Many elementary children have serious misconceptions about the meaning of the equal sign (“Fostering Relational Thinking while Negotiating the Meaning of the Equals Sign”, Molina & Ambrose, Teaching Children Mathematics, Sept. 2006). Most of them think it means “put the answer here” or “do the adding” or whatever operation was involved in the equation. I personally have had students, when I wrote something like 5 = 2 + 3 on the board, say, “Mrs. Norris, you wrote it wrong!” The comment is not surprising since they almost always see equations with the answer on the right.
Students tend to be uncomfortable when their notions of the equal sign are challenged. “Backwards” equations (as was 5 = 2 + 3) or ones such as 2 + 3 = 4 + 1 put students’ understanding at a disequilibrium as they struggle to make sense of what is being said in the equation.
I had the opportunity this week to work with five different classes around the concept of equality. The school had purchased (at my request!) a class set of student balances, along with a larger, demonstration-sized balance, which we used to represent equations. We began with some “regular” ones, and then moved on to showing 5 = 5, 12 = 12, etc.. Equalities like that, with no operation symbol at all, were a bit startling to most of the students, but they quickly understood the logic of such statements as we represented them on the balance scale. Since the scale only goes to 10 on each side, we explored how to represent double-digit numbers greater than ten using base-10 representation. Thus 12 was put on the balance as 10 and 2, 20 was put on as two 10, etc.
We represented and recorded many “backwards” equations, and then moved on to ones that had two numbers on both sides of the equal sign, or multiple numbers on each side. We explored multiplication by hanging multiple weights on a single number (e.g. 4 groups of 3 balanced with 10 and 2).
The drawback of using the scales is that you cannot represent subtraction. In most cases the children used the tool, then wrote the equation they had created. One girl wrote her equation first, 9 – 1 = 6 + 2 but then could not represent that on these simple balances. We will explore such extensions in further sessions.
Because the class was hands-on and very interactive, every student was engaged. There were many comments made about the balance system being “cool” and many questions about when we would use the scales again. And, seriously, don’t we want them eager to come back for more math?!
Children need many experiences with equations that are not in “regular” form if they are to build an understanding of the true meaning of equality. I encourage you to find ways to explore this concept, one that is a critical component of algebra, with your students.