The need to develop consistent theoretical frameworks for the teaching and learning of discrete mathematics, specifically of graph theory, has attracted the attention of the researchers in mathematics education. Responding to this demand, the scope of the Van Hiele model has been extended to the field of graphs through a proposal of four levels of…

The reconciliation of theories of concepts based on prototypes, exemplars, and theory-like structures is a longstanding problem in cognitive science. In response to this problem, researchers have recently tended to adopt either hybrid theories that combine various kinds of representational structure, or eliminative theories that replace concepts…

Two articles, Edelsbrunner and, Schneider (2013), and Nokelainen and Silander (2014) comment on Musso, Kyndt, Cascallar, and Dochy (2013). Several relevant issues are raised and some important clarifications are made in response to both commentaries. Predictive systems based on artificial neural networks continue to be the focus of current…

This paper explores alternative assessment methods from the perspective of categorizations. It addresses the technologies that support assessment. It discusses initial, formative, and summative assessment, as well as objective and subjective assessment, and formal and informal assessment. It approaches each category of assessment from the…

Naive observers typically perceive some groupings for a set of stimuli as more intuitive than others. The problem of predicting category intuitiveness has been historically considered the remit of models of unsupervised categorization. In contrast, this article develops a measure of category intuitiveness from one of the most widely supported…

Application of equality constraints on the categories of a variable is a simple and useful extension of multiple correspondence analysis. Equality is an easy way to incorporate prior knowledge. A procedure to deal with unequal category numbers and with subsets of variables is outlined and illustrated. (SLD)

This paper describes and evaluates the use of decision theory as a tool for classifying examinees based on their item response patterns. Decision theory, developed by A. Wald (1947) and now widely used in engineering, agriculture, and computing, provides a simple model for the analysis of categorical data. Measurement decision theory requires only…

The problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic is addressed. Scaling is obtained by solving a constrained entropy model. A Kullback-Leibler statistic is generated that operationalizes a measure for the strength of consistency among the sets of categories. (TJH)

An extended marginal homogeneity model is introduced for the analysis of square contingency tables with the same ordinal row and column classifications. The model hypothesizes that the ratio in each column of table delta will exhibit an exponential function of the column. (SLD)

A general approach is described for the analysis of categorical data when there are missing values on one or more observed variables. The method is based on generalized linear models with composite links. Situations in which the model can be used are described. (SLD)

Mathematical relationships between three-mode component analysis and stationary component analysis are explored. Theorems are presented giving constraints that must be satisfied for equivalency between component representations provided by the methods. In general, the two approaches give mathematically distinct representations. (SLD)

This paper proposes that patterns of proximity data that have been characterized in terms of asymmetric similarity may be alternatively characterized in terms of differential bias. An additive similarity and bias model is reviewed, and it is proposed that biases can be stimulus based as well as response based. (SLD)

An asymmetric version of J. Cohen's kappa statistic is presented as an appropriate measure for the agreement between two observers classifying items into nominal categories, when one observer represents the "standard." A numerical example with three categories is provided. (SLD)

In Guttman-type quantification of contingency tables and multiple-choice data (incidence data), the trivial solution because of marginal constraints is typically removed before quantification. Relevant formulas are presented for cases affected by the trivial solution and those that are not. (SLD)

A fairly large number of different methods exist for evaluating comparisons or null hypothetical contrasts. Because it is difficult to choose among them, this paper develops a 2 by 2 classification of the methods. The first dimension of the classification is decision-based error rate vs. experimentwise error rate. The second dimension is planned…