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.
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I really like the structure of your paragraphs. Both have great topic sentences. The advantages are very easy to identify and are supported by appropriate examples. I got a little lost in the middle of the second paragraph, because it seemed two disadvantages were mixed together, namely the difficulty teachers have learning to teach this way and the difficulty students have becoming accustomed to learning this way.
ReplyDeleteI agree with your disadvantages. I think that there might be a steep learning curve for teachers who want to teach this way. Research suggests that students may also struggle initially with this type of instruction, particularly some the older students in high school. As for the differences in the amount students might learn, I wonder if this is a disadvantage of many other forms of instruction as well. For example, in classrooms that teach only rules and procedures, many students get left behind and never really master the content. Do you think there is an instructional method that might decrease this gap in student learning?
I thought you did a wonderful job explaining what Warrington was trying to teach within her classroom and how she was accomplishing it. You gave great examples that helped me to see what you were explaining. I also completely agree that students who have been learning mathematics through algorithms for their whole life would really struggle to switch to this method of learning. They would be expecting the teacher to give them a "method" of accomplishing the tasks set before them, and would NOT like to be told to just "figure it out together" in class.
ReplyDeleteHowever, I don't feel that there are many teachers out there who necessarily think that this method of teaching does not work. It is true that it would be very difficult for students to switch to this way of learning, but I think that most teachers believe that it could be done. The problem is the time it takes to do so. Warrington had been teaching her students about fractions for five months, and there is no way a public school teacher could spend that much time on one aspect of mathematics with the core curriculum they have to follow.
You did a great job with using quotes and examples from the essay. "Students form and discover mathematical relationships on their own without being taught them explicitly" is a statement I believe could be good in some ways and bad in others. In the case of Warrington's class, I do not think this was a problem because Warrington facilitated the discussion well. However, in a case like Benny's this could be a bad situation. False mathematical relationships could be conjectured. Nice job on your post. A suggestion I have (for the visual presentation more than the content) for your blog is to break up your information into smaller sections/paragraphs. One long paragraph of words is kind of intimidating for someone to want to read. :)
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