Robert Shaw

Ecological mechanics: A physical geometry for intentional constraints

Abstract A proposal is made for a new discipline, ecological mechanics . This version of mechanics is complementary but not reducible to classical relativity, and quantum mechanics. Where traditional mechanics attempt causal analyses for all motions, ecological mechanics explicitly addresses the motions of living systems that exhibit goal-directedness. The shortcomings of the physical geometries underlying traditional mechanics are reviewed, and means are proposed for redressing their deficiencies for modeling the behaviors of intentional systems. This demands a new physical geometry that retains all the best features of the old ones but is extended to accommodate intentional acts. The new physical geometry combines a variant of Minkowskis space-time geometry with a (Cantorian) fractal geometry which reformulates Einsteins energy conversion law ( E = mc 2 ) and Plancks energy distribution law ( E = fh ) so that they apply, more realistically, to the scale of living systems. A new scaling technique called ecometrics , is introduced for accomplishing this feat. This approach assumes a symmetry operator which acts to ‘intentionalize’ causation and to ‘causalize’ intention so that perceptual information and action control processes are defined over a commensurate but dual measurement bases. The promise of ecological mechanics rests on the imputed discovery of a new conservation law which holds locally rather than absolutely. Empirical evidence is reviewed and graphically portrayed mathematical arguments are given that tend to support the hypothesis.

LD Youth and Mathematics: A Review of Characteristics

This represents the second in a series of three articles by John F. Cawley and his associates on mathematics and learning disabled students. Based on information gleaned from the literature as well as an extensive data pool collected by the authors, the present article includes an interpretative review of the characteristics of learning disabled youth as they relate to mathematics. The authors delineate the many facets of failure with which the learning disabled youngster is faced. A set of discriminators are specified for identification of certain subgroups of learning disabilities. Finally, the data presented are shown to provide insight into assessment procedures for youth with disabilities in mathematics. The final article in this series on mathematics will appear in the Spring issue of the Quarterly. The focus will be on problem solving and the application of mathematical skills and concepts to “real-life” situations.

MATH WORD PROBLEMS: SUGGESTIONS FOR LD STUDENTS

This represents the final article in a series of three articles by John F. Cawley and his associates on mathematics and learning disabled students. The authors present specific strategies for systematically developing problem-solving experiences for LD students in mathematical curricula. The information discussed in this article is based on the extensive data pool collected by the authors on the mathematics characteristics of LD populations and the program development that has resulted from their efforts.

Mathematics and Learning Disabled Youth: The Upper Grade Levels1

Compared to disorders in reading, spelling, and written expression, mathematics disorders have received little attention by learning disabilities specialists. However, in recent years, Cawley and his associates at the University of Connecticut have made significant contributions in this area through their research and development efforts with learning disabled populations. This represents the first in a series of three articles which will focus upon mathematics and learning disabled students. Specifically, this article presents a model for mathematics programming for handicapped youth that considers both the characteristics of the learner and the appropriate representation of mathematics. The second article will present an interpretive review of the literature and the characteristics of learning disabled youth and mathematics. The final article will focus on problem solving and the application of mathematical skills and concepts to “real-life” situations.

Affordances and infant learning: A reply to Horowitz

Abstract We reply to three major points made by F. Horowitz (1983, Developmental Review , 3 , 405–409) in her commentary on the ecological approach to infant knowing presented by E. Goldfield (1983, Developmental Review , 3 , 371–404). We first clarify the relation between perceiving and acting from an ecological perspective, and distinguish between affordances as environmental properties scaled to the perceiver/performer and representations as mental structures. We then present a critique of the process of association offered by Horowitz as an explanation of infant learning. Association fails to specify the constraints on what is learned, while the ecological process of noticing affordances, presented by Goldfield, considers such constraints. We conclude by presenting operational criteria for measuring affordances and evidence that perception is scaled to the perceiver/performer.

The role of perceptual organization in the depth perception of kinetic lattice displays

College student subjects were asked to judge perceived depth in computer-generated displays. In all displays, one lattice of points moved through a stationary lattice in either a rowwise or columnwise direction. No points of the two lattices ever touched. Two display variables, strain and shear, each had a significant effect on depth ratings. Shear, however, was only effective at the level of strain for which depth ratings were high. The results confirm earlier studies in which “topological breakage” information was found to affect depth perception. The outcome of this study suggests that principles of perceptual organization can influence the nature of effective breakage information.

Intentional Quantum Dynamics: Entangling Choices and Goals

An unresolved problem in psychology is prospective control, i.e., the question of how information about what is intended to be done can influence what is being done [22]. Over the past quarter century we have addressed this issue by working toward an intentional dynamics approach based on the Feynman path integral. From initial to final condition, i.e., from goal-selection to goal-satisfaction, the kernel of the integral’s transform, \(K(t_1,t_0)\), somehow propagates a path that solves a two-point boundary problem just as any constrained particle must. Here we treat choices at choice-points (including the initial, current, and final states) encountered along goal-paths as superpositions. Intention, or goal selection, is hypothesized to be just another word for entanglement whose path stability can be measured using quantum correlation. Also, we hypothesize that objects’ multiple uses (affordances) encountered along the way can be treated as superpositions that “collapse” as the goal-paths are successfully propagated. Under this approach, we hypothesize that intentional activities are made possible by the system’s entanglement dynamics – the progressive making and breaking of entanglements in order to stay on a goal-path.

The Pendulum Swings Back in Jr. High/ Middle School Math

Robert A. Shaw is an associate professor of education at the University of Connecticut. Richard A. Dempsey, chairman of the Department of Secondary Education at the University of Connecticut, is coordinator of this regular feature. 1 orces and combinations of forces act upon and influence the roles of mathematics in society today. They include: societal forces which are often referred to as needs and expectations of society; psychological forces which involve the needs of youth in

The Reliability and Validity of the Contemporary Mathematics Test

new instrument that is designed to measure the extent to which students have mastered course content in modern mathematics. The test series, which consists of two forms at five levels (Lower Elementary, Upper Elementary, Junior High, Senior High, and Algebra), yields total raw scores which can be converted to percentile ranks, standard scores, or stanines through the use of normative tables. According to the Manual, the test series measures results common to programs in contemporary mathematics. Emphasis is placed on understanding concepts and skills pertinent to solving problems in the areas of structure and number as well as