The Deficits of Students’ Orientation in Solving Proportion Problems, as Revealed through Task Modifications

BACKGROUND: Using the Activity Theory of education (Galperin, 1992; Talyzina, 2018), this article examines the students’ actions that constitute the early stages of forming the concept of ratios. The psychological analysis of mastery of this concept shows that it essentially depends on understanding the coordination of the changes of two independent values (area, velocity, density, etc.). OBJECTIVE: The present research considers differences in students’ operations with numbers on various tasks, based on their comprehension of ratio relations (direct and inverse proportions); these differences are revealed through posing certain modified tasks, but may stay unnoticed in regular tasks. The goal of the study was to identify the criteria for adequate assessment of the sustainability of the students’ orientation in modified tasks. DESIGN: A test of 15 tasks was designed based on Galperin’s classification of task variations: domain specific, logical, and psychological. The formulation of the tasks disguised the operations needed to achieve the right answer, and sometimes even prompted the wrong solution. There were 12 tasks on direct proportions — four sample and eight modified; and three inverse proportion tasks: one sample and two modified. One hundred sixty (160) students (5–6th grade, 11–13 years old) took the test in writing. RESULTS: The comparison of students’ performance on the sample and modified tasks showed significant differences. Modifications impaired the students’ performance on both types of proportion problems (direct and inverse). Logical and psychological modifications had the most impact on the quality of the students’ orientation and thus proved to be most indicative in terms of students’ orientation quality assessment. CONCLUSION: The data suggest the following: 1) that the concepts of proportionality which the students acquired from a regular school curriculum lack “generalization,” and 2) that students’ ability to apply the ratio concept is very sensitive to the way the word problem is presented. These findings are essential for evaluating students’ multiplicative thinking: their actual level of comprehension cannot be revealed through their performance on regular tasks.