The goal of this project is to identify Algebraic Knowledge for Teaching (AKT) that is useful to develop students’ algebraic thinking.  Through an exploration of U.S. and Chinese elementary expert teachers’ video-taped lessons, this project has discerned a type of AKT named Teaching through Example-based Problem Solving (TEPS), which is documented in an upcoming book with Routledge (Ding, 2021). This approach emphasizes engaging students in the process of working out an example task through pertinent representation uses and deep questioning (see Figure 1). With regard to representation uses, a teacher may situate a worked example in a real-world context (e.g., word problem), which can be modeled through “concreteness fading.” For deep questioning, a teacher may ask concept-specific questions to promote meaning-making and comparison questions to promote connection-making. TEPS has been illustrated through selected Chinese and U.S. lesson episodes (with 25 annotated video clips) on two fundamental mathematical ideas, inverse relations and the basic properties of operations.

TEPS has been developed based on several lines of research in this project. First, anchored by several cognitive principles recommended by the Institute of Education Science (IES) (Pashler et al., 2007), our project videos reveal cross-cultural differences in teachers’ use of worked examples, representations, and deep questions, which seems to align with different teaching focuses. In particular, Chinese lessons focused mainly on concepts and quantitative relationships while U.S. lessons focused on computation strategies. However, unique representations such as array models occur mainly in the U.S. but not Chinese lessons. This suggests an integration of cross-cultural insights, leading to our identification of the TEPS. 

Second, students’ assessment data also indicates cross-cultural differences in understanding the targeted topics, inverse relations and basic properties. When linking student learning to classroom teaching, we found that instructional quality in terms of teachers’ use of worked examples, representations, and deep questions significantly predicts students’ learning gains. This confirms our choice in developing TEPS based mainly on the Chinese lessons while supplementing with the US lesson insights. 

Third, when the identified TEPS components were shared with teachers, we found teachers in both countries demonstrated strong interests in learning from their international peers. In their re-taught lessons, our project intervention, we found that U.S. teachers widely incorporated the cross-cultural insights especially concreteness fading approach in their instruction. These findings inform the transferability of TEPS in enhancing elementary mathematics teaching.

As an integration of cognitive research and best international practices, TESP contributes to the implementation of the Common Core State Standards. Fundamental mathematical ideas like inverse relations and basic properties are systematically emphasized by the Common Core; however, there is a lack of sufficient guidance on how these fundamentals can be effectively taught in the field. TEPS narrows this gap by providing an evidence-based approach that can be used to develop students’ algebraic thinking and beyond.

TEPS also has a theoretical contribution. This approach takes the middle ground of two long-debated research areas: teaching through worked examples and teaching through problem-solving. As the selected Chinese and U.S. lessons demonstrated, these two debated areas can be seamlessly integrated into a lesson to support students’ initial learning of a mathematical concept. By actively engaging students in working out an example task that is situated in a word problem context, students can construct a mental schema to solve subsequent problems.

Broader impacts of this project include at least four aspects. First, participating teachers in both countries expressed strong interests in their international peers’ lessons, which lays a foundation for follow-up collaboration. After a Chinese classroom visit, some U.S. teachers shared their learning experiences with the field through a NCTM research symposium and other invited talks. Likewise, Chinese teachers published several action research studies based on the project data. Second, we disseminated TEPS to under resourced schools through several PD workshops in the School District of Philadelphia. Teachers outside of this project also expressed strong interests in the proposed approach and made deep reflections on their current curricula. Third, TESP and other project materials (e.g., student instruments) have been incorporated into Temple’s graduate and undergraduate math methods course, which activated preservice teachers’ curiosity about international classrooms and supported their deep learning of conceptual mathematical teaching. Finally, the upcoming book that systematically documents TEPS is expected to make sustained impact in the field. This book can be beneficial for anyone who is interested in research on elementary mathematics teaching including cognitive researchers, mathematics educators, post-graduate students, and inservice and preservice teachers.

Ding, M. (2021). Teaching early algebra through example-based problem solving: Insights from Chinese and U.S. elementary classrooms. Abingdon, Oxon: Routledge.

Pashler, H., Bain, P. M., Bottge, B. A., Graesser, A., Koedinger, K. McGaniel, M. et al. (2007). Organizing instruction and study to improve student learning (NCER 2007-2004). Washington, DC: National Center for Education Research.

Last Modified: 11/22/2020
Modified by: Meixia Ding