In von Glasersfeld's paper, he brings up the idea of "constructing knowledge", in opposition to "acquiring" or "gaining" knowledge. The difference between these two ideas may not seem very apparent at first, but von Glasersfeld makes it very clear that the two concepts of how we receive knowledge are quite different. When we talk about "acquiring" or "gaining" knowledge, it is implied that others who have some knowledge, such as in mathematics, can pass this knowledge on to others, and that is how we "acquire" the knowledge. However, von Glasersfeld believes that we "construct" knowledge in life. We cannot know the TRUE reality of the world around us, because everything we "know" comes from experiences that we have had, and how we have filtered them through our own eyes and minds. As we have these experiences, we begin to construct ideas and theories to how the world around us works; and in our case, how mathematics works. We then only change our ideas and theories when we run into a contradiction with another experience. These experiences can include listening to a lecture, doing homework, and experimenting in the world around us. This idea of "constructing knowledge" is like building a brick wall, one brick at a time; each experience we have is another brick to add to the wall. When a brick doesn't fit with the wall we have already constructed, we either have to break the wall down some so it does fit, or we throw the new brick away. Constructivism then becomes a powerful idea in all aspects of learning, and especially applies to mathematical teaching.
In my own teaching, constructivism would make me teach in a variety of ways. Given enough time, I would take each mathematical concept and help the students experience that concept in multiple ways. Taking ideas from the National Council of Teachers of Mathematics, I would give a small lecture to the students, followed by a hands on activity to illustrate the concept. Next, I would do a visual example either on the board or the overhead, including a graphing application if it was applicable. Finally, I would give the students their assignment early so that if they had any more questions, they could ask me in class before they took their homework home with them. This way, hopefully, students would be able to "construct" a correct knowledge of the concept I was teaching; and any contradictions to the ideas they had would be discovered and fixed within that class period. With this method, and with constructivism in mind, I think this will be an extremely powerful way of teaching mathematics to students.