Entry #5

Warrington's ideas about students constructing their own knowledge are very innovative and have many advantages. One of these advantages is that the exchange of ideas in the classroom is prevalent. Warrington uses an example about the problem 1 divided by 2/3. When she allowed the class to share how they had gotten different answers, the students could listen to their classmates and either agree that they had gotten the correct answer or find the mistakes they had made. Another remarkable advantage is that students form and discover mathematical relationships on their own without being taught them explicitly. When one girl was doing the problem about chocolate-covered peanuts, instead of using methods similar to her peers, she simply "doubled it and divided by one." She had discovered that the answer is the same even when the equation is doubled without anyone teaching her that relationship. The last main advantage is the "intellectual autonomy" that students develop. They begin to think on their own, like one girl who disagreed with the rest of her class about a certain division with fractions problem. The way that the other students had done the problem did not make sense to her, and she would not let it rest. She explained her reasoning, which happened to be correct, even though she was standing alone from her peers.
There are several disadvantages of Warrington's methods, which inhibits her ideas from becoming more widespread. One of the disadvantages is that generally teachers do not believe it works so they would not be willing to try it. Warrington said that most teachers think that certain algorithims are required for students to learn how to do more complex math. Similarly, if a school was to start teaching in a way that students would have to construct their own knowledge, the students would have to go through a major learning curve because they have never been expected to do that before. Warrington said the response, "I don't know how to do this," was not present in her classroom because the students knew they were expected to find a way to do a problem. For students who have not been learning this way, there would be major resistance to learning on their own. Another more specific disadvantage is that all of the students might not be learning and understanding as well as the top students. In the way Warrington described her classroom, students were allowed to say the answer when they got it and explain their method. Although some of the students were constructing their own knowledge very well, there could be some students who were not and were left confused.

Constructivism

Ernst von Glasersfeld discussed how knowledge is not transferred from one person, book, etc. to another person, but rather that the person interprets and senses the world around them. This is called constructivism. Glasersfeld talks about constructing knowledge rather than acquiring knowledge because he believes there isn't just one specific truth that we take in, but rather that we perceive what is happening around us and convert that into our own truth. Because no one perceives something in exactly the same way as someone else, what every person thinks of as truth is different. We each also go through different experiences so our knowledge of one topic or event could be completely different from another person's knowledge of that same topic or event. This is another way how different people have different ideas of what truth is. Experience plays a large role in not only supplying knowledge and our idea of truth, but also in changing it. We may think that the truth is one way of looking at a specific topic, but then our experience comes along and contradicts our idea of truth. We then have to change our idea of truth to match our new experience. This is how knowledge is constructed and not merely acquired through the transfer of information.

Constructivism can readily be applied in a mathematics classroom through the use of it in making sure students fully understand the idea being taught. Through the idea of constructivism and not being able to simply transfer knowledge from teacher to student, there must be a way to teach mathematics where students discover on their own and construct their own knowledge. Then the teacher needs to be responsible to test a student's knowledge and correct it. This will verify that the student's idea of knowledge and truth is the same as the teacher's knowledge and the rest of the math world's knowledge. I do not want to suggest one way to apply constructivism becauseI think there would be many possibilities of how to do this. Every student learns in a slightly different manner and would discover math knowledge in a different way. Students could make discoveries through doing practice problems in homework and having to explain how and why they did the things they did. This would provide them with a way to make an idea about how a certain algorithim is done. If the teacher sees a flaw in their explanation, he or she could give the student an example problem for the student to solve that will not give the correct answer if the student follows his or her own explanation. When the student solves and the problem and explains it again, he or she will have a new knowledge about that algorithim that matches more closely with that of the teacher. Experience would be changing the student's knowledge.