Otten, S., Herbel-Eisenmann, B. A., & Males, L. M. (March 2010). Proof in algebra
reasoning beyond examples. Mathematics Teacher, 103(7), 514-518.
This paper is written for the purpose of convincing its readers that the abilities of proof technique can be made at an earlier age than they are now. Taking a classroom of students from an early algebra class, the teacher attempts to help students learn about proof technique through fraction equivalence. By taking two seemingly different fractions and proving that they are in fact equal; this is accomplished by the cross-product rule and given a few examples that this works, the teacher asks the students how one might prove a more general fraction equivalence. The class then creates the fraction problem a/b = 2a/2b and then extends this to a/b = na/nb. Then by using the cross product rule that they know they show that anb = bna thus completing their proof. Having given enough evidence that they students created this solution on their own with little guidance the paper does indeed show that kids are able to develop an understanding of proofs earlier than thought.
While students are required to give proofs in Geometry they do not understand what they are doing when they do so and they do not develop the required abilities of analysis from their proofs, so this paper attempts to convince the reader that students are able to create proofs earlier than they do now. However, in this paper there is given only one example of kids doing such and in the example the teacher helped guide the kids in their thinking. Yes, the students came up with the solutions to the steps on their own, but the teacher led their thinking in the direction it needed to go; without her, this would not have worked.
From my own experience of proofs in Geometry I found that it was very difficult to do so because the teacher gave little more than one example, without several instances of what needs to be done, kids cannot do the work well on their own.
I also found this paper to be more or less useless; while it does give an example of early proof, it is simply "a vignette" and does not cover more than one classroom. Therefore the paper can lead to nothing but a hypothesis and does not show that it is indeed true in any concrete way. So I found this paper to be somewhat lacking in its presentation despite the interesting idea that is brought forth.
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I think you did a very good job at picking the point of the article and explaining this point. I gathered that the paper was showing that it is possible for children to start proof techniques early on. I felt that it was all understandable, and that you did proof read this before you published.
ReplyDeleteYou did a good job at sounding professional and not biased. I would have not guessed your response from just reading your summary so I felt it was unbiased.
"This paper is written for the purpose of convincing its readers that the abilities of proof technique can be made at an earlier age than they are now" This was a great topic sentence and you explained what this mean throughout. Again, I thought you were clear in this, so it made sense the way it is.
The thing I would have liked to learn more about the paper is how the teacher prompted this. If they had never seen a proof before, how would they know what it mean to prove something? It would be interesting to know how she went about this and to see if he/she was really letting them prove it, or again like teachers are doing now, just guiding them down the path to the right answer.
Overall, great job.
Based on your summary of the article, I share your concerns about the main point of the paper. Research has shown that students are often unprepared to learn about proof when they reach Geometry, and so I can understand why it might be a good idea to try to introduce proof earlier in the curriculum. However, it is unclear how this introduction to proof would make a difference in later proof writing and reading. Do you think it will make a difference?
ReplyDeleteI also agree with your stance toward the main point. I thought your reasons were very persuasive. Do you think there was anything salvageable in the paper? Did it help you think about how you might introduce proof earlier in the curriculum?
Too bad this paper was not helpful to you. You may want to look through other students' summaries of articles from the Mathematics Teacher to identify a more interesting paper for the next reading assignment.
That was a great summary! I am interested in the article even though you said it was somewhat useless, so you sold me! I was also impressed with your professional tone.
ReplyDeleteGreat job identifying a main point. Your blog was easy to read and understand. The paragraph was very organized and it flowed very well. I feel like I know what the article was attempting to point out even though I didn't read the exact article. I think it's great that this teacher was introducing the students to proofs at a younger age. It is great that they will have more experience. I wonder if the students would be able to do proofs on their own without the teachers help easier when they are in higher math classes. I also wonder if this early exposure really helps them be more independent in the long run or if they would still be dependent on the teacher to guide them throughout the whole proof. I would be interested to see this same class of students a few years down the road and see if they really understand more than those who first see proofs in a geometry class. Great post!
ReplyDeleteI felt that your summary was very good. I understood it very well, and thought the structure and logical flow was perfect. I thought you had a professional tone throughout the post, except for the end. I felt like you were a bit harsh on the article at the end. I wonder if you think there was something of value in the article at all? Maybe something you would eventually like to incorporate into your own classroom? Thanks for the great post!!
ReplyDelete