Below is a figure that had a tremendous impact on me when I saw it ten years ago in a paper written by Carl Wieman and Kathy Perkins (figure adapted from work done by Eric Mazur at Harvard). The figure shows two problems designed to test the same basic physics concept, namely how current flows in circuits. Determining the current through the various resistors in part (a) requires solving a pair of equations, and as a teacher a reasonable assumption is that if students can do the question correctly then they understand the concepts. However, what Mazur found is that many students could solve this challenging problem but were unable to solve the (apparently) much simpler conceptual problem in part (b) asking about how the brightness of the light bulbs will change when the switch S is closed.
The students were simply using a “recipe” to solve the problem given in (a), indicating that they would quickly become confused when attempting to apply their knowledge outside of the artificial realm of the problems that they had practiced. This realization played a significant role in my decision to spend three months as a Christine Mirzayan Fellow at the National Academics with the Board on Science Education , where I became convinced that active learning approaches could help students with these conceptually challenging topics.