The ability to solve physics problems that require multiple concepts from across the physics curriculum—“synthesis” problems—is often a goal of physics instruction. Three experiments were designed to evaluate the effectiveness of two instructionalmethods employingworked examples on student performance with synthesis problems; these instructional techniques, analogical comparison and self-explanation, have previously been studied primarily in the context of single-concept problems. Across three experiments with students from introductory calculus-based physics courses, both self-explanation and certain kinds of analogical comparison of worked examples significantly improved student performance on a target synthesis problem, with distinct improvements in recognition of the relevant concepts. More specifically, analogical comparison significantly improved student performancewhen the comparisonswere invoked betweenworked synthesis examples. In contrast, similar comparisons between corresponding pairs of worked single-concept examples did not significantly improve performance. On a more complicated synthesis problem, selfexplanation was significantly more effective than analogical comparison, potentially due to differences in how successfully students encoded the full structure of the worked examples. Finally, we find that the two techniques can be combined for additional benefit, with the trade-off of slightly more time on task.

We examine students' mathematical performance on quantitative "synthesis problems" with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking, formulation and combination of equations require conceptual reasoning; simplification of equations requires manipulation of equations as computational tools. Mathematical complexity is operationally defined by the number and the type of equations to be manipulated concurrently due to the number of unknowns in each equation. We use two types of synthesis problems, namely, sequential and simultaneous tasks. Sequential synthesis tasks require a chronological application of pertinent concepts, and simultaneous synthesis tasks require a concurrent application of the pertinent concepts. A total of 179 physics major students from a second year mechanics course participated in the study. Data were collected from written tasks and individual interviews. Results show that mathematical complexity negatively influences the students' mathematical performance on both types of synthesis problems. However, for the sequential synthesis tasks, it interferes only with the students' simplification of equations. For the simultaneous synthesis tasks, mathematical complexity additionally impedes the students' formulation and combination of equations. Several reasons may explain this difference, including the students' different approaches to the two types of synthesis problems, cognitive load, and the variation of mathematical complexity within each synthesis type.

A body of research on physics problem solving has focused on single-concept problems. In this study we use "synthesis problems" that involve multiple concepts typically taught in different chapters. We use two types of synthesis problems, sequential and simultaneous synthesis tasks. Sequential problems require a consecutive application of fundamental principles, and simultaneous problems require a concurrent application of pertinent concepts. We explore students' conceptual performance when they solve quantitative synthesis problems with varying mathematical complexity. Conceptual performance refers to the identification, follow-up, and correct application of the pertinent concepts. Mathematical complexity is determined by the type and the number of equations to be manipulated concurrently due to the number of unknowns in each equation. Data were collected from written tasks and individual interviews administered to physics major students (N = 179) enrolled in a second year mechanics course. The results indicate that mathematical complexity does not impact students' conceptual performance on the sequential tasks. In contrast, for the simultaneous problems, mathematical complexity negatively influences the students' conceptual performance. This difference may be explained by the students' familiarity with and confidence in particular concepts coupled with cognitive load associated with manipulating complex quantitative equations. Another explanation pertains to the type of synthesis problems, either sequential or simultaneous task. The students split the situation presented in the sequential synthesis tasks into segments but treated the situation in the simultaneous synthesis tasks as a single event.

We report a study on students' approaches to quantitative synthesis problems with varying mathematical complexities. Synthesis problems involve multiple concepts typically taught in different chapters. In this study, mathematical complexity is determined by the number and the type of equations that must be simultaneously solved. Students from a second year calculus-based physics course were randomly assigned to solve one of three synthesis problems varying in mathematical complexity: simple, medium, or complex. Results from extended written and interview responses revealed four major problem-solving approaches used by the students: trial-and-error, flawed reasoning, knowledgeable, and expert-like approach. Students solving the simple problem used all the four approaches, whereas those solving the other two mainly used the "trial-and-error" or "flawed reasoning" approaches. A common phenomenon is that many students could identify the appropriate concepts but failed to correctly apply them. Additionally, the students made similar mistakes on all the three problems.

This study compared the effect of two types of interventions on subsequent student performance with a target synthesis problem. Students either solved two single-concept problems (priming) or compared worked solutions across one of four different analogical reasoning conditions. These four conditions varied according to the type of examples compared (single-concept vs. synthesis) and structural similarity to the target problem. Taken together, the analogical reasoning conditions performed significantly better than control (d=0.31). However, there were no significant differences between the different analogical reasoning conditions, or between analogical reasoning and priming. Although student responses on the target synthesis problem were similar across conditions, their responses to the analogical reasoning tasks showed potentially useful differences in referenced concepts and response grain size, from generic to more precise.

Synthesis problems, which are problems requiring the application of multiple concepts such as energy conservation and kinematics, are common in physics curricula, and improving students' skills in solving such problems is typically a key instructional goal. Despite the prevalence and importance of synthesis problems, many students struggle with them more than with their single-concept counterparts. In order to identify possible bottlenecks on this task, we asked students to solve a problem synthesizing two different topics (including energy, momentum, circular motion, and kinematics) as well as a pair of single-concept problems involving the individual components, varying their sequential order (synthesis + single-concept versus single-concept + synthesis). We found that students' primary difficulties arose not only from the deficiency in applying the individual concepts but also from the inability to recognize the relevance of both concepts, which in some cases may be caused by one dominant concept overshadowing the other.

We report on the design and pilot evaluation of a simple natural language computer tutor that targets student difficulties with the concepts of force and motion. The tutor prompts students to respond in free-response natural language to questions that address the relationships between the directions of net force, velocity, and acceleration. To examine the effectiveness of the natural language format, we compared student performance on a previously validated force and motion assessment after tutoring via natural language and multiple choice formats. Natural language training with feedback, multiple choice training with feedback, and natural language training without feedback formats resulted in effect sizes of d = 0.60 (p = 0.07), d = 0.46 (p = 0.13), and d = 0.09 (p = 0.97) respectively versus a no-training control. In addition, a median split on course grades showed no significant aptitude-treatment interaction across training conditions. However, accounting for time spent on training, the multiple choice training was significantly more efficient. For the natural language format, an analysis of performance (62% identification of an initial student response), false positives, and typical student answer patterns suggest room for improvement and subsequent study.

This thesis explores two case studies into the use of short answers and self-explanation to improve student learning in physics. The first set of experiments focuses on the role of short answer questions in the context of computer-based instruction. Through a series of six experiments, we compare and evaluate the performance of computer-assessed short answer questions versus multiple choice for training conceptual topics in physics, controlling for feedback between the two formats. In addition to finding overall similar improvements on subsequent student performance and retention, we identify unique differences in how students interact with the treatments in terms of time spent on feedback and performance on follow-up short answer assessment. In addition, we identify interactions between the level of interactivity of the training, question format, and student attitudinal views of the respective trainings. The second case study focuses on the use of worked examples in the context of multi-concept physics problems – which we call “synthesis problems.” For this part of the thesis, four experiments were designed to evaluate the effectiveness of two instructional methods employing worked examples on student performance with synthesis problems; these instructional techniques, analogical comparison and self-explanation, have previously been studied primarily in the context of single-concept problems. As such, the work presented here represents a novel focus on extending these two techniques to this class of more complicated physics problem. Across the four experiments, both self-explanation and certain kinds of analogical comparison of worked examples significantly improved student performance on a target synthesis problem, with distinct improvements in recognition of the relevant concepts. More specifically, analogical comparison significantly improved student performance when the comparisons were invoked between worked synthesis exam-