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HOW VALID IS "VALIDITY?"
A DISCOURSE ON LEARNING HIERARCHIES


It is widely accepted that an adequate theory has to meet some general criteria. The focus of instructional theory is on the development of "principles of instruction bearing on the attainment of educational objectives" (Snelbecker, 1974). Snow (1973) has itemized a list of criteria for the evaluation of theory: explicit postulates and definitions, clear delimitation of its extent, logical consistency, consistency with experimental data, capacity for generation of hypotheses and and predictions, testability, parsimony, quantif iability, avoidance of unnecessary formalization, avoidance of unnecessary symbolization, avoidance of oversimplification and "vigilance." By "vigilance," Snow means that the central and non-essential features of the theory must be distinguishable.
Specific to instructional theory, Glaser and Resnick (1972) note five characteristics of "prescriptive theory for the optimization of learning":analysis of the task properties of a knowledge domain, diagnosis of the characeteristics of the learner, design of the instructional environment, assessment of specific instructional effects, assessment of generalized learning outcomes. Bruner (1966) also attempts to specify criteria for a theory or instruction. It "sets forth rules" concerning the most effective way of achieving knowledge or skill and determines the criteria and conditions for meeting them." DeCecco and Crawford (1974), by implication, suggest what might be an "effective way"; that is, the recognition that theories of teaching are subordinate to theories of learning and theories of development. Gropper (1976, p. 7) contends that:
"A theory [of instruction) that leads to an accommodation of differences in


subject matter and of differences among learners is likely to be the most generative." Atkinson (1972) acknowledges the task, learner and evaluation dimensions recognised by the other writers, but he notes "obstacles;" Atkinson asserts that "present learning models are inadequate," and that 'tpowerful instructional strategies must necessarily be adaptive; that is, they must be sensitive to a learner’s unique response history."
In general, then, the major questions to ask of an instructional theory are: does it specify what any learner would need to know in order to do a task? Does it attempt to match learner characteristics and capabilities to tasks? Can success or failure at a task be determined? Does it achieve the limits of its purported generality? Some of these questions are to be addressed to Gagne’s learning hierarchy theory.
Learning hierarchy theory has been described as a "task-analysis instructional theory" (Snelbecker, 1974). As such, it essentially identifies what is to be taught from a determination of what is to be done, and specifies how instruction should take place. Miller (1973) claims that the processes of task analysis are not strictly scientific
task description is a branch of technology dependent on invention rather than on "science" which is dependent on research and discovery in the classical sense (Miller, 1973, p. 5).
Further, the criteria of evaluation of task description should be "pragmatic" rather than conclusive. A second concern of Miller1s is that task analysis itself is " . . an art, and as an art is largely dependent for its excellence and utility on the expertise of the task analyst."
Glaser and Resnick (1972) present a somewhat different view of task analysis; it is "characterized by the descriptions of tasks in terms of the demands they place on such basic psychological processes as attention,
perception and linguistic processing. Further, since the individual’s capacities change over time, task analysis reflect current knowledge and assumptions concerning the processes available at different stages of learning or development" p. 209.
It remains to be seen whether the learning hierarchy fits into the framework of the art of task analysis of the practical world, or if it has any scientific, psychological reality.
Hierarchical theory was an attempt to account for generalized, complex learning or "productive learning" (Gagne, 1962). In general, a hierarchy is generated for a complex task by asking the following question iteratively? "What would an individual have to be able to do in order that he can attain successful performance on this task, provided he is only given instructions" (Gagne, 1962, p. 358). By this repetitive procedure, a complex task could be explained in terms of a number of simple tasks or behaviors. The theory in its initial formulation was reductionistic and behavioristic.
From the theory of learning, were derived explicit theories of instruction and intellectual development (Gagne, 1970a). It was claimed that "the most dependable condition for the insurance of learning is the prior learning of prerequisite capabilities," and "learning of relatively specific intellectual skills" is what constitutes "development. A further corollary of the learning theory is the assumption that if a hierarchical arrangement of skills could be arranged in a particular subject matter area, then that arrangement constitutes an optimal learning sequence.

The theory itself has been subject to review and refinement, and is much more circumscribed than was its initial formulation. For instance, what was referred to as "knowledge" (Gagne, 1962) has become "intellectual skills" (Gagne~ 1968). An intellectual skill is a capability that enables
the performance of a general set of tasks. It is only one of the "domains" of learning, (Gagne, 1972) others of which are motor skills, verbal information, attitudes, and cognitive strategies. Gagne (1972) claims that the distinction between domains is necessary to "distinguish parts of a content area that are subject to different instructional treatments;" and "for relating the instructional procedures of one subject matter to those of another." Moreover, domains "require different techniques of assessing their out comes." Thus domains are ways of categorizing learning outcomes within and across subject areas.
Learning hierarchies are applicable only to intellectual skills (Gagne, 1968) but not independently of other factors. These factors are first, the retrieval into short term memory of the subordinate skill prerequisite to the performance of a higher level skill, and second, the provision of "cues as to sequence," the order in which the subordinate skills must be combined (Gagne, 1974). One can discern that there is in the formulation, some recognition of complexity; there is some search for a mechanism of combination. Perhaps it has been concluded that continuity and practice are not merely insufficient for learning (Gagne, 1970), but they might not be necessary.
Confined to the domain of intellectual skills, the learning hierarchy is evidently safe, protected from the vagaries of new experimental paradigms on encoding, retrieval, retention (Gagne, 1970). But the conceptual status of the learning hierarchy is under scrutiny. Soulsby (1975) contends that the hierarchy is contrived; it is an unsuccessful attempt to make deterministic linkages between simple forms of learning and complex forms. Further, the concept "prerequisite" as defined and used by Gagne, makes the hierarchy "self-authenticating." Soulsby explains, using the example of chaining, one of the lower forms of learning:



There is in this connection a philosophical objection to the whole notion of a hierarchy as such; that the principle of a hierarchy is necessarily established by the nature of its description. For example, if chaining is defined in terms of sequences of stimulus-response connections, it necessarily follows from this definition that stimulus-response learning is prerequisite of chaining since by virtue of the definition it cannot be otherwise. p. 126
There is an analogous problem in relation to the construct "Vertical Transfer" (Gagne,1970).Soulsby1s objection is that a capability that is superordinate must always be learnt subsequent to one that is subordinate to it, otherwise it (the superordinate task) is not superordinate. What Soulsby is in effect saying is that the probability of vertical transfer must be zero when the subordinate capability is absent; there ought to be no question of the superordinate task being learned even at a slow rate when the subordinate capability is absent (Gagne, 1970a).
Phillips and Kelly (1975) concur with Soulsby when they point out that the experimentation done to validate hierarchies (White, 1974, for example) is "irrelevant." Their argument is that if a task is defined in a given ordering of sub tasks, it can only be recognized by that ordering, that ordering constitutes the "meaning"1 of the complex task. So a hierarchy is a "conceptual truth." Phillips and Kelly continue:
But of course Gagne does not realize this, and he sometimes makes the serious error of talking as if there was only a trend for mastery of a higher element in the hierarchy to be related to mastery of lower elements. pp. 361-362
What these conceptual analyses of Gagne's work reveal is that a hierarchy is true by implication of its definition and the definition of its components. Another observation that Soulsby and Phillips and Kelly have in common is their recognition that the notion of a learning hierarchy is applicable only to subject matter having appropriate structure, namely mathematics and the sciences. Phillips and Kelly assert that Gagne's theory is not a
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psychological theory at all: "The pursuer of Gagnean hierarchies needs the skills and knowledge of a physicist or mathematician rather than those of a psychologist" p. 362.
Soulsby reveals the oversimplification inherent in the theory and manifest in its corollaries, in his claim:
Gagne . . . sidesteps the difficult question of the relationship between the logic of the task and the psycho-logic of the learner by creating in the latter the image of the former. p. 122
Evidently, correct passage by a learner through a body of structured subject matter is taken by Gagne to be an indication of how humans develop and how humans learn skills.
To recapitulate and reflect on the criteria for assessing the adequacy of a theory; the hierarchical theory has been delimited to an area in which it might have predictive ability. The question is, can a theory of school learning afford not to address itself to verbal information--a major medium of instruction? (Kolb, 1967, Beeson, 1977)
The logical consistency of the theory has been questioned in the analysis of the meaning of a hierarchy. Is the literature of validation studies nothing but empiricist busy-work?
Finally the parsimony of the theory is in doubt. The cumulative theory of learning (Gagne, 1970) the notion that learning equals development has been challenged. Perhaps the answers lie in the data.
Studies on learning hierarchies have been extensive and reviews have been largely comprehensive. White (1973) summarizes the earlier results and problems. The studies were largely involved in the validation of hierarchies; which means it was attempted to show that to have a postulated higher order capability, the lower-order capability had to be present.
There are two main types of validation procedures, the psychometric and the experimental. In the psychometric procedure, a battery of tests of the terminal task and subtasks is given, and dependency relationships are determined by contingency table for a two task hierarchy (Resnick, 1973) or by the method of White and Clark (1973) for a more complex hierarchy. It has been claimed (Carroll 1973) that the only test of validity of a hierarchy is the experimental approach, because it gives evidence of transfer of learning from one capability to another, after instruction. white (1973) points out that the experimental approach is prohibitively unwieldly;
Let the Roman numerals I, II, III, . . . stand for elements which compose a postulated learning hierarchy. Suppose that it has been hypothesized that possession of II is essential for the learning of I. Then the rigorously logical method testing the truth of the hypothesis is:
1. Secure a sample of Ss.
2. Test their possession of element II.
3. Discard those who have II, and divide the remainder randomly into two groups
4. Teach one group element II, then I, teach the other group element I only.
5. Test both groups for possession of I and II. If no incidental learning has taken place, none of the second groups will possess II, and if the hypothesis is correct only the first group will contain members who possess I. Some members of the second group may acquire II through being exposed to the teaching of I, and some of them may have acquired I also. But if the hypothesis is correct, none of the second group will possess I without II.
One can see the complexity that results. It is claimed that for twenty member hierarchy, 74 experimental groups would be necessary. As a compromise, White (1974) suggests a paradigm that reduces some of the methodological problems, which were mainly small sample size, delay of testing, use of one item for each projected hierarchical relation. The paradigm is as follows:
1.Define, in behavioral terms, the final task of the hierarchy.
Derive the hierarchy by asking Gagne's question (What must
the learner be able to do in order to learn this new element, given only instructions?") of each element in turn, from the final task downwards.
3. Check the reasonableness of the postulated hierarchy with experienced teachers and subject-matter experts.
4. Invent possible divisions of the elements, so that very precise definitions are obtained.
5. Carry out an investigation of whether the invented divisions do in fact represent different skills.
6. Write a learning program for the elements, embedding in it test questions for the elements.
7. Have at least 150 suitably chosen subjects work through the program, answering the questions as they come to them.
8. Analyze the results using a test of hierarchical relationship to see whether any of the postulated connections should be rejected.
9. Remove from the hierarchy all connections for which probability under the null hypothesis is small.
It could be asked at this stage, what is really being validated? Is it the organization of the subject matter or the intuition of the "experienced teacher" or is an optimal path for learning being developed? Perhaps some answers can be obtained from specific research findings.
A Foundation Study
Gagne (1962) investigated the learning of the skill "finding formulas for sum of n terms in a number series according to a postulated hierarchy among seven ninth grade boys. It was found that no subject succeeded on a higher task who failed on a lower task. There were ten tasks in all, but there had been different levels of failure on five. Specific instruction using a learning program resulted in success at all tasks except by one subject. Gagne (1962) claimed that the results indicated that the analysis of knowledge as being the presence of underlying learning set abilities was correct, and further that these abilities could be given by instruction He states: "One cannot depend upon a measurement of general proficiency or aptitude to reveal much variability in the capabilities people bring with them to a particular task" p. 365. This statement by Gagne, could be taken to be an argument against innateness; and less as an argument against interactionism.



However, considerable effort has been spent in constructing learning hierarchies for tasks which purport to transform the level of development.
A Piagetian Task
Tasks deemed appropriate for young children have been tested by Resnick (1973) and others. However, the task investigated by Wiegand (1973) was given to sixth graders.
Wiegand (1973) felt that a hierarchically based learning program could be appropriate for learning a task in science that required logical thinking. Two tasks were administered, both of which involved inclined planes. The subjects N 30 had failed both tasks and were then assigned to three experimental groups. One group (DTR) was shown the task, was tested and
any errors were explained and retested. The two remaining groups (TR, T) had no demonstration and one of them had no explanation or retest. The DTR group did not differ from the TR group in either the criterion task or the transfer task, and.none of the groups differed on the initial test. Wiegand takes this to mean that the demonstration had no effect and that being tested alone was insufficient. Rather, it was the explanation of errors on tests that were hierarchically arranged, that was crucial. Since everyone who succeeded on the final task was later able to do the Piagetian task, Wiegand asserts that what was lacking was specific skills which could be learnt by instruction. Further, a bolder claim is made. "These results indicate that intellectual development has been brought about by the cumulative effects of the learning of concretely-referenced skills, rather than by the adaptation of structures of intellectual growth" (p. 349).
Almy (1973) issues caution, in the acceptance of Wiegand's generalization, noting that the ages of the subjects would not preclude their experience on the initial test from being a part of the general experience required to do the task. Almy’s proviso, notwithstanding, a major implication of the cumulative theory of learning is that there is no difference in the type of skill that a learner will use in attempting to do a task, but the difference is in the number of skills. According to the theory, instruction is the provision of experience to add skills. The limitation of this implication is that it takes from the learner, the capacity for constructive responding or for initiating. The implication for learning hierarchies is the same as for cumulative theory of learning
The Active Learner
The problem of the activity of the learner is relation to hierarchies has been addressed at the level of development and in respect to learning. McKeen and Eisenberg (1973) investigated learning sequences generated by college students with the hypothesis that students "exhibit and organize topics into valid learning hierarchies." For a terminal mathematical task, McKeen and Eisenberg constructed a "skeletal hierarchy," and using student responses with three successive groups, developed a final hierarchy with 72 hypotheses of task dependency. Two groups of high and low ability respectively, were instructed using the hierarchy . Eighty-five percent of the hypotheses were found to be valid for the high ability group, while 33% were valid for the low ability group. McKeen and Eisenberg (1973) claim that this is evidence that learning hierarchies generated with student input can be valid for a particular population. They do not describe the extent of modification that the skeletal hierarchy underwent. Further, that the hierarchy was less useful for the low ability group (as evidenced by scores and by test of hierarchical validity) deserves comment. In Gagne's (1970) formulation, it was proposed that every step should be made explicit, facilitating every student.
Siedl and McKeen (1974) in further pursuing student-generated hierarchies,


suggested that students with different levels of ability would construct different types of hierarchies, which would be appropriate to the levels of the groups that constructed them. On a task in matrix algebra, high and low ability groups worked on problem sets prepared by instructors. The problem sets were equivalent to the "skeletal hierarchy." Learning sequences were generated from the summaries of the activities of the students. Two refinements were done and the final hierarchies developed for low and high ability students were composed. The criterion for difference between hierarchies was set: "five or more of these comparisons was considered sufficient to conclude that the learning sequences themselves were different." The criterion was not met.
The results of the Siedl and McKeen (1974) and McKeen and Eisenberg (1973) investigations are inconclusive. In terms of the validation "technology" proposed by White (1974) these two studies have a number of flaws. Moreover, the initial skeletal hierarchy was constructed by the instructor and might have constrained the behaviors of the students, as subject matter is wont to do. However, the attempts, at least nominally, to reconcile the sequencing of the materials to the student's own organization of material and the instructor's perception of the interaction is perhaps a more realistic formulation of the teaching-learning process. (Jenkins, 1977)
Siedl and McKeen and McKeen and Eisenberg make no mention of verbal information in the hierarchies with which they were involved. Yet this has been a major theoretical concern in learning hierarchy theory. From Kolb (1967) to Beeson (1977), there has been the concern that hierarchies that contain verbal information tasks are bound to be invalid. Yet it is ironic that step 3 of hierarchy construction is "Check the reasonableness of the hierarchy with experienced teachers and subject-matter experts."

Perhaps step 9 should be revised; (Remove from the hierarchy all connections for which probability under the null hypothesis is small) and a search for the relationship between skills and verbal information be carried out. Gagne and White (1978) are aware of the difficulty of making the distinction between propositions and observable behaviors. They now propose a model within which links intellectual skills to propositions, images and episodes. Under this formulation, intellectual skills are the "learned memory structures that underlie the identification of concepts and the application of rules." Within the model, intellectual skills are not independent of verbal information, "they are connected in memory by relating propositions that link them with their subordinate skills, as well as with other memory structures."
In this new round of .theory, it appears as if Gagne is saying something different. flit may be that he has identified the locus of the "associations on which his theorising has been based. If that is the case then the "shift from S-R to information processing " (Scandura, Frase, Gagne, Stolurow, Stolurow and Groen) that he now agitates for might not be substantive. It could be that he has returned to the problem of the "acquisition of knowledge."
In summary, then, it has been argued that a theory which has had considerable impact on educational psychology has been found wanting on conceptual grounds. The theory has spawned a vast technology and methodology of self-validation (Airasian and Bart, 1973; White and Clark, 1973), yet perhaps the most realistic attempt to apply the theory to the educational process has been the least methodologically sound attempt. Exciting prospects for educational practice may lie in the synthesis of the learning hierarchy theory and its adversary theory, cognitive developmental psychology (Case, 1978).

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