Adding and subtracting vectors: The problem with the arrow representation

Abstract

A small number of studies have investigated student understanding of vector addition and subtraction in generic or introductory physics contexts, but in almost all cases the questions posed were in the vector arrow representation. In a series of experiments involving over 1000 students and several semesters, we investigated student understanding of vector addition and subtraction in both the arrow and algebraic notation (using i, j, k) in generic mathematical and physics contexts. First, we replicated a number of previous findings of student difficulties in the arrow format and discovered several additional difficulties, including the finding that different relative arrow orientations can prompt different solution paths and different kinds of mistakes, which suggests that students need to practice with a variety of relative orientations. Most importantly, we found that average performance in the ijk format was typically excellent and often much better than performance in the arrow format in either the generic or physics contexts. Further, while we find that the arrow format tends to prompt students to a more physically intuitive solution path, we also find that, when prompted, student solutions in the ijk format also display significant physical insights into the problem. We also find a hierarchy in correct answering between the two formats, with correct answering in the ijk format being more fundamental than for the arrow format. Overall, the results suggest that many student difficulties with these simple vector problems lie with the arrow representation itself. For instruction, these results imply that introducing the ijk notation (or some equivalent) with the arrow notation concurrently may be a very useful way to improve student performance as well as help students to learn physics concepts involving vector addition and subtraction.