Posted at Nov 16/2018 04:30PM by Stu 4:
This is my fourth attempt to write out a question, and I think I might finally have a better handle on what I want to ask. I am asking this question personally so that I can understand the usefulness of learning trajectories.

It seems to me that Ellis's learning trajectory diagram is too complex to encapsulate in a 30 page article. She could have likely written several articles (or even a book) just explaining the graphic on pg. 160. Her conclusions were relatively straight forward and easy to follow, but the inclusion of the learning trajectory and the 1-2 paragraph summary of each level was too much information for me to keep track of.

Personally, I felt that the complexity of the learning trajectory diagram detracted from the message of the article. My confusion only increased when Ellis repeatedly stated that this complex graphic should only be considered as representative of a small population and was certainly not exclusive.

Maybe I missed something. But I think that I would have gotten the same ideas from the article without the learning trajectory graphic and the inclusion of her codified structure.

In short, what is the purpose for including such a complex, 18-part diagram detailing possible permutations of ideas a student might move through as they interact with exponential functions? When she said what she meant in the discussion I felt like it was so much easier to follow.

Finally, I have heard the term learning trajectory before but am not familiar with it. From the way the author discusses them, it sounds like a popular thing to try and come up with. Are they all this complex? If so, how do we expect classroom teachers to absorb and implement them?

Posted at Nov 16/2018 04:39PM by Stu 4:

I think that my biggest question is whether or not her diagram was necessary to convey her ideas. Maybe this is more a question about organization and presentation of data in a way that is sensible to the reader....

Posted at Nov 17/2018 11:53AM by Stu 1:
Stu 4, I think you pose an interesting question here. I want to begin by stating that I too agree that the diagram presented is quite complex. I think what is important to note is that a researcher probably would not present a teacher with this model. Rather, I believe this diagram was created to demonstrate the coding scheme that the authors used to analyze students' progression through the learning trajectory. Further, in the beginning of the article, the authors mentioned that a learning trajectory does not describe the only path of learning or even the best path of learning. Rather, it presents a possible characterization of students' learning over time. I believe this diagram demonstrates this. Also, the authors note that students' covariation and correspondence views of exponential function coevolved as they progressed through tasks. This diagram also demonstrates this. I feel that learning trajectories are an important part of mathematics education research because they provide a model of students' initial understanding of concept, how this understanding evolves, and what interventions (mathematical tasks, tools, representations, and teaching actions) promoted the students' concept development. This is extremely useful for teachers as it provides them with insight into students' understanding of a concept, which can then help teachers adjust their pedagogical choices and actions. As far as the complexity of other learning trajectories, I will do more reading for my final project and get back to you! :)

Posted at Nov 17/2018 02:36PM by Stu 1:
I am having trouble unpacking this statement in the Confrey paper: "Ratios are never singular instances of a relationship between magnitudes but are constructed by objectifying and naming that which is the same across proportions. To recognize a ratio is to recognize the homogeneity of ratio across more than one instance" (pg 74). Can anyone help?

Posted at Nov 17/2018 03:21PM by Stu 5:
@Stu 1, this is what I'm thinking about those sentences on ratios. I think they're saying that ratios describe what is constant between different proportions. The "objectifying and naming" part reminds me of what we did in class. Once we named what we were looking for (e.g. a distance from y to the midpoint), we were able to describe the situation more clearly and see it represented in different situations. To them, recognizing a ratio is recognizing that the ratio stays the same in some situations even if the representation changes. Does that help?

But, I'm still iffy about what a proportion is. Proportions can compare a part to a whole, but I think they can mean more than that. So, a discussion about what proportion means might be helpful too.

Posted at Nov 18/2018 05:58AM by Stu 5:
Okay. I think I have a definition of proportion.

A and B are in proportion if there exists a constant X such that A=XB.

For this paper, it's more useful to think of that definition as X=A/B. For example, pi=(Circumference)/(Diameter).

Does this seem right?

Posted at Nov 18/2018 02:05PM by Stu 9:
@Stu 5, I agree with your answer to @Stu 1's question. To Confrey and Smith, a "Ratio" is akin to Pat's construct of Rate, i.e. a reflectively abstracted Ratio. They even cite the difference I believe.

Posted at Nov 19/2018 05:13PM by Stu 6:
@Stu 1, I am going to take a chance and try to put Confrey & Smith's ratio into my own words. On page 73 they state "ratio is viewed as a description of the invariance across a set of proportions." It's as if the student "sees" the ratio is constant when looking at similar (proportional) geometric shapes. This coincides with students thinking about (performing) splitting actions. The action of repetitive doubling is seeing the constant "double" as invariant across the mental image. Thus, pi is the constant ratio when looking at similar circles. Again, this is my very shaky stab at the concept.

Posted at Nov 20/2018 07:35AM by Stu 8:
I think I get what you're saying. I think to them ratio is necessary for similarity. I would say that Confrey and Smith argue that students conceive of a ratio as a mathematical object; if they think of a circle as a ratio, for example, they think of the invariance of proportion between the radius and circumference of all circles.