Blog Entry 5

There are many advantages to the style of teaching used by Warrington in this paper. Students were given the opportunity to explore the material with fellow students. This led to a lot of student interaction and the exchange of ideas in order to help them come up with the best solution. This helps the students to learn how to converse in mathematical discussions. Secondly, since they were not taught any new type of arithmetic they had to build off of the knowledge that they already had. By this time the students had a firm understanding of division and created understanding built on that solid foundation is good. Students also were able to adapted their ideas during the class discussion. Another advantage that I think is really good is that for me personally when I have struggled through the learning process and figured something out on my own then I have been able to remember and recall that knowledge much more easily than when I have simply had things explained to me. Lastly the students were able to strengthen their ability to solve problems. They started by estimating, and then they used this knowledge along with their knowledge of division of whole numbers to solve problems.

Along with these advantages comes many disadvantages that need to be considered. The first of which and probably the greatest disadvantage is that it took more time. If the students spend a week on this subject alone then it will cut into the amount of time that can be spent on other topics. The invert and multiply idea is very easy to teach very quickly. If every topic were taught with as much depth as the division of fractions was in Warringtons paper then you would not be able to cover nearly so much material in one year. Another possible disadvantage is that if this was not done correctly then students may never come to an understanding of how to come up with the proper solution. I know that there have been a few times in Math Ed 117 where I thought I had the right thought process and later found out that I didn't even have the right answer. So teachers teaching using the method discribed in Warringtons paper would need to know how to guide the students to coming up with the right answers and/or have a way of ensuring that the students were reaching the correct conclusion.

Blog Entry #4

Contructing knowledge is an action. Von Glasersfeld explains that you do not "gain" or "acquire" knowledge because these are terms that are too passive for explaining the act of increasing ones knowledge. He explains that the way that people increase their knowledge is through constructing it, and not through receiving it from someone else. Meaning that they experiment and find contradictions and through this process they find meaning or construct knowledge. Also, since knowledge is constructed it must be built on what we already know. So it is important to recognize that knowledge is built on or seen through our previous understanding.

As a teacher who believes in constructivism I would try to be aware of the way students learn. This would include giving the students plenty of opportunities to experiment in different mathematical contexts. I would try to help them find their way to various contradictions allowing them to revise and construct around these problems. Another part of being aware of the way students learn would be to understand the importance of knowing what the students have already learned. Since they will only build on what they have previously learned it is important that the way the teacher teaches is compatible with their current knowledge.