In contrast with the formal, analytical strategies developed by professional intermediaries, information seekers also use a variety of informal, heuristic strategies. These informal, interactive strategies are clustered together under the term browsing strategies. In general, browsing is an approach to information seeking that is informal and opportunistic and depends heavily on the information environment. Four browsing strategies are distinguished in this chapter: scanning, observing, navigating, and monitoring. The term browsing reflects the general behavior that people exhibit as they seek information by using one of these strategies.

Browsing is a natural and effective approach to many types of information-seeking problems. It is natural because it coordinates human physical, emotive, and cognitive resources in the same way that humans monitor the physical world and search for physical objects. It can be effective because the environment and particularly human-created environments are generally organized and highly redundant – especially information environments that are designed according to organizational principles. Browsing is particularly effective for information problems that are ill defined or interdisciplinary and when the goal of information seeking is to gather overview information about a topic or to keep abreast of developments in a field.

Introduction

Our second numerical laboratory assignment is the computation of an area under a given curve to a desired accuracy. Unlike the calculation of the areas of various polygons, the computation of the area of a circle, an ellipse, or the area under the curve y = 1/log x from x = 2 to x = 7, say, is not at all trivial and requires methods of integral calculus or numerical approximations. Such areas arise not only in a geometrical context but also in various applications in engineering, biology, and statistics. In keeping with our policy of making the material as accessible as possible to precalculus students, without sacrificing rigor, we shall somewhat limit the generality so that results can be proved by elementary means and generalizations pointed out.

We shall henceforth be interested in the computation of the area under the graph of a positive function y = f(x), from x = a to x = b, but limit ourselves for the time being to monotonic (increasing or decreasing) or convex functions.

Rectangular approximations

Although the ensuing analysis is carried out in terms of a general, positive, monotonic function f(x), a ≤ x ≤ b, it is advisable that the laboratory participants bear in mind a concrete example such as f(x) = 1/x, 1 ≤ x ≤ 2.

Introduction

In the spirit of the mathematical laboratory, special attention is accorded to the elimination of traditional mathematical tables, which students have always used as “black boxes,” without the faintest understanding of their origin and construction. Thus, these tables were antieducational tools that the students aquired without any mathematical enlightenment and used as “cookbook recipes.” Now that calculators and microcomputers are used in mathematical education, the danger arises of replacing one set of black boxes with another. Of course, we are not advocating the introduction of these new electronic black boxes merely for using, say, the logarithmic built-in function of the computer (or pressing the “log” key on a pocket calculator). What we do advocate is to teach students what is behind such built-in functions as part of the material covered in the mathematical laboratory. This subject fits naturally into the environment of the laboratory and reveals the “story behind the key.” The attainment of this objective is the subject of this chapter and the next.

We might ask whether students should be allowed to use the built-in functions before (and during) learning how they were built in. We feel that no harm can result from such a practice, so long as the students are told expressly that their “ignorant” use of the built-in functions is temporary. Before long, the secrets held by the computer keys will be revealed.

Built-in functions

The term computer library functions refers to the collection of built-in functions – sin x, In x, ex, arctan x, to name but a few – that were installed in the computer's permanent memory. These built-in functions, of course, are efficient approximations of the abstract mathematical entities they represent. By efficiency we mean that every evaluation is performed with utmost speed and yields all the correct significant figures that are available on the computing device used. The construction of such built-in functions usually entails lengthy, computationally expensive preparations, which, however, are carried out only once. The first preparatory step is to reduce, as much as possible, the interval [a, b] in which the given f(x) is to be approximated (examples of this strategy can be found in Sections 2.2 and 7.6). To guarantee the desired correct significant figures, we also must control the relative error in the approximation of f(x) in [a, b]. This issue will be discussed in Section 8.3.

We shall concentrate on polynomial approximations, making use of the results obtained in Chapter 7, and shall consider the possibility of constructing rational approximations in Sections 8.5 and 8.6. We start with polynomial approximation of trigonometric functions in the same spirit with which we treated In x in Chapter 7.

The mathematical laboratory

Accumulated experience has shown that early emphasis on algorithmic thinking, augmented by actual computing, is indispensable in mathematical education. Recognizing the cardinal importance of the individual, active involvement of every student in the computational work (as opposed to mere demonstration by the teacher), we advocate the use of mathematical laboratories equipped with microcomputers. Optimally, a special room should be set aside for the mathematical laboratory. Failing that, physics or biology laboratories can be used since they tend to create the proper atmosphere. A pair of students is assigned to each microcomputer, as to a microscope in a biology laboratory, and spends a few hours a week working with the microcomputer in the laboratory.

The mere presence of an increasing number of microcomputers in various educational institutions, even those at which a programming language such as True-Basic or Pascal is taught, in no way constitutes a new mode of teaching and learning. The full potential of microcomputers and proper courseware should be harnessed to improve the state of the art in education. Moreover, a new role will be played by the mathematics teacher when traditional “chalk-and-talk” methods are augmented by active participation as a laboratory instructor. The numerous advantages of such computer-aided teaching of mathematics are detailed in Section 1.2.

The laboratory work will center around specific assignments, or modules, to be carried out by the participants at their own pace.

Numerical Mathematics – A Laboratory Approach is a unique book that introduces the computational microcomputer laboratory as a vehicle for teaching algorithmic aspects of mathematics. This is achieved through a sequence of laboratory assignments, presupposing no previous knowledge of calculus or linear algebra, where the “chalk and talk” lecturer turns into a laboratory instructor. The computational assignments cover basic numerical topics that should be part of the mathematical education in the era of microcomputers.

In writing this book at the precalculus and pre–linear algebra level, we were mainly addressing an audience of four groups: first-year university students of mathematics, sciences, and engineering who have had no exposure to systematic calculus; students at teachers' training colleges who will be tomorrow's teachers of mathematics and computer science; superior high-school mathematics students; and scientific programmers at all levels. Various parts of this book were successfully tested on classes representative of each of these groups and subsequently modified. The material was received enthusiastically by high-school students who were members of Tel Aviv University's Math Club, some of whom are now faculty members of the School of Mathematical Sciences. The material was also welcomed by members of New York University's summer program for talented high-school students (held every summer at the Courant Institute of Mathematical Sciences and directed by Henry Mullish), and by several classes of in-service or future mathematics teachers at Tel Aviv University and at New York University.

Introduction

In this chapter we present an algorithmic approach to the solution of systems of linear equations, another typical subject for the mathematical laboratory. No knowledge of matrices, vectors, and their underlying theory is presupposed, and thus the laboratory participants can handle this material even before the study of linear algebra.

After the development of an algorithm for the solution of “naive” systems of linear equations, special attention will be paid to problematic cases in which unrealistic answers with huge errors might be obtained. In particular, we shall discuss reasons for loss of accuracy, sensitivity to minor changes in the data, pivoting, scaling, and computational efficiency. By elaborating on each of these points by means of appropriate examples, we hope to present this traditionally abstract mathematical subject in a concrete, practical way that will be more meaningful to many students.

Coefficient tables

Systems of linear equations arise naturally in many practical areas such as mixing liquids, work and power calculations, electrical circuit computations, and marketing problems. It is particularly useful to demonstrate the subject under consideration by means of 3 × 3 systems (three equations and three unknowns). Such systems are not too large and cumbersome, but nevertheless constitute a case in which a pattern is revealed. Occasionally, when it is necessary for clarity, 4 × 4 and 2 × 2 systems will also be used.

In this chapter, I consider the potential and actual reuse opportunities within UNIX. First, several methods are suggested that could increase the likelihood that the next submission matches an item in a small set of predictions offered to the user for review and reuse. All methods are applied to the UNIX traces, and the predictive “quality” of each method is measured and contrasted against the others. In the second part of the chapter, I investigate how well the reuse facilities supplied by the UNIX shell are used in practice.

Conditioning the distribution

In the last chapter, particular attention was paid to the recurrence of command lines during csh use, and to the probability distribution of the next line given a sequential history list of previous ones. We saw that the most striking feature of the collected statistics is the tremendous potential for a historical reuse facility: the recurrence rate is high and the last few submissions are the likeliest to be repeated.

One may predict what the user will do next by looking at those recent submissions. But there is still room for improvement, because a significant portion of recurrences are not recent submissions. Can better predictions of the user's next step be offered? This section proposes and evaluates alternative models of arranging a user's command line history that will condition the distribution in different ways.

The recurrence distributions of Section 5.4.2 were derived by considering all input for a user as one long sequential stream, with no barriers placed between sessions.

Humans are the most versatile of creatures, and computers are their most versatile of creations. Human–Computer Interaction (HCI) is the study of what they do together; in particular, HCI aims to make interaction better suit the humans. Computers contribute to art, science, engineering, … all areas of human endeavor. It is no surprise, then, that there is heated debate about what the essence of HCI is and what it should be. What is good HCI? The answer to this question will be elusive given that there is good engineering that is not art, good art that is not science, and good science that is not engineering.

It's easier to see what form of answer there can be by taking a quick excursion into another field. Imagine the discovery of a dye, such as W. H. Perkin's breakthrough discovery of mauve. Is it science? Yes: certain chemicals must react to produce the dyestuff, and the principles of chemistry suggest other possibilities. Is it art? Yes: it makes an attractive color. Is it engineering? Yes: its quantity production, fastness in materials, and so forth, are engineering. Perkin's work made the once royal purple accessible to all. Fortunately there is no subject “Human Chemical Interaction” to slide us into thinking that there is, or should be, one right view of the work of making or using, designing, standardizing, or evaluating a dye. Nevertheless, we appreciate a readily available, stunning color, used by an able artist, and one that lasts without deteriorating.