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Instructional principles gleaned from cognitive science play a critical role in improving classroom teaching. This study examines how three cognitive instructional principles including worked examples, representations, and deep questions are used in eight experienced elementary teachers' early algebra lessons in the U.S. Based on the analysis of 32 videotaped lessons of inverse relations, we found that most teachers spent sufficient class time on worked examples; however, some lessons included repetitive examples that also included irrelevant practice problems. Most teachers also situated new teaching in concrete contexts, which were faded into abstract representations. However, connections between concrete and abstract were not always made. The largest challenge was rooted in teachers' inability to ask deep questions that elicited students' deep explanations. Some teachers focused on key words and provided students with direct explanations. Implications are discussed. [ABSTRACT FROM AUTHOR]

This study examines how sampled Chinese and U.S. third and fourth grade students (NChina = 167, NUS = 97) understand the commutative, associative, and distributive properties. These students took both pre- and post-tests conducted at the beginning and end of a school year. Comparisons between students' pre- and post-tests within and across countries indicate different learning patterns. Overall, Chinese students demonstrate a much better understanding than their U.S. counterparts. Among these properties, the associative and distributive properties appear to be most challenging, especially for the U.S. students. By the end of grade 4, some Chinese students demonstrate explicit understanding of the associative and distributive properties across tasks; almost no U.S. students achieve a comparable level of understanding on these properties. Student understanding in different contexts also reveals cross-cultural differences. Chinese students tend to reason upon concrete contexts for sense-making, which is rare with U.S. students. Finally, the Chinese student sample shows clear growth of understanding across grades, but this is not seen in the U.S. sample. This understanding gap between the two countries is found to dramatically increase over time. Implications are discussed. [ABSTRACT FROM AUTHOR]