An Ordering-Theoretic Method to Determine Hierarchies Among Items

Abstract

EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT 1973, 33, 291-300. BART (in press) and Airasian and Bart (1971) introduced tree theory as an alternative measurement model with a boolean algebraic framework. Subsequent to those two statements tree theory was relabelled by Bart and Airasian so that the theory would more closely mirror its algebraic framework. Ordering theory has as its primary intent either the testing of hypothesized hierarchies among items or the determination of hierarchies among items. In ordering theory, response patterns for bivalued items are viewed as atoms in a boolean algebra with as many generators as there are items being considered. An ordering (formerly termed a tree) is the union of the obtained atoms and indicates the logical relationships among the items.Ordering theory shares with classical models the item response matrix, but does not use summation across subject rows to express in the form of the correct responses of a subject his standing on the trait measured. This is the first departure of ordering theory from older models of measurement. These models invariably assume that the trait measured is linearly ordered and can be measured with a simple additive model-e.g., a summative score (Gulliksen,

A class of k-modes algorithms for extracting knowledge structures from data

“…This procedure was developed out of ITA (Bart & Krus, 1973;Van Leeuwe, 1974), and it is aimed at uncovering the logical implications among the items in Q. Such logical implications form a quasi-order (i.e., a reflexive and transitive binary relation) on the set of items.…”

Investigating the Effects of Teaching Based on an Analysis of High School Students' Knowledge State of Concepts Associated with Astronomical Observation

“…즉 과학 개념의 위계는 상·하위 간 기능들 사이의 유기적 관련성과 순서를 제시함으 로써 학습목표 도달을 더욱 쉽게 이루어질 수 있게 한 다 (김범기 외, 1996). 과학 개념에 대한 위계 분석은 주로 서열화 이론 에 의해 분석되어 왔으나 (임청환, 1992;Bart, 1976;Bart and Krus, 1973), 최근에는 개념의 위계 구조뿐 만 아니라, 학습자의 지식상태까지도 알 수 있게 하 는 지식공간론의 활용 가능성이 제기되고 있다 (김석 천 외, 2007;김승동 외, 2002;박상태 외, 2005;변 두원 외, 2004;송하영과 김영신, 2010;윤마병과 김희수, 2010;Albert and Lukas, 1999;Albert et al, 1995;Doignon and Falmagne, 1998). Wilson(1985)은 개념의 위계를 학생에 의해서 학습되어지는 순서인 '심리적 계열'과 교수-학습 순 서인 '교수 계열', 학습 과제의 배열 순서인 '논리적 계열'로 분류하였다.…”

A Structural Approach to the Validation of Hierarchical Training Sequences. Task 1. Technical Report

“…However, the validation procedures used to examine the prerequisite-skills assumption are extremely time consuming and may not be suitable for broad scale application. (Guttman, 1944), multiple scalogram analysis (Lingoes, 1963), and the ordering tneoretic method (Bart & Airasian, 1974;Bart & Krus, 1973) has been faulted on similar grounds. None of these procedures provides a suitable statistical test for prerequisite relations (Airasian, Madaus, & Woods, 1975;Dayton & Macready, 1976;White, 1974 Proctor, 1970;White & Clark, 1973).…”