Hi Wendy,

Thank you for your interest!

We are running the project with future mathematics teachers in two programs, middle grades (Grades 4-8) and secondary grades (Grades 6-12). In the middle grades program, future teachers take a standard first calculus course and 2 mathematics for teachers courses. One on number and operation and one on algebra. The video clip we included is from early in the algebra course (so think second semester). Most of the number and operations course focuses on the meaning of multiplication with whole numbers shown in the clip, properties like commutativity, and how to think about multiplication in consistent was across whole numbers and fractions.  Students also study both types of division with whole numbers in the first course. At the beginning of the second course they extend the study of division from whole numbers to fractions. 

Future teachers in the secondary program, not shown in the video, complete most of a math major including a course in abstract algebra. They need a whole semester to go from whole number multiplication and division, to fraction multiplication and division, to proportional relationships and linear equations. We suspect that most secondary preparation programs do not devote adequate time to building this foundational content in a coherent way. 

Sybilla teaches the future middle grades teachers and I teach the future secondary teachers. I do not find much resistance from future secondary teachers. I open the course reminding my students that are being prepared to teach grades 6-12 and so are responsible for the content. I also tell them about an experience I had several years tutoring a 10th student who was having trouble with geometry homework not because he did not understand the geometry content, but because he did not understand fractions and division. Then I ask students to give word problems that illustrate fraction division. They find this hard, and I have convinced them that the course is important. 

Hi Miriam,

Thank you for the question. The courses in which we work are content courses. The preparation programs also include separate methods courses (we refer to them as pedagogy courses). As such, we do not focus as much in the courses on how students think, though students' ideas do come up. I would say the most prevalent and explicit pedagogical component is being able to give clear explanations for solutions to problems that make use of drawn models, especially double number lines and strip diagrams, to show quantities. The video clip includes one nice example of this. Future teachers are learning how to use such models to support their own thinking and as a resource for explaining their thinking to others. Most future teachers who come to our courses have only been taught rules for calculating with numbers. Reasoning with drawings is new for them.

Thanks! this is really interesting work!

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.

Hi Jeremy,

Thank you for the questions about the long-term trajectory of our work beyond the current project. I have answers for two of them.

With respect to curricula, in the short term, Sybilla is taking lessons from our research and incorporating them into her math for elementary teachers textbook, now in its 5th edition. We also think that a main implication of our approach for curricula is that teaching topics like fraction division in isolation is less likely to be successful than an approach like ours which attempts to address coherence across many topics in the multiplicative conceptual field. Ultimately, we would like curricula both for future teacher and for students to exhibit such coherence. 

We are interested in assessment. For the current project, we are investigating the sensitivity of the DTMR Fractions survey to detect pre-post shifts in profiles. There are pieces about this survey in EM:IP and the recently published JRME monongraph on psychometrics and mathematics education. The survey is based on diagnostic classification models, not IRT. One of the things that might come out of our project is refined ideas about core components of reasoning to assess, how to assess them (e.g., item design), and what kinds of psychometric models might be good matches.

Finally, with respect to district leaders, we do have as part of our dissemination plans to present to a conference of GA superintendents. We are interested in learning about what resonates with them.