Date of Award

Spring 1980

Project Type


Program or Major

Mathematics Education

Degree Name

Doctor of Philosophy


The principal result of this study is a logically and empirically valid Structure of Knowledge for Differential Calculus constructed using a methodology based on Learning Hierarchy Theory (LHT) and Order Theory (OT). A review of existing research into calculus learning is presented and properties of the strucutre of knowledge which address deficiencies in existing knowledge discussed. LHT is developed and techniques for the empirical validation and generation of hierarchies are discussed. Included in this discussion is a detailed analysis of the important properties of OT.

The generation of the structure of knowledge consisted of two phases: the logical analysis and the logical-empirical analysis. In the logical analysis, an examination of textbooks yielded nine content units, each of which was defined by a set of exercises which pertained to it. A structure, called the Hierarchy of Levels of Equivalence Classes, was then imposed on each set of exercises by using Task Analysis, the Prerequisite Relation and the Theory of Equivalence Classes. This structure implied the Hierarchy of Levels of Skills (HLS). The nine HLS formed the logical structure of knowledge for Differential Calculus. The hierarchies of levels produced are not hierarchies in the traditional sense, because there are no "arrows" connecting "boxes," but rather the prerequisite relation is defined by the assignment of exercises to levels. The logical structures were analyzed empirically using data collected during administration of a first semester college calculus course. The empirical procedure was based on OT and employed the prerequisite relation, the Theory of Equivalence Classes, and some ideas from the Theory of Directed Graphs. A Hierarchy of Levels of Representatives (HLR) was constructed for each HLS. Each HLR implied a logical prerequisite matrix which, once transitivity problems had been resolved, defined a partial order on the elements of the HLR. An OT analysis was performed on each logical prerequisite matrix. Tolerance levels of .05 and .10 were used. The OT analysis was used to pinpoint places where the structures failed to meet minimum levels of the theory-relevant properties: Task Dependency, Completeness, and Positive Transfer. Logical prerequisite matrices were validated at the .05 level and transfer ratios set at .6000. Unhypothesized relations of disconformity less than or equal to .10 were investigated. Logical reanalysis was used to resolve the pinpointed empirical difficulties. In order to resolve difficulties which were encountered in the analysis of calculus word problems, a theory or word problem solving was developed and used to transform original sets of word problems into sets which permitted meaningful logical-empirical analysis.

In general, the logical analyses were supported; even though many unhypothesized relationships were implied by the data. Transfer ratios were high, reflecting the design of items and the sequence of testing.

An algorithm which answers many of Gagne's concerns about the specification of content was the major contribution of this research to LHT. Standard tolerance levels of OT were found to be inadequate to control for disconformity consistent with the theory, and it was recommended that tolerance levels be set more liberally and that their role in future studies be expanded. The procedure developed also extends the traditional OT analysis so that one hierarchy with logical-empirical foundations is ultimately produced.