# Showing papers in "International Journal of Mathematical Education in Science and Technology in 2002"

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TL;DR: In this paper, the Sumudu transform of a special function f (t) with a corresponding sumudu transformation F (u) has been studied and the effect of shifting the parameter t in the function f(t) by τ on the transform F(u) is analyzed.

Abstract: This note discusses the general properties of the Sumudu transform and the Sumudu transform of special functions. For any function f (t) with corresponding Sumudu transform F (u), the effect of shifting the parameter t in f (t) by τ on the Sumudu transform F (u) is found. Also obtained are the effect of the multiplication of any function f (t) by a power of t and the division of the function f (t) by t on the Sumudu transform F (u). For any periodic function f (t) with periodicity T > 0 the Sumudu transform is easily derived. Illustrations are provided with Abel's integral equation, an integro-differential equation, a dynamic system with delayed time signals and a differential dynamic system.

109 citations

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TL;DR: The Bernoulli polynomials are generalized in this article and some properties of the resulting generalizations are presented, as well as a generalization of the polynomial generalization.

Abstract: The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.

76 citations

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TL;DR: In this paper, a spatial operational capacity model for the development of visual and spatial skills is proposed to assist in the design of appropriate learning activities and materials for specific scientific concepts using acceleration as an example.

Abstract: Students of all ages struggle with physics not only due to the complexities of the subject, but also due to inadequacies with their skills and knowledge of mathematics. The teaching of the physical sciences at school level presupposes that the foundation building blocks (e.g. position, length, angle and time) are put in place in mathematics classes at lower school levels. By the time the student reaches secondary school, the endeavour is to develop scientific concepts (e.g. velocity and acceleration) that are a combination of the foundation building blocks. The utilization of a spatial operational capacity model for the development of visual and spatial skills is proposed. It reflects the complexity of the interaction between physics/science and mathematics, and endeavours to assist in the design of appropriate learning activities and materials for the development of specific scientific concepts using acceleration as an example.

62 citations

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TL;DR: In this paper, the problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey the law of logistic growth and each predators functional response to the prey approaches a constant as the prey population increases is addressed.

Abstract: The present paper deals with the problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey the law of logistic growth and each predators functional response to the prey approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability are studied using eigenvalue analysis. The existence of bionomic equilibria has been considered. The problem of determining the optimal harvest policy is then solved by using Pontryagin's maximal principle. Finally, some numerical examples are given to illustrate the results.

37 citations

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TL;DR: In this paper, the authors set forth some salient results in the algebra of circulant matrices which can be used in time series analysis and provided easy derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions.

Abstract: This paper sets forth some salient results in the algebra of circulant matrices which can be used in time-series analysis. It provides easy derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions. A statistical test for the stationarity or homogeneity of empirical processes is also presented.

36 citations

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TL;DR: In this paper, an iterative method that approximates numerically the solution of f (x) = 0 is proposed. But the method requires only function and first derivative evaluations.

Abstract: Using Newton's method as an intermediate step, we introduce an iterative method that approximates numerically the solution of f (x) = 0. The method is essentially a leap-frog Newton's method. The order of convergence of the proposed method at a simple root is cubic and the computational efficiency in general is less, but close to that of Newton's method. Like Newton's method, the new method requires only function and first derivative evaluations. The method can easily be implemented on computer algebra systems where high machine precision is available.

33 citations

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TL;DR: This article provided baseline data in an area of the mathematics curriculum that is beginning to receive greater attention than previously and found improvement with grade in expressing probability numerically and in distinguishing conditional events, but no change in incidence of conjunction errors.

Abstract: The objective of this study was to provide baseline data in an area of the mathematics curriculum that is beginning to receive greater attention than previously Four survey items were completed by 2615 students in grades 5 to 11 Two survey items asked for estimates of probability or frequency for everyday events (A), (B), and their conjunction (A and B) Two survey items asked for estimates of probability or frequency for conditional events, (X|Y) and (Y|X) Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and in distinguishing conditional events, but no change in incidence of conjunction errors The relationships of responses to conjunction items with those to conditional items, and of both with responses to other items of basic chance measurement were considered Implications were related to interpretation of the results in terms of previous research and suggestions for educators

31 citations

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TL;DR: In this article, the authors have linked the use of computer algebra system to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.

Abstract: If the use of a computer algebra system (CAS) is to be meaningful and have an impact on students, then it must be grounded in good pedagogy and have some clearly defined goals. It is the authors' belief that an important goal for teaching mathematics with the CAS is that courses be designed so that students can become active participants in their learning experience, planning the problem-solving strategies and carrying them out. The CAS becomes an important tool and a partner in this learning process. To this end, here the authors' have linked the use of the CAS to an existing classification scheme for Mathematical Tasks, called the MATH Taxonomy, and illustrated, through concrete examples, how the goals of teaching and learning of mathematics can be set using this classification together with the CAS.

28 citations

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CINVESTAV

^{1}TL;DR: This study documents the students' use of Cabri-Geometry to work on tasks that led them to construct particular relationships and to explore and connect different mathematical themes or ideas.

Abstract: Recent curriculum proposals recognize that the use of technology plays an important role in students' learning of mathematics. What do students need (in terms of mathematics resources) in order to use technology as a mathematical tool? When does the use of technology become a powerful tool for students? To what extent do students' approaches to mathematical tasks, via paper and pencil, differ from technological approaches? Are these two approaches compatible? Are previous students' experiences transferable to technology approach experiences? These are fundamental questions that are important to investigate in order to identify and analyse the potential in using technology as tools to construct and understand mathematical ideas. This study documents the students' use of Cabri-Geometry to work on tasks that led them to construct particular relationships and to explore and connect different mathematical themes or ideas.

24 citations

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TL;DR: In this article, three level-implicit finite difference methods of order four are discussed for the numerical solution of the mildly quasi-linear second-order hyperbolic equation A (x, t, u)u xx + 2B(x,t, u, ux, ut ), 0 0 subject to appropriate initial and Dirichlet boundary conditions.

Abstract: Three level-implicit finite difference methods of order four are discussed for the numerical solution of the mildly quasi-linear second-order hyperbolic equation A (x, t, u)u xx + 2B(x, t, u)u xt + C(x, t, u)u tt = ƒ (x, t, u, ux, ut ), 0 0 subject to appropriate initial and Dirichlet boundary conditions. A new technique is introduced to obtain the stability range of the wave equation in polar coordinates. Fourth-order approximation at the first time level for a more general case is also discussed. The fourth-order accuracy of the method is demonstrated computationally by four examples.

23 citations

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Alma College

^{1}TL;DR: In this article, a team-oriented formal testing method was used in a mathematical modelling course taught during the Alma College intensive spring term to assess the students' acquisition of problem-solving and mathematical-thinking skills.

Abstract: Assessment of students' attainment in courses is often driven by the method of instruction. When mathematics is taught in the traditional style of lectures on theory coordinated with homework on standard problems, the testing is often oriented to reproducing the skills demonstrated by the instructor. If a more collaborative teaching method is used, how does one assess the students' acquisition of problem-solving and mathematical-thinking skills? The paper discusses a team-oriented formal testing method used in a mathematical modelling course taught during the Alma College intensive spring term.

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TL;DR: In this article, the pendulum equation θ + ⋋ 2 sin θ = 0 and two approximations for it were investigated and the Fourier series was derived to at least eleven decimal places.

Abstract: We investigate the pendulum equation θ + ⋋ 2 sin θ = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin θ do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are appropriately chosen very small and the time interval is short. On the other, we suggest that computationally, there is no advantage taking these approximations. We further justify this by employing an approach to deriving Fourier series approximations to the pendulum equation accurate to at least eleven decimal places. Students can generate highly accurate Fourier series solutions to nonlinear equations and thus concentrate on the qualitative aspects of the model rather than the computational difficulties.

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TL;DR: In this article, a formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved, which is applied to the study of the geometry of n-simplices.

Abstract: A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

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TL;DR: In this paper, a simple boom structure is analyzed and its reliability is assessed using Monte-Carlo simulation, assuming that four of the structural variables are random with a known joint probability distribution function.

Abstract: Using elementary engineering mechanics, a simple boom structure is analysed and its reliability is assessed. It is assumed that four of the structural variables are random with a known joint probability distribution function. From the structural analysis, an explicit expression for failure is obtained. The reliability of the structure is computed analytically from first principles of probability theory. Additionally, it is shown that Monte-Carlo simulation can be implemented more readily with results that compare favourably to the theoretical calculations. Thus, the power and utility of Monte-Carlo simulation are demonstrated. Specifically, the boom is analysed for different probability distribution functions for the underlying components. The analytical and numerical analysis contained in this application are appropriate for any upper level undergraduate course in applied mathematics, scientific computation or engineering reliability where the Monte-Carlo method is being studied.

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TL;DR: In this paper, the notion of uniform convergence at a point is proposed as an alternative to the Boas approach in establishing this and consequently, other results, which stay within the realm of a first proof course in classical mathematical analysis.

Abstract: It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take ƒ n (x) = x n on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a,b] is, nonetheless, continuous on a dense subset of [a,b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.

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TL;DR: In this paper, the authors used a mixture of tasks presented in either an open or closed form, and found that the success rate in performing the given tasks seems to depend on the degree of openendedness inherent.

Abstract: The use of short assessment tasks can provide valuable information about undergraduates' knowledge and understanding. However, it is known that there are gender-related differences in performance on certain types of objective tests, both among school pupils and university undergraduates. This article focuses on undergraduate learning, using a mixture of tasks presented in either an open or closed form. Although the success rate in performing the given tasks seems to depend on the degree of openendedness inherent, more unexpected is the consistent difference in achievement between men and women students.

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TL;DR: In this paper, the authors discuss the need to link the construction of the examination and evaluation of the exam scripts in terms of an existing classification scheme for mathematical tasks, the MATH taxonomy.

Abstract: The use of computer algebra systems (CAS) in the teaching and learning of mathematics is likely to have a minimal impact on the overall mathematics curriculum until the CAS becomes a legitimate tool to be used in mathematics examinations. The important issue is how the examinations can be written in a way that allows the student the appropriate use of a CAS and tests the students' understanding of the underlying mathematical concepts and problem solving strategies. To this end, the authors give concrete examples of such examinations and discuss the need to link the construction of the examination and evaluation of the exam scripts in terms of an existing classification scheme for mathematical tasks, the MATH taxonomy. The sample examination questions and proposed solutions are evaluated within this framework.

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TL;DR: In this paper, it was found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates, who generally have one or two semesters of calculus and some linear algebra as part of their curriculum.

Abstract: Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. Thi...

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TL;DR: In this article, the properties of two sequences generated by the recurrence relation G n+2 = 4G n+1 − G n, where n is the number of vertices in the graph.

Abstract: In this note properties of two sequences generated by the recurrence relation G n+2 = 4G n+1 − G n , are studied. It is shown that one of the sequences leads to a family of diophantine triplets. Some interesting properties of these sequences are also established.

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TL;DR: In this paper, a new necessary and sufficient condition for equality to hold is presented and then some interesting consequences and applications are discussed, as well as some interesting applications and applications of the condition.

Abstract: The well-known Frobenius rank inequality established by Frobenius in 1911 states that the rank of the product ABC of three matrices satisfies the inequality rank(ABC) U rank(AB) + rank(BC)- rank(B) A new necessary and sufficient condition for equality to hold is presented and then some interesting consequences and applications are discussed.

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TL;DR: In this article, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration, and two of the programs are animated.

Abstract: The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the graphical user interface have been specifically designed to help the student understand the processes behind these important introductory concepts. Each program has a series of demonstrations that show unusual, difficult or important cases.

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TL;DR: In this paper, the authors present teaching ideas designed to support the belief that students at all levels (preservice teachers, majors, secondary and elementary students) need exposure to nonroutine problems that illustrate the effective use of technology in their resolution.

Abstract: This article presents teaching ideas designed to support the belief that students at all levels (preservice teachers, majors, secondary and elementary students) need exposure to non-routine problems that illustrate the effective use of technology in their resolution. Such use provides students with rapid and accurate data collection, leading them to sound conjectures, which is a precursor to learning mathematical proof. Students will therefore learn that while technology can be an effective tool for investigating problems, the onus of providing convincing arguments and proofs of their conjectures rests squarely on their shoulders. The paper describes how a diverse group of students took advantage of the power of the TI-92 to enhance their chances of reaching this final stage of proof. A series of mathematical problems are presented and analysed with a keen eye on the appropriate integration of the TI-92. A student survey was used to inform the results. To conclude, several challenging, yet accessible, non...

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TL;DR: This article investigated gender-related effects in the Western Australian Calculus Tertiary Entrance Examinations for the period 1995-2000, three years before and three years after graphics calculators were introduced.

Abstract: The paper reports an investigation of gender-related effects in the Western Australian Calculus Tertiary Entrance Examinations for the period 1995–2000, three years before and three years after graphics calculators were introduced. Consideration is given to students' total examination scores, scores on questions grouped by curriculum component and students' actual use of graphics calculators on two questions from the year 2000 examination. Results show that over the six-year period the performance of girls—on the examination as a whole and in most curriculum components—was superior to that of boys at the lower end of the achievement scale, while boys recorded the best performance at the top end of the scale. The results are partially explained by the participation rate in calculus for girls being lower than that for boys. Where superior performance is recorded for girls it is frequently attributable to competence with analytic methods. Superior performance favouring boys typically occurred on questions wh...

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TL;DR: In this paper, a generalized solution of the adjoint of the Poisson equation is used to obtain the necessary boundary conditions, and the BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.

Abstract: Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.

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TL;DR: In this article, the authors describe the use of a spreadsheet in a mathematics teacher education course and show how the tool can serve as a link between seemingly disconnected mathematical concepts, and argue that revisiting elementary content in a technological context enables pre-service teachers to appreciate the role that conceptual knowledge can play in the development of spreadsheet-enabled pedagogy.

Abstract: The paper describes the use of a spreadsheet in a mathematics teacher education course. It shows how the tool can serve as a link between seemingly disconnected mathematical concepts. The didactical triad of using a spreadsheet as an agent, consumer, and amplifier of mathematical activities allows for an extended investigation of simple yet intriguing properties of whole numbers. The authors argue that revisiting elementary content in a technological context enables pre-service teachers to appreciate the role that conceptual knowledge can play in the development of a spreadsheet-enabled pedagogy.

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TL;DR: The Modelling Track at the Technische Universiteit Eindhoven as discussed by the authors was introduced in 1995 as a response to the employers' complaint about graduates' poor ability to apply their theoretical knowledge.

Abstract: In 1995, the Department of Mathematics and Computing Science at the Technische Universiteit Eindhoven introduced the Modelling Track in the Mathematical Engineering curriculum. It was a response to the employers' complaint about graduates' poor ability to apply their theoretical knowledge. In their first three years pairs of students collaborate on non-mathematically defined problems using mathematical methods. In 1998, the university decided to introduce Design Based Learning (DBL) in all its curricula, therewith complying with the needs of employers for graduates' applicative and collaborative abilities. DBL means professionalization, activation, co-operation, creativity, integration and multidisciplinarity. For Mathematical Engineering this innovation could signify the Modelling Track to incorporate the DBL features, e.g. working with larger groups or integration with other courses. By the year 2000, few changes have been realized. The question remains whether the original Modelling Track already fulfi...

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TL;DR: In this article, it is argued that the variance of two observations is easily calculated without the use of a sample mean, and that this sense of pairing may result in precision when computer programs are used for the calculation.

Abstract: The usual formula for variance depending on rounding off the sample mean lacks precision, especially when computer programs are used for the calculation. The well-known simplification of the total sums of squares does not always give benefit. Since the variance of two observations is easily calculated without the use of a sample mean, and the variance of a sample of n observations is the average of the variances of observations based on n(n-1)/2 distinct subsets of units of size 2 from the sample, it is argued that this sense of pairing may result in precision. Some other forms of variance are presented which provide some insight into it. The contribution of a new observation of variance is highlighted, which is important in sequential sampling. Notions are illustrated with examples.

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TL;DR: In this article, the application of the Thomas-Reiche-Kuhn sum rule to simple quantum-mechanical models and its apparent violation by the rigid rotator are discussed.

Abstract: We discuss application of the Thomas-Reiche-Kuhn sum rule to simple quantum-mechanical models and its apparent violation by the rigid rotator.

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TL;DR: In this article, the Mobius strip, torus and Klein bottle are used to graphically and analytically illustrate the differences between orientable and non-orientable surfaces.

Abstract: The Mobius strip, torus, and Klein bottle are used to graphically and analytically illustrate the differences between orientable and non-orientable surfaces. An exercise/laboratory project using the non-orientable Boy surface is included.

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TL;DR: In this article, a new transform proposed by Oyelami and Ale for impulsive systems is applied to an impulsive fish-hyacinth model and a biological policy regarding the growth of the fish and the hyacinth populations is formulated.

Abstract: A new transform proposed by Oyelami and Ale for impulsive systems is applied to an impulsive fish-hyacinth model. A biological policy regarding the growth of the fish and the hyacinth populations is formulated.