Leveraging Key Developmental Understanding of Early Proportional Reasoning Learning Trajectory Research in Modeling for Mooncakes

• In Event: Partnering With Practicing Teachers to Strengthen Inclusive Instruction Using Learning Trajectories

Abstract

Using Lesson Study with practicing teachers and a mathematics coach, mathematics educators introduced learning trajectories(LTs) research around early proportional reasoning and utilized Simon’s (2006) key developmental understanding to analyze students’ flexibility and efficiency in proportional reasoning situations. The teachers and researchers hypothesized that connecting to students' cultural and cognitive assets in the meaningful context of cooking for a holiday would support successful strategies for proportional reasoning. We analyzed the interplay between teachers’ focus on the LT and students’ modeling capacities to study how students used a modeling process to scale up specific recipes and subsequently create a general rule for scaling up. Celebrating a festival and making mooncakes for a party became the centerpiece of the modeling activity where two classes of third-grade students took a recipe and had to figure out how to make enough mooncake for their celebration.
Our research questions were: 1) In what ways does mathematical modeling, connected to students' cultural knowledge, provide opportunities to move students along a proportional reasoning learning trajectory? 2) How does a LT-focused Lesson Study support teachers’ and coaches’ learning as they teach and elicit students’ key developmental understanding?
The class was presented with a dilemma entitled Mooncake Modeling. The only information given was a recipe for 6 mooncakes. The teacher posed a modeling routine called ‘Notice and Wonder’ to engage students in making assumptions and defining some important variables to consider. Students proposed the following modeling question, “How can we make enough mooncakes for the whole class?” There were measuring realias like the measuring cups, teaspoons and liter containers. While the class explored the problem, the teacher seized a “catch and release” opportunity to capitalize on a mathematical moment. She sensed students were grappling with how the recipe yielded 6 servings and introduced the term “batch” to build collective knowledge around a composed unit. Students used an additive and a “build up strategy” that connected to the multiplicative understanding necessary for the scalar perspective (Carney et al., 2019). The teacher used the additive approach as well as the emerging multiplicative thinking from students to consider a generalized rule for scaling up the recipe. Much like other studies (Steinthorsdottir et al., 2009, Lobato et al., 2010), classroom observations demonstrate students moving back and forth between additive and multiplicative strategies in their early proportional reasoning using a build-up strategy for the ratio with addition, multiplication, or some combination of operations.
Using LTs in lesson planning, supported teachers in providing access to content for students who range across a diverse learning continuum, while maintaining the rigor of the mathematics. The use of LTs was critical in moving classroom discourse to a generalizable rule of calculating how many batches are needed for their party. Combining LTs with a modeling context provided scaffolds to support student success, elicit students understanding, and preview future learning of proportional reasoning. The use of LTs strengthened teachers’ ability to support, recognize, and celebrate student work, acknowledging their strengths in using mathematical knowledge.

Authors

• Jennifer M. Suh, George Mason University

• Laurie Capen, George Mason University

• Sara Birkhead, George Mason University