Blog Entry #5

In the article, "How Children Think about Division with Fractions," Warrington advocates that a constructivist approach to mathematics can work in a classroom. She takes the reader through the process of division with fractions with her 5th and 6th grade class to illustrate her point. One of the advantages that she found was it gave the students intellectual autonomy. This meant that the students each had their own choice in discovering their intellect of mathematics. An example of intellectual autonomy in her classroom was when the one student did not agree with all the other students about the answer to 4 2/5 divided by 1/3. This stemmed to a huge discussion and debate over the right answer, in which each student had their own right and ability to edit their answer according to what they believed from the other students. This is a wonderful way in which Warrington's constructivist approach allows deeper understanding to take place in mathematics. Another advantage is that the students are able to increase their problem solving abilities because they are not relying on the teacher to teach them algorithms and the right answers, but instead, they are developing mathematics on their own and building upon their previous knowledge. In the first three problems that Warrington gave, she described the answers the students were giving her, and these answers came from the students' confidence in their mathematical abilities because they were building off of what they already knew. These abilities can be used in all subjects in school and in learning for the rest of their lives.
Although the article concluded with success in Warrington's classroom, I saw a few disadvantages as I was studying the situation. Firstly, I think the time that was spent teaching these students fractions was way too long! It said in the paper that she had been working with them on fractions for months, and when I learned them we did not spend that much time on fractions. This way is just too time consuming for everything that students need to learn. And frankly, I did not learn fractions this way and yet I am still able to do math at a university level, so therefore I must not be completely lacking in what I am learning, so the year that they spent learning this will put them behind in learning more mathematics, and more complicated mathematics, if learned in this way, will take even longer, and that time will not be found in schools. Another disadvantage I saw was that some students were figuring this out on their own. For all we know, there could have been many quiet students who never said anything, and never actually figured it out on their own, but instead just took their classmates' word for it, just as they would have taken the teacher's word for it if this had been a traditional classroom. So I am not sure if Warrington's approach really solved all the "problems" that she finds in traditional classrooms. In conclusion, I think her approach is a style that should be experimented with in more classrooms before it is truly accepted as greatness.

Blog Entry #4

Von Glasersfeld believes that all people construct the knowledge that they have. They do not gain or acquire knowledge, but construct and build their knowledge through a filter in their minds, shaped by the experiences that they have had in their lives. Therefore, knowledge is a theory because it is always changing because people's experiences are always happening that change and tweak the knowledge that they have previously constructed.

If I believed in constructivism, as a math teacher, I would emphasize the importance of understanding - or of seeing how it is that my students are filtering the knowledge that I am trying to teach. I would do this by having a few minutes at the end of each class period where they turn to their neighbor and each teach each other what they have learned that day. I would walk around and try to catch as many people as possible during their teaching descriptions. This would help the students, in a non-threatening way, share the knowledge they have constructed and give me a chance to catch any mistakes they have taken place in their understanding.