The core of the ordinary work of the research mathematician consists essentially in experimenting, conjecturing and proving. The main task of the one who tries to learn what mathematics is and how to become in any degree a mathematician should be to look at the mathematician's way to act when he or she is at work and to try to imitate and follow the same steps. Therefore teaching, and learning, mathematics should be essentially a continuously interactive occupation between the three main elements involved: teacher, learner, mathematics.

Experimentation in mathematics has become, through the presence of the recent technological instruments something very different, much easier, more interesting and more fruitful than it never before has been. One can experiment numerically, algebraically, visually,... with such an ease as it was never dreamed of a few years ago. And the possibilities of experimentation are improving very quickly and will continue doing so in the future. Our abilities to conjecture the presence of very deep and rich mathematical interrrelations are in this way becoming greatly expanded through the simple exercise of improving and and refining more and more our experiments until we can be practically sure that the theorems we guess through our repeated experiments have to be true.

The capabilities of the tools we now have in order to truly prove such guesses are also becoming more powerful. Calculations and analytical and algebraic manipulations that could have taken many hours, and even years some time ago are now done in seconds. The possibility to hand over to different programs even the task of proving the conjetures at which we arrive through our experiments is a reality in some disciplines and soon will be expanded to many others.

For this reasons it is quite likely that in a rather near future a great part of the mathematical activity in research, teaching and learnig, around any particular mathematical topic, will consist of the following tasks, in every step supported by the tools we now have at our disposal:

1. to design with imagination, and guided by the wisdom gathered by us or by others, experiments that can direct us in the exploration of the subject we are considering
2. to try to make some initial guesses about the mathematical relationships that lye perhaps hidden under our experiments, our calculations, our graphical images, under the structures of different types we can discover
3. to confirm or to refute our conjectures, and to perfect them through more refined experiments, now more accurately guided through the previous explorations
4. to try to prove our final conjectures, perhaps by means of some type of automatic method

Just about everyone who has used a computer algebra system to teach or do mathematics has, on occasion, been surprised and frustrated at the "simplified" results returned by the system. For this reason it is important to have a basic understanding of what is "under-the-hood" of the system you are using. Such knowledge makes it easier to obtain the results you want, and to understand those you don't.

As one of the principal authors of the Derive© computer algebra system and its predecessor, muMATH?, I would like to pass on some of the insights I have gained over the last 25 years implementing these systems. I will discuss the evolution of the principles and methodologies Derive uses to automate the simplification of mathematical expressions. Among the topics covered will be the definition and universality of simplification; the special difficulties presented by the calculus functions, infinity, and its inverse; the three phases of the simplification process (parsing, expansion, and reduction); the relative merits of top-down vs. bottom-up simplification; and that most elusive goal: the commutivity of simplification and substitution.

Computer algebra systems are powerful tools, but they are no substitute for understanding the mathematics underlying the problem you are trying to solve. In short, if you are using Derive, or any computer algebra system, results should be viewed with a healthy skepticism and verified if at all possible.

New technologies are rapidly changing our ways of communication and the art of teaching as well as extending ways of learning. Communication is an essential part of mathematics education and mathematics teacher education. Communications through technology can support the goals of mathematics education courses and programs, and can promote the professional development of participating teachers and educators. This talk is about:

My experience about Internet-supported lessons for mathematics teacher students; The development of Internet-based materials which can be used beside lessons and seminars in mathematics education and mathematics teacher education; The creation of a "virtual private network" between the university and a punch of schools and the intended collaboration between teachers and researchers. .

© 2001   ACDCA - Austrian Center for Didactics of Computer Algebra