Share Cognitive Science Implications for Teaching Mathematics

A constant is a symbol which represents the same number throughout a given discussion. (This is a stipulative definition)

A variable is a symbol which may represent any value throughout a given discussion. (This is a stipulative definition)

In the equation 3x + 7 = 0, it is clear that 3, 7 and 0 are constants and x is a variable.

In the equation ax + b = c, it is less clear which are constants, and which are variables. One convention is that letters near the end of the alphabet (x, y, z) are used to represent variables while letters near the beginning of the alphabet (a, b, c) are used to represent constants. But this is only a convention and is not always followed.

Consider this stipulative definition: A linear equation in one variable is an equation which can be written in the form ax + b = 0 where a and b are real numbers and a is not 0.

We expect that you understand that x is a variable and both a and b represent fixed but unspecified real numbers. Thus, x is a variable and both a and b are constants.

Textbooks and websites introduce the concept of variable at the beginning of their discussion of Algebra. When I served on textbook selection committees I would insist on definitions of variable and constant which were equivalent to the above. That simple requirement excludes many textbooks and websites.

The definitions and explanations presented above are simple enough that no further simplification is required. Avoid reference material which fail to use definitions for variable and constants equivalent to those presented above.