S. H. Erlwanger's essay Benny's Conception of Rules and Answers in IPI Mathematics gives a striking example of the necessity for Mathematical understanding through the example of one clever -- but ill informed -- boy named Benny. By looking at Benny's progress in the IPI program, he is successful and quick, though when Erlwanger interviews with him he finds astonishing minunderstandings saying things such as 2/1 + 1/2 = 1. However, despite Benny's confusion, Erlwanger attempts to help the boy grasp mathematics for several weeks and Benny does indeed realize that something like 29/10 = 2.9; but Benny is still under the impression that 2/1 + 1/2 equals a whole. This then exemplifies that an understanding in mathematics must be built early on, or it is very hard to teach later.
Erlwanger further argues that teacher participation is required. That the teacher must be aware of the understanding that each student has. In IPI, "[Benny's] teacher can only become aware of his problems if he chooses to discuss them with her." And in this method of learning the teacher is more of a robot simply correcting answers by the key regardless of anything else. Benny realizes this himself and complains that "they have to go by the key... what the key says." To be sure, if the teacher isn't actively involved in teaching the curriculum then their is no understanding with their 'learning', because when one doesn't know something, the person with experience helps make it clear and the same is true for math.
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Kyle, you did an excellent job giving examples to the main point that an understanding in mathematics must be present in order to have learning take place. I like that you even described what happened after Erlwanger tried to help Benny. That even though it helped a little, Benny still had little understanding because he hadn't learned it early on.
ReplyDeleteI had a hard time at first finding what you thought the main point was. The introduction flowed into the reasoning so my only suggestion is to make a stronger introduction statement that clearly states your main point.
Thanks, Haley Bly :)
Thank you, your second paragraph was clearly written. It had a clear topic sentence. The word "learning" in single quotations should be in double quotations and maybe I would have elaborated more on what you meant by learning. I understood what you were trying to say but it took me a few times over to understand it. Maybe you could clarify it earlier in your paragraph when you discuss that a teacher needs to have understanding of what the children understands.
ReplyDeleteThis paragraph was perfect. It even related everything back to the original article and the example that Erlwanger used with Benny. I also like how you related it to all teaching in general, not just math.
ReplyDeleteI really liked the flow of your blog entry. It is very easy to read.
ReplyDeleteI would define what you meant by a "mathematical understanding". We just wrote blogs last time about two different types of understanding. Which type are you referring to?
Also, a main idea seems to be introduced in the last sentence of the first paragraph. And the introduction of more main ideas seem to continue in the second paragraph. Try to get it all rolled into one topic sentence.