It is clear that technology can enhance the learning environment in quite different ways. Observations I have made in my own classrooms, as well as student comments in their evaluations and in taped discussions, have led me to draw the following conclusions. These statements should be regarded, however, as conjectures that need further investigation by the mathematics profession as we move into classrooms that are more and more technologically sophisticated.
- For new material, technology can help students master concepts and formalism simultaneously.
- For material that was previously learned intuitively and informally, technology can bridge the gap between intuition and formalism.
- For material that has been learned well at an elementary level, technology can help students explore more advanced material creatively and independently.
There are themes that ran through all the courses, and I can only repeat here what has been discussed at length in our profession. Technology allows students to investigate examples more easily and enhances visualization. Instead of working through one example with paper and pencil in the course of an hour, the student can generate six or eight with the computer -- and with the dynamic geometry software, thousands. Patterns can emerge from the examples. Students are much more able to see the concepts behind the formalism and the theory.
Primarily, however, the lab experiences changed the dynamics of the courses. A carefully constructed syllabus became a hindrance for each course -- the unpredictability of the lab experience meant that I had to be prepared to discard my lesson plans for the day and respond to their comments and questions. I had to ask myself what my basic goal was in each class and be flexible about whether a list of theorems could be covered -- I refocused on a sparse collection of fundamental concepts in each course. Students felt empowered by their own discoveries, and they began to provide at least as much energy to the classroom as I did. The students were quick to discern that they had more control of the learning environment and responded to my pleasure in the change.
One of my classroom "triumphs" came during a problem set presentation in geometry. I had inadvertently written out a problem incorrectly for the class. The student who was to present it came to the board and said meekly, "I think this problem is not correct." When I said that on the surface it had seemed OK to me, she stomped her foot and shouted, "I know this is wrong!" She proceeded to show the class the counterexample she had constructed with Geometer’s Sketchpad . This indeed is success in the classroom!