Posted at Sep 14/2018 02:43PM by Stu 4:
I have been thinking about the difference between partitive and quotative division, which the article talks about at the bottom of page 18. I'll be honest, I had to look up the definitions again because it has been a long time since I studied models for performing operations as an undergrad.

Quotative division, or repeated subtraction, can be used to explain how Ann came up with the correct answer when she had a speed-length with which to determine the time it takes to go over and back. Ann keeps a mental tally of how many times she uses the measuring stick to divide up the interval, which gives her a time in seconds.

She runs into difficulty when she has to use the partitive method of division, which is the fair sharing model. How can 200 feet be evenly shared between 6 seconds? This is the mental process she would need to go through to conceive of speed. Recognizing that 33 1/3 feet is the distance that can evenly be shared between each of the six seconds is not something she could come up with. She wanted a ruler with which she could perform repeated subtraction.

Bill inherently and subconsciously understood both methods of division, but had likely never explicitly studied the models before. Thus, it makes sense to me that he could not differentiate the methods in his mind; they existed under a single umbrella of "division." He lacked the depth of understanding in models of operations to fully explain to Ann anything more than "divide two quantities."

Posted at Sep 15/2018 12:10PM by Stu 1:
Ann's use of speed-lengths is similar to the idea of a measuring stick that we have been discussing in Pathways. The speed-lengths are what Ann used to measure time. When she was asked to do the reverse process (i.e. finding speeds) she could no longer use this measuring stick that she had created for herself.

Posted at Sep 15/2018 12:23PM by Stu 9:
@Stu 4, did he lack the understanding, or was his understanding so strong that he believed a single example was enough to communicate to Ann the distinction between the operations? bottom of page 18

Posted at Sep 15/2018 03:11PM by Stu 4:
@Stu 9, I don't think he lacked the understanding. But I think the understanding was subconscious. I don't think he would have been able to articulate the difference in division methods without resorting to putting the formula on the board and talking about dividing on both sides to isolate a quantity.

As a 6th grader, I don't think Ann would have been capable of understanding inverse operations yet. Thus, the method Bill favored was beyond Ann's comprehension. Bill needed another method to explain the situation, but could not readily find one.

In the second full paragraph of page 20, the authors note that Bill has a "consistent difficulty in using everyday language to discuss mathematical ideas." This further reinforces the idea in my head that Bill is thinking about operations on an algebraic equation, rather than how the contextual quantities are being split up in a given situation. In other words, he is thinking about a mathematical process rather than a model to perform a real-world task.

Do my ideas seem coherent? I want to make sure that I'm getting my point across...

Posted at Sep 15/2018 03:57PM by Stu 7:
@Stu 4 I totally agree with what you are saying. I think this ties what we were saying in class to Bill's thought process. His thoughts became automatic because maybe he never had to struggle with the idea of what division means in these different models. It just came "naturally" as the article says. Therefore, breaking his own thought process down was much more difficult when trying to teach it to someone else, in this case someone who couldn't articulate his thoughts without more steps in between.

Posted at Sep 17/2018 07:18AM by Stu 5:
@Stu 7

I never heard of partitive and quotative division until now. When explaining the meaning of division, I think I switch between the two types assuming they mean the same thing. I think Bill has a similar problem. The two ideas are so connected in his mind that he doesn't recognize that Ann is only thinking of division in the quotative sense.

Posted at Sep 17/2018 03:39PM by Stu 7:
@ Stu 5, Yes what you said is exactly what I mean. Bill doesn't understand why Ann's ideas don't lead her to see all of the connected meanings that he has constructed because he maybe never struggled with it.

Posted at Sep 17/2018 04:00PM by Stu 5:
The main focus of this article is that communication between student and teacher (or between anyone) is /hard/. My previous summary focused on Ann’s ways of thinking and how Bill interacted with her. But, I never mentioned their inability to understand what the other was trying to explain. As said in the beginning of the article, people in a conversation may think that they understand each other even when that is not the case. When viewing the videos, I focused on specific interactions rather than an overall interpretation of what was occurring. Does anyone else feel like they had this problem?