Mathematical Equality

I feel like the meaning of the equals sign is misunderstood by students everywhere. Each year, I start every math class the same way. I draw an equals sign on the board and ask students what it means. Sometimes I get the response I am looking for (it means the two sides are the same), and sometimes students tell me that the equals sign separates the “problem” from the “answer.”

                See, we go into kindergarten and are given a bunch of worksheets that have problems like “2 + 3 = _____” and most of us put a 5 in the blank. “Two plus three equals blank” conditions us to think that the left side of the equals sign is the problem, the right side is the answer, and the equals sign just separates the two of them. Problem = Answer. Really, the equals sign is stating that two plus three is the same as five. We use that blank space to represent the notion of an unknown, as if we were saying 2 + 3 = x and then x = 5.

                This misconception leads to some common errors when solving equations, like 3x + 5 = 17.

   3x + 5 = 17

= 3x + 5 - 5 = 17 - 5

= 3x = 12

= 3x/3 = 12/3

= x = 4

The above work states that every first expression is the same as all the following expressions: 3x + 5 = 17 = 3x + 5 + (-5) = 17+ (-5) = …  At some point there is a statement that 17 = 12 = 4, which of course is not true. This stems from the fact that students view the equals sign as the bridge from the beginning of the problem to the end, not a statement of fact. (Instead, when solving, all four of those equals signs on the left end of the solution should be removed. There’s no “bridge” between the equations, each line simply represents independent statements).

                There are other issues I've found with the notion that the equals signs splits up the problem and the answer. Students who are able to solve 5x + 8 = 23 struggle to solve 23 = 5x + 8. Or, when solving the equation 9x + 5 = 17x - 4, students will subtract the 17x from both sides so that the x's are on the left side, even though I think subtracting the 9x is much easier. 

                I’ve found that clarifying this fact early in a classroom pays dividends long term. It makes the steps involved in solving equations more logical, it allows students to understand the relationship between inputs and outputs in function notation, and, most importantly, it gets students to think about the mathematical statements they are making while writing their work. These are all little things that make life easier in the long run, for them and for me.

                I wasn't great at math when I was in school. I was never told, explicitly, what the equals sign meant, and I would put the equals sign in-between every line when solving equations, like I did above. The first math class I took in college corrected that mistake, and I felt foolish. However, when it was clarified to me, other things began to fall into place as well. I hope to do that for my students a bit earlier.