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 (usingî,ĵ,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.

We investigate student confusion of concepts of electric and magnetic force. At various times during a traditional university-level course, we administered a series of simple questions about the direction of force on a charged particle moving through either an electric or a magnetic field. We find that after electric force instruction but before magnetic force instruction most students answer electric force questions correctly, and we replicate well-known results that many students incorrectly answer that magnetic forces are in the same direction as the magnetic field. After magnetic force instruction, most students answer magnetic force questions correctly, but surprisingly many students incorrectly answer that electric forces are perpendicular to electric fields, as would happen if a student confused electric forces with magnetic forces. As a further indication of interference between electric and magnetic concepts, we also find that students’ responses depend on whether electric or magnetic force questions are posed first, and this effect depends on whether electric or magnetic force was most recently taught

We study student understanding of the direction of the magnetic force experienced by a charged particle moving through a homogeneous magnetic field in both the magnetic pole and field line representations of the magnetic field. In five studies, we administer a series of simple questions in either written or interview format. Our results indicate that although students begin at the same low level of performance in both representations, they answer correctly more often in the field line representation than in the pole representation after instruction. This difference is due in part to more students believing that charges are attracted to magnetic poles than believing that charges are pushed along magnetic field lines. Although traditional instruction is fairly effective in teaching students to answer correctly up to a few weeks following instruction, especially for the field line representation, some students revert to their initial misconceptions several months after instruction. The responses reveal persistent and largely random sign errors in the direction of the force. The sign errors are largely nonsystematic and due to confusion about the direction of the magnetic field and the execution and choice of the right-hand rule and lack of recognition of the noncommutativity of the cross product.

We report on five experiments investigating response choices and response times to simple science questions that evoke student "misconceptions," and we construct a simple model to explain the patterns of response choices. Physics students were asked to compare a physical quantity represented by the slope, such as speed, on simple physics graphs. We found that response times of incorrect answers, resulting from comparing heights, were faster than response times of correct answers comparing slopes. This result alone might be explained by the fact that height was typically processed faster than slope for this kind of task, which we confirmed in a separate experiment. However, we hypothesize that the difference in response time is an indicator of the cause (rather than the result) of the response choice. To support this, we found that imposing a 3-s delay in responding increased the number of students comparing slopes (answering correctly) on the task. Additionally a significant proportion of students recognized the correct written rule (compare slope), but on the graph task they incorrectly compared heights. Finally, training either with repetitive examples or providing a general rule both improved scores, but only repetitive examples had a large effect on response times, thus providing evidence of dual paths or processes to a solution. Considering models of heuristics, information accumulation models, and models relevant to the Stroop effect, we construct a simple relative processing time model that could be viewed as a kind of fluency heuristic. The results suggest that misconception-like patterns of answers to some science questions commonly found on tests may be explained in part by automatic processes that involve the relative processing time of considered dimensions and a priority to answer quickly.

We examine student understanding of the magnetic force exerted on a charged particle and report three findings from a series of tests administered to introductory physics students. First, we expand on previous findings that many students believe in "charged" magnetic poles and find that although students may answer according to a model where a positive charge is attracted to a south pole and repulsed by a north, these students may not believe that the poles are charged. Additional models produce identical answer schemes, the primary being magnetic force parallel to magnetic field. Second, the representation format affects responses: students answer differently when the magnetic field is portrayed by a field source vs. by field lines. Third, after traditional instruction improvement in student performance is greater on questions portraying field lines than for questions portraying field sources.