The purpose of this qualitative study is to investigate middle-school students? understanding of their notation for ratios and to determine, through semi-structured task-based interviews, possibilities for using numerical comparison problems to promote their growth in understanding. The issue of concern is that students use fractional representations of ratios as a convenient notation for solving missing-value problems, but when they use this notation to solve numerical comparison problems, they are unable to interpret and compare the ratios as intensive quantities.
Patterns are reported for students? notation, their problem-solving strategies, their expressions of extensive and intensive quantities, and their use of contextual elements from the problem. A model of ratios and fractions based on Venn diagrams and a general model of ratios and other number-type domains provide the framework for charting students? activities and explanations.
Conjectures based on the data include an association between crossmultiplication and decontextualization and a hierarchy of number types that illustrates students? relative ability to interpret the number types as intensive quantities. Conclusions from the study include recommendations for using numerical comparison problems to give students at different sublevels of quantitative reasoning a stronger conceptual foundation for ratio-related topics. The study demonstrates that students in middle school can make progress in the short term at solving numerical comparison problems using comparisons based on both extensive and intensive quantities, but in response to this short-term intervention, the students demonstrated limited transfer of knowledge across problems.
