Not Everything Is Proportional: Effects of Age and Problem Type on Propensities for Overgeneralization

TLDR

Overreliance on proportional reasoning is common in elementary and secondary mathematics, leading many students to apply proportional methods across diverse domains. The study examined how the misapplication of proportional reasoning develops with age and educational experience. Researchers administered a paper‑and‑pencil test containing proportional and nonproportional arithmetic problems with a missing‑value structure to 1,062 students from grades 2 to 8. Students overapplied proportional methods when inappropriate, with error rates rising through grade 5, declining from grade 6 onward yet remaining substantial in eighth grade, and varying by the underlying nonproportional model type.

Abstract

Previous research (e.g., De Bock, 2002) has shown that-due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education-many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of proportional reasoning with the age and the educational experience of students. A paper-and-pencil test consisting of several types of proportional and nonproportional arithmetic problems with a missing-value structure was given to 1,062 students from Grades 2 to 8. As expected, students tended to apply proportional methods in cases in which they were clearly not applicable. Although some errors of overapplication were made in the 2nd grade, their number increased considerably up to Grade 5 in parallel with the growing proportional reasoning capacity of the students. From Grade 6 on, students started to distinguish more often between situations when proportionality was applicable and when it was not, but even in 8th grade, a considerable number of proportional errors were made. The likelihood of error varied with the type of nonproportional mathematical model underlying the word problems.