The Equal Sign: What It Really Means

A Tweet I read a few days ago led me to a blog post by Nancy Q. Smith titled Equality. In her post, Nancy talks about helping her first graders understand the meaning of the equals sign. The blog reminded me of one of the Starting Points I included in the current fourth edition of About Teaching Mathematics.  It’s titled “The Equal Sign: What It Really Means,” and this blog is an excerpt, focusing mostly on the section I included that addresses how I use the children’s book, Quack and Count by Keith Baker, in kindergarten and grade 1.

Students love Quack and Count. They are engaged and captivated by the rhymes and the illustrations of seven ducklings slipping, sliding, leaping, diving, and finally flying away. The first combination introduced in the book is 6 plus 1, shown with six ducklings on the left-hand page and one duckling on the right-hand page. The text reads:
Slipping, sliding, having fun
7 ducklings, 6 plus 1

My typical lesson plan is first to read the book just for students to enjoy. Then I revisit the book and, for each spread, have the students count the ducks on the left, then on the right, and then altogether. For each combination, I write an equation on the board or on chart paper to represent the rhyme and illustration mathematically. For the combination above, for example, I’d typically write 6 + 1 = 7. But after my own experiences and becoming aware of research about the equals sign, I changed that and now write the equation as 7 = 6 + 1.

The first time I taught the lesson with this change gave me a firsthand experience with what the research reported. I was teaching the lesson in a first grade class. After I wrote 7 = 6 + 1, I read the equation aloud. Russell, one of the students, frowned and emphatically shook his head back and forth.

“No,” he said, “you wrote it backwards. It’s supposed to go six plus one equals seven.” It was as if Russell had leapt off the pages of the research articles.

“Yes, that’s another way I could write the equation,” I said and recorded Russell’s suggestion.

Then, for each of the other combinations in the book, I wrote the equations both ways. At the end of the rereading of the book, I had written two columns of equations.
7 = 6 + 1          6 + 1 = 7
7 = 5 + 2         5 + 2 = 7
7 = 4 + 3        4 + 3 = 7
7 = 3 + 4        3 + 4 = 7
7 = 2 + 5        2 + 5 = 7
7 = 1 + 6         1 + 6 = 7

Also, as I read each equation aloud two ways, I alternated using “equals” and “is the same as” for the equal sign, hoping to develop understanding of its correct meaning.

About the Meaning of the Equal Sign
Plus and minus signs are operation symbols, indicating an action to be performed to the numbers at hand. The equal sign, however, is a relational symbol, not an operation symbol. No action is associated with it. Instead, it describes the relationship of equivalence between two expressions, a state of being, not the result of an action.

What often gets in the way of developing relational thinking is that students are used to focusing on computation to get answers, and also to read equations from left to right the way they read sentences. They’ve had much practice with both, so it’s no surprise that they see the equal sign as an indication that they need to write an answer. The suggestion above for using the children’s book Quack and Count is helpful, but it’s not sufficient for helping students broaden their concept the equal sign so that they understand that it’s an indicator of equivalence, not a signal that “the answer comes next.” For additional suggestions for older students, see About Teaching Mathematics  (Math Solutions, 2015), Starting Point 7, pages 32–35.

Another Helpful Resource
To build students’ understanding of equivalence and the equal sign, I rely on some of the lessons from Lessons for Algebraic Thinking: Grades K–2 (Math Solutions 2002), a book I coauthored with Leyani von Rotz (Math Solutions, 2002). Chapter 11, “Two Handfuls” is especially useful.