Philipp, R. A., and Vincent, C. (2003). Reflecting on learning fractions without understanding. ON-Math, Vol 2, No. 2, 1-6.
The main idea of this article, “Reflecting on Learning Fractions without Understanding,” is that students learn less if they are taught procedural followed by conceptual, compared to students who learn more from only being taught conceptual instruction. Routine procedures actually inhibit the learning of concepts. This idea was apparent in a study done in a classroom of only conceptual understanding methods. The experiment involved the teacher giving a purely procedure-based lesson about converting between improper fractions and mixed numbers. The experiment was video-taped, students given an assessment, and then interviewed. Students realized that this was a different teaching method as the class went along. A few weeks later, the conceptual side of converting fractions was taught and similar procedures followed. After calculating the results of the assessments, it was apparent that students retained a far greater deal of information after a long period of time when taught conceptually rather than procedurally. When some students were taught procedurally followed by conceptually, results showed that students’ understanding dropped incredibly. One student, Rachel, said she does not really figure out the problems herself when she tries to do problems procedurally; she is just following the way the teacher tells you to figure it out. On the other hand, when she solves problems conceptually, she understands. This is what she had to say: "So when I figure that out, it's easier, and, um, once I figure it out, it's, it stays there 'cause I was the one who brought it there. So, and it is just easier to do when you figure it out yourself, instead of having teachers telling you." When Rachel was asked to solve a problem in the interview, she first attempted it with the procedure taught by her teacher. She arrived at the wrong answer. When told so, she tried the same problem her own way, arriving at the correct answer. When asked why she solved the problem procedurally first, she said it was because that was the way she learned it first. Thus, the conclusion was made that procedure followed by conceptual teaching is not as effective as purely conceptual teaching.
I thought this was an incredible finding. I do not, however, agree that teaching conceptually is the ultimate method of teaching out there. Obviously, as the article outlines, strictly conceptual teaching is better than strictly procedural teaching. I think that the best way for students to learn is by teaching conceptual and procedural hand-in-hand. This is different from what was discussed in the article because the two types of teaching were done weeks apart from each other. We have already discussed why purely conceptual teaching may not work – it takes more time, not all the material gets covered, not all the students get involved, etc. If some procedural is taught along with the conceptual, then students will really understand the material (from the conceptual part of the lesson) and they will know how to quickly arrive at answers (the procedural part of the lesson), and thus they will do well in school and get throughout the required amount of material. For me, I like it when teachers have taught the conceptual side of concepts so that I can truly know what is going on, followed by the procedural side of things so that I might be able to compute answers quickly and efficiently. I think the key part of this idea is for teachers to teach how the conceptual and procedural connect to each other. This will result in optimum understanding, in my opinion at least.
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This article is really interesting! You did a great job in your first paragraph! The topic sentence was crystal clear and you brought the rest of your paragraph together nicely. You had a lot of specific examples which helped me understand the article better, and you didn't drag on that I got bored. You kept a great tone throughout the whole article and it seemed like you understood your article. Great job!
ReplyDeleteBibliography feedback: italicize the journal title; remove "vol"; italicize the first 2; remove ", No."; put parentheses around the second 2
ReplyDeleteI thought that your summary of the article was very helpful. You had a clear topic sentence that identified the main idea of the paper. You also had a lot of great description in your paragraph that gave a good sense for the content of the paper. I think you could probably have done without the quote, because it was difficult to read and didn't seem to add any more than a quick paraphrasing would have accomplished.
I liked the clear stance that you took toward the main idea of the paper. I am also persuaded that incorporating both types of understanding--procedural and conceptual--are important for learning mathematics. I think this is why Skemp included both in his definition of relational understanding. However, as this paragraph points out, how to introduce and develop both types of understanding may not be easy. In other words, it is not clear that introducing both conceptual and procedural knowledge at the same time will lead to the learning of both of these. In fact, research has shown that often students blow off the conceptual part if they know that they are going to be taught the procedure soon. How do you think you can avoid this happening in your own teaching?
I loved your use of the quote in your summary, that alone caught my attention and makes me curious to read your article. I thought your summary was very well done but because of its length I got a little confused between the procedural and conceptual examples. I agree with what you said in your second paragraph, I think it is most important to have students learn conceptually, but to also connect this learning to procedural methods after they have already learned the material.
ReplyDeleteI thought this article seemed very interesting. You did a really nice job of sounding scholarly. It was a very informative blog that made me want to read the article.
ReplyDeleteIt was maybe a bit long, but I think you did a good job of making me want to keep reading anyways.