Otten, S., Herbel-Eisenmann, B. A., and Males, L. M. (2010). Proof in Algebra: reasoning beyond examples. Mathematics Teacher, 103(7), Page 514.
The goal of the authors of this article was to convey the idea that proofs could be utilized in beginning algebra classes, not just geometry. Through a case study of an algebra one class, the authors demonstrated how the cross-multiply rule is generally taught, and then presented an alternate version in which the use of proof could easily have been inserted into the discussion. Usually, students only encounter proofs when they come to geometry; however, the use of proofs in algebra can greatly assist students in their comprehension of difficult concepts. Also, the discovery of why rules work in algebra, will also lead students to a greater capacity to like math because it's no longer "a collection of procedures and rules" that students feel they have no ownership over, rather it provides the students the opportunity to "realize that mathematics is a state of mind characterized by inquiry and a thirst for justification."
Although I agree with what the article is arguing I feel that, due to the fact that they invented a hypothetical situtation instead of actually testing out their idea, they did not quite present as strong as an argument as they could have. It is slightly ironic to me that the article is written about proving things, as opposed to just showing examples, but then the authors present an example instead of setting out to prove their point. I do like the ease at which the teacher, in the hypothetical situation, was able to construct a proof. It really didn't take that much more time to generalize the cross-multiply situation to show that it works for all situations. Once again; however, I have a problem because of the fact that the cross-multiply rule is an easy situation to come up with a proof for. What about more complicated situations such as solving systems of equations or how we came up with the quadratic formula? In these situations it could take quite a bit more effort to show why they work. In summary, proof of things in algebra is a good thing that teachers should look into, but at the same time, it might just be better to provide students with examples.
You did a good job recognizing the authors' main idea of proofs and giving support evidence from the article. The reader was able to clearly tell from your writing how the authors supported their idea and the application in the classroom.
ReplyDeleteI am not sure about the connection between proofs and the comprehension of difficult concepts. This idea is briefly mentioned, but I would have liked to see some more detail about this connection. I say this because in linear alegbra many times my teacher would run through a proof, but it honestly just made me more confused about the subject on hand.
Scott,
ReplyDeleteI really liked how you had a strong topic sentence with the main point of the article and then clearly explained what the article argued about. I felt I understood the point of the article without having to actually read the article. However, I feel at the end when you quoted from the article that it wasn't really necessary. You could have paraphrased and gotten the same meaning across.
Thanks for your post!
I liked how your topic sentence was the first of your entry, you immediately stated the main point of the article. I thought you flowed well in your writing and summarized the article well. I might have talked a little more about other points in the article rather than focusing on the one cross multiply example. But maybe this was the only example given. I am curious to read this article, it is a very interesting idea to use what students learn is geometry, proofs, and apply it to broadening the understanding of algebraic concepts.
ReplyDeleteYour paragraph was really easy to follow. I like the important points that you chose to summarize because it made your paragraph interesting and clear. I would like to know more about how the authors plan to implement proofs in general, rather than just in one specific case. They may not talk about this in the article, but if so, I think it would be a beneficial area to focus on.
ReplyDeleteYou did a nice job summarizing the article in your first paragraph. In your topic sentence, I might change the word 'could' to 'should'. Of course you could use proofs in algebra and it sounds like the author thinks you should. Then the stance you take in your second paragraph would be more clear.
ReplyDeleteI felt your second paragraph was focused. I do wonder though, why do you agree with the authors? Your last sentence also makes me feel that you might not agree with the authors.
Very well written! I'm curious about reading this article now because I've read a couple of blogs about it already. It seems rather obscure to me that the author wouldn't test their theory and opted for a hypothetical situation. I'm glad that you pointed that out. You did a really good job letting your voice ring through your writing.
ReplyDelete