摘要:

Incorporating multiple representations enhanced through technology is a theme in reformed mathematics curricula. While the need for students to use and reason on multiple representations is widely accepted, the research‐based knowledge on how to best accomplish this goal is just beginning to emerge. Current cognitive theories indicate that students’ knowledge of and preference for various representations is crucial to the decision process. This project develops and validates an instrument for determining student preferences for multiple representations of functions and establishes base‐line data for future research. The results indicate that students do have preferences for representations of functions and that these preferences vary between contextualized and non‐contextualized settings. Furthermore, students’ preferences between these two settings are more similar for students in a graphics calculator‐based calculus course than for students in a traditional calculus course. If a student has a cognitive preference for a representation, it is likely that the strongest connections between representations and between concept and representation are constructed to and from the student's preferred representation. Thus, attempts to situate the development of mathematics in context and attempts to enable students to approach mathematics as a sense‐making activity may do well to build from students’ preferences for representations.

ISSN:0020-739X    

DOI:10.1080/0020739980290101

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

ISSN:0020-739X    

DOI:10.1080/0020739980290102

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Most mathematicians, responsible for teaching solution methods for partial differential equations, would claim that an exact solution is always preferable to a numerical solution. This paper throws doubt on this view and shows, by considering a particular example, that the distinction between an exact solution and a numerical solution to a partial differential equation is not as clear as is often claimed. We demonstrate this by obtaining the exact solution for the torsion problem for a rectangular corner section. It appears the ‘exact’ solution cannot be written in closed form but necessarily involves the solution of an infinite system of linear equations which, except in certain special cases, can only be solved numerically. Thus, computing the exact solution can involve as much numerical effort as in computing the numerical solution. However, the exact solution contains algebraic ‘forms’ not present in the numerical solution which can be used to analyse special cases. This solution, which will be of particular interest to those working in the area of solid mechanics, is described in detail. The solution technique will attract the interest of those looking for non‐trivial applications of the use of Fourier series in the solution of linear partial differential equations.

ISSN:0020-739X    

DOI:10.1080/0020739980290103

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

This paper provides an insight into the levels of understanding of probability concepts of Malaysian secondary school pupils and the relationship among their probabilistic understanding and the independent variables of sex, form and mathematical ability. Information related to the levels of understanding of probability concepts is essential to curriculum design in this branch of mathematics. A study was conducted on a representative sample of 1266 form four and six (lower) pupils from fifteen secondary schools located in the suburban district of Klang, Selangor Darul Ehsan, Malaysia. A probability concepts test was administered and data collected and analysed to obtain factual information regarding the understanding of probability concepts among Malaysian pupils. This paper discusses the performance of pupils on intuitive and taught probabilistic concepts, describes the establishment of a probability concepts hierarchy and considers probabilistic understanding in relation to the independent variables sex, form and mathematical ability. A scale compromising three levels was set up. The results of the distribution of levels showed that generally Malaysian pupils were at Level 2. The study also revealed that boys, form six pupils and pupils with a higher mathematical ability performed better than their counterparts. Finally, several implications for educational practice are suggested.

ISSN:0020-739X    

DOI:10.1080/0020739980290104

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Three straightforward methods of solution of a simple hypersingular integral equation are presented. While two of the methods, presented here, rely on the inversion of Cauchy‐type singular integral operators, the third method uses the inversion of a simple Abel's operator with a weak singularity. All the three methods lead to the known form of the solution of the hypersingular integral equation considered.

ISSN:0020-739X    

DOI:10.1080/0020739980290105

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The objectives of the study were to (1) develop an instrument for measuring the mathematical thinking of high school students and (2) determine the effects of the class level and student's gender on the mathematical thinking of high school students in Nebraska. The subjects of the study were 239 high school students (grades 9‐12) selected from 18 Central Nebraska high schools participated in the study. The mathematical thinking instrument was developed by the researcher and administered to all subjects of the study in April 1994. Analyses of data collected indicated that significant differences in mathematical thinking of high school students existed in favour of the higher class level in all cases with the exception of grades 11 and 12. The mean score of the 11th grade students was significantly higher than that of the 12th grade students. In addition, no significant differences were found between the mean scores of male and female high school students at any class level or on any of the six characteristics of mathematical thinking dealt with in this study. The researcher recommends that more studies dealing with mathematical thinking be conducted to improve the measuring instrument in order to employ it for the discovery of students’ specific weaknesses in the development of their mathematical thinking. These studies will provide educators with guidance for developing effective curricula, methods and materials to remedy such weaknesses.

ISSN:0020-739X    

DOI:10.1080/0020739980290106

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Just as in the common triangle, solving the trihedral angle means getting some of the data and inferring the rest mathematically. Most such problems are solved by solid geometry and by spherical trigonometry. Here we present an approach which facilitates the handling of such problems. Moreover, the two basic theorems (namely, the law of sines and the law of cosines) of spherical trigonometry are derived as relatively simple by‐products of the general approach. Spherical trigonometry is a simple and vivid example of non‐Euclidian geometry and also a vital tool in navigation, astronomy and geodesy, etc. So it is worthwhile to consider it in the early stages of mathematical education.

ISSN:0020-739X    

DOI:10.1080/0020739980290107

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The first purpose of this paper is to describe different series which converge at varying speeds to Euler's constant, 7. Limits are set for the errors of approximation in these series. Our second aim is to illustrate the scope for numerical exploration of these and similar results by means of graphical calculators, spreadsheets or other computer programs. Such facilities are now increasingly easy to use and widely available to mathematics students. Exercises or project work based on this material serve to motivate the ideas of limits, size of error, order of convergence and methods of numerical approximation, and are a valuable means of developing students’ skills of investigation and analysis.

ISSN:0020-739X    

DOI:10.1080/0020739980290108

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

We give an intuitive physical explanation showing how the Neumann and Dirichlet boundary‐value problems for Laplace's equation can be rewritten as integral equations by the use of single‐ and double‐layer boundary potentials. The ideas presented here are representative of some of the basic ideas of potential theory.

ISSN:0020-739X    

DOI:10.1080/0020739980290109

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The monitoring of the errors resulting from the solution of a set of linear algebraic equations is conventionally achieved using either the mathematical standard perturbation theory for ill‐conditioning or using the computer studies route through interval arithmetic. In this paper an alternative is proposed based on continuous monitoring of the errors in each operation, and the three methods are compared. The comparisons address the issues of how reliable the error estimates are with increasingly ill‐conditioned equations, and how much extra workload is involved in the error computations. Cases where such monitoring can be misleading are discussed with one very poignant example based on Gauss‐Legendre quadrature illustrating this argument.

ISSN:0020-739X    

DOI:10.1080/0020739980290110

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor