In Warrington's "How Children Think about Division with Fractions" she describes many different positive things that come from teaching through constructivism, without teaching the mindless algorithms often involved in mathematics. One of the advantages that she described was that the children were learning to think for themselves. She recalls one time when one of her students went against both her classmates and even her teacher to state what she believed to be the correct answer. She was confident in her beliefs and figured that the way she got her answer was correct and made sure the class knew. Another advantage she points out is that the children were able to work with each other and think of their own ways to see patterns. When she shared the story problem of seeing how many 1/2 lb bags you could make out of 5 3/4 pounds of peanuts, one girl realized that if you double everything, easier numbers are available to do the math with. Simply comparing and seeing that her answer was right helped her to become confident in her thinking style. I agreed with both of these advantages and one advantage I saw was that, the children were really learning how to do these. Down the road they would not forget the "rules" because they put in the effort to figure them out by themselves.
Although the advantages are strong, there are a few disadvantages to the system. One clear disadvantage, is that the students were not told the right answers. Although this may seem like a great idea, because the students would have to be confident in what they thought, they could never really know if what they were doing was completely perfect. An example of this is 4 2/3 divided by 1/3. This problem is a little bit more challenging, and the students wondered when it was all said and done which answer was correct. Without a concrete answer, they just had to accept what they were thinking. ALthough a majority of the students probably did understand, there is a great chance that some of the students did not completely understand, and that would be what they always remembered. So, there are definite disadvantages to the system as well.
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I really liked your first paragraph. You had a strong topic sentence. The advantages were described clearly and illustrated with an example from the paper. I think that you captured many of Warrington's ideas very well.
ReplyDeleteI appreciate your concern about the children not having the correct answer available to them. I remember how I liked looking in the back of the book and checking my answers. Often incongruencies between my answer and the answer in the back of the book would cause me to go back and check my work or look for a new approach. I also remember that sometimes there were wrong answers in the back of the book, or correct answers in a different form that I did not recognize as being equivalent. That led to a lot of frustration. In addition, I've noticed that because I relied so heavily upon the answers in the back of the book, I find myself lacking confidence and not being very clever about how to check my answer when the correct answer is not readily available. This makes me wonder if having immediate access to answers is a two-edge sword.
I agree with what you said. I had the same perspective. I do think though that the students need to work it out them selves for a while first and maybe even never be told the right answer but if the teacher knows it's wrong, just ask different questions that maybe contradict the answer so they'll search for a different one.
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