[ = ] Is Equal To

Is the equation “3 + 2 = 5” completely clear? Is there just one way to understand it? Do all educated adults understand the equals sign in the same way? Theoretically, an equation symbolizes a perfect equivalence or interchangeability; that is, an equals signs tells us that the two expressions flanking it stand for one and the same thing. The notion of equality, when described this way, seems so simply that straightforward that it would seem hard to imagine any other way of interpreting it. And yet there is another side to the notion of equality, and it comes out of a native analogy that we will call “the operation – result analogy”.

In this alternate interpretation, the left side of an equation represents an operation, while the right side is the result of the operation. This is a naïve analogy in which equations are tactly likened to processes that take time, and it crops up in situations that have nothing to do with school or mathematics, and which influence everyone, including young children. For instance:

Point at + cry = obtain a desired object

Vase + knock over = shards of glass on the floor

Mud + hands = mess

DVD + DVD player + remote control = watch a movie

Chocolate + flour + eggs + mix + bake = cake

Cheese + lettuce + tomato + bread = sandwich

3 + 2 = 5

Here, the equals sigh is a symbol that links some sort of action in the world to its outcome, and it can be read as “gives” or “yields” or “results in”. When seen that way, “3 + 2 = 5” is not the statement of an equivalence at all; rather, it expresses the idea that the process of adding 3 and 2 results in 5.

The ideas of interchangeability and operation-result are different. The second point of view clearly embodies an asymmetrical conception of equations, in which the two sides play different roles, one side always standing for a process and the other always representing its outcome. To write “5 =5” would be incompatible with this viewpoint, since no process is indicated. Likewise, writing “7 – 2 = 8 – 3” is also troublesome, since now there is no result. And lastly, writing “5 = 3 + 2” would be disorienting, because the operation and its result occupy the wrong sites. Indeed, many first- and second-graders understand equality in just this fashion, insisting that “5 = 3 + 2” is “backwards”, and that “7 -2 = 8 -3+ makes no sense because “after a problem there has to be an answer, not just another problem.” Since even back at “5 = 5+, replacing it with something such as “7 – 2 = 5”

The operation-result naïve analogy guides children before they encounter the concept of equivalence, because the notions of a process and its result are familiar even to toddlers. These notions are close cousins to the notions of cause and effect, as well as to the idea that certain means have to be used to reach certain ends.

Although today’s children may acquire a fairly decent understanding of equality in elementary school, coming up with the symbol “=” took a long time for humanity as a whole. A symbol for equality in mathematics first appeared only in the year 1557, in a book by the Welsh mathematician Robert Recorde. He wrote:

“I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: ==, because noe 2. thynges, can be moare equalle.”

The word “gemowe” means “twins”, and the “twinnedness” of the upper and lower horizontal lines was intended to symbolize the general idea of equality. The fact that a symbol for equality too such a long time to occur to anyone, even though mathematics has existed for at least two millennia, reveals that it is far from a self-evident notion.

Although for many adults today the idea that “equality equals equivalence” may seem obvious in a mathematical context, it doesn’t follow that the operation-result view of equality has disappeared from their minds. In fact, people often write down, and read aloud, equations in a way that reflects their unconscious understanding. For instance, the reading “4 + 3 = 7”, many people will say “for plus three makes seven”, whereas for “ 7 = 4 + 3” they might say “seven is the sum of four and three”. If education always resulted in equations being seen as statements of interchangability, then by the end of high school, the operation-result view of the equals sign would surely have disappeared for once and for all. The order of the two sides is an equation would be completely irrelevant, and both ways of writing an equation down would elicit exactly the same commentary. However, this turns out not to be the case. …. Pages 407-409