Great Ideas of Primary Mathematics

Background

Variables

At the start of the 20th century, variables were defined as a quantity that could assume an infinite number of values or a generalized number. An example of a situation in which a variable is used in this way is an expression such as

3x – 5x – 24

because the x in the equation can represent an infinite number of values. However, between the late 1950's and 1980's there was a move to refine this terminology. Variables began to be defined as a symbol that represented an element of a set ¹. The set could be the real numbers, or the rational numbers, or the integers, or the whole numbers, or something else, according to context. I will define a variable as a symbol standing for any element of some set. If we were to be completely correct when defining a variable, the set should be defined, but in reality it usually is not. Sometimes the set can be determined from the context of the problem but other times, if vague, the set can be unclear. The shift in definition is not a radical change but rather a refinement.

Variables can be used in many different contexts. Some examples of specific contexts that I will discuss with my students are as follows:

• A variable can represent a quantity, such as area. When discussing area we might use the letter A for the variable. However, A can represent something completely different in different situations-it could be the area of a rectangle, or of a hexagon or of a circle, or whatever shape is being discussed.
• Variables may be used in equations to express a relationship between quantities. For example the area of rectangle (A) can be computed as A=bh where b is the base and h is the height of the rectangle.
• A variable is used to form equations or expressions in which one is representing a specific situation. This is the context of a variable mostly used in this specific unit (examples to follow).
• Often textbooks and many teachers define a variable as an unknown that we are looking for, but in reality here also, the variable is a quantity that can vary in some set. What distinguishes the variable as unknown from other uses of variables is, that instead of just using the variable to represent a quantity, we are asking a question and some value of the variable will be the answer. For example in 2x – 8 = 0, we are asking, is there any value of x in the set that makes this equation true. So x is varying in a set, and we are asking: can it take a value that makes the equation valid. For this particular equation, if we assume that we are in the set of whole numbers or integers or rational numbers, then x would equal 4. However, if we defined the set to be numbers between 15 and 20 there would be no value that x could be to make that equation valid.

Equals sign

The other content-related issue when approaching expressions is the misconception of the equals sign. Several mathematics education researchers remark that there is an operational view of the equals sign as well as a relational view ². Most of our students have been overexposed to the operational view, which states that students see the equals sign as representing an action needs to be performed and something needs to be written as an answer. The adoption and prevalence of this in the classroom, is partially due to the fact that most elementary students learn arithmetic in this manner and the equals sign becomes associated with finding an answer from a very early age. However, to be successful in middle and high school algebra, and conceptually understand the equals sign as meaning equivalence, students need to be exposed from an early age to the relational view ³. This view presents the equals sign as something signifying both sides having the same value, or in other words the equals sign meaning "the same as." Because I am a high school algebra teacher I have no control over how my students have been exposed to the equals sign in earlier grades. However, because I do not know their background and conceptual knowledge of the equals sign, it is even more imperative to present it as a topic of discussion. I will talk with my students about the meaning and how we can learn to accept that one does not always want a computational answer, while simultaneously talking about when it does signify a computation.