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Aki Murata and Chana Stewart

This set of lesson examples demonstrates effective uses of magnets, number lines, and ten-frames to implement practice standards as first graders use place value to solve addition problems.

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Aki Murata and Karen Fuson

The framework of Tharp and Gallimore (1988) was adapted to form a ZPD (Zone of Proximal Development) Model of Mathematical Proficiency that identifies two interacting kinds of learning activities: instructional conversations that assist understanding and practice that develops fluency. A Class Learning Path was conceptualized as a classroom path that includes a small number of different learning paths followed by students, and it permits a teacher to provide assistance to students at their own levels. A case study illustrates this model by describing how one teacher in a Japanese Grade 1 classroom assisted student learning of addition with teen totals by valuing students' informal knowledge and individual approaches, bridging the distance between their existing knowledge and the new culturally valued method, and giving carefully structured practice. The teacher decreased assistance over time but increased it for transitions to new problem types and for students who needed it. Students interacted, influenced/supported one another, and moved forward along their own learning paths within the Class Learning Path.

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Laura Bofferding, Melissa Kemmerle and Aki Murata

Focusing on differentiation, three teachers push all their kindergartners toward recognizing relationships among combinations.

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Aki Murata, Laura Bofferding, Bindu E. Pothen, Megan W. Taylor and Sarah Wischnia

This study investigated how elementary teachers in a mathematics lesson study made sense of student learning, teaching, and content, as related to using representations in teaching multidigit subtraction, and how changes occurred over time in their talk and practice. The lesson-study process paved a group talk path along which teacher talk shifted from superficial to deeper consideration of student learning. By providing a context in which interactions of diverse ideas drove teacher learning, lesson study facilitated teachers making connections between the craft knowledge of teaching and scholarly knowledge. Individual teacher talk paths varied within the group path, and one teacher's learning path and the interaction of different learning paths is discussed.