Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

Interdisciplinary education – a predator–prey model for developing a skill set in mathematics,biology and technology

The science of biology has been transforming dramatically and so the need for a stronger mathematical background for biology students has increased. Biological students reaching the senior or post-graduate level often come to realize that their mathematical background is insufficient. Similarly, students in a mathematics programme, interested in biological phenomena, find it difficult to master the complex systems encountered in biology. In short, the biologists do not have enough mathematics and the mathematicians are not being taught enough biology. The need for interdisciplinary curricula that includes disciplines such as biology, physical science, and mathematics is widely recognized, but has not been widely implemented. In this paper, it is suggested that students develop a skill set of ecology, mathematics and technology to encourage working across disciplinary boundaries. To illustrate such a skill set, a predator–prey model that contains self-limiting factors for both predator and prey is suggested. The general idea of dynamics, is introduced and students are encouraged to discover the applicability of this approach to more complex biological systems. The level of mathematics and technology required is not advanced; therefore, it is ideal for inclusion in a senior-level or introductory graduate-level course for students interested in mathematical biology.     

Materials by design and the exciting role of quantum computation/simulation

It is now well-recognized that we are witnessing a golden age of innovation with novel materials, with discoveries important for both basic science and device applications—some of which will be treated at this Workshop. In this talk, we discuss the role of computation and simulation in the dramatic advances of the past and those we are witnessing today. We will also describe the growing acceptance and impact of computational materials science as a major component of materials research and its import for the future. In the process, we will demonstrate how the well-recognized goal driving computational physics/computational materials science—simulations of ever-increasing complexity on more and more realistic models—has been brought into greater focus with the introduction of greater computing power that is readily available to run sophisticated and powerful software codes like our highly precise full-potential linearized augmented plane wave (FLAPW) method, now also running on massively parallel computer platforms.We will then describe some specific advances we are witnessing today, and computation and simulation as a major component of quantum materials design and its import for the future, with the goal—to synthesize materials with desired properties in a controlled way via materials engineering on the atomic scale. The theory continues to develop along with computing power. With the universality and applicability of these methods to essentially all materials and properties, these simulations are starting to fill the increasingly urgent demands of material scientists and engineers.     

Introducing computational thinking through hands-on projects using R with applications to calculus,probability and data analysis

The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts.     

Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.     

Undergraduate computational science and engineering education: A SIAM Working Group Report

Computational Science and Engineering (CSE) is a rapidly growing multidisciplinary area with connections ot the sciences, engineering, mathematics and computer science. CSE is a legitimate and important academic enterprise, even if it is yet to be formally recognized by a number of institutions. The undergraduate arena is the most important segment of the educational pipeline, since it prepares teachers for the high school environment, invigorates students to pursue graduate studies in cutting edgetechnical fields, and produces a vast number of future employees for industry and the “knowledge-based” economy. Thus it is critical that CSE curricula and programs are a viable option for every undergraduate student. This presentation introduces the SIAM Working Group report on undergraduate CSE education. The particular focus here is on background and the varying nature of programs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Effects of a Mathematics Infusion Curriculum on Middle School Student Mathematics Achievement

Increasing mathematical competencies of American students has been a focus for educators, researchers, and policy makers alike. One purported approach to increase student learning is through connecting mathematics and science curricula. Yet there is a lack of research examining the impact of making these connections. The Mathematics Infusion into Science Project, funded by the National Science Foundation, developed a middle school mathematics‐infused science curriculum. Twenty teachers utilized this curriculum with over 1,200 students. The current research evaluated the effects of this curriculum on students' mathematics learning and compared effects to students who did not receive the curriculum. Students who were taught the infusion curriculum showed a significant increase in mathematical content scores when compared with the control students.     

Enriching Science and Math Through Engineering

This case study reviewed the collaborative efforts of university engineers, teacher educators, and middle school teachers to advance sixth‐ and seventh‐grade students' learning through a series of project‐based engineering activities. This two‐year project enriched regular school curricula by introducing real‐world applications of science and mathematics concepts that expanded opportunities for creativity and problem‐solving, introduced problem‐based learning, and provided after‐school programming (for girls only) led by engineering students from the local university. This engineering education initiative showed significant impact on students' (1) confidence in science and mathematics; (2) effort toward science and mathematics; (3) awareness of engineering; and (4) interest in engineering as a potential career. With regard to gender, there were no significant differences between boys' and girls' responses. The girls' confidence in their own skills and potential, however, was significantly more positive than the boys' confidence in the girls. These results gave rise to new questions regarding mentor/mentee relationships and the overall effect of “girls only” mentoring.     

Using the Tower of Hanoi puzzle to infuse your mathematics classroom with computer science concepts

This article suggests that logic puzzles, such as the well-known Tower of Hanoi puzzle, can be used to introduce computer science concepts to mathematics students of all ages. Mathematics teachers introduce their students to computer science concepts that are enacted spontaneously and subconsciously throughout the solution to the Tower of Hanoi puzzle. These concepts include, but are not limited to, conditionals, iteration, and recursion. Lessons, such as the one proposed in this article, are easily implementable in mathematics classrooms and extracurricular programmes as they are good candidates for ‘drop in’ lessons that do not need to fit into any particular place in the typical curriculum sequence. As an example for readers, the author describes how she used the puzzle in her own Number Sense and Logic course during the federally funded Upward Bound Math/Science summer programme for college-intending low-income high school students. The article explains each computer science term with real-life and mathematical examples, applies each term to the Tower of Hanoi puzzle solution, and describes how students connected the terms to their own solutions of the puzzle. It is timely and important to expose mathematics students to computer science concepts. Given the rate at which technology is currently advancing, and our increased dependence on technology in our daily lives, it has become more important than ever for children to be exposed to computer science. Yet, despite the importance of exposing today's children to computer science, many children are not given adequate opportunity to learn computer science in schools. In the United States, for example, most students finish high school without ever taking a computing course. Mathematics lessons, such as the one described in this article, can help to make computer science more accessible to students who may have otherwise had little opportunity to be introduced to these increasingly important concepts.     

A comparison of analytic and knowledge-based approaches to closed-shop scheduling

This paper compares three different approaches to scheduling in a closed-shop environment, making the case for a knowledge-based approach. A manufacturing example from the food industry is used as a vehicle for the presentation. The first approach attempts to find an optimal solution using a mixed integer linear programming formulation, but the size of the problem renders this approach impractical. The second approach uses a spreadsheet program to obtain feasible solutions, but imbedded assumptions in the heuristics used allow it to be used only for simple demand patterns. The third approach employs expert systems technology. It includes several heuristics and takes all constraints into consideration. The solution obtained may not be optimal, but computational tests suggest that it is far superior to both spreadsheet and manual approaches.     

Agent‐based computational models and generative social science

This article argues that the agent‐based computational model permits a distinctive approach to social science for which the term “generative” is suitable. In defending this terminology, features distinguishing the approach from both “inductive” and “deductive” science are given. Then, the following specific contributions to social science are discussed: The agent‐based computational model is a new tool for empirical research. It offers a natural environment for the study of connectionist phenomena in social science. Agent‐based modeling provides a powerful way to address certain enduring—and especially interdisciplinary—questions. It allows one to subject certain core theories—such as neoclassical microeconomics—to important types of stress (e.g., the effect of evolving preferences). It permits one to study how rules of individual behavior give rise—or “map up”—to macroscopic regularities and organizations. In turn, one can employ laboratory behavioral research findings to select among competing agent‐based (“bottom up”) models. The agent‐based approach may well have the important effect of decoupling individual rationality from macroscopic equilibrium and of separating decision science from social science more generally. Agent‐based modeling offers powerful new forms of hybrid theoretical‐computational work; these are particularly relevant to the study of non‐equilibrium systems. The agent‐based approach invites the interpretation of society as a distributed computational device, and in turn the interpretation of social dynamics as a type of computation. This interpretation raises important foundational issues in social science—some related to intractability, and some to undecidability proper. Finally, since “emergence” figures prominently in this literature, I take up the connection between agent‐based modeling and classical emergentism, criticizing the latter and arguing that the two are incompatible. © 1999 John Wiley & Sons, Inc.     

Pseudo‐Spectral Methods for the Laplace‐Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One

The inversion of the Laplace‐Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high‐order accurate pseudo‐spectral approach, applicable to closed surfaces of genus one in three‐dimensional space, with a view toward applications in plasma physics and fluid dynamics. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017     

Exploring hurdles to transfer : student experiences of applying knowledge across disciplines

This paper explores the ways students perceive the transfer of learned knowledge to new situations – often a surprisingly difficult prospect. The novel aspect compared to the traditional transfer studies is that the learning phase is not a part of the experiment itself. The intention was only to activate acquired knowledge relevant to the transfer target using a short primer immediately prior to the situation where the knowledge was to be applied. Eight volunteer students from either mathematics or computer science curricula were given a task of designing an adder circuit using logic gates: a new context in which to apply knowledge of binary arithmetic and Boolean algebra. The results of a phenomenographic classification of the views presented by the students in their post-experiment interviews are reported. The degree to which the students were conscious of the acquired knowledge they employed and how they applied it in a new context emerged as the differentiating factors.     

Finite-Temperature Coarse-Graining of One-Dimensional Models: Mathematical Analysis and Computational Approaches

We present a possible approach for the computation of free energies and ensemble averages of one-dimensional coarse-grained models in materials science. The approach is based upon a thermodynamic limit process, and makes use of ergodic theorems and large deviations theory. In addition to providing a possible efficient computational strategy for ensemble averages, the approach allows for assessing the accuracy of approximations commonly used in practice.     

Fundamentals of Computational Conformal Geometry

Computational conformal geometry is an inter-disciplinary field between mathematics and computer science. This work introduces the fundamentals of computational conformal geometry, including theoretic foundation, computational algorithms, and engineering applications. Two computational methodologies are emphasized, one is the holomorphic differentials based on Riemann surface theory and the other is surface Ricci flow from geometric analysis.     

An Ethnoscience Approach to Curriculum Issues for American Indian Students

This paper describes a course offered to teachers of American Indian students, which focused on the development of culturally relevant activities as part of the science and mathematics curricula. In response to the concern that American Indian students do not find meaning in the curriculum, these activities were embedded in a holistic approach to the curriculum, and the informal science and mathematics of the culture were linked with the traditional school science and mathematics. Informal results suggest that the development of these connections will help American Indian students make sense of what they are learning, both in the context of the culture and in the context of school science and mathematics.     

Mental Computation of Students in a Reform-Based Mathematics Curriculum

Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study, fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with “invented” algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.     

The emergence of computational sociology

The world of science has undergone a major transformation by virtue of technological innovations in computing and information proessing. Sociology is one site in which this change is being played out. Our basic aim is to set out a revised image of any modern science, within which we can conceptualize and discuss the role of a newly emergent subfield we term computational sociology. Specifically, we expand the familiar two‐component model of a science, featuring a theoretical and an empirical side, to include a computational component. We show how the three components interrelate in a triangular system in which empirical data analysis, theoretical explanation and computer simulation link the three components. We close our paper with a brief discussion of how one new development in computation relates to concepts of sociology, an instance of the hybrid character of computational sociology.     

Computational approaches to parameter estimation and model selection in immunology

One of the significant challenges in biomathematics (and other areas of science) is to formulate meaningful mathematical models. Our problem is to decide on a parametrized model which is, in some sense, most likely to represent the information in a set of observed data. In this paper, we illustrate the computational implementation of an information-theoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.The approach is illustrated by modelling LCMV infection using a family of models based on systems of ordinary differential and delay differential equations. The models (which use parameters that have a scientific interpretation) are chosen to fit data arising from experimental studies of virus-cytotoxic T lymphocyte kinetics; the parametrized models that result are arranged in a hierarchy by the computation of Akaike indices. The practical illustration is used to convey more general insight. Because the mathematical equations that comprise the models are solved numerically, the accuracy in the computation has a bearing on the outcome, and we address this and other practical details in our discussion.     

Revisiting an Ancient Problem Through Contemporary Discourse

This article presents a computer-mediated discourse on the Pythagorean equation in a university classroom of preservice and in-service teachers. It shows how the use of a spreadsheet as a two-dimensional modeling tool enables students to conjecture the general solution to the Pythagorean equation. A computational approach to the ancient problem improves the well-known off-computer discussion. The environment also provides a kind of a visualization of Fermat's Last Theorem.