The Role of Examples in Teaching

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“A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.” (Halmos 1985, p. 63)

Suppose you want your students to know that a function has an inverse if and only if it is a bijection (both one-toone and onto). You could state the theorem, perhaps prove it, and work some related problems. Or, you could ask them to explore a set of carefully chosen examples, creating an opportunity for students to observe the relationship. We observed a college discrete mathematics class in which the second approach was taken. Students examined a set of nine functions to determine which functions had inverses; the functions were chosen to challenge assumptions about functions and their properties. Students determined whether the functions were injective (one-to-one), surjective (onto), or both (bijective). Data from students provided insight that only the functions with inverses were bijective. This type of mathematical activity served to review function concepts and provide opportunities for making significant mathematical observations, which can then be explored further or proven.