Future Directions

This book documents, from a cross-cultural perspective, how worked examples can be unpacked deeply by engaging students in the problemsolving process in elementary mathematics classrooms. As explained in Section 1.3, this approach is different from both a traditional approach to worked examples and a pure problem-solving approach with minimum teacher guidance. Future studies on the TEPS approach or its components can be expected to contribute to both cognitive science and mathematics education research. Given that the TEPS approach draws more heavily on Chinese lesson insights, it is important to follow up on teacher implementation of this approach in cross-cultural settings. As mentioned above, in my project, we introduced the TEPS components to teacher participants who then re-taught their lessons. Our video analysis of some lessons indicates both successes and challenges (Ding et ah, 2021, April). For instance, we found that teachers were more skillful in implementing representational strategies than asking deep questions (especially follow-up questions). We will continue this line of video analysis to explore how this approach functions in the classrooms of expert teachers after they receive some short exposure to the ideas documented in this book.

Future research is needed to explore how the TEPS approach is applicable in average in-service teachers’ classrooms. In my project, all the teacher participants have been evaluated as expert teachers according to either local or national standards. Their strong interest in TEPS demonstrates that this approach holds promise; however, their follow-up implementation suggests the need for further support. Perhaps a short summer workshop is not sufficient to establish new teaching habits. More sustained support from researchers will be the key to finding out the limits and potential of this approach. Now with the book available to the field, researchers may discuss the TEPS approach with teachers through workshops while teachers can then use this book as an instructional resource for ongoing support beyond the workshops. In this sense, it is also possible to explore how the proposed approach, as documented in the book, plays a role for supporting teacher learning and can be adapted to their mathematics teaching routines.

Future research might also explore how TEPS can provide learning support to preservice elementary teachers (PSTs). PSTs are future teachers who are just starting their professional learning cycle in college classrooms. On the one hand, PSTs do not have entrenched instructional behaviors that are resistant to change; on the other hand, PSTs often do not have sufficient prior knowledge in mathematics teaching, which could, in turn, hinder their understanding of this targeted approach. As such, it would be interesting to explore whether PSTs learn this targeted approach differently from expert or experienced teachers and how PSTs may be better supported in learning the instructional insights gleaned from our cross-cultural best practices. This direction of future research and practice may also be applied to novice teachers who are just starting their teaching career.

Despite the promise of the TEPS approach documented in this book, I should acknowledge again that this book focuses on worked examples rather than other related elements such as practice problems. Although high-quality teaching of worked examples is a necessary condition for effective learning, it is not sufficient if the established worked example- effect is not reinforced and enhanced through subsequent practice. In this sense, there is a need to explore how example tasks may be effectively shifted to practice problems with careful choices of tasks and instructional design. Existing research indicates that “variation” (Huang & Li, 2017) and “coherence” (Cai, Ding, & Wang, 2014) are key aspects of Chinese lessons. These techniques are evident in textbook design (Ding & Li, 2010; Ding & Li, 2014; Li et al., 2008) and were observed in our project lessons during the teaching of worked examples. In fact, our project teachers who watched the worked example portion of these lessons expressed interest and curiosity in what happened during the follow-up practice in their international peers’ lessons. Moving forward, it might be fruitful to explore how teachers could transition worked example tasks to practice problems with variations and coherence and how teachers may provide feedback to students to reinforce the coherent underlying concepts through varied practices. The TEPS approach documented in this book therefore exists as a starting point for future research and not as a fully formed end.

In conclusion, I envision that the TEPS approach has great potential for supporting meaningful and deep teaching of early algebra and beyond. To digest and implement this approach, teachers might need collective support from mathematics educators, researchers, professional developers, and textbook designers. Instructional changes can be expected only with continuous collaboration and ongoing research. Consequently, children’s mathematical thinking, especially in the area of algebra, can be enhanced with the support of sustained collective effort.

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