Student and teacher reasoning on rational numbers, multiplicative structures, and related topics have been heavily researched by the international community. Despite all of this work, it remains a challenging area for students to learn based on data from international studies (PISA, TIMSS). My research group has compiled a massive database on this research (n>500 studies) and have conducted syntheses across the five strands of multiplication and division: area and volume; fractions, ratios and proportion, and rate; decimals and percent; and scaling and similarity. A pool of assessment items from these studies has also been accumulated. In this lecture, I will report on how the syntheses were conducted, describe the definitions of terminology proposed as a result of the review, summarize the empirical findings and provide information on how others might access the database. In addition, I will report on our approach to using the database to propose a set of learning trajectories for each subarea as warranted by a review of research and how we are linking these to assessment formats to build
diagnostic measures of student progress.

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