Warrington

In Warrington's paper she describes her experience of creating fifth and sixth grade class environments in which the students develop understanding of division of fractions without first being given algorithms from the teacher. The results were fascinating and as far as the reader can tell from the article, very successful.

Some advantages of this teaching style as demonstrated in the article are the fact that the students' minds are free to explore and develop without being "shackled" by rules. Another clear advantage is demonstrated by the girl who disagreed with the rest of the group. Warrington describes her as having "intellectual autonomy", which is considered something positive out in the world. Overall the most important advantage is the extended understanding they develop as they work together and can't give up and ask the teacher. They build off of each others' ideas until what that come up with makes sense. After long discussions and heavy thinking, they develop deep and long-lasting understandings of fractions.

A disadvantage could be that some students might lead the discussions; ones who are further along or simply better at division of fractions might continue to grow and construct knowledge while the students not quite as proficient might just tag along and listen to the others without ever grasping how to do it on their own. I also question if the group could have come to a confident conclusion that was incorrect without knowing. If it weren't for the autonomic girl who pointed out that the last part was wrong, they might have stuck with their previous answer; so not being given the correct answer could be a problem.

Constructivism

In his "Learning as a Constructive Activity" von Glasserfeld discribes the concept of constructivism. I believe he uses the term construct because we gain knowledge so experience so it is constantly building upon what we already know, and it grows and grows and adds upon itself as we try to come to know truths. As we experience different situations, we adjust what we "know" to fit what we come to learn during these experiences. Certain conditions of course are required. We must be actively participating and working to come to solutions on our own. It is true however that we can never be sure as to whether what we know if absolutely correct because everything is relative and there is no way to know the absolute truth of anything. All we can do is try to come up with the best ideas we can from different problems we encounter.

In my classroom I want to begin lectures by presenting problems that I haven't yet taught them how to do. I want to give them an oportunity to try to figure it out own their own first or at least gain a greater concept of what the problem is asking for and what kind of information they would need to complete it. I hope that this will help my students construct their knowledge in such a way that in the future they will not always need a teacher to tell them how to do everything, but rather they can solve problems on their own.