I recently read the Common Core standards for Math for grade 6, which is where the concept of variable appears to be introduced. **The standard in question** reads:

Write, read, and evaluate expressions in which letters stand for numbers.

I tweeted this out, and one person responded with this observation:

@davidwees in d/dx[f(x)]=mx+b …d, f, dx, f, x, m, and b all have completely different meanings, and only m and b “stand for numbers”

— John Chase (@mrchasemath) October 27, 2013

Clearly this definition of variable is limited, and possibly misleading.

How do you define variable with your students? How do you introduce the topic?

There is clearly some confusion in the minds of the authors of the CCSS math. 3x + 2 is an algebraic expression, but x is not a variable. All individual items in an expression are the values of quantities (usually measurements). Some of the values are fixed in advance, eg 2, some of them are represented by letters, eg x, which are placeholders for their values. In a simple equation, say 3x + 2 = 5, the x is the placeholder for a value which makes the two sides of the equation equal. The word ‘variable’ is a description of the nature of the value of a quantity, basically when we are interested in the effects of many different values on something. So, in an equation of the form y = 3x + 2 we are interested in the values of the quantity y for various values of the quantity x. In this situation only (which includes functions) is the word ‘variable’ meaningful. Unfortunately I still have an affection for the definition : A parameter is a variable constant!!!!!

The Australians have a useful practice: there, in mathematics education, they call ‘variables’, ‘pro-numerals’, on the analogy of nouns and pronouns. ( It would be better if they called them ‘pro-numbers’, of course, which is what they are, instead of perpetuating the confusion between numbers and the symbols that we use to represent them.)

John Chase and Howard Phillips raise a fundamental issue: the lack of a consistent, logical, information-bearing naming and symbol system for most of the important concepts in mathematics.

So a good question might be: suppose we were starting over. What would be the best system of names and symbols for mathematics?

I nominate two things for starters: adopt the number-naming system of many East Asian languages (“ten-one, ten-two, ten-three … two-tens-one …”, and indicate exponentiation with an explicit operator rather than through typology (and call it “self-multiplication”).