Student Understanding of Cross Product Direction and Use of Right-hand Rules: An Exploration of Representation and Context-dependence

Students in introductory physics struggle with vector algebra and with cross product direction in particular. Some have suggested that this may be due to misapplied right-hand rules, but there are few studies that have had the resolution to explore this. Additionally, previous research on student understanding has noted several kinds of representation-dependence of student performance with vector algebra in both physics and non-physics (or math) contexts (e.g. Hawkins et al., 2009; Van Deventer, 2008). Yet with few exceptions (e.g. Scaife and Heckler, 2010), these findings have not been applied to cross product direction questions or the use of right-hand rules. Also, the extensive work in spatial cognition is particularly applicable to cross product direction due to the spatial and kinesthetic nature of the right-hand rule.;A synthesis of the literature from these various fields reveals four categories of problem features likely to impact the understanding of cross product direction: (1) the type of reasoning required, (2) the orientation of the vectors, (3) the need for parallel transport, and (4) the physics context and features (or lack thereof). These features formed the basis of the present effort to systematically explore the context-dependence and representation- dependence of student performance on cross product direction questions.;This study used a mix of qualitative and quantitative techniques to analyze twenty-seven individual think-aloud interviews. During these interviews, second semester introductory physics students answered 80-100 cross product direction questions in different contexts and with varying problem features. These features were then used as the predictors in regression analyses for correctness and response time. In addition, each problem was coded for the methods used and the errors made to gain a deeper understanding of student behavior and the impact of these features. The results revealed a wide variety of methods (including six different right-hand rules), many different types of errors, and significant context-dependence and representation-dependence for the features mentioned above.;Problems that required reasoning backward to find A&ar; (for C&ar;=A&ar; xB&ar; ) presented the biggest challenge for students. Participants who recognized the non-commutativity of the cross product would often reverse the order ( B&ar;xA&ar; ) on these problems. Also, this error occurred less frequently when a Guess and Check method was used in addition to the right-hand rule.;Three different aspects of orientation had a significant impact on performance: (1) the physical discomfort of using a right-hand rule, (2) the plane of the given vectors, and to a lesser extent, (3) the angle between the vectors. One participant was more likely to switch the order of the vectors for the physically awkward orientations than for the physically easy orientations; and there was evidence that some of the difficulty with vector orientations that were not in the xy-plane was due to misinterpretations of the into and out of the page symbols. The impact of both physical discomfort and the plane of the vectors was reduced when participants rotated the paper. Unlike other problem features, the issue of parallel transport did not appear to be nearly as prevalent for cross product direction as it is for vector addition and subtraction.;In addition to these findings, this study confirmed earlier findings regarding physics difficulties with magnetic field and magnetic force, such as differences in performance based on the representation of magnetic field (Scaife and Heckler, 2010) and confusion between electric and magnetic fields (Maloney et al., 2001). It also provided evidence of physics difficulties with magnetic field and magnetic force that have been suspected but never explored, specifically the impact of the sign of the charge and the observation location.;This study demonstrated that student difficulty with cross product direction is not as simple as misapplied right-hand rules, although this is an issue. Student behavior on cross product direction questions is significantly dependent on both the context of the question and the representation of various problem features. Although more research is necessary, particularly in regard to individual differences, this study represents a significant step forward in our understanding of student difficulties with cross product direction.