Proportional Relationships Common Core Algebra 1 Homework

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Hi Kevin,

Thank you for the questions. I will answer them in reverse. For the current project, we are not trying to  connect teachers' preparation to student learning. That is down the road. Next fall we will be conducting a study comparing future teachers at UGA and at a second institution in Georgia that also prepares teachers, but not using our multiplication-as-measurement approach. A main measure we will be using the DTMR Fractions survey, developed as part of an earlier NSF project. That survey measures components of reasoning needed to solve fraction arithmetic problems in terms of measured quantities.

As for your second question, we are simply trying to understand how our measurement perspective can help future teachers begin to construct a coherent view of topics related to multiplication. Our ideal goal is for them to understand that whole number multiplication, arithmetic with fractions, proportional relationships, and linear equations are all tied together by an analysis of units and groups of units. Further topics related to geometric similarity and statistics can also be integrated.

The main learning theory we draw from is diSessa's knowledge-in-pieces (KiP) that emphasizes incremental coordination and refinement of diverse knowledge elements. In our case those elements include things like prior associations with terms like "multiply" and "division," meanings for the equal sign, etc. At the moment, it looks likes aspects of coordination class theory (a subset of KiP) will be useful for gaining insight especially into the interview data we are collecting as part of the project. The conjecture is that in order for future teachers to develop a coherent view of multiplication as measurement, they need to develop ways to perceive the multiplicand, multiplier, and product amount in diverse settings. This maybe akin to developing different concept projections and more elaborate chains of inference for establishing the equal-sized groups structure. Other learning theories, such as Steffe and colleagues' radical constructivist accounts of how children learn fractions have some ideas useful for us, but we have not yet found theoretical constructs like schemes to be particularly helpful in understanding the moment-to-moment reasoning we observe.

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