Semantics of Variables — Cousino Math

I was talking to @misscalcul8 about representations of linear inequalities. She viewed the solutions of the inequalities as regions in the coordinate plane, and was intrigued by representing the solutions as intervals on the number line. This reminded me of the various semantic roles that variables play.

So far, these four roles are all that I have been able to identify.

Fixed, Unknown Value

This is the initial idea of a variable that we present to students. Here, a variable has a particular, but unknown value. We can learn the value of the variable by solving an equation. For example, under this interpretation, the equation \(2 x + 7 = 21\) translates to the question “What number when doubled and added to 7 equals 21?” By solving the equation, we discover the value of \(x\) to be 7.

Whenever a variable is seen to have a fixed but unknown value, our goal is to learn the actual value. It is as if the value of the variable, usually \(x\), is hidden and locked away. We may retrieve the value by employing algebra. This conception of a variable works well for equations with exactly one solution, but can be extended to other situations. For equations with more than one solution, like \(x^{2}=4\), we cannot deduce the true value of \(x\). But we can conclude that the correct value is either \(2\) or \(-2\). Similarly, inequalities will not typically yield the one true value but are only able to restrict the possibilities of what the true value can be. For example, the solution to \(x^{2}\leq 4\) is exactly one of the two values \(-2\) and \(2\), rather than being any value.

When dealing with two or more variables, the fixed but unknown value interpretation of variables is workable but not profitable. Take \(y = x + 3\) as an example. Here, the idea of a graph is inconceivable. If the correct values are \((x, y)= (-3, 0)\), then there cannot be another other values of \(x\) and \(y\). Instead of a line on the coordinate plane, the solution here is a single point. Granted, if we know nothing about the variables beyond the equation, the set of possible solutions is \(\left\{ (t, 2 t + 3)\mid t\in\mathbb{R}\right\}\). But according to this view, only one point, among the possible solutions, is the correct solution.

In summary, we start teaching algebra by introducing it to students as “solving for the unknown”. By doing so, we instill in them the view that a variable is merely a placeholder for a fixed and, as yet, unknown value. This is a reasonable mental model of algebra for those new to the subject. However, this model does not last very long. We quickly want to move to the idea of variables as jointly varying quantities when tackling relations with more than one variable, and in particular functions.

Varying Value

Anyone familiar with algebra is probably most familiar with the idea that the value of a variable may vary. A fixed variable is “faithful” to its value, whereas a changing variable cannot be forced “to commit” to one value or another. Seeing a variable as being free to change subsumes the fixed, unknown interpretation. Solving equations or relations now can be understood as saying “If this original equation \(x^{2}=4\) is true, then it must be the case that either \(x=2\) or \(x=-2\).”

The richness of variables as varying quantities is seen with two or more variables. Graphs on the coordinate plane demand that the input variable be permitted to take on many different values, one for each point on the graph. Despite students’ struggles, combining graphical and algebraic perspectives reveals a treasure of insight. Even in college algebra, we can ask deep questions like “as \(x\) increases, does \(y\) increase, decrease, or remain constant?” or “as \(x\) increases, does the rate of increase of \(y\) accelerate, decelerate, or remain the same?”

Despite being a familiar and fascinating perspective, variables as changing quantities is rather sophisticated and is probably not the way to introduce students to elementary algebra.

Fixed, Arbitrary Value

One might instead call these kinds of variables parameters. In the point-slope form of a line \(y = \color{blue}{m} x + \color{blue}{b}\), the symbols ‘\(y\)’ and ‘\(x\)’ are varying quantities; and ‘\(\color{blue}{m}\)’ and ‘\(\color{blue}{b}\)’ are parameters. They aren’t values which are thought to change with \(x\) and \(y\). While they can change, these parameters are seen as having values independent of the other variables. Like the fixed, unknown variables, parameters are seen as placeholders, which are to be replaced by their value in specific contexts.

Indeterminate Value

Indeterminate variables appear in more theoretical mathematics, when variables don’t have to represent for anything. For most people, equality between polynomials is where this idea can be seen. We can say that the two polynomials \(p(x) = 2 + 5 x\) and \(q(x) = 5 + 2 x\) are equal when \(x = 1\). But we also say that \(p(x)\) and \(q(x)\) are different polynomials. In one sense, we have that \(p(x)=q(x)\) if and only if \(x = 1\); and in the other sense, we have that \(p(x)\neq q(x)\). The difference lies in the role of \(x\) (and the role of ‘=’). If the variable is a fixed or varying quantity, then the equation \(p(x)=q(x)\) is a prompt to solve for \(x\) in order to find what value(s) of \(x\) makes this equation true. If the variable is indeterminate, then it doesn’t represent a number. So \(p(x)=q(x)\) is a statement about the algebraic equivalence of the two polynomials. Equality holds in this case if we can algebraic manipulate \(p(x)\) to look exactly the same as \(q(x)\), or vice-versa. But if there are essential algebraic differences, which is the case for \(p(x) = 2 + 5 x\) and \(q(x) = 5 + 2 x\), then we have that \(p(x)\neq q(x)\). For polynomials, two polynomials are equal if and only if their coefficients, in standard form, are equal.

In other words, \(p(x)=q(x)\) asks “are these curves identical?” when \(x\) is indeterminate, and “do these curves intersect?” otherwise. There are more, deeper examples such as polynomials in modular arithmetic, formal power series, and generators for free objects. But it is sufficient to show that this meaning gives us a mathematical way of saying that two functions can intersect, and yet still be different functions, such expressive power is justification enough for many.

Conclusion

Can you think of another role or that there should be fewer roles? Please let me know what you think in the comments below.