摘要:

At some level most of us teach or use algebra, and for many teachers it has become second nature to such an extent that we know what to do and do it, possibly in our own way, using a set of axioms and rules with which we are familiar. We have done what all good students do—we have made the work our own. Students may face slightly different axioms from different teachers but these broadly amount to the same once the work is known. For students unfamiliar with the ordinary algebraic manipulations some of the slight variations from one teacher to another can become a separate mystique, but more importantly students need to develop their own way of doing algebra. In this article we propose a project for secondary teachers in training designed to introduce them to a ‘strange’ algebra in which they may find themselves, initially strangers but which has the same ‘old’ laws with which they are familiar. The new algebra absorbs integration and differentiation operators as elements in the field and so can be used to solve some differential and integral equations. Having said all this about a new algebra we note that those familiar with the work of Mikusinski [1] and Erdelyi [2] will readily see how these systems are related to this algebra. A deeper underpinning of this algebra depends on results in distribution theory [3].

ISSN:0020-739X    

DOI:10.1080/0020739920230101

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

The paper compares several approaches usually utilized by students to obtain a value for the acceleration of a uniformly accelerated motion starting from experimental data. Two elementary methods are studied and compared with the least squares method.

ISSN:0020-739X    

DOI:10.1080/0020739920230102

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In a recent issue of this journal, Proll presented a graph partitioning problem to illustrate how tighter formulations of integer linear programs (ILP) result in reduced computational effort. This paper presents a large number of other problems where tighter formulations of the associated ILPs lead to similar benefits thus providing a more comprehensive framework for discussing ILP formulations in industrial engineering and mathematical modelling courses.

ISSN:0020-739X    

DOI:10.1080/0020739920230103

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

We start with an innocuous enough looking puzzle, and indicate its myriad of solutions, covering en route most of the ideas of elementary probability. A main message is that meaningful answers in statistics can only result from explicit reference to reasonable assumptions, and the tendency to isolate models from reality (arising from an abstract mathematical tradition) threatens sterility in the discipline. The paper completes with the authors letting their academic hair down, and generalizing the simple example to involve advanced material in an elementary manner.

ISSN:0020-739X    

DOI:10.1080/0020739920230104

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This research study investigates the concept of function developed by students studying English A‐level mathematics. It shows that, while students may be able to use functions in their practical mathematics, their grasp of the theoretical nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop prototypes for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, is not stressed and proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sin*, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. Investigations reveal significant misconceptions. For example, threequarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and threequarters thought that a circle is a function. This reveals a wide gulf between the concepts as perceived to be taught and as actually learned by the students.

ISSN:0020-739X    

DOI:10.1080/0020739920230105

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This paper presents the derivation and the physical meaning of the general fourth‐order linear differential equation (with sectionally continuous derivatives) of the deflection curve and its general formulation and solution as a multipoint boundary value problem. An algorithm is presented in which shearing forces, bending moments, deflections, critical load and natural frequency are calculated for (non‐)uniform beams and columns, with arbitrary lateral and axial, (dis)continuous distributed load functions and concentrated loads, on (intermediate) (elastic) supports.

ISSN:0020-739X    

DOI:10.1080/0020739920230106

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

This paper reports on the attitudes of 901 primary school teachers in England and Wales towards the new national curriculum in mathematics, as well as English and science. It was found that most teachers perceive themselves to be competent to teach this new curriculum in mathematics (and English) but not in science. It is argued that this is because the national curriculum in mathematics represents a codification of existing content and practice. A comparison with a text widely used in the mathematical preparation of primary school teachers provides supporting evidence for this view. With the exception of computer‐based mathematics and increased attention to probability and statistics (data handling), the national curriculum in mathematics reflects the curriculum which has been taught in most primary schools for two decades.

ISSN:0020-739X    

DOI:10.1080/0020739920230107

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

After a first playful encounter with fractal geometry I realized very quickly that the notion of dimension is very important within this discipline. Looking more precisely I noticed that this notion is, moreover, extremely difficult.

ISSN:0020-739X    

DOI:10.1080/0020739920230108

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

In this paper we consider a vehicle routeing problem in which trucks are assigned streets from which to collect house refuse. We discuss the construction of simple networks which aid in the solution of the garbage collection problem. After the construction of the networks, we consider two solution procedures.arabicThe vehicle routeing problem is formulated as an integer linear programming problem, ILP. The ILP formulation is appropriate here because the decision for a particular truck to collect or not to collect house refuse along a particular street is captured by use of binary variables which can take the values zero or one. In this case we use the computer package VINO1(visual interactive nonlinear optimization) to solve the ILP. The VINO package uses Lotus 1‐2‐3 or Symphony as an interface (i.e. input and output medium).arabicWe use graph theory results to solve the vehicle routeing problem. In particular we use the result that a graph is Eulerian (i.e. each arc of the graph can be traversed exactly once) if and only if each node (vertex) has an even degree (the degree of a node is the number of arcs incident with it) to determine the minimum total distance that all the trucks must travel in order to collect all the house refuse. This value and the Eulerian result is then used in constructing the routes for each truck.

ISSN:0020-739X    

DOI:10.1080/0020739920230109

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor

摘要:

How was it that the physical science emerging in the seventeenth century ‘fitted’ so conveniently into the form of traditional algebra? Greek and early modern numerical and algebraic concepts, embodied in their respective notations, call for examination. The concept of number replacing that of the Greeks, which is at the base of traditional modern algebraic concepts, is whole and self‐determining, in contrast to the variegated wholeness and monodic dependence of the latter.

ISSN:0020-739X    

DOI:10.1080/0020739920230110

出版商:Taylor & Francis Group    

年代:1992

数据来源: Taylor