Five different studies that involve reflection on problem solving are described in the following chapters. In the first investigation, we revisit the categorization study of Chi et al. (1981), which was conducted with only eight introductory physics students, with a large number of introductory students in a classroom setting. The original Chi study came up with some excellent qualitative results that could perhaps be compared to more quantitative data. Categorization may be interpreted as a form of reflection in that students must think about how to solve the problem in order to properly categorize it. It is possible that first-year physics students will exhibit reflection on a higher level than simply describing “surface” features of the problem. We also compare the categorization by calculus-based and algebra-based introductory physics students with physics graduate students and faculty from a previously conducted study (Singh 2009). We find that calculus-based students seem to categorize better than algebra-based students, possibly from reduced cognitive load based on a more structured understanding of the role between mathematics and physical concepts. In addition, we find that there is an overlap between introductory students and graduate students, suggesting that we may not automatically consider graduate students as experts or introductory students as novices.
The second study examines reflection directly as an individual exercise in the format of a recitation activity whose goal is to diagnose one’s own errors, as well as transfer of this knowledge to an isomorphic problem. Here, a rubric is designed based upon heuristic considerations in order to properly evaluate students’ self-diagnostic approaches. We investigate the level of scaffolding necessary for students to effectively self-diagnose and we also investigate whether transfer of knowledge occurs from a self-diagnosed quiz problem to a paired isomorphic problem on a midterm exam. We will examine the concepts of near and far transfer
(Etkina and Mestre 2004) by examining two sets of problems in this context. We find that in terms of physics content, students with minimal scaffolding may benefit from experiencing more cognitive engagement with regard to transferring the knowledge to a paired problem, but if the problem is sufficiently difficult, students may be unable to learn from minimal scaffolding. In contrast, superficial review of one’s work may require no cognitive engagement and may not be effective for learning. We also find little improvement on problem-solving approaches, suggesting that one or two interventions are not enough to improve problem solving habits.
The third study observes whether advanced undergraduate students automatically reflect and learn from their mistakes. In this study, undergraduate students in advanced quantum mechanics are asked the same question twice during two different exams. The rubric designed in the second study is adapted for evaluating students in this situation. We find that one may not assume that advanced undergraduate students have implicitly learned to reflect on their work as a result of further experience in formal coursework on even a direct transfer, and in fact exhibit tendencies that may be considered novice-like.
The fourth study examines reflection as a peer exercise in the form of actively discussing homework problem solutions in an in-class recitation setting. We seek to find out whether orienting the recitation around this activity will cause students to more frequently invoke basic elements of effective problem solving strategy, as well as the effect on exam grades. We find that while no quantitative difference occurs on the final exam score between students who participated in the peer exercise and students who did not (although there is some evidence that those who participated in the peer exercise were somewhat weaker at the beginning of the course), basic problem-solving strategies such as drawing diagrams seem to be encouraged by the peer activity.
The last study investigates students’ epistemological views towards problem solving. We adopted a survey (Cummings et al. 2004) based upon the more general Maryland Physics Expectations Survey (Redish et al. 1998) and added several of our own questions which focus on several topics, one of which is reflection on problem solving. One goal is to compare graduate students with algebra-based and calculus-based introductory students and faculty. The graduate students’ responses are compared when they answer these survey questions about graduate-level problem solving and problem solving in introductory physics. We focus on each of these groups’ responses on individual questions and analyze cases where major differences are found between groups. The results focus on graduate students and seem to indicate two themes. First, graduate students are seen as in a transition from novice-like expectations and approaches to those of experts. Second, graduate students often have less expert-like views towards graduate- level questions than towards introductory-level questions. In addition, interviews suggest that there are different reasons for novices and experts to express views that do not seem expert-like, and therefore answers to the survey questions should be treated more carefully in light of this.