This project explores how elementary teachers can teach arithmetic that prompts students’ algebraic thinking. Guided by cognitive research, we analyzed videos of Chinese and U.S. expert teachers’ lessons on inverse relations and properties of operations. Insights suggest an “example-based problem solving” approach, documented in an upcoming book for the field.
Algebra readiness is recognized as an important gatekeeper to future success in mathematics. However, many U.S. students are ill-prepared for the study of algebra, indicating a great challenge facing elementary teachers in preparing students’ for their entry into algebra. The goal of this CAREER project is to identify, from a cross-cultural perspective, essential algebraic knowledge for teaching (AKT) that will enable elementary teachers to better develop students’ algebraic thinking. Focusing on two fundamental mathematical ideas emphasized by the Common Core State Standards—inverse relations and properties of operations—this study explores AKT based on integrated insights of the U.S. and Chinese expert teachers’ classroom performance.
The identification of AKT in this project is innovative because it is the very first study to seek AKT focusing on fundamental mathematical ideas from a cross-cultural perspective. Note that the targeted fundamental mathematical ideas are early algebra topics emphasized by the CCSS. To identify AKT, the project is guided by high-quality cognitive research-based recommendations and on expert teachers’ actual classroom practice in a cross-cultural setting.
Our video analysis indicates cross-cultural differences. An integration of video insights (more heavily from Chinese lessons) suggests the example-based problem solving approach including four major components with the first two related to representation uses and the latter two deep questioning:
- Situating a worked example in a real-world context
- Modeling the real-world context with concreteness fading
- Asking concept-specific questions to promote meaning-making
- Asking comparison questions to promote connection-making
We shared these findings with the project teachers who re-taught their lessons. An analysis of the US lessons shows greater successes in implementing insights about representations than deep questioning.