### Learning from Teaching Teachers: A Lesson Experiment in Area and Volume with Prospective Teachers

#### Abstract

Learning from the practice of teaching teachers is a promising approach for mathematics teacher educators to attain the extensive knowledge and abilities needed for their roles. As mathematics teacher educators ourselves striving to develop across our professional lifespans, we used one such approach, known as a lesson experiment, to investigate how our instruction affected prospective elementary teachers' conceptual understandings of area and volume. Classroom evidence included audio recordings of ten prospective teachers’ work on lesson activities as well as written work on three post-assessments. Our analysis for the lesson experiment consisted of two phases, first determining to what extent the learning goals were achieved and then evaluating our hypotheses and instruction for how they supported or hindered the learning goals. Findings revealed the prospective teachers enhanced their understandings of area and volume, yet some unexpected complexities in their thinking arose. The lesson experiment led to instructional recommendations for improving the lesson and enhanced our knowledge as mathematics teacher educators. Implications include the value of lesson experiments as a reflective process to help mathematics teacher educators develop over time as well as practical advice for helping prospective teachers enhance their understandings of area and volume.

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