Erlwanger believes that understanding the reasoning behind the rules is fundamental in learning mathematics. Benny was a sixth-grade student who had been using the IPI method of math since second grade. The IPI method is basically following example problems and then doing similar problems in order to learn a specific rule. Benny has been successful, even one of the top of his class, but when questioned, Benny states incorrect rules. Repeatedly Erlwanger mentions how Benny was only interested with the rules of math, in this case fractions. Benny had been discovering "magical" rules since he began the program in second grade. Benny thought they had been invented by someone and made to be the ultimate standard for doing mathematics. Erlwanger wants us to know that relational understanding is the most beneficial. We need to know the how and the why of the rules in order to completely understand the rules and be able to use them consistently. Benny said he was on a wild goose chase to find the answers in the key because he thought there were multiple right answers, and he was just trying to find the right one. The IPI program had led him to teach himself incorrect rules about fractions and decimals that he blindly followed. He didn't understand that there should be one correct answer. Benny also thinks that all the rules are set in stone, and that there is only one way to do something. If he had a relational understanding, however, he would be able to find more than one way to do a problem because he would understand how the rule worked.
The whole concept of relational understanding and knowing how a rule works is still valid today. When we know how a learned rule works, we can apply it in new situations and not just in the specific case where it was learned. We also might be able to apply a different rule to a given situation because we fully understand how the situation works. There definitely could be more students like Benny who, for one reason or another, end up teaching themselves incorrect rules but get by with high grades. This should also be applied today. Simply because a student has a high grade does not mean they understand the material and are able to get the correct answer all of the time. Benny was one of the top students in his class, but he had gotten by without someone noticing that he didn't actually understand the material. Today we should be more aware of this and catch it at the beginning. By the time Erlwanger talked to Benny, he had been doing this for four years and could not change his thinking even when Erlwanger went back and tried to teach him. We could prevent cases like Benny if we could assure that the students relationally understand the rules, which could be difficult to do.
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Very thorough. :) I especially liked how you really tied the two articles together and tried for a cohesiveness between the two. One thing that I would have done is been more assertive. It feels to me that the comments, although valid, just weren't quite as supported as I would have liked. Really great summary!! :)
ReplyDeleteYou came up with a different main point for this article than I did, and yet you provided so much evidence from the paper that you convinced me that your main point is just as valid as the one I noticed. Good job! Also, I think your main point is incredibly relevant to today's mathematics teaching, and for that reason, I like yours better than mine.
ReplyDeleteI think that one thing that could have made your Paragraph 1 better is a clearer organization of ideas. As I have already admitted, your argument was overwhelmingly persuasive. And yet I would have appreciated a clearer logical flow in your argument. In other words, could you organize the evidence in such a way that it was clear why the first point was first, why the second point was second and how it fit with the first point, etc.?
Great job and excellent ideas. It really got me thinking. I think a little bit of work could be done with organizing but I find that just a minor detail with how good this was.
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