# Showing papers in "International Journal of Mathematical Education in Science and Technology in 1994"

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TL;DR: A survey of the reported research about students' errors, difficulties and conceptions concerning elementary statistical concepts is presented in this article, where the authors focus on other statistical concepts, which have received little attention hitherto.

Abstract: This paper presents a survey of the reported research about students’ errors, difficulties and conceptions concerning elementary statistical concepts. Information related to the learning processes is essential to curricular design in this branch of mathematics. In particular, the identification of errors and difficulties which students display is needed in order to organize statistical training programmes and to prepare didactical situations which allow the students to overcome their cognitive obstacles. This paper does not attempt to report on probability concepts, an area which has received much attention, but concentrates on other statistical concepts, which have received little attention hitherto.

155 citations

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TL;DR: In this article, the method of Kanwal and Liu for the solution of Fredholm integral equations is applied to certain linear and nonlinear Volterra integral equations of the second kind.

Abstract: The method of Kanwal and Liu for the solution of Fredholm integral equations is applied to certain linear and nonlinear Volterra integral equations of the second kind. Some equations considered by other authors are solved in terms of Taylor polynomials and the results are compared.

121 citations

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TL;DR: In this paper, the Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations (PDEs) is demonstrated with respect to three different PDEs.

Abstract: The Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations.

114 citations

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TL;DR: This article illustrates how MATLAB can be used in a numerical analysis course to enhance the teaching of: Newton's method, Gaussian elimination, Chebyshev approximation, least squares polynomials, error analysis for numerical differentiation, adaptive quadrature, Runge‐Kutta methods, and the solution of Laplace's equation.

Abstract: Today's software offers more for numerical analysis than just programming. The software MATLAB can be used to do things the traditional way; writing loops; branching using logical decisions and invoking subroutines. Now a larger programming environment is available; graphics and built in subroutine libraries. These features are influencing the way numerical analysis is taught. MATLAB is based on lists and many algorithms can be streamlined by taking advantage of this structure. Graphical output for interpolation, curve fitting and the solution of differential equations is easily produced by manipulating these data structures. This article illustrates how MATLAB can be used in a numerical analysis course to enhance the teaching of: Newton's method, Gaussian elimination, Chebyshev approximation, least squares polynomials, error analysis for numerical differentiation, adaptive quadrature, Runge‐Kutta methods, and the solution of Laplace's equation. Our students have enjoyed MATLAB, and had a better experienc...

50 citations

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TL;DR: The work on epistemology and pedagogy is detached from a personal experiential basis of teaching, and thus it is in conflict with the well established principle that knowledge construction (and this includes mathematics knowledge as well as knowledge of mathematics epistemological knowledge) is a product of personal problem-solving activity.

Abstract: Current teacher education programmes suffer from a lack of attention to the three crucial components of teachers’ knowledge: mathematics content, episte‐mology, and pedagogy. As a result, they cannot achieve the desired quality in teachers as was envisioned by the current mathematics education leadership. Teachers’ mathematics knowledge is far from being satisfactory even in terms of the standards for high‐school mathematics. The work on epistemology and pedagogy is detached from a personal experiential basis of teaching, and thus it is in conflict with the well established principle that knowledge construction (and this includes mathematics knowledge as well as knowledge of mathematics epistemology and pedagogy) is a product of personal, experiential problem‐solving activity. The effort of teacher education programmes must centre on these three components of teachers’ knowledge base. In particular, teachers’ knowledge of mathematics should be promoted and evaluated in terms of mathematics values, not spe...

26 citations

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TL;DR: In this paper, the use of graphics calculators by students taking a traditional calculus test was studied by analysing the examination scripts of a representative sample of 37 first-semester students studying calculus.

Abstract: The use of graphics calculators by students taking a traditional calculus test was studied by analysing the examination scripts of a representative sample of 37 first‐semester students studying calculus. Analysis of students’ responses showed that, contrary to expectations, the calculator was under‐utilized by most students, both in terms of the amount and the way in which it was used. The capacity of the students to deal simultaneously with graphical and algebraic information from two independent sources, seemed to be the main obstacle to effective use. Finally, when students use a graphics calculator to help them answer questions in a traditional calculus examination, the examiner can often gain more insight into the students’ true level of understanding than if they did not use it.

22 citations

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TL;DR: In this paper, the authors link Pythagorean triples and recurrence relations in a way which reveals an underlying pattern of "arrowhead curves" in the form of an arrowhead curve.

Abstract: Pythagorean triples and recurrence relations occur in many parts of mathematics. This paper links them in a way which reveals an underlying pattern of ‘arrowhead curves’,

19 citations

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TL;DR: In this article, a stochastic model for two interacting species with a prey-predator relationship in the presence of self-interaction is considered, where the parameters describing the interactions have been perturbed by a wideband stationary stochastically process.

Abstract: In this paper a stochastic model for two interacting species with a prey‐predator relationship in the presence of self‐interaction is considered. The parameters describing the interactions have been perturbed by a wideband stationary stochastic process. Here we have discussed the stability behaviour of this model.

18 citations

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TL;DR: In this paper, the authors present an analytic proof of Young's inequality by the application of the mean value theorem for integrals known from a first course in real analysis and use the heuristic strategy of analogy, which is a constructive source of discovery.

Abstract: Our aim is to present a completed form of Young's inequality. We will give an elementary analytic proof of this inequality by the application of the mean value theorem for integrals known from a first course in real analysis. Moreover, to facilitate understanding, the heuristic strategy of analogy, which is a constructive source of discovery, will be used.

16 citations

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TL;DR: In this paper, the role of visualization of mathematical concepts as an influence on the processes of formation of conjectures and mathematical thinking is explored, with a specific theme in mathematics: polygonal numbers.

Abstract: In this study we explore the role of visualization of mathematical concepts as an influence on the processes of formation of conjectures and mathematical thinking. First, we look at some of the obstacles in a computer environment which arise from the type of environment employed, with relation to a specific theme in mathematics: polygonal numbers. Second, in another computer environment, using the same topic of mathematics, we consider the influence on the processes of mathematical thinking.

14 citations

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TL;DR: In this article, the authors consider the way that mathematics should be understood and propose that this understanding is essential and should be at least part of the grading process for all students in mathematics courses.

Abstract: We consider the way that mathematics should be understood. It is this understanding which is essential and which should be at least part of the grading.

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TL;DR: The formal teaching of fractions remains a well-established part of the elementary school mathematics curriculum, beginning in grade four, in spite of the continuing existence of pertinent questions regarding its value in comparison to other mathematics topics.

Abstract: The formal teaching of fractions remains a well‐established part of the elementary school mathematics curriculum, beginning in grade four, in spite of the continuing existence of a number of pertinent questions regarding its value in comparison to other mathematics topics. The low achievement rates among students in fractions knowledge and computation has led to some scepticism as to the validity of the defences made of fractions teaching. It does not appear that the current advocacy made for the extensive teaching of fractions to children is based on empirical evidence. This instruction, instead, seems to be perpetuated by the forces of tradition and logic, rather than by experimental data. Certain key issues regarding this instruction beg for a clearer resolution by researchers in mathematics education. In addition, an input by teachers as to the usefulness of fractions in children's everyday lives and in the learning of higher‐order mathematics should be given greater consideration. Investigations of t...

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TL;DR: In this article, the use of spreadsheets in introducing students to the concept of a limit of a sequence is demonstrated and the possible computer-based scenario as the enhancement of the teaching/learning process of calculus is exemplified.

Abstract: The use of spreadsheets in introducing students to the concept of a limit of a sequence is demonstrated and the possible computer‐based scenario as the enhancement of the teaching/learning process of calculus is exemplified. It is shown how the spreadsheet's operational capability assists visualizing the Bolzano‐Cauchy principle of convergence and leads eventually to the possibility of employing computer technology in deciding the convergence of positive series.

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TL;DR: This paper illustrates how electronic spreadsheet programs, such as Lotus 1‐2‐3 and Microsoft Excel, can be used for numerical analysis by considering the nature of spreadsheets with particular reference to their precision and accuracy.

Abstract: This paper illustrates how electronic spreadsheet programs, such as Lotus 1‐2‐3 and Microsoft Excel, can be used for numerical analysis. It begins by considering the nature of spreadsheets with particular reference to their precision and accuracy. It continues with an extended example taken from matrix manipulation which uses both the Jacobi and Gauss‐Seidel iterative methods for solving sets of simultaneous equations. The spreadsheet approach allows users to compare the details of how the two methods work. The paper concludes with a restatement of the advantages of spreadsheets for numerical analysis.

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TL;DR: A Markov chain process is applied to the major states through which one would proceed in order to effect the study of a given real system and through this measures for the difficulty of this study are obtained.

Abstract: We apply a Markov chain process to the major states through which one would proceed in order to effect the study of a given real system and through this we obtain measures for the difficulty of this study. The assumptions we make about the process of the study, although they are generally acceptable, are by no means standard and therefore proper modifications to our basic model may be necessary in several cases. Our model gives an important theoretical frame, but its application in practice depends upon the successful calculation of the transition probabilities involved in each particular case.

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TL;DR: In this paper, the authors present a simple but elegant computer verification method for the Goldbach conjecture, and propose two implementation approaches (via C and Maple) to this method, which is easy to implement and easy to understand.

Abstract: In class, students are often curious about the fact that any even integer bigger than 4 is the sum of two prime numbers. They usually cannot understand why the obvious fact cannot be proved mathematically. From the computational point of view, unless we find an even number which cannot be expressed as a sum of two prime numbers, we cannot claim the Goldbach conjecture is true. So, the most important thing is to try all the even integers to check if they conform to the Goldbach conjecture. Unfortunately, most of the modern verification methods need deep mathematical ideas which usually cannot be understood by undergraduate students. In this paper, we present a simple but elegant computer verification method for the Goldbach conjecture, and propose two implementation approaches (via C and Maple) to this method. Compared with other verification methods, ours is easy to implement and easy to understand. But of course, at present we may not expect this method, or probably any other existing methods, to be able...

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TL;DR: In this article, the authors present a simple unification of the Voigt functions K(x,y) and L(x and y) through Fourier and Mellin transforms resulting in connections with the error functions, the parabolic cylinder functions and, ultimately, the Whittaker functions.

Abstract: This paper aims to present a simple unification of the Voigt functions K(x,y) and L(x,y) through Fourier and Mellin transforms resulting in connections with the error functions, the parabolic cylinder functions and, ultimately, the Whittaker functions. Analogues of related functions are highlighted. The Voigt integrals will follow as natural consequences for analytical evaluations and uses.

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TL;DR: In this article, the authors obtained a theorem which generalizes some known radii results concerning linear combinations of analytic functions in some well-known classes studied in literature, and used it to obtain a theorem that generalizes the radii of linear combination of functions.

Abstract: The object of this paper is to obtain a theorem which generalizes some known radii results concerning linear combinations of analytic functions in some well‐known classes studied in literature.

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TL;DR: In this paper, a geometrical method and an analytical method of predicting the distribution of primes and non-primes based on visual information obtainable from a matrix map of divisibles is described.

Abstract: There are three famous unsolved mathematical problems in number theory, namely the theory of partitions, Fermat's 'Last Theorem', and the prime number theorem A geometrical method and an analytical method of predicting the distribution of primes and non-primes based on visual information obtainable from a matrix map of divisibles is described Past investigations tend to concentrate on properties of the prime numbers The author feels that much information could be gathered by studying the distribution of both prime and non-prime numbers using a matrix map Both methods give accurate, deterministic mathematical models of the distribution of primes and non-primes globally The only problem is that there is no end to the prime number series and thus the prime number theorem remains unsolved

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TL;DR: In this article, it is argued that many mathematical procedures are more easily and more visually represented on a spreadsheet than in a programming language, and three mathematical investigations are proposed which illustrate the transparent nature of a spreadsheet.

Abstract: Spreadsheets are essentially mathematical creations, nevertheless their use in mathematical education is very limited. Almost certainly students of mathematics spend more time using programming languages than spreadsheets. It is argued that many mathematical procedures are more easily and more visually represented on a spreadsheet than in a programming language. Three mathematical investigations are proposed which illustrate the transparent nature of a spreadsheet.

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TL;DR: In this article, a simple proof is given for the following inequality by the name of "covariance inequality": cov(f(X), g(X))≥0 The case of equality is investigated in detail, and a few applications regarding some inequalities, variance reduction and correlation are provided.

Abstract: Let X be a random variable and f(t), g(t) be two non‐decreasing real functions. A simple proof is given for the following inequality by the name of ‘covariance inequality”: cov(f(X), g(X))≥0 The case of equality is investigated in detail, and a few applications regarding some inequalities, variance reduction and correlation are provided. Finally, generalization of the inequality in different directions is discussed.

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TL;DR: In this paper, an international comparison of Hungarian and Finnish pupils' conceptions of mathematics teaching was made. But the Hungarian pupils placed greater emphasis on exact teaching methods which strive for understanding according to pupils' capabilities, they also stressed computational aspects of mathematics, such as rapid performance, correct answers, memorization of rules and a belief in proper procedures.

Abstract: This paper reports on an international comparison of seventh graders’ conceptions of mathematics teaching. A questionnaire survey was made of the conceptions of a total of 200 pupils in Hungary and Finland. The main results were, as follows: The Finnish pupils were more in favour of calculation‐centred working, where the teacher is always in control of the learning process than their Hungarian counterparts. The Hungarian, in contrast to the Finns, placed greater emphasis on exact teaching methods which strive for understanding according to pupils’ capabilities, they also stressed computational aspects of mathematics, such as rapid performance, correct answers, the memorization of rules and a belief in the existence of proper procedures.

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TL;DR: In this article, a didactic model with which we can teach new mathematical concepts is presented, which is a three-part approach with the following stages: conceptual modelling; abstraction and formalization of mathematical concepts; applied modelling.

Abstract: It is of our belief that modelling must constitute one of the essential components of teaching mathematics in secondary education And it is worth noting the fact, that while this process has attracted interest from at least the early 1980s, few researchers and educators believe in the teaching of mathematics through the modelling process Prompted by the problems we have spotted in the curriculum of Greek secondary education, we have developed modelling‐orientated teaching, a didactic model with which we can teach new mathematical concepts This model is a three‐part approach with the following stages: conceptual modelling; abstraction and formalization of mathematical concepts; applied modelling In this paper we also quote elements from the international discussion about the role and place of modelling in mathematical education, a discussion that still goes on In the last part we present some of the conclusions of wider research, in the context of which we have developed modelling‐orientated teaching

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TL;DR: In this paper, a further investigation of the subexponential series is presented, and relevant connections with substantially more general results, available already in the mathematical literature, are also pointed out.

Abstract: A further investigation of the so‐called sub‐exponential series which was considered recently by Chorlton and Clarke, and also Glaister, is presented. Relevant connections with substantially more general results, available already in the mathematical literature, are also pointed out.

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TL;DR: The purpose is to assist students in seeing uses and applications of max‐min principles in real life, and to improve reading comprehension and problem‐solving skills.

Abstract: The problem‐solving method with handout material is an instructional method, which includes handout material prepared by taking Polya's problem‐solving stages. The handout material consists of a short review of the theorems related to max‐min word problems, general procedure for solving max‐min word problems, comprehension guides, and problems with unfinished solution steps. The purpose is to assist students in seeing uses and applications of max‐min principles in real life, and to improve reading comprehension and problem‐solving skills.

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TL;DR: In this article, it is shown how geometric transforms (probability generating functions) can be used to study the expected number of tosses until HTHT...HT.

Abstract: It is shown how geometric transforms (probability generating functions) can be used to study the expected number of tosses until HTHT.. .HT.

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TL;DR: In this paper, the triangular numbers are generalized in a natural way and a discovery-type approach is used to find properties of these numbers, and generalized combinations that arise are connected to the Stirling numbers of the first kind.

Abstract: The triangular numbers are generalized in a natural way and a discovery‐type approach is used to find properties of these numbers. The numbers are connected to the Pascal numbers and generalized combinations that arise are connected to the Stirling numbers of the first kind.

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TL;DR: Four exemplary error analysis methods are presented here in a wider context, and their relevance to a range of quadrature methods, from the elementary Simpson's rule to the recent non‐interpolatory rule of Ehrenmark is considered.

Abstract: Methods for error analysis in numerical quadrature are often presented to students in a ‘one‐off style which seems to bear little relevance to the rest of the course, and relies on apparently analytic trickery for its success. Four exemplary error analysis methods are presented here in a wider context, and their relevance to a range of quadrature methods, from the elementary Simpson's rule to the recent non‐interpolatory rule of Ehrenmark, is considered, giving a general alternative to the special analysis of Ehrenmark which encompasses his case into the general framework.

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TL;DR: The rational canonical form theorem is one of the fundamental results in undergraduate linear algebra as discussed by the authors, and every author attempts to make the details easy to follow, which is the case in this paper.

Abstract: The rational canonical form theorem is one of the fundamental results in undergraduate linear algebra. As a result, every author attempts to make the details easy to follow. Presented here is an alternative proof in three short and relatively simple steps, based on a generalization of a proof of the Jordan canonical form theorem. To complete a short treatment of the rational canonical form theory, a proof of the uniqueness of the rational canonical form is also presented. Finally it is shown that the preceding techniques can also be used to generalize another short proof of the Jordan canonical form theorem by Filippov.

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TL;DR: The model is applied in various situations of teaching, but the main application is in the construction of a semantic distance suitable for analysing students’ difficulties with quadrilaterals.

Abstract: In this paper a model of semantic difference between various mathematical terms used in the classroom is built. As a basic tool in the construction of this model the concept of codification, which is a mapping σ of a space M of meaning into a set A of terms or expressions, is used. Given a set ? of codifications (which may belong to the teacher or to the pupils), a semantic distance μS is defined in ? by the formula: μS (s,t) = sup {d(x,y)|x, y?M with σ 1(x) = s, σ 2(y) = t for some σ 1, σ 2?S} where d is a distance function in M. In particular the ‘semantic width’ of a term t, WS (t) = μS (t, t), is generally different from zero. The model is applied in various situations of teaching, but the main application is in the construction of a semantic distance suitable for analysing students’ difficulties with quadrilaterals. In this respect a ‘Boolean’ representation of types of quadrilaterals is combined with a ‘tree‐like’ representation related to Aristotle's γ?vη, and as a result, a larger semantic space i...