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.
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Annalee,
ReplyDeleteI agree with your thoughts about problem solving. This teaching style promotes students to learn how to solve problems for themselves and you nicely showed that through examples from the article. Furthermore, I liked how you said it applies to other subjects and our lives. I had not considered that before and it is a good point to make, especially because that benefit seems to have long-lasting consequences.
I wonder the impact it would truly have on those quieter students. Even if they were not participating in the classroom discussion, I think it is important how they are considering different students' ideas. The quieter students eventually have to decide for themselves which of their peers' ideas are correct and then go with it, and that process in itself is thought-provoking enough for students to learn from.
Nice post! Thanks for all your ideas :)
I really enjoyed reading your blog entry. I can tell that the article raised some issues that you care about, and that you have spent some time thinking about these issues.
ReplyDeleteI think that you have done a nice job of capturing some of the advantages that Warrington believes come from her way of teaching. These advantages are described well and illustrated with examples, which helps clarify the points you are making.
I also appreciate your concern about some aspects of her instruction. You're right that there are greater time requirements for this type of instruction. However, I wonder if your own experience is an appropriate benchmark for how much time is required to learn fractions. My guess is that many of your classmates did not develop the same level of proficiency with fractions that you did. Also, as this class has probably demonstrated, your ability to compute using fractions doesn't necessarily imply that you have an adequate understanding of fractions and fraction operations. For me, just knowing how to perform the computations doesn't mean that you know math. You also need to be able to understand why the computations work, when they are appropriate, and how to apply them to a wide variety of situations. I'm not saying that we need to spend months and months on fractions. I am questioning, however, if the current amount of time we dedicate to teaching fractions is adequate.
I really enjoyed your analysis of this article and thought you did a great job thinking about the advantages and disadvantages to this teaching method. I agree completely with you that although this type of teaching has merit, it needs more testing and even more development. I like your example that you did not learn fractions this way, yet you are still able to do math at a university level. This is a very good point.
ReplyDeleteWhen you brought up the point that when there were disagreements in the classroom it left room for students to have their own right answer, I had some flashbacks to when I was learning math. When I thought I had a right answer and another student tried to disprove me, I had a hard time accepting that and at times I didn’t! I think that this autonomy might be a disadvantage at times in a classroom.
I agree with what you said about how Warrington spent too much time on division with fractions. Too many other things would have had to been cut from the curriculum to do this, and I am not too sure it all worth it. I also agree that some students would just slide by, not taking the effort to really understand and just taking their classmate's word for it instead. One suggestion I have for your post, more for the visual effect rather than content, is to break your information up a bit. One large paragraph without any white space is rather intimidating for someone to want to read. :)
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