When I was a lad studying mathematics, the equality sign seemed particularly simple: The stuff on the left is equal to the stuff on the right. However, I have since been developing a much more sophisticated perception of the simple little sign. It can indeed be a troublesome symbol, not because of its own meaning but because of how we tend to treat it.

### Arithmetic

Consider this problem, which most any elementary school student can solve:

- (2 + 2 = ?)

The “answer”, naturally, is 4. However, this carries a few assumptions:

- We are operating in base 10.
- We are being asked for the simplest expression of the value.

The first assumption is a bit of a throwaway. The second, however, is crucial. The first encounter most of us have with the equality sign is that the goal is to take some complicated expression on the left of the equality sign and represent it as a single value. Here’s a mathematical reality: There is no single solution to the problem above. Any other expression that evaluates to 4 is a suitable answer:

- (2 + 2 = 3 + 1)
- (2 + 2 = 2 cdot 2)
- (2 + 2 = 4i^{2+2})

The expression (4) is perhaps the simplest way to communicate its underlying value, but it’s hardly the only way.

In this case, then, we’re asking children to do something beyond what the sign itself means: Do some stuff on the left, then put the result on the right. (This observation comes to me from Robert Jacobson in a Google+ discussion.)

### Algebra

Think about how you’d approach this problem:

- (2x + 4 = x – 1)

If you’re like the standard high school algebra student, you’re taught to move the variable to the left and the rest of the stuff to the right:

- (2x + 4 – x = x – 1 – x)
- (x + 4 = -1)
- (x + 4 – 4 = -1 – 4)
- (x = -5)

This conflicts with how it’s done in arithmetic. Now, instead of juggling only on one side of the equality sign, we’re now juggling on both sides. Additionally, the arithmetic stuff that used to happen solely on the left side now happens mainly on the right side; the left side is for algebraic stuff.

In both cases, there’s an implied directionality. In arithmetic, the expressions go on the left and the values go on the right. In algebra, the variables go on the left, and the arithmetic, which is then simplified down to a value, goes on the right. The similarity is that the ultimate goal is to have a value by itself on the right side of the sign, but the processing that leads up to that is different.

### Puzzles

There are a few puzzles making the rounds on the Internet that use the equality sign in a nonstandard way. For instance, solve this puzzle:

- (1235 = 0)
- (2356 = 1)
- (8112 = 2)
- (6790 = 3)
- (3689 = 4)
- (8080 = ?)

The answer is six; the point is to count all the closed loops on the left.

Such puzzles generally use the equality sign to indicate a function relationship: The number on the right (be it a value, a shape, a string, or something else) is the input value, and the right side shows the numeric output. The claim with some of these puzzles (such as the one above) is that preschoolers have an easier time with them than adults, the implication being that adults have lost the ability to think outside the box. If this claim is true, I would argue that adults have attached consistent semiotic meaning to numerals and equality signs while preschoolers haven’t crawled into the box yet in the first place. Because preschoolers have no consistently fixed meaning, they find whatever pattern seems relevant for a given problem. My own preschooler, for instance, knows his numerals, but he hasn’t yet realized that order matters. Seeing a number like 1235, he might read it off as 1, 2, 3, 5 or as 5, 3, 2, 1 or even out of order entirely, such as 1, 5, 3, 2 or 2, 1, 5, 3. (All of this assumes, naturally, that it’s true that preschoolers have less trouble with this sort of problem.)

### Computer Programming

While puzzles often use the equality sign as a generic function evaluator with the value placed on the right, computer programming languages tend (albeit not universally) to do the opposite. Some languages have an ambiguous equality sign: In conditional statements, it functions to compare the values of two expressions and returns TRUE if the values are the same, FALSE otherwise; in non-conditional statements, it functions solely to assign the value of the expression on the right to the variable specified on the left. In such a language, both uses are shown in: if (x * 4 = y – 3) then x = x – 10

However, another common occurrence in computer languages is to disambiguate by using == rather than = for the comparison, so that = always means “evaluate the stuff on the right and assign that value to the variable on the left.”

Furthermore, Object-Oriented languages tend to use the equality sign to assign either simple numeric values or object complexes. This runs contrary to something that students are supposed to learn in geometry: The difference between equality and congruence.

### Geometry

Two line segments that have the same length are not equal: They’re congruent. Two triangles with the same angles and side lengths are likewise not equal: They’re congruent. At the high school level, equality is reserved for single-dimension values: The lengths of two congruent line segments are equal.

In Object-Oriented Programming, at least the languages I’m familiar with, two objects with the same values in all characteristics are equal.

Now, this isn’t necessarily a contradiction. Two line segments that have the same length but that are situated in different places don’t have all the same characteristics. Students in geometry tend to struggle particularly with the notion that a line segment is congruent with itself, something which might come up when proving, for instance, that the two triangles created by the diagonal of a parallelogram are congruent. So whether there’s an actual contradiction between geometry and Object-Oriented Programming relies on whether two line segments in the exact same space are equal or merely congruent.

### Summary

This has been an overview of the ways I’ve seen the equality sign being used. At its heart, the sign means exactly what I said at the outset: The value of the expression (or perhaps object) on the left is equal to that of the right. The various assignments and implications about directionality are added on during the education process and are not inherent to the symbol itself. However, I can’ t help but think that, by sidling from one usage to another, frequently without clear explanation to the students, we’re further obscuring the mathematical learning process.

aniya