Proper Use of Examples Part I

Examples are an important instructional device used by every mathematics teacher. However, I fear that many of us (myself included) present the wrong examples and use them to illustrate the wrong things. Furthermore I am absolutely convinced that many students do not know the purpose of examples and consequently use them improperly.

For this essay I will concentrate on those examples which are illustrations of a mathematics creature as defined in a formal definition. Such examples are generally a part of the initial discussion and explanation of the definition.

Definitions in mathematics are stipulative definitions as opposed to lexical definitions.

A synonym for lexical definition is extracted definition because it is extracted from common actual usage. Extracted definitions have a truth value—they can be true or false. Beginning algebra students are generally familiar with extracted definitions only. These are the definitions found in a common dictionary (lexicon).

Definitions used in mathematics are very different. Definitions in mathematics are always stipulative definitions. They are stipulative in the sense that they specify usage rather than report usage. Stipulative definitions do not have a truth value. They are neither true nor false—they just are! Early algebra students are generally completely unfamiliar with stipulative definitions.

A definition in mathematics does not announce what has been meant by the word in the past or what it commonly means now. Rather it announces (stipulates) what will be meant by the word (or term) in the present work, argument, or discussion.

Because mathematics definitions are stipulative we must depend on them much more explicitly than we do with lexical definitions of words used in casual conversation. For that reason examples are not to be used as a replacement for the definitions in mathematics. This is a lesson that must be learned by many authors, teachers and all others involved in development of instructional material. This is a hard lesson we should teach our students. We should help the student to understand that examination of a bunch of examples will not reveal the stipulative definition. Rather the stipulative definition must be used to determine if the example is indeed the creature defined in the definition.

Suppose the stipulative definition of a jeok is presented and is followed with a bunch of examples. The natural inclination of students is to look at the examples and to use that experience to determine if a newly encountered creature is indeed a jeok. After all, that technique works well enough for identifying dogs (never mind that a hyena is not a dog). This technique lacks the precision required in mathematics and does not acknowledge the stipulative nature of mathematics definitions. In the study of mathematics, and mathematics communication, there must be a dependence on the stipulative definitions.

Examples do indeed help the student understand complexities and nuances of a definition. Students frequently overlook those illustrations of complexities and nuances. Better constructed examples and more discussion could alleviate the problem.

More importantly, examples should be instructional devices designed help the student determine if he/she understands (or does not understand) the definition. Students must be taught to use examples in this manner; it is not natural.

If an example is to be used by the student to evaluate his/her understanding, then we should construct examples to further that goal. In a discussion of the examples the student should be guided through activities which encourage (or force) the student to learn how to use a stipulative definition.

When a student learns to study and use examples in this manner that student will be assured that he/she understands the definition. Furthermore the student will be gaining experience in the use of stipulative definitions. That skill is useful during the construction of an argument whether it is a mathematical argument or is part of some other discipline.

If we teach students to use examples in this way we have taught them something about mathematics, but we have also provided them with a thinking skill which can be used in any part of life. I consider that to be a desirable outcome.