In Mary Ann Warrington's paper, "How Children Think about Division with Fractions", she gives examples from her class of students at the Atrium School in Watertown, MA, to illustrate the fact that children can develop their own mathematical formulas and ideas when you give them the opportunity to discuss them within the classroom. Warrington believes strongly that "children construct knowledge on the basis of what they already know," and that the best way to teach them is to let them teach themselves through discussion in the classroom. Warrington does not even tell her class the correct answers, but lets them guide themselves to discovering the concepts themselves. The greatest example she gives is when she asked the class what 4 and 2/5 divided by 1/3 was. Many believed that it was 13 and 1/15, but one student disagreed and brought up the idea of 13 and 1/5. As Warrington states: "This response was not only logical and mathematically correct but also a shining example of the autonomy that children develop when encouraged to think for themselves." Giving children the opportunity to think this much and develop their own mathematical ideas and theories can be quite a wonderful way to allow children to explore the world of mathematics.

However, there are flaws with teaching this way. The first is the time constraint. In public schools, this method of teaching would not work in any way. Warrington had actually been teaching fractions to these students for FIVE months, which, in a public school, is the time given to teach MANY different topics. Second, the issue arises, what if that one student hadn't had those thoughts, and nobody had thought that 4 and 2/5 divided by 1/3 is 13 and 1/15 was wrong? Would Warrington give them the correct answer, or would she let them go on thinking the way they were? Third, what about those who just don't have the ability to think about mathematics in this way. For some, math just does not ever click. And last, Warrington said that "children *can *and *do* invent ways to do sophisticated mathematics," but sometimes those ways are just plain wrong. Take Benny for an example. He had many ways of doing math, but only some of them were actually logical and correct. Warrington's method of teaching is a wonderful way of letting children discover the world of mathematics, but it would really only work in a private school institution, and never in a public school with the curriculum they have to cover.

Kevin,

ReplyDeleteI totally agree with the fact that Warrington's method takes WAY to long! Classrooms cannot spend months and months and months on fractions! And if she wants all math to be taught this way, more complicated math will just take even longer than these fractions took to learn, and that time just does not exist!

One thing I think you could take into account is that Warrington did not set up her classroom like Benny's classroom was set up. Warrington was listening to everything the students were saying! Therefore, if no one had come up with the right answer, I truly believe she, and any other teacher, would have intervened. Benny's classroom did not have a teacher listening to everything he was doing. That teacher just looked at his answers at the end of each unit. Thus, I am forced to disagree with you that Warrington would ever let her students construct false knowledge.

Thanks for your post!

Your thought about this teaching style allowing students to explore the world of mathematics is nicely worded. I like the idea of exploration because that is what the students are doing; as they are given new material, they explore to figure out what methods and rules apply and can be formed. Furthermore, your paragraph nicely lead up to this final idea.

ReplyDeleteI wonder if this teaching style could truly never work in a public school. I understand the time commitment, but I believe that it could be done. I do not think private versus public schools are the issue at hand, it is the support of the school, parents, co-teachers and students that really make the difference.

I think that you have identified many of the objections that people have with Warrington's method of teaching. However, your objections are not new to Warrington--I'm sure she's heard every one of them several times from other teachers. Interestingly enough, she keeps teaching this way. My guess is that she has pretty good responses to these objections. Can you guess what they are?

ReplyDeleteI really appreciated your passion and the depth of your comments in this entry. I would like to encourage you, however, to follow the guidelines for the entries more closely. Your argument would have been strengthened if you had divided your thoughts into two paragraphs with appropriate topic sentences for both. In fact, I think you could have made all of the same points using the suggested format, but done so in an even clearer way.

I enjoyed reading your response. I agree that Warringtons method is extremely time consuming and in secondary schools it would be impossible to take time to wait for students to give answers rather than just tell them. I do think that because Warrington was teaching elementary grades that it would be a little more realistic for public schools to implement her teaching methods.

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