The technique of proof plans, is outlined. This technique is used to guide automatic inference in order to avoid a combinatorial explosion. Empirical research to test this technique in the domain of theorem proving by mathematical induction is described. Heuristics, adapted from the work of Boyer and Moore, have been implemented as Prolog programs, called tactics, and used to guide an inductive proof checker, Oyster. These tactics have been partially specified in a meta-logic, and plan formation has been used to reason with these specifications and form plans. These plans are then executed by running their associated tactics and, hence, performing an Oyster proof. Results are presented of the use of this technique on a number of standard theorems from the literature. Searching in the planning space is shown to be considerably cheaper than searching directly in Oyster's search space. The success rate on the standard theorems is high. These preliminary results are very encouraging.
SUMMARY. The object of the study was to examine the statistical techniques available for the analysis of process-product studies involving non-randomised quasi-experimental designs, and to demonstrate the practical effects of their use on the data from the Teaching Styles study (Bennett, 1976). Of particular concern were the ' unit of analysis ' or aggregation problem, and the differential effects of treatment grouping by cluster and factor methods.The original grouping of teachers into formal, informal and mixed styles was investigated using a latent class model for the 38 binary questionnaire items. Convincing evidence of three overlapping latent classes was found. The comparison of latent classes in terms of pre-test gain scores was examined using a series of variance component models, allowing for correlation of children within the same class. Differences among classes were altered by the probabilistic clustering of the latent class model compared to the original findings, and the significance of the differences was reduced when the correlation among children was allowed for.INTRODUCTION IN the four years since the publication of Teaching Styles and Pupil Progress (subsequently abbreviated to TS) there have been rapid developments in the statistical methods available for the analysis of complex data. While these developments are still in their early stages, it is already clear that they will have an important influence on the analysis of large-scale educational research studies. Two of these developments are particularly important for the analysis of educational data from surveys and observational studies: the development of latent class models for clustering nonhomogeneous populations, and the development of unbalanced variance component (' mixed ') models for nested and cluster sampling structures.The objects of this article are to describe the application of these modelling procedures to the Teaching Styles data, to report the conclusions drawn, and to compare these conclusions with those found in the original analysis. Implications for future research studies are also discussed (for statistical detail see Aitkin et al., 1981). In the re-analysis, two main questions were considered:
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