The purpose of this study is to investigate how two 7^th grade students with different concepts of ratio reason about change in covarying quantities in quadratic function problems and how these differences affect the process of expressing algebraic equations. To this end, we selected two 7^th grade students who had no formal instruction in quadratic functions and four clinical interviews (each of which lasted about 90 minutes) were conducted every week from December 2019 to January 2020 with each of them. Our analysis revealed that one student whose conception of ratio was at the level of additive comparison attended to differences in amount of change in quadratic problem situations and interpreted the problems additively by repetitively adding the differences, whereas another student who operated at the interiorized ratio level could perceive the problem situations as the differences in amount of change per unit time, which resulted in their distinctive ways of expressing quadratic functions as equations. We have finalized our study by suggesting an explanatory hypothesis about students’ ratio concepts that seem highly pertinent to their ability to interpret quantitatively and to express equations of quadratic functions.

Keywords비,내면화된 비,내재화된 비,양화,양적 추론,변화율,이차함수,ratio,internalized ratio,interiorized ratio,quantification,quantitative reasoning,rate of change,quadratic function