In the problem 5x + 4 = 9, what exactly does the “equal” sign mean? Before you answer, let me tell you a story.
Years ago, my oldest son was struggling with a simple algebra problem, and I was trying to help him. (I was not yet a teacher at the time.) He didn’t know where to even start approaching the problem. Then suddenly, I had an epiphany of how to explain it to him.
“When solving a simple algebraic equation,” I told my son, “think of the equation as a balance. Envision a scale. If you do something to one side of the equation, you need to do the same thing to the other side. The equal sign (and here’s the epiphany) tells you that one side must be the same value as the other side.”
Slowly, a look of understanding came over his face. He subtracted 4 from both sides. Then, he divided both sides by 5. From 5x + 4 = 9, he solved for x = 1.
It was great to see him succeed. But with my epiphany, I also started to understand why I had such anxiety and underachievement as a math student in school. In fact, I realized that I had grown up with a dysfunctional relationship with the equal sign.
As a student, I incorrectly thought the equal sign meant: “here comes the answer.” What I lacked was a true conceptual understanding of mathematical equivalence. In reality, the equal sign refers to equality or “the same value as.” Just as four quarters are the same value as one dollar, 5 + 4 is the same value as 9. And 5x + 4 = 9 is an equation to be balanced.
A faulty interpretation of a symbol as basic and common as the equal sign can unsettle a child for many years. So can other misconceptions, like the belief that multiplying always increases a number. (This interpretation does not work well with fractions, like 1/4 x 1/4 = 1/16.) Or that when you “reduce” a fraction you are decreasing its size. (Replace “reduce” with “simplify,” and 2/4 = 1/2 makes a lot more sense.)
When these kinds of early misconceptions are not addressed, they can expand, weaken confidence and create what is commonly known as
Procedural Knowledge vs. Conceptual Knowledge
As a teacher, I’ve listened to parents’ concerns and frustrations about how their children are learning math these days. They want to help their struggling children but are “baffled” by the material. They talk about their own math instruction and how they fluently and efficiently solved math problems using “procedural knowledge”—a set of steps, actions or procedures.
What I try to explain is that these days, we teach for conceptual knowledge. Our students don’t just memorize times tables. They’re expected to understand and explain their mathematical reasoning.
We want kids to demonstrate deep understanding through multiple representations, which include drawings, area models and word problems. Procedures and shortcuts are also taught, but not until the student can explain why those shortcuts work. We like our students to get the right answer, but we need them to know why it’s the right answer.
The reason for teaching conceptual understanding is to help students to see connections between the math they’re learning and the math they already know. This can be especially empowering for kids with learning and thinking differences. It prepares them to solve the real-world problems they will face in the future.
Yes, math is being taught differently today. It may be a little more
difficult for parents at times, but it definitely can be better for kids.
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