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The well attended March 1994 HIse workshop in Amsterdam was a very lively con­ ference which stimulated much discussion and human-human interaction. As the editor of this volume points out, the Amsterdam meeting was just part of a year-long project that brought many people together from many parts of the world. The value of the effort was not only in generating new ideas, but in making people aware of work that has gone on on many fronts in using computers to make mathematics more understandable. The author was very glad he attended the workshop. * In thinking back over the conference and in reading the papers in this collection, the author feels there are perhaps four major conclusions to be drawn from the current state of work: 1. graphics is very important, but such features should be made as easy to use as possible; 2. symbolic mathematical computation is very powerful, but the user must be able to see "intermediate steps"; 3. system design has made much progress, but for semester-long coursework and book-length productions we need more tools to help composition and navigation; 4. monolithic systems are perhaps not the best direction for the future, as different users have different needs and may have to link together many kinds of tools. The editor of this volume and the authors of the papers presented here have also reached and documented similar conclusions.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
The goal of the project Human Interaction in Symbolic Computing (HISC) which took place in 1994–1995 at the Research Institute for Applications of Computer Algebra (RIACA) in Amsterdam was to investigate a variety of techniques and paradigms which could lead to better user interfaces to symbolic-computation systems (current and future).
Norbert Kajler

The ACELA project: aims and plans

Abstract
The most visible aim of the ACELA (architecture of a computer environment for Lie algebras) project is the production of a state-of-the-art interactive book on Lie algebras; state-of-the-art mathematically as well as in its interactive potential. While we have chosen this as a worthwhile and challenging goal by itself, this target also serves as a concrete milestone for our longer-term aims, offering a realistic and far from trivial testing ground for our ideas.
Arjeh M. Cohen, Lambert Meertens

Active structured documents as user interfaces

Abstract
Mathematicians manipulate complex abstract objects and expect some help from the computer in this task. A number of systems have been developed for that purpose. The early developments focused on methods and algorithms for numerical and symbolic computations, without paying too much attention to the user interface of systems using these algorithms. Other tools have been developed for helping computer users to prepare mathematical documents. This trend is illustrated by the famous TEX system (Knuth 1984) that most mathematicians use nowadays. Here again, the emphasis was put on the algorithms and on the quality of the result, but the language provided to the user is not very user-friendly, although very powerful.
Vincent Quint, Irène Vatton, Jean Paoli

Direct manipulation in a mathematics user interface

Abstract
The user interface problems of existing mathematics systems are well known and are discussed in detail elsewhere (see, e.g., Kajler and Soiffer 1998).
Ron Avitzur

Successful pedagogical applications of symbolic computation

Abstract
At the Education Program for Gifted Youth (EPGY) we have developed a series of stand-alone, multi-media computer-based courses designed to teach advanced students mathematics at the secondary-school and college level. The EPGY course software has been designed to be used in those settings where a regular class cannot be offered, either because of an insufficient number of students to take the course or the absence of a qualified instructor to teach the course. In this way it differs from traditional applications of computers in education, most of which are intended to be used primarily as supplements and in conjunction with a human teacher.
Raymond Ravaglia, Theodore Alper, Marianna Rozenfeld, Patrick Suppes

Algorithm animation with Agat

Abstract
Algorithm animation is a powerful tool for exploring a program’s behavior. It is used in various areas of computer science, such as teaching (Rasala et al. 1994), design and analysis of algorithms (Bentley and Kernighan 1991), performance tuning (Duisberg 1986). Algorithm animation systems provide a form of program visualization that deals with dynamic graphical displays of a program’s operations. They offer many facilities for users to view and interact with an animated display of an algorithm, by providing ways to control through multiple views the data given to algorithms and their execution.
Olivier Arsac, Stéphane Dalmas, Marc Gaëtano

Hypermedia learning environment for mathematical sciences

Abstract
Computers play an essential role in research and education in applied mathematics and the natural and technical sciences. Graphical interfaces have made it easier to use computers, so that nowadays many educational and research problems can be conveniently solved with existing mathematical software and hardware. Graphical object-oriented programming environments such as HyperCard (Apple Computer 1987) and ToolBook (Asymetrix 1991) have made it possible to easily integrate text, graphics, animations, mathematical programs, digitized videos and sound into hypermedia (Ambron and Hooper 1990, Jonassen and Mandl 1990, Jonassen and Grabinger 1990, Kalaja et al. 1991, Nielsen 1990). Typically, hypermedia programs contain large amounts of data. Fortunately, these can be put on CD-ROMs.
Seppo Pohjolainen, Jari Multisilta, Kostadin Antchev

Chains of recurrences for functions of two variables and their application to surface plotting

Abstract
When generating curves or surfaces of closed-form mathematical functions, usually the most time-consuming task is function evaluation at discrete points. Most programs (among them most of the existing computer algebra systems) achieve this by straightforward evaluations of linearly sampled points through whatever numerical evaluation routines the particular system provides. More specifically, most programs use evaluations of the following form:
$$G\left( {{{x}_{0}} + n{{h}_{x}},{{y}_{0}} + m{{h}_{y}}} \right) for all n = 0, \ldots ,N,m = 0, \ldots ,M $$
for some given two-dimensional function G(x, y), starting points x 0, y 0 and increments h x, h y. For example, the following loop is used inside Maple’s plot 3d function (Char et al. 1988):
$$\begin{gathered} xinc : = \left( {xmax - xmin} \right)/m; yinc : = \left( {ymax - ymin} \right)/n; x: = xmin; \hfill \\ for i from 0 to m do \hfill \\ y : = ymin; \hfill \\ for j from 0 to n do z\left[ {i,j} \right] : = f\left( {x,y} \right); y : = y + yinc od; \hfill \\ x : = x + xinc \hfill \\ od; \hfill \\ \end{gathered} $$
Olaf Bachmann

Design principles of Mathpert: software to support education in algebra and calculus

Abstract
This paper lists eight design criteria that must be met if we are to provide successful computer support for education in algebra, trigonometry, and calculus. It also describes Mathpert, a piece of software that was built with these criteria in mind. The description given here is intended for designers of other software, for designers of new teaching materials and curricula utilizing mathematical software, and for professors interested in using such software. The design principles in question involve both the user interface and the internal operation of the software. For example, three important principles are cognitive fidelity, the glass box principle, and the correctness principle. After an overview of design principles, we discuss the design of Mathpert in the light of these principles, showing how the main lines of the design were determined by these principles. (The scope of this paper is strictly limited to an exposition of the design principles and their application to Mathpert. I shall not attempt to review projects other than Mathpert in the light of these design principles.)
Michael Beeson

Computation and images in combinatorics

Abstract
Combinatorics has always been concerned with images and drawings because they give interpretations of enumeration formulae leading to simple proofs of these formulae, and sometimes they are themselves central to the problem. Even if some small example drawings do not contain all elements of the proof, they are often useful to guide the intuition.
Maylis Delest, Jean-Marc Fédou, Guy Melançon, Nadine Rouillon

Erratum to: Design principles of Mathpert: software to support education in algebra and calculus

Without Abstract
Olivier Arsac, Stéphane Dalmas, Marc Gaëtano

Erratum to: Algorithm animation with Agat

Without Abstract
Michael Beeson

Backmatter

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