Institutional Norms and Policies That Influence College Mathematics Professors in the Process of Changing to Reform‐Based Practices

This case study was an investigation of the role of the institutional culture of a university in the process of changing to reform‐based practices for two college mathematics professors. A framework is presented for identifying and analyzing institutional norms and policies that are present and those lacking in supporting faculty efforts toward reform. The primary categories considered in the framework are the teaching and learning context, the professional community, and the university's reward system. This framework is applied to the cases, and findings indicate that institutional norms in the forms of priorities for how time is allocated to professional responsibilities (as part of the teaching context), and colleagues and administrators who understood and shared the professors' goals for reform (as part of the professional community) strongly influenced the professors' efforts toward change. Additionally, a reward system that recognized efforts to improve teaching and learning was critical in establishing a culture that promoted change. However, despite the fact that both professors were in the same mathematics department, the nature of the influence of these factors was not uniform. The institutional culture indeed had multiple layers that required examination for each professor's individual context. The findings help highlight the importance of institutional context for both K‐12 and college level teaching and learning. The nature of these influential factors and the institutional layers were discussed for each case, along with implications for other institutions.     

Enhancing management's understanding of operational research models

The problem of the gap between management's understanding of a model and the model builder's insight is a large one. This problem continues to challenge the practice of operational research and plague contemporary modelling efforts. This paper offers an approach to enhancing a manager's understanding of the model with respect to an application. It sets out a procedure whereby (a) the reasons for the management's judgements are uncovered, and (b) the sources of disagreement with the recommendations obtained from the model are identified. The manager, learning from the model, may then change his or her judgements. Alternatively, the model builder may modify the model parameters or assumptions, leading to convergent understanding. The procedure is based on the framework of social judgement theory. An illustrative example is offered in the context of facility planning in the cement manufacturing industry.     

Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence

This study presents how the introduction of a metaphor for sequence convergence constituted an experientially real context in which an undergraduate real analysis student developed a property-based definition of sequence convergence. I use elements from Zandieh and Rasmussen's (2010) Defining as a Mathematical Activity framework to trace the transformation of the student's conception from a non-standard, personal concept definition rooted in the metaphor to a concept definition for sequence convergence compatible with the standard definition. This account of the development of the definition of sequence convergence differs from prior research in the sense that it began neither with examples or visual notions, nor with the statement of the formal definition. This study contributes to the Realistic Mathematics Education literature as it documents a student's progression through the definition-of and definition-for stages of mathematical activity in an interactive lecture classroom context.     

The Determination of Approximately Optimal Policies in Markov Decision Processes by the Use of Bounds

In the general area of Markov decision processes, a lot of attention has been given to deriving upper and lower bounds for approximating the optimal performance level. These are, in themselves, not useful unless they can be used to derive an approximately optimal policy. The existing literature does this specifically in the context of the computational methods being used at the time. However, irrespective of the method used, a straight application of one step of Howard's policy space method will give the desired results.     

How neutral networks influence evolvability

Evolutionary algorithms apply the process of variation, reproduction, and selection to look for an individual capable of solving the task at hand. In order to improve the evolvability of a population we propose to copy important characteristics of nature's search space. Desired characteristics for a genotype–phenotype mapping are described and several highly redundant genotype–phenotype mappings are analyzed in the context of a population‐based search. We show that evolvability, defined as the ability of random variations to sometimes produce improvement, is influenced by the existence of neutral networks in genotype space. Redundant mappings allow the population to spread along the network of neutral mutations and the population is quickly able to recover after a change has occurred. The extent of the neutral networks affects the interconnectivity of the search space and thereby affects evolvability. © 2002 Wiley Periodicals, Inc.     

A network-based futures method for strategic business planning

Many strategic decisions in business are made in a context which the decision makers perceive as uncertain, complex and opaque. A method, based on Rhyne's field anomaly relaxation technique, is described of generating a network of states which characterise the environment or context in which strategic decisions are to be made. These states represent possible future conditions for the business, and knowledge of them allows improved strategic understanding and decision making to be achieved. This paper describes the method, using a representative real-life application to illustrate the process.     

Role of design-philosophies in interfacing manufacturing with marketing

Product design involves the mapping of a product's marketing attributes to its engineering and manufacturing features. In practice, it is a non-trivial task for design engineers to determine an appropriate mapping between a product's marketing and manufacturing attributes so as to generate a product design that satisfies customer-needs while being feasible to produce within the technical and financial constraints of the firm's manufacturing domain. In this paper, we hypothesize that in a mature industry, the expertise that guides designers in this mapping process exists in the form of an engineering design-philosophy that governs the design of products in that industry. Clearly, if we are able to discern the design-philosophies that exist in an industry, they can then be used to channrl designers' creativity to be congruent with the commercial objectives of firms in that industry. Using real-world data from the automobile industry, we show the existence of design-philosophies and illustrate methodologies for identifying and interpreting them. We also describe predictive models that allow managers to exploit the knowledge available in these design-philosophies and accurately specify a new product's manufacturing attribute values, given market input regarding the product's desired features. This total approach should save firms valuable product designing time and enable them to generate products that can be successfully produced and sold.     

The generation of a social welfare function under ordinal preferences

Against the background of recent impossibility theorems, the paper establishes a number of rules for the formation of social preferences to ensure the generation of a Bergson-Samuelson social welfare function of the desired ‘individualistic’ form. It demonstrates that such welfare functions can indeed exist under conditions of simply ordinality of individual preferences and a lack of interpersonal comparability, for the general n-person case with only weak restrictions upon individual preferences. A valuable tool of social welfare theory for the single-profile case is therefore restored under a wide set of conditions. By examining Kemp and Ng's justification for their A3 condition in this context, we are able also to highlight a number of key differences between the conditions satisfied by many political constitutions and the requirements for achieving the positive existence of a Bergson-Samuelson social welfare function. In particular the paper focusses here on the weighting systems on individual preferences that are consistent with generating an individualistic welfare function and examines the information content of individual ordinal preferences which must be utilized if the desired welfare function is to result.     

Linking the individual forgetting context with customer capital from a seller's perspective

Customer capital is a result of interaction between an organization and its customers. Customers change their characteristics, including addresses, behaviour and preferences; but as the customer requirements change, basic beliefs or processes that is, things that individuals take for granted at an implicit and an explicit level of knowledge, must also change. This paper aims to identify the role of an individual forgetting context on customer capital in today's organization through an empirical study of 229 sellers (front-line contact people) in the Spanish optometry industry. Two structural equation models have been used, resulting in the conclusion that before obtaining an up-to-date memory, it is necessary to identify new ways of doing and interpreting things, which in turn result in a shift in relations that favour the customers.     

The Influence of Incompressibility on the Microstructure of Dielectric Elastomers

Dielectric elastomers (DE's) are considered in recent times for the development of so called smart actuators. This relatively soft material is deformed under the influence of an external electric field. It is an ideal candidate for application in which complex surface deformations are desired. To increase the deformation range the insertion of material inclusions seems to be a good alternative. In the current work a mixed three field formulation in proposed to handle the quasi incompressibility of DE's for numerical simulations. A numerical analysis is performed in which the mixed formulation is compared with a standard displacement formulation. The results show the importance of the consideration of the nearly incompressible behaviour for the case of heterogeneous DE's. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Globalization, social development, and operations research: an exploration of strategies for success

This paper argues that in today’s world order, policy-making takes precedence over analytical and technical models that have historically formed the mainstay of Operations Research. The structural and human dimensions of this transition is discussed in the form of six Policy Imperatives aimed at the creation of a world order that makes it more humane and less vulnerable to failure. It is proposed that though OR occupies a prominent role in developmental initiatives, in order for its successful implementation in meeting social goals, the context in which it is invoked must be conducive to social transformation. An earlier version of this paper was presented at the International Conference on Operations Research for Development (ICORD) held December 27–30, 2002 at Anna University, Chennai. The author would like to acknowledge the helpful comments of anonymous reviewers. He would also like to thank Professor K.S.P. Rao of Anna University for his gracious assistance.     

A three-dimensional progressive damage model for fiber reinforced composites with an implicit-explicit integration scheme

This contribution proposes a fully three dimensional “continuum damage model” (CDM) to describe the interlaminar and intralaminar failure mechanisms of transversely isotropic elastic-brittle materials under static loading. The constitutive model is derived from an energy function with independent damage variables for each damage mode. The evolution law is based on energy dissipation within the damage process, taking into account the critical energy release rate to weaken the effect of mesh dependent outcome. The onset of damage can be predicted with Cuntze's failure mode concept [1] as well as with Hashin's failure criteria. In this model linear stress decreasing is assumed. In addition, an implicit-explicit integration scheme, first proposed by Oliver [3] for isotropic damage models, is adapted to increase the stability and robustness of numerical simulations and to decrease the computational cost of material failure analyses. By comparing the results from implicit-explicit integration schemes and standard implicit integration schemes, a high level of agreement is found. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reversible dynamics in discrete dissipative systems

The random walk model of Brownian motion is an example of a stochastic system which exhibits intrinsically irreversible behaviour. In spite of this, a simple discrete version of the model has been shown to harbour dynamics which are reversible and are described by a discrete form of Schrödinger's equation. The reversible dynamics appear as second order effects in this diffusive model, and the usual relationship between macroscopic irreversibility and microscopic reversibility is itself reversed. This will be discussed in the context of the `Brussels' school' on irreversibility.     

Mathematica™ in context

Aspects of the problem of teaching introductory undergraduate mathematics are considered in the context of both an increased participation rate in higher education as well as increasingly sophisticated computational technology. In particular, some of the changes in student and governmental expectations of course outcomes are canvassed, and an ongoing project initiated as a response both to these changes and to the availability of modern computational algebra systems that have sophisticated user interfaces is described. The project's aim is to develop students' mathematical understanding by undertaking practical laboratory work focused on applications that are perceived by students to be relevant to their social context and employment aspirations.     

A model of incentive contracts for just-in-time delivery

A mathematical model is used to analyze how to structure on-time delivery incentives in a contract between a buyer and a single supplier of raw materials when early shipments are forbidden. The buyer's choice of incentives takes the supplier's cost-minimizing response to incentives into account. The least cost incentive a buyer can select is specified by a probability of on-time delivery and an incentive scheme to achieve that probability. These optimal solutions are characterized without specifying the flow time distribution. A Method of selecting incentives that can help buyers improve on-time delivery performance is provided; however, the limitations of incentives are also considered. Achieving exactly 100% on-time delivery is shown to be non-optimal and only feasible under specific conditions. When management can not specify the shortage cost, their selection of a desired probability of on-time delivery allows for the determination of an imputed shortage cost.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Book Review

An Introduction to Difference Equations. Second Edition by Saber N. Elaydi, New York: Springer—Verlag, 1999. ISBN 0-387-98830-0. $54.95. Gone are the days when difference equations arose mainly in the context of sections of flows or as finite difference approximations to PDE's. Today difference equations have come into their own, both as objects of intrinsic mathematical interest and as dynamical models in their own right. Discrete models form an important part of dynamical systems theory independently from their continuous cousins. In Saber Elaydi's book dynamicists have the long awaited discrete counterpart to standard textbooks such as Hirsch and Smale (“Differential Equations, Dynamical Systems, and Linear Algebra”). The first edition of this book appeared in 1996. The second edition includes substantial new material including appendices on global stability and periodic solutions, a section on applications to mathematical biology, and a new chapter entitled “Applications to Continued Fractions and Orthogonal Polynomials”. Additional material on Birkhoff's theory now appears in the chapter on asymptotic behavior. The initial chapter covers first order equations, including equilibria, cobwebbing, stability, cycles, and the bifurcations of the discrete logistic equation. Chapter 2 moves on to higher order linear equations and briefly treats the difference calculus (for an in—depth treatment, see “Difference Equations: Theory and Applications. Second Edition” by Ronald E. Mickens, New York: Van Nostrand Reinhold, 1990). The subsequent chapters include systems of difference equations, stability theory, Z—transforms, control theory, oscillation theory, asymptotic behavior, and applications to continued fractions and orthogonal polynomials.

The chapters are composed of short sections, each of which ends with a nice selection of exercises. Answers to the odd—numbered problems appear in the back of the book. The core chapters include sections of applications to various fields such as population biology, economics, and physics. Several famous examples and topics are treated in the applications, including Gambler's Ruin, the Nicholson—Bailey host/parasitoid model, the heat equation, and Markov chains. Many discrete models are noninvertible, yet as many frustrated modelers know, most of the old standard treatments of linearization and the Stable Manifold Theorem., coming as they do from the context of sections of flows, require invertibility. Commendably, Elaydi avoids the needless assumption of invertibility in his stability theorems, and also in the Stable Manifold Theorem. However, invertibility is assumed in the Hartman—Grobman Theorem, where indeed it is necessary to establish conjugacy between the map and its linearization (see “An Introduction to Structured Population Dynamics”, CBMS—NSF Regional Conference Series in Applied Mathematics, Vol. 71, SIAM, Philadelphia, 1998 by J. M. Gushing, for an example of a noninvertible map for which the conjugacy fails. Readers may be interested to know that in this reference a weaker version of the Hartman—Grobman Theorem is proved that does not require invertibility but does establish the desired correspondence between types of hyperbolic equilibria in maps and their linearizations.)

This book is in Springer's Undergraduate Texts in Mathematics series and is indeed a very readable and appropriate text for advanced undergraduates or beginning graduate students. According to the author, the main prerequisites for such a course are calculus and linear algebra, with basic advanced calculus and complex analysis needed only for some topics in the later chapters. This is true; however in most situations the book would be best appreciated by students with a bit more mathematical maturity than is engendered by today's calculus and beginning linear algebra courses.Elaydi's book is a valuable reference for anyone who models discrete systems. It is so well written and well designed, and the content is so interesting to me, that I had a difficult time putting it down. I am pleased to own a copy for reference purposes, and am looking forward to using it to teach a senior topics course in difference equations.     

Hilbert's metric and the analytic hierarchy process

This paper explores some of the properties of Hilbert's projective metric as a measure of closeness between two ratio scales in the context of the Analytic Hierarchy Process. Smallperturbation arguments are used to contrast the sensitivity and the distributional behavior of this metric with the more traditional Euclidean distance function, in situations where the paired comparison of alternatives is subject to random perturbations, and priorities are estimated either by Saaty's eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbert's metric has surfaced which allows for the construction of confidence regions for an underlying priority vector. These regions are seen to enjoy good coverage properties.