Blog Entry #5

The practice of teaching mathematics without telling students the procedures or even the right answers has many advantages. First, students’ interaction and exchange of ideas plays an essential role to the their acquirement of knowledge. The students are able to convince each other of their reasoning or why a certain answer is not feasible; additionally, they build off each others’ ideas in order to come up with the best solution. Secondly, the students are able to adapt and apply their knowledge. The students were able to use what they knew about whole number division and apply it to division of fractions. Furthermore, students modified or adapted their ideas during the class discussion. Next, students develop the skill of problem solving. When first figuring out division of fractions, they started by estimating, and then they used that knowledge in conjunction with their division of whole numbers knowledge in order to solve the problem. Lastly, students are taught to think of themselves. They come up with their procedures and methods for division of fractions, rather than relying on their teacher as the source for invert and multiply. Overall, this teaching method appears to have many advantages for the students.
However, there are also disadvantages associated with this teaching style. It takes longer for the class to go over the material than maybe a more traditional teaching style would. The students spent at least a week working with division of fractions; whereas, the invert and multiple rule could be taught easily in a day or two. As a result, it appears the students would not get through as much material in a year. Additionally, there is a lot of responsibility on the students. In Ms. Warrington’s class the students volunteered answers and participated in “intellectual bantering,” but if students did not participate or did not feel comfortable in the classroom, then this teaching style would be impossible. Therefore, you need willing students and a positive classroom in order for this style to be successful. Lastly, there is great responsibility on the teacher too. The teacher needs to know how the children are thinking, which will take a considerable amount of time and effort. Also, the teacher needs to know when to intervene in a discussion or just let the students struggle. If the teacher is not aware, then the students can easily make incorrect assumptions and get off the path of learning accurate mathematics. This teaching style does appear quite advantageous, but the disadvantages need to be considered too.

Blog Entry #4

The term constructing knowledge comes from von Glaserfield’s opinion that we gain knowledge through our experiences; our current knowledge is the result of our past experiences. As we have new experiences, we are constantly adding to or adapting our knowledge. Therefore, our knowledge is a “theory” because it can never be complete, since we are constantly having new experiences that change it.
One implication of constructivism for teaching mathematics is that the teacher needs to ask questions in order to assess her students’ understanding. According to constructivism, everyone learns in a different way because we all have had different past experiences, and therefore, we modify or add to our individual experiences. As a result, it is necessary for the teacher to accurately assess the students’ learning. The way for teachers to do this is to ask questions in order to assess their individual understanding. This can be done by asking certain students to justify their answers or by asking similar questions. We learn from constructivism that even if students have the right answer, they might not understand the concept accurately. Therefore, teacher involvement is necessary and one way this can be done is through questioning the students and their understanding of the concept.