Abstract:

Textbooks in all fields of study have changed very little over the years and decades past. They remain static treaties of topics that authors prescribe with as large an audience as possible. This is particularly the case in Mathematics and science where the active participation of the learner is crucial. In this paper, the author explores new solutions to these old problems. One of the insights revealed by contemporary research in cognitive psychology is that mathematical and scientific concepts are distinctly personal. These concepts are constructed by each student in terms of her or his own background, rather than transferred as a finished body of knowledge from instructor to student. This means that the curriculum must be designed with tools to support the learner's own construction of learning.

The ever advancing and increasing power of computational technology offers unique opportunities to design such a curriculum. In light of these advances, genuine teaching/learning environments that integrate important ideas, such as participation, visualization, and integration, are possible.
Participation refers to the active involvement of students in the construction of their own knowledge. This process recapitulates the development of certain scientific ideas by formulating key problems in ordinary language, and encouraging students, through class discussion and computer exploration, to discover what is needed to refine that language and thereby to reconstruct the appropriate formalisms. Participation, in this sense, puts students more in control of the pace and, to some extent, the direction of their learning. It is obvious that students are better prepared to understand the answers to the questions that they themselves ask. I encourage students to experiment and to ask their own questions. In order for this to work, a delicate balance has to be struck. It is necessary to give students the tools they need to formulate and ask their questions without over-structuring the activity so that their involvement becomes superfluous.
Visualization refers to the formation of stable and coherent mental pictures of abstract constructions and processes in mathematics and science. The ability to visualize constructions and processes requires practice and experience. The best students enter their courses with refined and exercised habits of visualization, and these students usually progress smoothly and quickly. But many students are forced to spend much of their time "discovering" the right pictures. And with the accelerated pace of many introductory courses, this often frustrates them. Instructional technology with its heuristic tools can provide real support for visualization.
Integration is the process by which students build bridges connecting principal ideas of different disciplines, and place them into a unified conceptual context. So it has to do with synthesis and organization.
The author explores these ideas through a number of demonstrations of interactive exploration in basic as well as advanced topics in mathematics and science.

Keywords:

Active Learning, Interactive, Exploration, Mathematics, Science, Instructional Technology.