Search

Some of the conceptual qualitative ideas needed to test nonlinear models empirically and to modify them are described. Relationships among these ideas and computer applications are also examined to elucidate the general process of nonlinear modeling. Two examples are presented along with a discussion of bifurcation, catastrophe, and maximum likelihood estimate methods. The first example concerns administrators' responses to innovation and uses a verbal description of events. The model is developed based on variables such as the amount of voluntary effort committed to the innovative project and the level of funding agreed to by the institution. An equation consistent with the hypotheses is presented. The second example starts with a mathematical model of promotions within an organization and shows how to go beyond the verbal statements. It is concluded that many observed phenomena in institutions are suggestive of nonlinear dynamics models. A number of standard types of dynamic behavior are well understood mathematics (catastrophe, periodicity, stochastic effects) and may be used to construct plausible models. (Author/SW)

The applicability of catastrophe theory to research in higher education is considered, with several problems that typically appear in the literature presented in a theoretical framework. A catastrophe model is attempted for each. The nature of mathematical modeling and the relationship between modeling continuous systems and discontinuous systems are assessed. Characteristics that must be present in a system for it to be suitable for a catastrophe theory based model are bimodality, divergence, discontinuity or catastrophe, hysteresis, and inaccessibility. Attention is also directed to interpreting a catastrophe surface. Four issues pertaining to the use of catastrophe theory as a tool for modeling behavior in the social sciences are as follows: its foundations of the theory in mathematics and natural philosophy; the assumptions needed for its application; details surrounding specific applications; and the attitudes, style, and intellectual honesty of its advocates. Hypothetical examples for applying the theory are considered: faculty vitality as a function of promotion difficulty and percent of junior faculty hired, teacher expectations as a function of student communication and performance, and student attrition as a function of previous academic performance and educational aspirations. Based on the literature on catastrophe theory, it is suggested that the bridges between the theoretical applications and those actual applications where theory can be tested through quantification have begun to be built. It is concluded that the usefulness of the theory may come from the ability of its models to provide more complete, and possibly more precise, description than appears possible with theories describing continuous behavior. Additionally, graphs based on the theory provide good conceptual tools. A bibliography is appended. (SW)

This paper discusses how mathematics can be interpreted as an integral part of technological planning and decision making, and how mathematics therefore operates as an integral part of technology. Three aspects of mathematics in action are presented. The paper provides for the theoretical task of grasping mathematics in action and identifying how mathematics supports a technological imagination, how it establishes possibilities to investigate particular aspects of possible technological constructions, and how mathematics becomes installed in society and starts operating as part of technological devices. It is also pointed out that an understanding of mathematics in action is crucial for interpreting basic aspects of social development. (Contains 66 references.) (KHR)

A powerful way of understanding something new is by analogy with something already known. An analogy is defined as a mapping from one structure, which is already known (the base or source), to another structure that is to be inferred or discovered (the target). The research community has given considerable attention to analogical reasoning in the learning of science and in general problem solving, particularly as it enhances transfer of knowledge structures. Little work, however, has been directed towards its role in children's mathematical learning. This paper examines analogy as a general model of reasoning and discusses its role in several studies of children's mathematical learning. A number of principles for learning by analogy are proposed, including clarity of the source structure, clarity of mappings, conceptual coherence, and applicability to a range of instances. These form the basis for a critical analysis of some commonly used concrete analogs (colored counters, the abacus, money, the number line, and base-ten blocks). The final section of the paper addresses more abstract analogs, namely, established mental models or cognitive representations that serve as the source for the construction of new mathematical ideas. A reference list contains 78 citations. (MKR)

Multilevel international data have been released from the Third International Mathematics and Science Study (TIMSS), providing an opportunity to apply multilevel modeling techniques in educational research. In this paper, TIMSS factors are classified in fixed and random categories according to the project design. Classifying fixed and random factors may help clarify the relevance of hierarchical linear modeling for TIMSS investigations. Relevant methods for the multilevel data analyses are discussed in light of the fundamental relationship between traditional dummy variable regression and hierarchical linear modeling. (SLD)

The Common Knowledge: Pittsburgh (CK:P), a technology-based project, introduced the Internet into all levels of the Pittsburgh Public Schools during 1993-97. This is a case study of the ideology, strategies, and process of the CK:P project describes the project's activities, examines the project in light of school-reform literature, and uses its experience to develop a conceptual framework for discussing such reform efforts. The language of chaos theory is used to describe the behaviors observed in the project and argues that the behaviors exhibited, as a result of implementing the CK:P project, are nonlinear, dynamic, and similar at every entry--individual, school, and district--of magnification. Although the behaviors appear to be random, chaotic, and unpredictable, patterns or points of stable attraction exist within the randomness. The project experience suggests that at least four elements must be present for school reform to occur: (1) the individual or cultural change is irreversible; (2) the change is internalized by the majority; (3) institutional shifts occur to support and sustain the change; and (4) the change conforms to an agreed-upon standard. The value of applying chaos theory to school reform is that it is a holistic process for analyzing complex systems. Through mathematical modeling, chaos theory looks at systems globally while addressing local variables. It attempts to replicate a system's complexity by considering both the existing turbulence and that generated when introducing an agent of change. A glossary and four figures are included. (Contains 19 references.) (LMI)

These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student's learning. Together the chapters demonstrate that teaching with any model requires care and attention to achieve conceptual clarity. Teachers need to act entirely within in the world of the model, only later adding other models to round out the mathematical picture. The teaching of integers is a key point for maintaining students' confidence that mathematics is a subject that makes sense. These chapters demonstrate how use of models can assist in this goal. [For the complete volume, "Exploring the Integer Addition and Subtraction Landscape: Perspectives on Integer Thinking. Research in Mathematics Education," see ED612853.]

We noticed that when workers try to explain their mathematical practices to inquisitive outsiders, breakdowns in communication arise. We present here an example in which a worker spontaneously uses metaphors and models to facilitate explanation and communication. We analyse these, drawing on Lakoff & Johnson (1999) and Lakoff & Nunez (2000) in substance and theoretical approach. We suggest that metaphors and cultural models ground associations between academic and workplace discourse genres, and point out how sensori-motor groundings of the "basic" metaphors may afford gesture and image-schema which free discourse from formal mathematical language. In general we see breakdown repairs as being built through cultural models that extend beyond local mathematical genres which situate mathematics within academic or workplace contexts. [For complete proceedings, see ED489597.]

Computer Algebra Systems (CAS) can be used to help understand the behavior of cubics and other families of curves. The analysis of the parameters associated with a family of curves can give insight into the behavior of the family and can be used within the context of a mathematical model to determine a policy for the implementation of the model in a real-world setting. Examples of this use are included from a calculus-based analysis of a mathematical model and the behavior of a model described by a differential equation. The case is made that the ability, afforded by the use of CAS, to keep the parameter in symbolic form enables an analysis that goes beyond the standard analysis that may be done in the standard situation using numerical coefficients. (MM)

The purpose of the study was to compare two regression-based approaches for measuring educational effectiveness in Tennessee high schools: the mean residual approach (MR), and a more general linear models (LM) approach. Data were obtained from a sample of 1,011 students who were enrolled in 48 high schools, and who had taken the Comprehensive Tests of Basic Skills (CTBS) in grade 8 and the Preliminary American College Test Plus (P-ACT+) in grade 10. Regression models were developed for predicting P-ACT+ scores from CTBS scores. In the MR approach, mean residual P-ACT+ scores were calculated for each high school from a pooled model. The LM approach included effect-coded high school dummy variables as predictors. Indicators of educational change, called Achievement Index (AI) values, were dveloped for each school; AI values were also calculated where race, coursework and grades, or entering achievement level was included in the models. The results show that the LM approach is more sensitive in identifying outlier schools and was less hampered by sample size constraints than was the MR approach. Statistical significance and identifying violations of the assumption of parallel slopes were also easier to determine. Five graphs and one table illustrate the discussion, and there are two references. (Author/SLD)

Classical test theory, item response theory, and generalizability theory all treat the abilities to be measured as continuous variables, and the items of a test as independent probes of underlying continua. These models are well-suited to measuring the broad, diffuse traits of traditional differential psychology, but not for measuring the outcomes of school learning. Discrete latent structure models offer a powerful and promising alternative. Abilities can be modeled as partially ordered sets of discrete states (at a minimum, "nonmastery" and "mastery") and may be linked according to an asymmetric "prerequisite" relation. Narrower, simpler abilities may be combined into broader, more complex abilities. The various possible outcomes of performing a task can be modeled as a partially ordered set of task performance states. Abilities and task performances are clearly distinguished from one another, and more than one ability pattern may permit successful performance of a given task. Subtasks need not be modeled as conditionally independent given ability. The mapping from ability states to task performance states shows clearly what a given test can and cannot measure, and what may be inferred from a given pattern of test performance. These models for ability and task performance, together with the mapping between them, may be augmented with a suitable model for measurement error (misclassification) to complete an alternative framework for scoring, analyzing, and interpreting test performance. This framework has the potential to solve significant measurement problems inherent in performance testing and other applications. (Contains 12 figures.) (Author/SLD)

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 hierarchies--and reasons are given for the higher efficiency of hierarchical theories, including the k-d tree. A description of the k-d tree includes its structure, computation rates, and balancing (branching); its applications to human cognition, including a comparison with Piaget's notions of equilibrium and cognitive stages of development; application to memory and forgetting theories; covergent and divergent thinking processes; logic paths and decision-making; and the function of sleep. A concluding discussion compares human and computer processing of information, and raises questions related to the hierarchical structure of brain activity. It suggests that the k-d model from information science may have a strong relevance to the study of human cognition, particularly regarding the study of memory and sleep, while at the same time allowing for vast differences between a computer and the human mind. (JB)

The critical role of computers in scientific advancement is described in this panel report. With the growing range and complexity of problems that must be solved and with demands of new generations of computers and computer architecture, the importance of computational mathematics is increasing. Multidisciplinary teams are needed; these are found in most advanced and industrial laboratories, but rarely in universities. The existing educational opportunities are not producing the required personnel to meet substantial shortages. Therefore, the panel strongly recommends increased federal support for: (1) research in computational mathematics, methods, algorithms, and software for scientific computing; (2) the development of interdisciplinary research teams; (3) the establishment and continued operation of a suitable research infrastructure for the teams; (4) graduate and post-doctoral students directly involved in the research of some interdisciplinary team; and (5) young researchers and cross-disciplinary visitors. In the second section, research opportunities in a number of mathematical areas are described. New modes of research are discussed next, followed by comments on educational needs and a final section on funding considerations. Appendices contain a list of related reports, information on laboratory facilities for scientific computing, and letters and position papers. (MNS)

This paper presents an alternate perspective for utilizing the action/process/object framework when discussing student conceptions of function. This perspective extends previously used frameworks by incorporating student conceptions that deal with functional properties and situational contexts, but differs in the analysis of the nature of the functional properties that are considered. The hypothesis is that students not only use notions of set-based correspondences and transformations to think of functions as mathematical objects, but functions are also conceived as mathematical objects that possess (or fail to possess) growth properties of specific functions. These properties include symmetry, linearity, and continuity. It is argued that all students who develop a structural view utilize functional properties, but a focus on different kinds of properties produce different conceptions of function. (Author)

Despite the fact that one of the strengths of two-year institutions is their ability to serve the diverse educational needs of their local communities, a single measure of community college effectiveness, the transfer rate, has come to dominate. Because this single measure of success can be used for both internal institutional review, and for public accountability, a major debate concerning how the transfer rate should be calculated has ensued. Two major initiatives, one overseen by the National Effective Transfer Consortium, and the other by the Center for the Study of Community Colleges, are underway to develop a nationally accepted definition of a transfer rate. Each of the two projects examines whether the transfer outcome has been achieved for a defined group of students, either those who entered at the same time (entering cohort) or those who left at the same time (exiting cohort). Both studies qualify who is represented in the potential pool of students, require a minimum number of hours of college-level work to be included in the cohort, and reject classifying students by curricular intent. Neither projects' definition should be endorsed in its current form. Instead, the definitions should be modified in the following ways: (1) multiple transfer rates should be considered, with perhaps one primary rate and several secondary measures; (2) the primary rate should be a pure outcomes measure utilizing an exiting cohort; (3) the entering cohort measurement should be considered as a secondary rate; (4) the transfer rate of vocational students should be reported separately; and (5) student intent should be considered only for a secondary rate. (JMC)

Given the current and widespread practical interest in mathematical understanding, particularly with respect to higher order thinking skills, curriculum reform advocates in many countries cite the need for teaching mathematics with understanding. However, the characterization of understanding in ways that highlight its growth, as well as the identification of pedagogical actions that sponsor understanding, represent continuing problem areas. This paper presents the theoretical basis for a companion document, entitled "The Characteristics of the Growth of Mathematical Understanding." It shows a theory of mathematical understanding that is based on the consideration of understanding as a whole, dynamic, levelled but non-linear process of growth. A model for the theory is represented by eight embedded rings, each representing a level of understanding activity attainable for any particular topic by any specific person. The levels are illustrated by tracing the building of an understanding of the concept of fractions. Beginning with the initial cognitive stage, the levels are called: (1) primitive knowing; (2) image making; (3) image having; (4) property noticing; (5) formalizing; (6) observing; (7) structuring; and (8) inventizing. The application of the theory to teaching is discussed. (MDH)

The planning and management of educational systems has been examined widely in the last decades despite the lack of a standard research method and a unified underlying theory. Research based on quantitative analysis yields data that may be processed using the graphic capabilities of computers. The combination of numerical data and computer graphics results in a visual, realistic presentation of scientific results. New computer software must be developed that will facilitate information analysis and visual presentation. Interactive video is an example of a video-computer technology that provides quick access to visual information, computer organization and control over database information, and use of that information. Qualitative information may be represented quantitatively by using a functional statistical model made up of factor and graph models. The information is then in a form that is easily assimilated by educational administrators, teachers, as well as other scientists. (DB)

To solve problems of causality and measurement in educational research, this paper combines seven variables into a proposed model of educational productivity on achievement tests. The authors first review psychological models of educational production that relate learning to aptitude and environment, and note that these models do not allow for interaction between the aptitude and environment variables. The authors then describe productivity models in economics and demonstrate that such models are explicit and testable and do allow for interaction among variables. An educational model emulating the economics models is proposed to explain academic achievement on standardized tests. In this model of educational productivity, an equation combines the variables of student ability and motivation, instructional quantity and quality, home and classroom environments, and age. According to the authors, the interaction among the model's variables may explain a number of problems in educational research, including the lack of demonstrable results from instructional innovation or compensatory education, the weak correlations between achievement and instructional time, and the confusion in findings relating aptitude to teaching methods. The paper's final section stresses the need to test the proposed model on long-term empirical data. (RW)

This study analyzes the effectiveness of a "guaranteed tax base" (GTB) as a reform measure designed to reduce the traditional heavy reliance of school financing on local property tax revenues and to help equalize per pupil expenditures across districts. Such measures call for matching locally raised tax dollars with state aid and consequently lowering the price of expenditures to some school districts. Previous studies by Feldstein (1975) and Ladd (1975) estimate school district expenditure equations using cross-sectional data from Massachusetts. Their results indicate a fairly large price effect on expenditures. The author's study of the price effect over a 5-year period of a GTB currently in place in Michigan rejects this approach as inadequate apart from an analysis of variables over time. Variables considered in the Michigan study include the price effect of the GTB on school districts, representative households, and state and federal aid. Results of a pooling of data for all years for all 451 school districts studied indicate that the effect of the GTB plan on expenditures in Michigan has been so small as to be of no policy significance. These results run counter to the earlier findings of Feldstein and Ladd and suggest that their results should perhaps be rejected. (JBM)

Contends that, though much attention is given in mathematics to the problem of sensitizing people to the need for identifying elements of a situation and representing them symbolically, little or no effort is devoted to the problem of helping them express the relationships among these elements. (11 references) (MKR)

Presents a summary report of the Algebra Working Group's discussion subgroup on secondary algebra at the Seventh International Conference on Mathematical Education held in Quebec City, Canada in August 1992. Three appendixes include the reactions to the keynote addresses in this issue. (MKR)

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)

Describes the use of time-lapse photographs of a running cat as a model to investigate the concepts of function and derivative in a college calculus course. Discusses student difficulties and implications for teachers. (MKR)

Discusses diagrams and problem solvers' constructions of a problem situation, how the way of drawing changes during the problem-solving process, and the meaning and sense of diagrams. (Contains 15 references.) (MKR)

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 applies to formal and informal instructional settings and play contexts. (HTH)

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 past." (HTH)

Explains traditional notions of efficient behavior; applies them to libraries and information centers; discusses how we measure efficiency; describes data envelopment analysis, a mathematical model designed for use with nonprofit organizations to measure efficiency through a weighting scheme; and discusses its use in school media centers. (Contains five references.) (LRW)

The current status of item response theory (IRT) is discussed. Several IRT methods exist for assessing whether an item is biased. Focus is on methods proposed by L. M. Rudner (1975), F. M. Lord (1977), D. Thissen et al. (1988) and R. L. Linn and D. Harnisch (1981). Rudner suggested a measure of the area lying between the two item characteristic curves (ICCs). Both signed and unsigned indices have been suggested, which may differ where the ICCs cross. Lord proposed asymptotic chi square tests for the equality of the "a" and "b" parameters as well as tests for each of these parameters separately. Thissen has more recently suggested a successive model fitting procedure where models to be tested can include those that constrain one or more of the three parameters for an item to be equal for two groups. The method developed by Linn and Harnisch calibrates the item only for the total group; the observed performance of the focal group is then compared to that expected from the IRT model. Issues raised by these methods are related to accuracy of item calibration, effects of estimation error, the problem of widely different ability distributions, goodness of fit, and problems associated with tests that are somewhat speeded or with data sets with many omits such as may occur on difficult tests under formula scoring instructions. Recent uses of parameter estimation methods based on marginal maximum likelihood methods using the BILOG program are discussed briefly. (TJH)

Previous economic models suggested that, due to the high channel capacity of cable technology and its capability of pricing television programs directly to the viewers, its diffusion would promote specialized "narrow appeal" programming designed to serve relatively small audience segments. While this has clearly occurred, these models do not account for other developments, such as cable's function as an aftermarket or foremarket for increasingly expensive dramatic programming, the frequent repetition of programs on cable, and the extreme contrasts in audience sizes and production budgets that characterize cable and broadcast programs. This paper summarizes relevant data and then offers a modified "program choice" model in order to explain these phenomena. In this version, decision-makers not only select among program types, but can vary production expenditures and exhibit programs more than once. Under these assumptions, direct pricing mechanisms and new channel capacity are shown to induce a monopoly cable operator not to fragment audiences by offering more specialized programs, but to increase production investments for "mass appeal,""lowest common denominator" programs. Twelve figures are included, and a discussion of the effects of a shift from advertiser to pay support on continuous demand and production functions is appended. (14 references) (Author/EW)

The symposium, "Taming the Rasch Tiger: Using Item Response Theory in Practical Educational Measurement," was organized to deemphasize the technical complexities of item response theory (IRT) and to show the audience how IRT can be used in practical educational measurement. Four papers from the symposium are summarized and comments are made on the uses of the Rasch model. The following points must be considered when using items from item pools that are scaled using the Rasch model: (1) the question of fit to the model is an empirical question to be established and not a theoretical answer to be assumed; (2) achievement test items will not easily fit the Rasch model unless the item pool is divided into a large number of independent sets, one for each specific objective; (3) the interpretation of the numbers produced by the application of the model is important and not the numbers generated; and (4) when proposing to use the Rasch model in generating test scores, recognize that one should understand the model and its implications in order to make sensible interpretations. (JAZ)

This paper argues that no single measurement strategy serves all purposes and that applying methods and techniques which allow a variety of data elements to be retrieved and juxtaposed may be an investment in the future. Item response theory, Rasch model, and latent trait theory are all approaches to a single conceptual topic. An abbreviated look at each technique explains its particular application. Standardized achievement tests which are prepared by publishing companies are criticized because they evaluate broad areas of curriculum without measuring the particular curriculum of a school district. With the availability of item banks, many individuals develop local criterion referenced tests. This process is time consuming and produces results which over time may not be desirable. As more data becomes available, the emphasis should shift from a particular measurement technique to the development of schemes to examine data drawn from various sources, reflecting a variety of perspectives. Facing the limitations of any testing technique, recognizing the limitations of data collection processes, and applying additional solutions to information problems promises new frontiers to the measurement community. (JAZ)