Erlwanger supports this in his study with Benny. Benny is one of the more successful students in the IPI program, but when Erlwanger interviews him he finds misconceptions that Benny developed through the program. Some of the evidence Erlwanger presents is Benny's understanding of adding fractions and multiplying decimals. Benny has different logic when it came to this and to him he was correct, even though he wasn't. Benny merely developed his own methods for explaining his answers to his satisfaction.
Erlwanger also argues that the IPI program places heavy emphasis on finding the answers rather than on the process or the understanding behind it. As a result the programmed form of IPI was making it a hunt for answers the teacher wants just to get them correct. Thus, in Benny's case Erlwanger states that a "mastery of content and skill" does not imply understanding.
One of the arguments Erlwanger presents that I think is beneficial to today is that the aim to teaching mathematics should be to "free the pupil to think for himself." He goes on to say that students should be provided with opportunities to discover patterns in numerical relationships, they should realize the independency of their actions in reasoning, seeking relationships, making generalizations, and verifying discoveries. One other support Erlwanger provides is that mathematiccs should be a subject in which rules come from concepts and principles.
I think that this is applicable to today because I believe that most of mathematics taught in school are very instrumental in that instead of deriving rules from mathematical concepts, students are taught the rule without the concept. I think if the students are allowed to experiement independently and then share with others their ideas about their discoveries, it will create more of a relational understanding between them.
I completely agree with your view on how mathematics should be taught in school. I was actually one of those kids who would have to have a mathematical rule or concept completely explained and proven to me in order to accept it. The only I question is what happens to those students who don't have the mental capacity or simply just don't care about understanding why things in math work the way they do?
ReplyDeleteI really like your ideas about what happens in schools today. I also think that far too much of today's teaching is instrumental, which causes problems like the ones Benny has. You effectively and fully explained Erlwanger's ideas that are still applicable today. I agree that making your own discoveries is very important. The only thing that I would work on is maybe having a topic sentence of your second paragraph that relates all of your ideas together and expanding your own thoughts a little bit more. Otherwise, I thought your argument was good.
ReplyDeleteI think that you have identified the main point of Erlwanger's paper and captured much of the argument he uses to discredit IPI. I had a little trouble understanding your topic sentence, particularly the end where you described the flaw. I also think that you did a nice job of identifying some of Benny's problems. I could have used more explanation, however, about what allowed Benny to develop those flaws. For example, you state that Benny was "successful" but lacked understanding. How could this happen? What was it about IPI that enabled him to do this? How could he get so many correct answers on the quizzes and still not understand?
ReplyDeleteI found it interesting your point on how students will develop a more relational understanding if they first discover on their own, then share with others what they learn. I believe that could very well be the case. As to improvement, I noticed that your first paragraph was about the flaws of the IPI program, and your second paragraph was about how students need to learn to think for themselves. Maybe there could have been more of a connection between the two paragraphs to make your overall point clearer. Great job on you post, I enjoyed reading it.
ReplyDeleteI really appreciate your ideas on how mathematics should be taught today. I believe that self-discovery grants students with a much greater understanding and sense of accomplishment. However, part of me is concerned that this teaching style may only work in an ideal situation, where all students are dedicated and committed to learning. Unfortunately, many students today need teachers to help them develop and discover their desire to learn.
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