So we can summarize radical constructivism as a focus on construction of our understanding of all phenomena that we seem to experience, including both our interaction with physical objects and with the ideas of ourselves and of others. Of course this construction that an individual makes is not done in complete isolation. No radical constructivist would deny that if, for example, an apple should fall on our head, then something has happened which is, at least in part, external to us (an apple fell). The point is that we cannot know anything about this external happening. All we can know is that our head hurts, that we saw (and perhaps heard) something which we interpret to be an apple in motion, and that there is a fresh apple lying on the ground. The rest of our knowledge about this event is our interpretation. The same holds for mental reality. In thinking about a function, for example, we might be trying to make sense out of the relation between time and the position of our apple, and we might even construct a formula that gives this position for any given time within a certain range. But we talk with others about these constructions, before, during, and after we make them and what we say must be compatible with what others are saying. In other words, although I cannot really be sure that my understanding of the function that describes the motion of the apple is the same as yours, we can --- and do --- insist that our respective understandings fit with each other, that there should be some agreement between what the two of us say about this function. It might even be that my construction of my understanding of this function (which we express, for short, as my construction of this function) comes as a result, at least in part, of interactions with you (e.g., conversation, listening to you lecture, reading a book you wrote, etc.)

It is this social factor that leads to a relation between the construction of knowledge and cooperative learning (CL). Obviously, if interaction with others is part of the construction of knowledge then a learner should have as much interaction as possible with the instructor and with other learners. It is the goal of having a two-way interaction between teacher and student that has led me to change my teaching style by replacing the lecture in which I do all of the explanation by discussions in which both the student and I try to express the ideas we are constructing. Similarly, the goal of multi-directional interaction among students is one of the reasons I use cooperative learning in all of my teaching and curriculum development.

So, in my teaching of collegiate level mathematics courses, I put the students in groups at the beginning of the semester and they remain in the same group for the entire course. The reason for this is that it takes time for a group to develop as the members learn how to interact with each other in the most productive ways, and I don't want to change a group just as it becomes productive.

Usually, I am using computers and the study of each mathematical topic begins with the students, working in their groups in a computer lab, performing certain tasks on the computer designed to foster their making certain mental constructions which research and analysis suggests can lead to their constructing an understanding of the topic. Instruction on exactly how to perform the tasks is minimal, partly because this stimulates interaction with the teacher and within the group. After working on the lab, the students come to class where I try to get them to reflect with their group on the experiences they had in the lab, and to work ooperatively to perform, with pencil and paper, other tasks, this time more closely related to standard mathematics problems and exercises.

The constructions of mathematical knowledge that students made in the lab and the classroom are reinforced through exercises they do for homework. Their knowledge is evaluated through examinations. All of their homework is done in their groups and each group submits a single assignment. About half of the testing is done in groups and half as individuals. I generally give