observed difficulties (Study 2 in: Halford, Maybery & Bain, 1986 and Halford & Leitch, 1989). The two questions, however, had been addressed separately. Thus, there is only indirect evidence that the observed interference at rehearsal and articulation is capacity-based. In addition, the developmental aspect (i.e., whether capacity changes with age) of the specific hypothesis for the relation between reasoning and capacity had not been addressed. In conclusion, it can be said that the problem for the relation between reasoning and capacity, although shared by neo-Piagetian theories, has been approached from quite different perspectives and with different means. These differences concern both the treatment of the capacity construct and the understanding of the processes involved in reasoning. 47 The capacity constructs, as defined in the discussed theories range from global nonspecific architectural constraints of the cognitive system to characteristics of particular specifiable structures or processes. It can be argued that these are, in fact, different levels of generality in specifying one and the same reality. There is, however, a well-known trade-off between scope and precision in theory building. From this point of view, theories and models that treat capacity as characterizing particular processes are more promising when the goal is to specify the \"bottleneck\" of a particular function. Thus, in considering the capacity constraints on reasoning the important task is to define the processes that are central to the reasoning activity. Such an attempt has been undertaken in Halford's theory and, to a certain degree, in Chapman's structural-functional model. Both models concentrate on processes that operate at the executive phase of the solution of a task and deal with task demands imposed by the formal properties of the reasoning demanded by the task. In both cases the process of solving the problem is presented as an application of an available inferential scheme to the material of the task. Capacity characterizes the \"read off' process, or the quantitative features of the inferential scheme. Although important, and probably sufficient for explaining the solution in certain cases, these processes are only a part of the reasoning activity. In novel situations, for example, the inferential scheme has to be constructed and it can be argued that it is the success of this constructive process that determines the exhibited higher form of reasoning. From this point of view, the \"bottleneck\" is situated at the input phase of the solution and characterizes the active construction of the problem space. 48 Both Chapman's model and Halford's theory allow for interpreting the quantitative dimensions at this phase in terms of the same processes applied to the executive phase. That is, according to Halford, the ability of integrating the task dimensions in unitary representation would determine the complexity of the mental model that is built. In Chapman's terms, the number of operatory variables that can be assigned values simultaneously determines the level of understanding with which the problem would be approached. The influence upon the solution of the capacity limits at the two phases, however, has not been studied empirically as yet, partially due to the difficulties associated with the separation of these phases in the experimental situation. In summary, as outlined in Table 1, the empirical evidence for the relation between reasoning and capacity postulated in the neo-Piagetian theories can be divided as coming from three sources. The first group comprises correlational studies that provide data for an association between performance on reasoning tasks and on tasks designed to measure capacity. There are two main problems with this type of evidence. The first one is the validity of the measurement tasks. The second is that the relation between performance on the two types of tasks might be explained as an artifact of correlated age changes. In other words, establishing the association does not prove that it is capacity-based and does not test the proposal that capacity acts as a necessary but not a sufficient condition for the development of reasoning. The second source of evidence comprises studies that use statistical techniques for testing in a specific way the necessity-but-not-sufficiency condition. Chapman and Lindenberger (1989), for example, used the procedure of prediction analysis that allowed 49 for generating and testing specific predictions for the distribution of subjects in the cells of the contingency table according to the type of relation between the measured factors. Although this method has certain advantages in comparison with the correlational analysis, it does not avoid the problem of measuring capacity with \"direct\" measures and does not allow for a more detailed analysis of the nature of the capacity constraints of the different processes involved in reasoning. The studies that comprise the third source of evidence for the relation allow for separating the influence on performance of the capacity limits of the compartment processes involved in reasoning through the use of dual-task methodology for an experimental disruption of the solution process at different points. In addition, part of the problems associated with the independent measurement of capacity are avoided through the use of design-based procedures for detecting capacity-limited performance. So far, the variance in performance attributable to particular compartment processes has been studied separately from the question of whether performance on a particular kind of task is capacity- or data-limited. In addition, by avoiding the problem of measuring individuals' capacity, the experimental approach has lost the means for testing the hypothesis for the necessity-but-not-sufficieney character of the relation by means of prediction analysis. -The study reported below is aimed at providing evidence for the operation of capacity as a necessary but not sufficient condition for the development of reasoning, by combining the advantages and avoiding the shortcomings of the approaches outlined so far. 50 Table 1 Sirmmary of the empirical evidence. Type of evidence Studies Advantages Weaknesses Correlational Johnson & Pascual-Leone, 1989; Morra, Moiso & Scopesi, 1988;deRibaupierre& Pascual-Leone, 1979; Case, 1985 (Ch. 4-11) Testing the (weak) prediction: if capacity were a developmental constraint, then there should be an association between levels of capacity and reasoning 1. Based on \"direct\" measures of capacity. 2. Does not explore causal relation. 3. The magnitude of correlations does not exceed the one that is usually found between cognitive tasks. Prediction analysis/ Binomial test Morra, Moiso & Scopesi, 1988; deRibaupierre & Pascual-Leone, 1979 Stewart & Pascual-Leone, 1992; Chapman, 1987; Chapman & Lindenberger, 1989; Pachevetal., 1993. Test of the specific (strong) hypothesis: Capacity is a necessary but not sufficient condition for the development of reasoning. 1. Based on \"direct\" capacity measures, which have questionable validity; 2. Does not allow testing for multiple constraints. Experimental Case, 1985; Maybery, Bain and Halford 1986; Halford, Maybery and Bain 1986; Leitch, 1989; Halford & Leitch, 1989. 1. Avoids the use of \"direct\" capacity measures. 2. Allows for testing multiple constraints. 1. Does not test the \"specific\" (strong) hypothesis directly. 51 Chapter 3 Hypotheses and Method Objectives and Hypotheses The review of the literature identified two sets of problems associated with the hypothesis that capacity is a necessary but not sufficient condition for development of reasoning. The first set concerns the empirical approach to the claim that there is an age-related increase in the capacity characteristics of the processes involved in reasoning. The second set of problems has to do with the nature of the constraints. There is no agreement whether the difficulties associated with solving a reasoning problem are due to limits in some general capacity of the system, to inability for memorizing the premises, or to inability for coordinating all the necessary information in an inferential scheme. The present study was designed to address questions relevant to both problem areas. With regard to the empirical approach to the relation between reasoning and capacity, it was established that most of the studies relied on procedures involving \"direct\" measures of capacity and task-analytical schemes with questionable validity. This resulted in rather rough estimates for the influence of capacity limits on reasoning, and the procedures very often involved averaging across performance or capacity measures in order to obtain statistically significant outcomes. In addition, the studies using the alternate approach to measuring capacity by means of a dual-task procedure focused on obtaining capacity-limited performance of subjects from a single group and did not address the questions about age-differences in capacity. 52 Thus, the first objective pursued with the study was to apply Hunt and Lansman's (1982) dual-task performance model to the question of age-related differences in resources. As a reminder, the model required at least two levels of difficulty of the primary (reasoning, in the present case) task and yielded an index of capacity-limited performance for the more difficult version. The need to distinguish between different age-groups poses the requirement of more than two difficulty levels of the task. In addition, these levels should differ only in their demands of resources. To meet these requirements, a matrices-completion reasoning task was chosen for the study. The task had the logical structure of addition or multiplication of classes and relations. The level of difficulty was manipulated by increasing the number of attributes that defined the classification criterion. This increase in the number of attributes did not affect the way of presenting the task and the way the task was to be approached. Assuming that there is an age-related increase in processing resources, a particular prediction can be stated about the pattern of results from the application of Hunt and Lansman's procedure. If age-groups of subjects are presented with versions of a reasoning task, which vary in their quantitative characteristics only, then one should expect to detect capacity-limited performance at different points of the \"task scale\" for the different groups. In particular, it was expected that younger subjects would exhibit capacity-limited performance at the easier levels of the task. 53 The second objective that was pursued with this study was to explore the constraints imposed by the capacity characteristics of multiple processes involved in the reasoning activity. . The review of the capacity concepts proposed in the four theories established considerable differences in the understanding of capacity constraints on reasoning. These discrepancies were partially due to the different conceptualizations of the critical processes involved in the reasoning activity. The authors had focused on specifying the limitations of processes associated with maintenance of information in short-term (or working) memory, and processes at the executive part of the solution. Despite these differences, the theories are similar in concentrating on a single process. The view of capacity constraints proposed in the present work differs from these theories in assuming that reasoning, as a complex activity, will depend on the capacity characteristics of the different processes that comprise it. The exact processes to be included in the solution depends upon the material and way of presenting a task. Thus, it is possible to have tasks equivalent in logical structure that need to be solved by applying different processes. It is also possible to have tasks, differing in logical structure, the solution of which is carried out by the same processes. The latter is the case with the task that was chosen for the present study. The constant format of presenting the task allowed for including both additive and multiplicative classification by varying only the attributes that determined the classification criterion. It was assumed that subjects had to carry out the following processing steps in order to solve each problem: (1) identification of the relevant to the task attributes; (2) construction of a model for the correct answer; 54 (3) comparison of the model to the proposed items and choice of an item. Thus, performance on the task could depend upon the capacity characteristics of each of these \" processes. One method for studying the influence of a particular process on the overall performance, as discussed in the first chapter, is the dual-task method of selective distraction. In brief, the method consists in disrupting the operation of particular processes by administering a specific concurrent task (e.g., additional memory load in order to disrupt the operation of the short-term store). A similar approach was used in the present study. Instead of applying a specific secondary task, however, the compartment processes were disrupted by administering one and the same secondary task at different time-points of a trial. In particular, the administration of the secondary task at the beginning of a trial was supposed to introduce additional load and to disrupt the process of identifying the relevant task attributes; the administration of the secondary task in the second half of a trial was aimed at disrupting the processes at the executive part of the solution. This modification allowed for combining Hunt and Lansman's procedure with the task of selectively disrupting the operation of the separate processes involved in the solution of the reasoning problem. In addition, the subsequent test of the easy-to-hard prediction allowed for inferences about the capacity characteristics of the disrupted process: if performance on the secondary task predicted performance on the hard reasoning task, then the disrupted process would be, most likely, approaching its capacity limits. The consequences of insufficient capacity could be then estimated by comparing 55 ) performance on the primary task alone with performance of the primary task in dual-task conditions at the same level of difficulty. The results of these comparisons bear direct relevance to the second objective, stated above. If the limits in the capacity characteristics of the separate processes did constrain reasoning, then performance of the reasoning task in the dual-task conditions should be worse than performance of the reasoning task alone at the same level of difficulty. These were the expectations about the results from this part of the study. Method The measurements in the study combined the requirements of two dual-task paradigms. These are: the easy-to-hard version of the dual-task paradigm (Hunt & Lansman, 1982) for detecting capacity-limited performance, and the selective distraction paradigm for assessing the effect of an interfering task on reasoning performance. The y two methods were combined in a single procedure and shared several measures but involved different analyses. Participants Participants were recruited from three public elementary schools in Abbotsford School District, British Columbia, and from the \"Kids Club\" at U.B.C. Child Care Services. Only students for whom a parental consent was obtained were approached. Overall, eighty-six subjects between 6 and 14 years participated in the study. Three groups were formed with approximately equal number of boys and girls in each 56 group. Group 1 included 31 children from the age range 6 — 8 (M= 83.45 months; SD = 5.94). There were 15 girls (M= 82.33 months; SD = 5.60) and 16 boys (M= 84.50 months; SD = 6.24) in this group. Group 2 included 28 children from the age range 9 ~ 11 (M= 117.71 months; SD = 9.82), with 16 girls (M = 120.63 months; SD = 10.66) and 12 boys (M= 113.83 months; SD = 7.27). Group 3 included 27 subjects from the age range 12 --14 (M= 149.74 months; SD = 6.63). There were 14 girls (M= 151.00 months; SD = 6.58) and 13 boys (M= 148.38 months; SD •= 6.68) in this group. The following criteria were taken into account when selecting the age groups for the study: (1) subjects had to be able to solve the easiest versions of the reasoning task; (2) the selection of subjects and groups had to ensure sufficient variance with regard to (the theoretically assumed) available resources within and between groups. With respect to the first criterion, additive and multiplicative classifications are characteristic for the stage of concrete operations and beyond (cf. Inhelder & Piaget, 1964). Thus, the children targeted in the study included those at the stage of concrete operations and formal operations. The second criterion was met by sampling children from the whole range around the ages of 7, 10, and 13. Equipment The presentation of stimuli and the recording of most of the responses were under the control of a NEC/ProSpeed/SX 20 portable computer. The experiment was programmed using the MEL 1.0 Integrated system (Schneider, 1990). 57 Tasks The primary task was a matrices completion classification task. In this type of task, subjects are given a set of elements in a multiplicative table layout, where all spaces but one are filled. Subjects have to identify the criterion (or criteria) according to which the elements are divided into two (or more) subclasses and complete the matrix by filling in the last space. The logical structure of the task corresponds to addition (in the case of one classification criterion) and multiplication (in the case of more than one classification criteria) of classes and relations. The psychological analysis, however, is complicated by difficulty in discriminating between different ways of solving the problem. Piaget, in his extensive study of classification and seriation in children (Inhelder & Piaget, 1964), points out that several conditions of the operational solution are met by the perceptual configuration of the matrix for the elements already given. Thus, it is possible to find the correct result not through operational reasoning but by extending the graphic properties of the given elements through following the vertical and horizontal symmetries in the matrix arrangement (pp. 153-154). In the present experiment, the attempts to ensure uniform solutions of the tasks included: 1) targeting age groups for which the operational solution is characteristic; 2) manipulation of the way of presenting the elements of the task, so that the influence of perceptual factors is reduced in certain experimental conditions. For each trial of the present experiment, subjects were first shown on the top left part of the computer screen an incomplete matrix in the form of a square (8 cm x 8 cm) 58 with four cells (4 cm x 4 cm each). The lower right cell was empty and the other three cells contained geometrical shapes that formed a pattern. Subjects were instructed to analyze the pattern and to try to figure out what the contents of the empty cell should be, in order to complete the pattern. The duration of the exposure of the incomplete matrix was set to 5 seconds (Phase 1). Next, four 4 cm x 4 cm squares, approximately! cm apart, appeared on the top right part of the screen. These squares were labeled A, B, C, and D, and contained geometrical shapes that represented possible elements for the empty cell of the square from Phase 1. Subjects were instructed to choose the element that corresponded to their solution and to indicate the choice by saying aloud the letter label1 of the cell (Phase 2). The choice was entered on the keyboard by the experimenter. The display of this second screen was terminated upon receiving a response or after 30 seconds if no response was entered. In fact, none of the subjects exceeded the 30 seconds time-limit of this phase. Subjects' choices, the accuracy of the choices, and their latency were recorded automatically by the computer. Finally, the termination of Phase 2 initiated the display of a third screen that contained the material from Phase 1 and Phase 2. Subjects were asked to explain their choice. The experimenter recorded on paper the number of relevant attributes referred to in the explanation for each trial. This number was later used to calculate a performance index, which was intended to capture eventual difficulties that subjects might have with the identification of the relevant task variables. Subjects were given 30 seconds for a justification of their answer. The next trial started after the 30 seconds had elapsed or after the experimenter entered a command signifying the end of the trial. 59 This basic structure of a trial appeared in modified versions across the different conditions of the experiment. The level of task difficulty was manipulated by varying the number of attributes defining the classification criterion. Tasks were designed at four levels of difficulty, that is, one-, two-, three-, and four-attribute tasks. The following attributes were used: shape (square, rectangle, circle, ellipse, or triangle), color (black or white), size (small or large), number (the maximum number of shapes was 3), and orientation (only rectangles, ellipses and triangles were combined with the orientation attribute; they were flipped 90 degrees to the right or to the left). There were two conditions of each trial, differing in the display of the four possible answers at Phase 2. In one of the conditions, referred to as the \"Perception\" condition here, subjects made their choice in the presence of the incomplete matrix. In the second, \"Memory\" condition, the four possible solutions were presented alone. This manipulation was aimed at ensuring that subjects would engage in active analysis during Phase 1. Examples of the three events of a trial for the two conditions are given on Figures 4 and 5. Concurrent performance on the primary task included the trial events described above and a secondary task administered at different points of a primary task trial. The secondary task was a manual reaction (pressing a key) to a tone signal. A series of 1000 Hz tones, of 100 ms duration each, were administered at different points of the task and the subjects had to respond to each by pressing the \"Z\" key on the keyboard. For all trials that involved an administration of a tone, as well as for the series, which was designed to measure subjects' criterion reaction time, subjects were instructed to keep a left-hand finger on the \"Z\" key. 60 Criterion reaction time was measured in a separate series, at the beginning of the experimental procedure. The series consisted of 25 four-second trials. Within each trial the tone was administered at one of four time points: 500, 1000, 1500, or 2000 ms. (There were 7 signals for the 1000 ms tone-onset point and 6 for each of the other tone-onset points). The order of trials was random and controlled by the computer. The secondary task in the concurrent performance conditions of the experiment involved the administration of the tone signal at the beginning of Phase 1 (400 ms after the initial display of the primary task), or at the beginning of Phase 2 (500 ms after the presentation of the second screen), or at the beginning of both screens (500 ms and 500 ms respectively for Phase 1 and Phase 2). Measures and Design The purpose of the study was two-fold: 1) to provide evidence for capacity-limited performance at different levels of difficulty of the task for the different age groups, and 2) to provide a test for capacity limitations at different points of the reasoning process, thus allowing inferences about possible multiple quantitative constraints on reasoning performance. These two aims, coordinated in the study, pose certain requirements to the design and to the measures necessary for the tests. Hunt and Lansman's (1982) procedure for detecting capacity-limited performance requires repeated measures of primary task performance at at least two levels of difficulty, Figure 4. A n example of a trial for the \"perception\" condition, three-attribute task. The attributes that define the classification criterion are: shape (square or circle), color (black or white), and size (smaller vs. larger). 63 for dual-task conditions and alone. Similarly, secondary task measures include performance alone and in dual-task conditions. The correlational character of the test poses the requirement of a fixed order of experimental events for each subject. Further, the second objective requires that there are several dual-task conditions within each level of difficulty. Finally, both methods are more reliable when there are multiple measurements for each data point. In the present study, subjects from each age group were given the reasoning (primary) task at three levels of difficulty. Pilot testing revealed that the youngest subjects could not solve correctly the four-attribute problem, thus, Group 1 was given the task at one-, two-, and three-attribute level of difficulty. A ceiling effect characterized the performance of subjects from the other two age-groups at the one-attribute level of the task. That is why, Group 2 and Group 3 were administered the task at the two-, three-, and four-attribute level of difficulty. There were two task conditions for the administration of the trials within each level of difficulty: the Perception and Memory conditions, mentioned in the description of the task! The first one resembles more closely the standard format for presenting matrices tests (e.g., Raven's \"Progressive Matrices\"). The Memory condition was introduced here to force and control for a more uniform way of solving the problem. In brief, the presentation of the four possible answers without the initial incomplete matrix gives subjects no alternative but to form a-model'of the answer and to try to memorize it at Phase 1. At the same time, if subjects are consistent in their approach to the solution one 64 should expect parallel (though, not necessarily equal) results, as indexed by the secondary and primary task measures of performance, in both conditions. Each Level of difficulty x Task condition block contained four conditions, designed to meet the requirements for comparing the effects of distraction at different points of the reasoning process. The trials of the \"No-distraction\" condition presented subjects with the primary (reasoning) task only. The results obtained from these trials provided the measure of performance on the primary task alone, which was necessary for the Hunt and Lansman's test, and also served as a base for estimating the effect of distraction on reasoning performance. The other three conditions contained concurrent performance of the primary and secondary task trials. \"Phase 1 distraction\" involved administering the tone signal at the beginning of the first screen; in \"Phase 2 distraction\" trials the tone was administered at the beginning of the second screen; in the \"Full distraction\" trials it was administered at the beginning of both. Performance indices from these trials provided measures of concurrent performance on the primary and secondary tasks! To meet the requirement for multiple measurements for each data point, there were three trials for each \"level of difficulty\" by \"task condition\" by \"distraction condition\" cell. Thus, subjects from Group 1 received 24 one-attribute task trials (12 trials in the Perception condition and 12 trials in the Memory condition), 24 two-attribute task trials, and 6 three-attribute task trials (the tasks at the highest level of difficulty were given in the No-distraction condition only). Subjects from Groups 2 and 3 were given 24 two-attribute task trials, 24 three-attribute task trials, and 6 four-attribute task trials. j 65 Since the main interest was in the quantitative characteristics of the solution process, subjects were instructed about the type of problem they would be given. Thus, the presentation of the trials was blocked according to the level of difficulty and the condition of presenting the task (Perception or Memory condition). Within each block, the trials of the No-distraction condition were presented first, and then the nine trials of the three dual-task conditions followed in a random sequence. The order was determined by a random draw before the experiment and was the same for all participants who received these blocks. Further, to minimize the differences in the impact that the attributes have on the difficulty of the task (for example, shape may be more readily utilized as a classification criterion than orientation), the number of tasks with a particular attribute or a combination of attributes was set to be approximately equal within a block. The same approach was adopted in determining the geometrical shapes for a trial: approximately equal number of squares, rectangles, triangles, circles, and ellipses, appeared within a level of difficulty block. The trials for the four-attribute level of difficulty task were the exception here. The inclusion of the relative orientation as part of the classification criterion restricted the choice of geometrical shapes to triangles, rectangles and ellipses. Finally, an equal number of A, B, C, or D correct responses appeared within each level of difficulty block. Subjects were explicitly warned not to expect that the correct answer would always be one and the same. The order of attributes, shapes, and responses, as they appeared across the trials of a block was determined by means of a random draw before the experiment and was fixed for all participants that received this block. Tables 2 to 5 show 66 Table 2 Trials for the one-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses. Trial number Task condition block Distraction condition Criterial Attribute Shapes Correct response 1 \"Perception\" No distraction shape circle, square D 2 \"Perception\" No distraction color triangle B 3 \"Perception\" No distraction orientation ellipse A 4 \"Perception\" Phase 2. number circle C 5 \"Perception\" Full orientation rectangle A 6 \"Perception\" Phase 1. size square B 7 \"Perception\" Ful l shape circle, triangle D 8 \"Perception\" Phase 1. color rectangle A 9 \"Perception\" Phase 2. orientation ellipse D 10 \"Perception\" Phase 2. size square C 11 \"Perception\" Phase 1. number triangle C 12 \"Perception\" Full shape circle, square B 13 \"Memory\" No distraction size square C 14 \"Memory\" No distraction shape triangle, ellipse D 15 \"Memory\" No distraction number circle C 16 \"Memory\" Ful l color square B 17 \"Memory\" Phase 2. number ellipse A 18 \"Memory\" Phase 2. orientation triangle C 19 \"Memory\" Phase 1. size rectangle A 20 \"Memory\" Full shape ellipse, triangle B 21 \"Memory\" Phase 2. color square D 22 . \"Memory\" Phase 1. number rectangle A 23 \"Memory\" Full orientation ellipse D 24 \"Memory\" Phase 1. shape triangle, circle B 67 Table 3 Trials for the two-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses. Trial Task Distraction Criteria! Attributes Shapes Correct number condition block condition response 25 \"Perception\" No distraction shape, color square, circle C 26 \"Perception\" No distraction color, orientation rectangle A 27 \"Perception\" No distraction number, color circle D 28 \"Perception\" Phase 1. number, size circle B 29 \"Perception\" Phase 2. size, orientation triangle D 30 \"Perception\" Ful l number,orientation ellipse B 31 \"Perception\" Full size, color square C 32 \"Perception\" Phase 2. color, orientation triangle B 33 \"Perception\" Phase 1. number, shape circle, square A 34 \"Perception\" Phase 1. shape, orientation rectangle, ellipse A 35 \"Perception\" Ful l size, shape circle, triangle C 36 \"Perception\" Phase 2. number, size rectangle D 37 \"Memory\" No distraction size, color circle C 38 \"Memory\" No distraction shape, orientation triangle, rectangle B 39 \"Memory\" No distraction number, shape ellipse, circle B 40 \"Memory\" Phase 2. color, orientation rectangle A 41 \"Memory\" Phase 1. size, shape square, rectangle D 42 \"Memory\" Full number, color triangle D 43 \"Memory\" Full size, orientation circle B 44 \"Memory\" Phase 2. number,orientation triangle C 45 \"Memory\" Phase 1. shape, color square, rectangle A 46 \"Memory\" Ful l number, size ellipse C 47 \"Memory\" Phase 2. shape,orientation ellipse, triangle D 48 \"Memory\" Phase 1. size, shape square, circle A 68 Table 4 Trials for the three-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses. Trial number Task condition block Distraction condition Criterial Attributes Shapes Correct response 49 \"Perception\" No distraction number, shape, color square, circle D 50 \"Perception\" No distraction color, orientation, shape triangle, ellipse C 51 \"Perception\" No distraction size, color, shape ellipse, rectangle B 52 \"Perception\" Phase 1. number, color, size circle , C 53 \"Perception\" Phase 2. number, color, orientation rectangle A 54 \"Perception\" Full color, size, orientation ellipse B 55 \"Perception\" Ful l number, shape, orientation rectangle, ellipse C 56 \"Perception\" Phase 2. color, shape, orientation triangle, ellipse D 57 \"Perception\" Phase 1. number, shape, size circle, square A 58 \"Perception\" Phase 1. shape, size, orientation rectangle, triangle B 59 \"Perception\" Full number, shape, size ellipse, rectangle D 60 \"Perception\" Phase 2. number, size, orientation rectangle A 61 \"Memory\" No distraction size, color, shape square, circle A 62 \"Memory\" No distraction number, shape, color circle, rectangle D 63 \"Memory\" No distraction number, color, size triangle C 64 \"Memory\" Phase 2. color, size, orientation rectangle B 65 \"Memory\" Phase 1. color, shape, orientation rectangle, ellipse A 66 \"Memory\" Full shape, size, orientation ellipse, triangle C 67 \"Memory\" Phase 2. number, size, orientation rectangle D 68 \"Memory\" Ful l number, color, size square A 69 \"Memory\" Phase 1. number, shape, orientation triangle, ellipse B 70 \"Memory\" Ful l number, shape, color circle, triangle B 71 \"Memory\" Phase 2. number, color, orientation rectangle, triangle , D 72 \"Memory\" Phase 1. size, color, shape square, circle C 69 Table 5 Trials for the four-attribute level of the reasoning task: conditions, attributes, shapes, and correct responses. Trial number Task condition block Distraction condition Criterial Attributes Shapes Correct response 73 \"Perception\" No distraction number, shape, orientation, color ellipse, rectangle B 74 \"Perception\" No distraction number, shape, orientation, size ellipse, rectangle A 75 \"Perception\" No distraction orientation, size, shape, color rectangle, triangle C 76 \"Memory\" No distraction number, color, orientation, size triangle D 77 \"Memory\" No distraction number, shape, orientation, color ellipse, triangle B 78 \"Memory\" No distraction orientation, shape, color, size triangle, rectangle D 70 the characteristics of the trials for each level of difficulty, with respect to order, task condition, distraction condition, attribute, geometrical shape, and correct response. Two dependent measures characterized performance of the reasoning (primary) task. One measure was the accuracy of the solution as recorded at Phase 2 of each trial. The measure presented the percentage correct (accurate) of choices from the three trials of each condition cell. Thus, failure to provide a correct answer to any one of the three tasks was assigned a score of 0%; one correct answer out of three Was assigned 33%; two / correct solutions — 67%; all three--100%. The second measure, attribute identification, was derived from children's justification of their answers. After making their choice at Phase 2, subjects were asked to explain their choice. The explanations were then scored by assigning one point for each attribute that was correctly included in the justification. The sum of the points from the three trials of a particular condition cell was obtained. This sum was then converted into the percentage of the total number of attributes that formed the classification criteria for the trials of the particular condition cell. The different levels of difficulty of the task required the coordination of a different number of attributes for achieving a correct response. Thus, presenting the scores as a percentage of the correctly identified attributes allowed comparisons of performance across the different levels of difficulty. Both primary task measures were treated as indices of concurrent performance or performance alone according to the conditions of the trials from which they were extracted. Secondary task measures, however, were derived from different tasks. 71 The measure of performance on the secondary task alone (criterion reaction time) was derived from a twenty-five trial series, that was administered as a separate task at the beginning of the procedure. The measure was calculated as the mean reaction time for the trials after the tenth trial of the series. Misses and reactions under 200 ms. were excluded from the calculation. Concurrent performance of the secondary task was estimated on the basis of the dual-task trials of the three distraction conditions. All measures were calculated as mean of the reaction time of the three trials that formed a condition cell. The mean of the Phase 1 and Phase 2 distraction was used as performance measure in the Full distraction condition, where the secondary task was introduced twice ~ at the beginning of both phases. In cases where subjects did not respond to a tone signal, the mean was calculated on the basis of the remaining trials. Procedure Subjects were tested individually in one session, approximately half an hour long, in a quiet room. Subjects sat at a table and faced the computer directly. The front end of the computer was set at about 20 cm from the edge of the table, so that there were about 60 cm from the subject's face to the screen, and the wrist of the subject's left hand could rest comfortably on the table when a left-hand finger was set on the \"Z\" key of the keyboard. The experimenter sat to the subject's right and controlled the events of the experiment through the keyboard with his left hand. 72 Subjects were briefly explained the purpose and the procedure for the study.. The study was presented as being similar to a computer game and aimed at revealing \"how well people manage to do several things together\"; the procedure was presented as consisting of two simple tasks that the subjects had to perform either separately or together. At the end of this preliminary phase, subjects were reminded that they may discontinue their participation at any moment and were invited to try the tasks. The trials, designed to provide a measure of the criterion reaction time, were administered first. Subjects were instructed to position their finger on the \"Z\" key, to listen for the tone signal and to press the key in response, without trying to anticipate the signal. After completing the series, subjects were acquainted with the reasoning task and given training trials. The stmcture of a trial was explained, using printed copies of four trials, one for each level of difficulty. Subjects from Group 1 were shown examples of one-, two-, and three-attribute tasks, while Group 2 and Group 3 were given examples of tasks from all four levels of difficulty. Subjects were explicitly told what they were supposed to do at each phase of a trial, what questions would be asked, and how they were supposed to answer these questions. Subjects were then asked to solve each one of the four tasks from the appropriate level of difficulty and to justify their choices. When there were mistakes, the tasks were explained again. Training consisted of eight trials from the one-attribute level of difficulty administered through the computer. The first set of four presented the Perception condition, the second set ~ the Memory condition. The duration of the displays for the first trials from both sets was controlled by the experimenter. These trials were used to 73 explain to subjects the task one more time and to illustrate the difference between the Perception and Memory conditions. The other three trials from both sets followed the way of presentation of the test trials. None of the tasks used for explaining the procedure and for training were repeated in the test series. After the training, subjects were warned that the testing trials were to begin and the No-distraction trials from the Perception condition of the lowest level of difficulty (one-attribute tasks for Group 1, and two-attribute tasks for Groups 2 and 3) were administered first. The nine dual-task trials for the Perception condition at the same level of difficulty followed. Then the two blocks of the Memory condition at the same level of difficulty were administered. This order was followed for the blocks at the other levels of task difficulty. Subjects were explicitly warned what kind of tasks to expect before each block. For the dual-task trials, the importance of solving correctly the reasoning task was emphasized. Subjects were told to respond to the tone \"as soon as they heard it\", but that the important thing was \"to be accurate in finding the correct shape\". All subjects understood the tasks and participated willingly. The pauses between the blocks provided the necessary time for relaxation and rest. The whole procedure took between 30 and 40 minutes per subject. 74 Chapter 4 Results Association between Concurrent Performance on the Secondary Task and Primary Task Performance According to Hunt and Lansman's model, capacity-limited performance on a primary task (and the individual differences in resources, which determine different levels of success on this task) is reflected in the association between performance on the primary task, performed alone, and the secondary task, performed concurrently with an easier version of the primary task. When there is a significant correlation between the levels of performance on the two tasks, this is an index that individuals are performing near-the-limit in the dual task situation. It also means that a decrement in performance on the primary task is due to limits in their available resources. Further, statistical significance should hold even when eventual variance associated with measures of performance on the easy version of the primary task, performed alone, and the secondary task, performed alone, are partialled out of the correlation. Thus, the appropriate index for capacity- limited performance on the primary task will be the partial correlation between performance on the hard version of the task, performed alone, and performance on the secondary task, performed concurrently with an easier version of the primary task, when performance on the easy-primary and secondary tasks (both performed alone), is partialled out. This model and the statistical test related to it were used here to provide evidence for age related changes in available resources, utilized for the solution of a reasoning 75 problem. It was argued, that if there are age-related differences in capacity then the capacity-limited performance on the primary task would be detected at different levels of task difficulty for the different age-groups. The index of capacity-limited performance was the significant partial correlation between performance on the primary task (alone) and performance on the secondary task (performed together with an easier version of the primary task), with the relevant predictors1 partialled out. Two such kinds of tests were performed on the data for each age-group of subjects. The first one explored the relation between two successive levels. For each age-group there were two successive levels of the test: the correlations and partial correlations between the measures at the one-attribute and two-attribute levels, and between the two-attribute and three-attribute levels, for Group 1; the correlations and partial correlations between the measures at the two-attribute and three-attribute levels, and, three-attribute and four-attribute levels, for Groups 2 and 3. The second kind of test seeks to detect the association between two non-consecutive levels of difficulty (non-successive levels tests). One such test was performed on the data for each age-group: the correlations and partial correlations between performance on the two measures at the one-attribute and three-attribute levels of the task for Group 1; the correlations and partial correlations between the measures at the two-attribute and four attribute level for Groups 2 and 3. 'One of the predictors is always the secondary task, performed alone. The second predictor varies according to the dependent measure used in the test. Thus, when accuracy is the index of primary task performance, the second relevant predictor is the accuracy score on the easier version of the primary task. When the number of identified attributes is the index of primary task performance, the second predictor is the number of identified attributes on the easier version of the primary task. .76 These two kinds of tests were applied to the data for each age group and each task condition (Perception vs. Memory) separately. In addition, the relation was explored separately for each dependent measure of primary task performance and for each distraction condition (i. e., Phase 1 distraction, Phase 2 distraction, Full distraction), which yielded a secondary task concurrent performance score. The statistical analysis was carried out in two steps. First, the correlations between the target variables were obtained and their significance was evaluated according to a more conservative ( 2-tailed, a=.01) criterion. Second, only relations that satisfied this criterion were explored further. Partial correlations for the variables were obtained, with the relevant predictors partialled out. Group 1 The results of the procedure for the data from Group 1 are shown on tables 6 and 7 (for the Perception and Memory conditions, respectively). The results for the Perception condition (Table 6), are quite straightforward. First, consider the character of the established relations. All significant correlations (and, in fact, most of those that did not reach the significance criterion) between performance measures oh the secondary and primary task are negative. This indicates that greater latencies on the reaction-time task (that is, less spare resources for the execution of the secondary task) are associated with lower performance scores (decrement in performance) on the primary task. This result is consistent with the assumption that the amount of available resources constrains reasoning. 77 Table 6 Capacity-limited performance tests for Group 1, perception condition. Test Measure Correlations Partial correlations accuracy attributes identification accuracy attributes identification Successive levels tests: Secondary (l-attribute,concurrent) to Primary task (2-attribute., alone) Distraction phase 1. phase 2. full -.0756 .2478 -.1977 .1185 -.4811** -.1964 -.4907** -Successive levels tests: Secondary (2-attribute, concurrent) to Primary task (3-attribute, alone) Distraction phase 1. phase 2. full -.7258** -.5759** -.5459** -.5581** -.5875** -.4413 -.6960** -.5448** -.4231* . -.5443** -.5529** Non-successive levels tests Secondary (1-attribute, concurrent) to Primary task (3-attribute, alone) Distraction phase 1. phase 2. full -.1492 -.0984 -.0925 -.0513 -.2730 -.3663 -Note, that only the correlations that exceeded the criterion value ( .4556) were explored further. N=3l * p<.05 **p