Ask a general person on the street to define mathematics, and they’re likely to say something about manipulating or combining numbers.
Ask a mathematician to define mathematics, and they’re likely to talk more about patterns. Here, for instance, is the first sentence of the definition provided by MathWorld: “Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined.” (Weisstein, Eric W. “Mathematics.” From MathWorld–A Wolfram Web Resource.)
I remember when Sudoku became all the rage. A friend said it was like mathematics, but wasn’t really because you didn’t need math to do it. Their thought process was basically “Numbers, so math, but no operations, so not math.”
The mathematician’s more likely thought process about Sudoku would be, “Numbers, so boring math, but patterns and logic, no operations, so more interesting math.”
Math education in early grades has traditionally focused on computation. Even when more complex or abstract concepts are covered, it usually feels like that’s in the service of aiding in computational automaticity and understanding.
The heart of algebra, meanwhile, is about exploring patterns. It is both a tool in itself and a bridge from the concrete world of arithmetic to the abstract world of what mathematics has evolved into.
It is a conceptual shift that often occurs without comment. I think this lack of comment contributes to student frustration: They thought that “mathematics” was one thing, and then over a fairly short span of time it becomes something else entirely.
Algebraic notation also lives in an Uncanny Valley where it looks like natural language, with letters both familiar and foreign, but it has its own rules:
- Algebraic notation is not necessarily evaluated from left to right.
- Capital and lower case; roman and italic — these are all different.
- Letters have no intrinsic meaning.
The first of these rules also applies to expressions just involving numbers, but it becomes more complex in application when expressions can’t be resolved. The order of operations dictates evaluation, based on the operator level.
The top level is exponentiation. While this technically includes radicals and logarithms, both of these have notation which contains implied grouping. The middle level is multiplication, and includes division. The bottom level is addition, and includes subtraction.
The second of these rules can be especially confusing and, in this era of auto-correction, frustrating. If I begin a sentence in Word with “x=3-x”, it will be “corrected” to “X=3-x”. This is different statement. The first one involves a single variable, x, and the equation can be solved to \(x=3/2\). The second one involves two variables, x and X.
We can take it to a more pedantic level: Variables are in italics and constants are in roman. So \(\text{e}\approx2.718\) should not be italicized, and integrals should be written as \(\int 2x\,\text{d}x = x^2 + C\) (where the operator \(d\) is not italicized). However, this convention is inconsistently followed.
The third of these rules seems to contradict the second, but the key is internal consistency. \(X, x, t,\) and so on are interchangeable in the Big Picture, but \(X\) and \(x\) mean different things within the same context.
If we want to discuss a function such that the output is the square of the input plus four times the input minus three, we could do it different ways: \[y=x^2+4x-3\]\[f(t)=t^2+4t-3\]\[😂(🤷)=🤷^2+4🤷-3\]
However, within the same context, we have to be consistent. So while \(y=x^2+4x-3\) expresses the same relationship as \(f(t)=t^2+4x-3\), it’s a different relationship than \(y=X^2+4x-3\).
This is absolutely and understandably confusing, and is one of the best arguments that the decision to use letters in mathematics (instead of creating new symbols) was a mistake.
At the same time, though, these are not decisions about algebra-the-science, these are notational conventions. They are conventions that are much newer than the notions behind algebra.
And there are newer conventions. Here’s how algebra might appear in Excel:
In Excel, mathematical expressions are clearly marked with an equal sign at the beginning. Expressions can involve either cell references (as here) or cell names.
When cell references are used, these cell references change when moved or copied:
This is because, in algebra, what’s relevant for naming is internal consistency.
Algebraic notions appear throughout computer programming, but the preference there (as in physics and elsewhere) is for variable names to have some sort of semantic importance.
The semantic arbitrariness of variable naming within algebra is a source of confusion for students, but it’s important for an understanding that \(A=bh\) and \(A=lw\) capture the same mathematical truth. I see many students getting caught up on the “correct” formula for the area of a rectangle, but both of those are correct. So is \(z=xy\), for that matter, so long as we specify that meanings of each of the variables in the context.
This confusion is also plain when discussing linear equations. In the US, we typically write \(y=mx+b\), but \(y=ax+b\), \(y=a+bx\), and \(y=ax+c\) are also in use, and are even the standard in some places in the world.
Ultimately, a linear equation describes a locus involving the two foundational operators: Multiplication and addition. Take any arbitrary number, multiply it by a given value, add a different given value, and consider the result.
In order to capture this with minimal ambiguity, we need to refer to each value by something consistent. A programmer might well choose “output = change_rate * input + initial_value”. This, too, is a valid way to write a linear equation.
A physicist may choose \(s_f(t) = r\times t + s_o\). Also valid.
The key is: The notation captures a relationship between values in a way that is meaningful to anyone who can read the notation. The specific notation is only relevant to the degree that it can be understood.
The true Heart of Algebra is the ordered representation of relationships. It is a bridge towards the art of mathematics, which is taming the chaos of the universe into some level of coherence.
Using consistent letters for specific relationship roles allows us to capture more abstract connections. Here’s one common convention:
- Use late-alphabet letters (\(x, y, z, t, u, v, w\)) for true, real-number variables
- Use early-alphabet letters (\(a, b, c, d\)) for parametric or constant variables
- Use mid-late-alphabet letters (especially \(i, k, m, n, p, q\)) for integer variables, especially in number theory
- Use early-mid-alphabet letters (\(f, g, h\)) for function names
- Use Greek letters (especially \(\alpha, \beta, \theta, \gamma\)) for angle measurements
This is just one convention, but it helps keep writing organized. It doesn’t give variables a specific meaning, but rather gives them a meaning related to their role. Even within this convention, \(f(x)=x^3+cx\) and \(g(t)=a_1t+t^3\) express the same relationship.
This understanding is important to facilitate the shift from the concreteness of arithmetic to the increasingly abstract realms of higher mathematics. To the extent that a student can pass through algebra classes in high school