# 5

In Mary Ann Warrington's paper about teaching division of fractions describes many advantages of teaching constructively. Teaching constructively, requiring the students to figure out mathematical procedures rather than simply teaching them, fosters inventive thinking to find the answer. This was demonstrated by the student who unintentionally used the invert and multiply rule. Another advantage she mentioned was the intellectual autonomy demonstrated by the students. When all of the students had agreed on an answer for a difficult problem, one girl refused to accept that answer because she did not think, or understand how, it could be correct.
Despite the benefits mentioned by Mrs. Warrington, there are also some disadvantages. A class room where everyone understands exactly what is going on is hard to imagine, probably because everyone has been that confused person at some point. For this reason, it seems likely that some of the students in the class room were just listening to other students and going by what they say rather than think about it for themselves. They may not understand and may be given the answers by the other students, which defeats the purpose of teaching constructively.

#4

In von Glasersfeld's theory of constructivism he uses the term constructing knowledge rather than acquiring knowledge. He does this because a student is not given knowledge but rather builds on what he or she already knows. Everything we know is constructed from things we observe and experience in life. For this reason one can not simply acquire information but must be able to relate it to something that they already have concluded to be true.

This is important for teachers today because teachers need to be able to teach in such a way that students can construct knowledge properly. One way to do this would be to relate new math concepts to past concepts that have already been mastered. This will allow students to be able to build on concepts that they have already found to be true. This is a constructivism way of teaching because it promotes constructing knowledge from interpretations that we have already made from experience.