Memo #2

Part I: Summary
All three of Lamon's articles focus on proportional reasoning, recognizing it in students work, helping students obtain it and becoming better at teaching it. Chapter one of Lamon's book discusses the topic of the books and breaks down proportional reasoning and what it means to understand it. Chapter two focuses changing student thinking, especially in terms of multiplicative thinking. Even when discussing fractions at the elementary level, Lamon emphasizes that teachers must connect that a number means nothing unless it is compared to something else (5 people in car vs. 5 people in a stadium). Lamon furthers this discussion by bringing up the topics of unitizing and norming. In later chapter Lamon studies the actual work of students on a variety of problems. Dissecting their proportional reasoning and seeing how their responses connect to what they may know and the gaps in what they need to know was also very insightful.
Lowery's article was discussing an activity that students would do help them develop a conceptual understanding in proportional reasoning. The actual activity is mentioned below in  question 3.

Part II: Reading Questions
1. What is proportional reasoning?
According to Lowery, "Proportional reasoning is the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree," (Lowery 1). Throughout the Lowery and Lamon readings, examples assessing proportional reasoning have been considered. Proportional reasoning is said to be expressed if students can use a comparative relation (observable in ratios, fractions, etc) to determine the outcome of a similar scenario. As stated by Lamon, developing a proportional reasoning requires skills/reasoning in multiple areas of cognitive thought processes. Students should have skills and knowledge of partitioning, relative thinking, quantities and change, unitizing, an understanding of rational numbers, and a sense of ratios. Because obtaining a keen proportional reasoning involves development in many areas, Lamon admits "the answer is, that we cannot say, in a very concise way what proportional reasoning is, nor can we say how a person learns to reason proportionally," (Lamon 6).

2.What are the central concepts and connections for teaching proportional reasoning?
As mentioned above, Lamon had a harder time defining proportional reasoning because there are so many aspects to consider when teaching for proficiency in proportional reasoning. Tying into last week's reading, the entire network of knowledge gets stronger and more cohesive as more connections are made between bits of knowledge. The same idea applies for proportional reasoning; as students acquire skills in partitioning (and realize a unit can be changed because of this), quantities and change (realizing a question can be posed based on relative or absolute conditions) and so forth in areas of ratios and rational numbers, their proportional reasoning will strengthen.

3. What are recommendations for teaching this topic for understanding?
One of the most emphasized recommendations was for teachers to help their students get comfortable with a unit being a value other than one. Helping students realize that a unit doesn't necessarily have to be one cupcake or one pizza or one soda can will allow them to make inquiries on their own and discover other ways of solving a problem. Also, teachers should state that there are multiple ways to solving a problem and encourage students to explain what they did in multiple representations (a picture, symbols, tables, etc.) this will help the teacher to identify areas of need for each student. Of course teaching with multiple representations is necessary, even more so with proportional problems because so much is involved when solving them.
One useful tool that stood out to me was the Lowery Proportional Activity that involved finding the heights of Mr. Small and Mr.Big using jumbo sized paper clips and smaller sized paper clips. This activity gave the student a drawing of Mr.Small and asked the student to find out how tall he was using the jumbo paper clips as the unit. Mr. Big's height in jumbo paper clips was already given. Next the student was given the smaller paper clips and was asked to determine the height of Mr. Small by using the small paper clip as the unit of measure. Finally, the student was asked to predict the height of Mr.Big in terms of the smaller paper clip. This activity seems like a good activity for developing a proportional reasoning because the student is able to use manipulatives, physically change a unit and measure a drawing without having to deal with the messiness of numbers/rulers/complex units. This activity is very basic but gets to the conceptual understanding of proportionality so as a teacher I would want to use or develop a similar activity that would get students thinking outside the box and focusing on something other than getting the right answer.
Common student difficulties often lay in the ways in which a question was asked. For instance, a question that was asked in the Lamon article involved drinking 75% of a case of Coca-Cola. Student responses were all over the place because students had different ideas of what a case of coke consisted of (24 cans, 12 cans or a 6-pack). I think, especially in the introductory stage of teaching proportionality, I would state the unit being used for the problem and slowly have students progress to working on problems where the unit is not as obvious.
Another common misconception is student thinking of a question too literally. For instance, "two trees  of different heights both grow 3 feet over some time, which one grew more?" can be thought of differently. I would want to make sure my students knew the difference between absolute or relative conditions.