We propose a framework for improving accuracy, fluency, and retention of basic skills essential for solving problems relevant to STEM introductory courses, and implement the framework for the case of basic vector math skills over several semesters in an introductory physics course. Using an iterative development process, the framework begins with a careful identification of target skills and the study of specific student difficulties with these skills. It then employs computer-based instruction, immediate feedback, mastery grading, and well-researched principles from cognitive psychology such as interleaved training sequences and distributed practice. We implemented this with more than 1500 students over 2 semesters. Students completed the mastery practice for an average of about 13 min =week, for a total of about 2-3 h for the whole semester. Results reveal large (>1 SD) pretest to post-test gains in accuracy in vector skills, even compared to a control group, and these gains were retained at least 2 months after practice. We also find evidence of improved fluency, student satisfaction, and that awarding regular course credit results in higher participation and higher learning gains than awarding extra credit. In all, we find that simple computer-based mastery practice is an effective and efficient way to improve a set of basic and essential skills for introductory physics.

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In experiments including over 450 university-level students, we studied the effectiveness and time efficiency of several levels of feedback complexity in simple, computer-based training utilizing static question sequences. The learning domain was simple vector math, an essential skill in introductory physics. In a unique full factorial design, we studied the relative effects of "knowledge of correct response" feedback and "elaborated feedback" (i.e., a general explanation) both separately and together. A number of other factors were analyzed, including training time, physics course grade, prior knowledge of vector math, and student beliefs about both their proficiency in and the importance of vector math. We hypothesize a simple model predicting how the effectiveness of feedback depends on prior knowledge, and the results confirm this knowledge-by-treatment interaction. Most notably, elaborated feedback is the most effective feedback, especially for students with low prior knowledge and low course grade. In contrast, knowledge of correct response feedback was less effective for low-performing students, and including both kinds of feedback did not significantly improve performance compared to elaborated feedback alone. Further, while elaborated feedback resulted in higher scores, the learning rate was at best only marginally higher because the training time was slightly longer. Training time data revealed that students spent significantly more time on the elaborated feedback after answering a training question incorrectly. Finally, we found that training improved student self-reported proficiency and that belief in the importance of the learned domain improved the effectiveness of training. Overall, we found that computer based training with static question sequences and immediate elaborated feedback in the form of simple and general explanations can be an effective way to improve student performance on a physics essential skill, especially for less prepared and low-performing students.

Students in introductory physics courses sometimes struggle to correctly break down a single vector into its components when provided only with an arrow, a magnitude, a reference angle, and a coordinate system. Students struggle further when asked to break down a vector in an inclined coordinate system, such as the weight vector of a box on an inclined plane. Varying the placement of the angle consistently affects student error and response patterns across four physics student populations: algebra-based mechanics, algebra-based E&M, calculus-based mechanics, and calculusbased E&M. This suggests that student difficulties with trigonometric vector components are persistent and pervasive, even across different introductory physics courses, and are far below the requisite near-perfect accuracy needed for such fundamental skills. Student error and response patterns are discussed for both problem types.

Through extensive testing and interviews of sophomore, junior, and senior engineering students in a Materials Science Engineering course at The Ohio State University, we found that these students struggle with many skills necessary for their coursework. Often these "essential skills"were prerequisite to the course and little to no instruction time was spent on them. Online training was developed to attempt to improve these skills. Students participated in the training several times over the term, with each assignment taking 10-20 minutes and consisting of 10 questions. Students were allowed unlimited attempts on each assignment and were required to achieve mastery (80% or better) for full credit. Training covered a wide range of topics: interpreting log plots and log scales, using metric prefixes for various conversions, estimating typical values of common material properties, employing dimensional analysis, and operating equations when given variables in mixed units.Unlike the success achieved by the log plots training, most of the topics saw little and insufficient improvement as a result of training, despite the basic nature of the skills. Future improvements to the training will focus on determining which factors will help to convince students of the importance of mastering these prerequisite skills.