Posted at Sep 15/2018 04:28PM by Stu 7:
I will try and get this discussion started by saying that the article covered what I wrote about but went in a different direction. I discussed in my essay that Bill "did the work for her" in attempting to make the connection between distance, time, and speed; whereas the article seemed as if it explained what Bill was trying to get Ann to understand. It also included what Ann would have had to "see" in his explanations in order to keep up with his explanations. The paper describes to the reader what Ann would need to understand in order to conceptualize what Bill was attempting to portray. I, on the other hand, was arguing that it was important for Ann to make these connections on her own with teachers there to provide guidance rather than have the connection handed to her.

Posted at Sep 16/2018 05:40PM by Stu 2:
Yes, I agree with your statement Stu 7. But also, with the guidance of Bill's explanations, Bill's use of mathematical language was mostly the downfall in the lessons. As the book mentioned, Bill understood in his own natural way the connection between speed, time, and distance through the operation of division. And because he was so static on not having her reflect on her methods of solving for a quantity, Bill assumed that if she could do the mechanics, then she "must" have the correct conceptual understanding behind the operation. She could have definitely understood the connections on her own, but only if he would have clarified her reasonings behind the math rather than moving onto the next example.

Posted at Sep 17/2018 08:39AM by Stu 3:
I think in Bill's comment that he thought Pat's suggestions about where Ann would struggle was excessive was the start of the issue. He didn't see the problems as different and was completely baffled that she so quickly came up with division in the first problem type and that even with heavy guidance wasn't seeing the 'correct operation' for the second problem. In my understanding of the article what was needed was to spend more time before ever even starting the problems set helping Ann to understand the relationship between distance, rate, and time and using contextualized words to describe them as opposed to a focus on operations. Bill did not understand the difference between using contextualized words to describe the situation and the operations (as to him all the context was in the correct choice of operation). In his frame of mind I am not sure having her reflect on her methods would have changed much. He saw what she was doing as equivalent to what he was thinking. He needed to first understand the difference in what their use of division on this situation meant in order to offer any guidance that Ann would be able to follow.

In general, I hadn't really focused on anything different then the article (except that I was more focused on Ann to explain what was happening then on Bill). The article however definitely expanded on the situation around what we observed and explained in much greater depth what I had been trying to communicate.

Posted at Sep 17/2018 09:29AM by Stu 8:
I agree that overall, the article explained many ideas that I had but with more elaboration and further ideas that I had not previously considered. I think the context of knowing Bill's background knowledge, the purpose of the study, and the preparation involved beforehand provides an opportunity to elaborate on and articulate much more thoroughly some ideas about what was happening in the lessons.

On the other hand, I think there were certain details not mentioned in the article that I thought to be important. The likely reason for this, in my opinion, is that the primary aim of the article was to give lay readers (or beginning math ed readers) as well as experts in the field an overarching idea of the first two lessons and their shortcomings in helping Ann to understand rate as they were reflected on, rather than to point out this and that detail, analyzing how it may have contributed to or taken away from a lesson. An example of one of these details was when Bill did not implore Ann to use units in excerpt 3 when she says "I would take forty, right?". While the article reflects on this excerpt as focusing on "Ann's reasoning and her understanding of the quantities", I think this along with other examples provides evidence of focusing more on arithmetical calculations, though I acknowledge that there was more emphasis on conceptual reasoning in this excerpt than some other portions of the lesson to be sure.