Promoting reinvention of a multiplication-of-fractions algorithm: A study of the Learning Through Activity research program

Whereas proficiency in performing the canonic multiplication-of-fractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifth-grade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplication-of-fractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.