This six month study investigated the similarities and differences between 137 fourth graders' understanding of and reasoning about multiplication, division, and proportion tasks. All students were administered a pretest mid-year and a posttest at the end of year. A sample of 18 students was interviewed individually on tasks involving reasoning about multiplicative and proportional relationships. This study provides evidence that students who are encouraged to use invented strategies for multiplication and division based on number relationships have a better understanding of the meaning of those operations and are more successful in extending their knowledge to proportional reasoning tasks than are those students who are taught conventional procedures exclusively. Contains 42 references. (Author/NB)