The newly-published paper on the study of how students learn to solve isomorphic problems based on the same principle indirectly illustrates the difference between understanding physics, and merely understanding about physics. In it, the authors describe, based on other studies, of groups of students given two types of problems: (i) problems that looks, on the surface, to be different, but actually are based on the same principle and (ii) problems that, on the surface, looks similar, but are based on different principles. They now make a similar test, but this time, giving students 2 sets of problems, one solved, and another one that is analogous to the solved problem. The students are then asked to solve the second problem.
Abstract: In this study, we examine introductory physics students’ ability to perform analogical reasoning between two isomorphic problems which employ the same underlying physics principles but have different surface features. Three hundred sixty-two students from a calculus-based and an algebra-based introductory physics course were given a quiz in the recitation in which they had to first learn from a solved problem provided and take advantage of what they learned from it to solve another problem (which we call the quiz problem) which was isomorphic. Previous research suggests that the multiple-concept quiz problem is challenging for introductory students. Students in different recitation classes received different interventions in order to help them discern and exploit the underlying similarities of the isomorphic solved and quiz problems. We also conducted think-aloud interviews with four introductory students in order to understand in depth the difficulties they had and explore strategies to provide better scaffolding. We found that most students were able to learn from the solved problem to some extent with the scaffolding provided and invoke the relevant principles in the quiz problem. However, they were not necessarily able to apply the principles correctly. Research suggests that more scaffolding is needed to help students in applying these principles appropriately. We outline a few possible strategies for future investigation.
But what is fascinating and totally reflects that I have been arguing, is a couple of paragraph in the paper's text:
It is well known that two physics problems that look very similar to a physics expert because both involve the same physics principle do not necessary look similar to the beginning students. Research has shown that when physics experts and novices are given several introductory physics problems and asked to categorize the problems based upon similarity of solution, experts tend to categorize them based upon the fundamental physics principles (e.g., conservation of mechanical energy, Newton’s 2nd law, etc.) while novices tend to group them based upon the surface features such as pulley or inclined plane. Similarly, when a group of introductory physics students and physics faculty were asked to rate the similarities between different pairs of problems, it was found that for problem pairs which only involve surface similarity but employ different principles students were more likely to rate them as similar compared to the faculty members. The different patterns that experts and novices discern in these problems reflects the difference between the ways in which the knowledge structure of experts and novices is structured and how they exploit it to solve problems. The fact that experts in physics have a well-organized knowledge hierarchy where the most fundamental physics principles are placed at the top, followed by layers of subsidiary knowledge and details, facilitates their problem solving process, allowing them to approach the problems in a more effective and systematic way. It also guides the experts to see the problems beyond the surface features, and makes the transfer of knowledge between different contexts easier.You may read the entire paper and what was tested upon. I'll copy the conclusion of the paper here:
In summary, deliberately using isomorphic worked-out examples to help students transfer what they learned from one context to another can be a useful tool to help students understand the applicability of physics principles in diverse situations and develop a coherent knowledge structure of physics. For introductory students, such well thought-out activity could provide a model for effective physics learning since the idea of looking at deep similarities beyond the surface features is enforced throughout the activity. It is possible that students will become more facile at the analogical reasoning processes if practice and feedback are constantly provided to them. The greatest benefit may be achieved if similar activities are sustained throughout the course over different topics and the coherence of physics as well as the importance of looking at the deep features of the problems is consistently explained, emphasized, demonstrated, and rewarded by the instructors.What is important here is that it takes knowledge and skills to be able to look at a problem, and break it down into its relevant components, especially in figuring out what are the relevant principles involved. This is what is lacking when you simply present physics either via examples, or via analogy, without the mathematical formalism and without a clear presentation of the central principle. This is what is done in pop-science books, which isn't bad since that is what you have to do to present such material to the general public, but you are not teaching physics here.