Definitions in mathematics play several roles:
- They facilitate communication.
- They provide the foundation for new learning. All new learning requires a foundation of prior knowledge.
- They are essential to learning mathematics.
- They are the core element of every simple concept.
- They promote understanding.
- They facilitate applications.
The five common types of definitions studied by scholars are: Persuasive, Precising, Theoretical, Lexical, and Stipulative. A complete discussion is available in this PDF.
The age-old adage that mathematics is very precise refers to the terms and concepts rather than computational accuracy. In support of this idea, terms in mathematics are defined with very precise stipulative definitions.
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! Beginning students are generally completely unfamiliar with stipulative definitions. Therefore, teachers must help students to work successfully with mathematics 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.
Stipulative definitions cannot be reliably learned by repeated exposure to instances of the definition. They must be memorized.
Learning a stipulative definition does not end with memorization. Memorization is only the very first step in an extended process to learn the definition. After multiple posts which address the benefits of learning mathematics definitions, I will detail the complete learning process.
My discussion of six benefits of mathematics definitions will begin in the next blog.