Instructors of physics often use examples to illustrate new or complex physical concepts to students. For any particular concept, there are an infinite number of examples, thus presenting instructors with a difficult question whenever they wish to use one in their teaching: which example will most effectively illustrate the concept so that student learning is maximized? The choice is typically made by an intuitive assumption about which exact example will result in the most lucid illustration and the greatest student improvement. By questioning 583 students in four experiments, I examined a more principled approach to example selection. By controlling the manner in which physical dimensions vary, the parameter space of each concept can be divided into a discrete number of example categories. The effects of training with members of each of category was explored in two different physical contexts: projectile motion and torque. In the first context, students were shown two trajectories and asked to determine which represented the longer time of flight. Height, range, and time of flight were the physical dimensions that were used to categorize the examples. In the second context, students were shown a balance-scale with loads of differing masses placed at differing positions along either side of the balance-arm. Mass, lever-arm length, and torque were the physical dimensions used to categorize these examples. For both contexts, examples were chosen so that one or two independent dimensions were varied. After receiving training with examples from specific categories, students were tested with questions from all question categories.
Successful training or instruction can be measured either as producing correct, expert-like behavior (as observed through answers to the questions) or as explicitly instilling an understanding of the underlying rule that governs a physical phenomenon. A student's behavior might not be consistent with their explicit rule, so following the investigation of their behavior, students were asked what rule they used when answering questions. Although the self-reported rules might not be congruent with their behavior, training with specific examples might affect how students explicitly think about physics problems.
In addition to exploring the effectiveness of various training examples, the results were also compared to a cognitive theory of causality: the contingency model. Physical concepts can often be expressed in terms of causal relations (e.g., a net force causes an object to accelerate), and a large body of work has found that people make many decisions that are consistent with causal reasoning. The contingency model, in particular, explains how certain statistical regularities in the co-occurrence of two events can be interpreted by individuals as causal relations, and was chosen primarily because it of its robust results and simple, parsimonious form.
The empirical results demonstrate that different categories of training examples did affect student answers differently. Furthermore, these effects were mostly consistent with the predictions made by the contingency model. When rule use was explored, the self-reported rules were consistent with contingency model predictions, but indicated that examples alone were insufficient to teach complex functional relationships between physical dimensions, such as torque.