Iterative Algebra and Dynamic Modeling

A Curriculum for the Third Millennium

Preface

This book is aimed at high school and college students who have developed facility in pre-calculus mathematics and are now interested in exploring the interaction of algebra with computer technology. It is also aimed at teachers who seek to enlarge, enrich, and apply the curriculum they teach by reconciling it with the technology that is having such profound impact on our lives and civilization. And finally, it is aimed at "miscellaneous readers" who may have completed their study of algebra some years ago but remain curious about the interaction of computer technology with the world of mathematics.

The book's medium for linking mathematics and technology is a spreadsheet program called Excel.® However, the procedures we describe in Excel can be implemented by any medium that facilitates "iteration." Such media include graphing calculators and simple programming languages, as well as other spreadsheet programs. Our use of Excel is based on a desire to provide a transparent programming tool and convenient graphical interface - while still challenging the reader to demonstrate a mathematical understanding of the procedures being used. The book's final chapter also makes use of a simulation package called Stella® that enables the user to formulate iterative procedures in an icon-based format.

The format of the book is geared toward "active learning." Most of the Figures are based on spreadsheet programs that the reader is asked to construct in Excel, or in some other medium that provides graphical representations of data. Many of the spreadsheets in these Figures include cells with dark boundaries. These cells contain numerical values (parameters) that can be changed - whereupon Excel recalculates its outputs and re-draws the corresponding graphs. Teachers are urged to consider this as a way of injecting a dynamic quality into their students' mathematical thinking and learning.

Intermingled with the text are "transitional problems" that call on the reader to fill gaps in the book's presentation. Some of these problems address mathematical techniques central to the ideas being developed. Each chapter ends with a collection of "Problems, Exercises, and Projects" that address related mathematical topics and applications at a range of levels. For readers familiar with calculus, there are occasional starred problems that set forth connections between iterative algebra and calculus-based techniques.

The goal of the book is to enrich the standard algebra curriculum by taking account of the computer's capacity for rapid and reliable iteration - thereby bringing the subject of al-jabr into the computer age. To reinforce the notion that our conception of algebra (and indeed of mathematics) is subject to change, Chapter I includes some discussion of algebra's roots in Babylonian reckoning, Greek geometry, and the Arabic formulation of al-jabr. Against this background, we go on to develop a broad range of mathematical topics and applications that have traditionally been regarded as "calculus-based."

Among the reasons for making such tools and applications accessible at the high school level is their connection to issues of "global change." Accordingly, many applications are drawn from ecology, economics, demography, and related subjects. The models so developed are intended to place contemporary issues concerning the global system into an analytical framework, one in which the issues confronting our society and civilization may take on a more coherent form.

Return to IADM Main Page.

©1999