The main point that Erlwanger is trying to get across is that with the IPI program, Benny just searched to find that rule that would get him to the correct solution that the program was looking for. Benny’s purpose in completing the IPI exercises and assessments was to find the rule, not necessarily to know the reason behind his procedures. Erlwanger argues that this way of reviewing and assessing the students causes the students to come up with their own rules for what they think the procedure is because the students were not taught the concepts behind the procedures. Since this is an individualized program, the students are expected to work mainly on their own. This seems to be causing the problem, especially for Benny. Benny is doing what is necessary to pass the assessments and do well, but he is not discussing with his teachers or peers what he has learned, so his crucial mistakes were not caught early on. Because the students have too much independence, they make up their own rules that fit the examples they see in the IPI programs, and they think this is true for all other problems involving the same mathematical concepts. Erlwanger explains throughout his argument that just because a student can master the skill and content does not mean that they have the correct understanding.
In today’s mathematical teaching, it is still important to remember to make sure that the students understand the concepts that are behind the procedures. When students only understand the procedures, they may be able to master a certain area of mathematics, but they will not be able to make connections using the information they had previously learned. For all mathematical content, it is always important to have the underlying concepts so that one can understand fully the procedures that are taking place. Math does make sense as long as the correct concepts are understood as well as the procedures.
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Thanks for the thoughts. I found that same argument about what Erlwanger was trying to say. I believe in Benny's case, because of the nature of the program he developed his own ways of forming the answers which brought up the argument that getting the right answers doesnt imply understanding. I also agree with teaching the concepts behind the procedures. I can look back on my schooling and realize that that is exactly what did NOT happen with me. I totally learned the rule without the reasoning and i think that that should be emphasized in teaching today. thanks for your blog.
ReplyDeleteI agree wholeheartedly with what you said here and agree that Benny has been misinformed. The necessity for mathematical understanding is so crucial and the lack of it is why I think so many people have a dislike of math.
ReplyDeleteIn your first sentence though you said Erlwanger's main point was how Benny misunderstood and then you continued in your paper to generalize to other students, so I think you didn't stay consistent there, but that's it. Awesome.
I loved how you spoke about understanding the concepts behind procedures, and that this leads to a congruity of all mathematics, because you can then see the connections between the different procedures and concepts. I think this is definitely one of the most important things that needs to be emphasized when teaching mathematics. I would like to hear more about how this could be done though, and what techniques could be used to teach for the understanding of the students. Great Job!
ReplyDeleteYou did a nice job providing evidence and specific examples from the paper.
ReplyDeleteI think it may have been clearer if you combined the first two sentences. I also might have added that just looking for rules causes problems, which is what you go on to provide evidence of.
Also, I interpreted the last sentence of the first paragraph as another main idea. I maybe would have put that in the beginning.