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This investigation examined ordered and equivalence relations among hierarchically arranged fraction identification tasks. The study investigated whether hierarchical ordering among fraction identification problems reflects the replacement of simple rules by complex rules. A total of 456 middle-class second-, third-, and fourth-grade children were asked to identify fractional parts of sets of objects. Latent class techniques revealed that children applied rules that were adequate for simple problems but had to be replaced to solve more complex problems.

A computer program (ALT) is presented as an aid in the analysis of cross-classified categorical data. Program ALT can accept a maximum of 500 variables as data and construct a frequency table of from 1 to 8 dimensions. Each variable may have as many as 8 categories. In addition, ALT can combine, recode, and create formal variables at the user's discretion.

Report SummaryThis report e4-m ed-vt developing the technology necessary to conduct cost effective and efficient validations of the sequencing of instruction used in the training of military occupational specialties.The specific objective ee~er-t-this-te-nhTr1cal-repo-rt was to validate the hierarchical ordering of task domains.A total of 317 subjects were tested on two algebra skill domains,'constructed from the Precision Measuring Equipment Curriculum of the Air Force Advanced Instructional System. 4 Latent structure techniques recently ivelupW U, Lc" Goodman *t-the-Unv-frsity of Chicago were used to validate the hypothesized ordering between domains. The first step in the analysis was to construct a set of models representing hypotheses about the tasks under examination. The models developed for use in the present analysis assumed three basic classesof individuals for tasks in an hypothesized domain. These classes included masters of the skill represented in the domain, non-masters, and individuals in transition between non-mastery and mastery. Non-masters were characterized as failing all items in the domain, and masters as passing all items.Transitional individuals were assumed to respond inconsistently in a manner congruent with the assumption that they were still in the process of acquiring the concept or rule underlying mastery of the tasks in the domains under examination. Models asserting that tasks were in the same domain were compar d to models asserting that the tasks were hierarchically ordered. f IA Texas Instrument 745 terminal purchased for the project was used in testing the extent to which the hypothesized models accurately represented the observed performance of the subjects. The analysis revealed two hierarchically ordered domains. The Independence-Equiprobability Model 24The Model of Symmetry 24Asymmetrical Equivalence Models 25 An Ordered Relation Model 27Results 28 Within-Domain Results 28 Between-Domain Results 28Discussion 32 Reference Notes 34References 37 Appendix A 38 The second is that subordinate skills in a hierarchy are prerequisite or necessary to superordinate skills (Gagne, 1977), and the third is that prerequisite skills mediate transfer for superordinate skills (Gagne , 1977). !IiThe present project is designed to investigate research questions related to the testing of these hypotheses for the purpose of establishing guidelines that can be used in the development of a technology for hierarchy validation.

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