To develop complex problem-solving skills, students need to learn to develop a coherent story about what the situation in the problem is, what the mechanism is, and what physics principles are appropriate to apply. This can be challenging for students who have been successful in science through memorizing answers and simply plugging numbers into equations. And it can be challenging for instructors who get a class full of students who “just want to be given the answers.” But taking the easy way out—matching students’ desires rather than their needs—does them and the scientific community a serious disservice.

As discussed in the earlier papers in this series,1 we use math in physics differently than it’s used in math classes.2 In math classes, students manipulate equations with abstract symbols that usually have no physical meaning. In physics, we blend conceptual physics knowledge with mathematical symbology. This changes the way that...

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