Student performance and consistency on dot product tasks in mathematics and physics contexts

As part of an ongoing project exploring student understanding of vectors in physics and mathematics, we discuss multiple-choice and free-response data from vector product surveys administered in introductory physics courses. In each course, students determined the magnitudes and directions of vector products in mathematics and physics contexts. We focus here on tasks related to dot products. Performance depends on the context. For example, while calculating the magnitude of the dot product in math and physics contexts, performance ranges from 28% to 59%. Furthermore, students do not always answer consistently – correctly or incorrectly – between two contexts or on the magnitude and direction questions within one context, ranging from 15% to 58% consistency across tasks. We coded students' free-response explanations and identified four main computational strategies that students used to calculate magnitudes – using a trigonometry-based definition, decomposing and multiplying corresponding components, scalar multiplication, and addition. Trigonometric definitions are the most common approach, but do not always correspond to a correct multiple-choice selection. Overall, a correct calculation does not guarantee a correct interpretation of the type of quantity (scalar or vector). To better understand the nature of student reasoning in these tasks, we are exploring variations in response times, inspired by the dual-process theories of reasoning framework.