P. J. Kellman, P. Garrigan, and T. F. Shipley's theory of 3-dimensional object interpolation asserts that existing data, as well as logical considerations, support the view that an identical contour interpolation process underlies the interpolation of partially camouflaged and partially occluded objects (modal completion and amodal completion,…

P. J. Kellman, P. Garrigan, & T. F. Shipley presented a theory of 3-D interpolation in object perception. Along with results from many researchers, this work supports an emerging picture of how the visual system connects separate visible fragments to form objects. In his commentary, B. L. Anderson challenges parts of that view, especially the idea…

Proposes a framework for discussing the nature of the relationships between contexts and the formation of mathematical knowledge through model and field-of-experience concepts. Supports and illustrates this framework with references to curricular innovation and educational research. (Contains 31 references.) (Author/ASK)

The aim of this series of four articles is to look critically, and in some detail, at the primary strategy approach to written calculation, as set out on pages 5 to 16 of the "Guidance paper" "Calculation." The underlying principle of that approach is that children should use mental methods whenever they are appropriate, whereas for calculations…

Comments on van Geert's mathematical model of Vygotsky's zone of proximal development, in this issue. Supports van Geert's use of a nonlinear model, noting that linear statistical models overlook variability in psychological phenomena. Discusses the time asymmetry in van Geert's model, which does not account for a subject's "developmental…

A brief survey of models in dealing with various types of understanding is given. Then a hybrid model, which proved inadequate for describing understanding, is outlined. Finally, four levels of understanding are discussed: intuitive, procedural, abstract, and formal. The concept of number is used to illustrate these levels. (MNS)

Argues that Vygotsky's "zone of proximal development" (ZPD) can be redefined as a nonlinear dynamic model. Presents a mathematical reformulation that generates a variety of developmental patterns and has the capacity to show under which conditions unwanted or unsuccessful developmental paths may occur. Illustrates how this reformulation…

Describes a covariational, rather than correspondence, approach to functions that emphasizes rate of change. Proposes three ways of understanding rate of change in relation to exponential functions. (Contains 41 references.) (Author/MKR)

This paper discusses a model of information storage and retrieval, the k-d tree (Bentley, 1975), a binary, hierarchical tree with multiple associate terms, which has been explored in computer research, and it is suggested that this model could be useful for describing human cognition. Included are two models of human long-term memory--networks and…

Mastering the basic number combinations involves discovering, labeling, and internalizing relationships, not merely drill-based memorization. Counting procedures and thinking strategies are components, and it may be that using stored procedures, rules, or principles to quickly construct combinations is cognitively more economical than relying…

The author first corrects Baroody's description of the network retrieval model for basic number facts, in which facts are stored in memory and retrieved as needed. He then indicates weaknesses in Baroody's argument. (MNS)

Building on Bloom's, and subsequent educational taxonomies, a larger, synthetic, theoretical basis, called the process hierarchy theory, is developed. Its relationship to general systems theory and its metatheoretical context are described. A general methodology for empirically testing aspects of the theory using mathematical modeling is explored.…

The focus is on the metacognitive awareness of ten high-achieving high school pupils in mathematics in Denmark and England and their understanding of their cognitive learning processes and strategies. Mainly unstructured focus group interviews investigate how they explain that they learn a mathematical concept that is new to them. I develop the…

We first discern three different sources to describe the non-modelling-friendly environment in Taiwan: the background of mathematics teachers and students, examinations and textbooks. Under such unfriendly circumstances, how one can implement the teaching and learning of mathematical modelling is explored. In this paper, we focus on the analysis…

This article examines the potential contribution of latent trait models to the study of intelligence. Nontechnical introductions to both unidimensional and multidimensional latent trait models are given. Multidimensional latent trait models can be used to test alternative multiple component theories of test item processing. (Author/CTM)