Does "=" equal understanding?

Do your students understand what the equal sign means?

Take a moment to reflect on that question.

Nearly every classroom teacher, when asked this question by the Student Work Study Teacher (SWST), answered “Yes!” or, “I sure hope so!” It’s probably reasonable to assume that student understanding of this concept is something that’s taken for granted by many educators. From an early age, students are shown “=” embedded in mathematics questions: from simple addition subtraction in the primary grades to more complex algebra and numeration in the junior and intermediate grades. In many ways, success in mathematics operations hinges on the idea of equality in a sentence -- simply put, knowing that 2 + 2 equals 4. If students are doing well in math, it reasons that they’ve got a firm understanding of this concept.

To check this understanding in junior and intermediate classrooms, the class is always give the same question. Students are asked to write down the problem on their paper or white board and to solve it.

4 + 8 = __ + 5 or 4 + 8 = __ + 7

Below is a representative sampling of some the answers students gave to this question.

In each classroom this task was given, anywhere from 70-100% of the class wrote an incorrect answer in the blank space. This usually comes as a surprise and/or shock to the classroom teachers:

"After some discussion with the SWST about how students would answer this question (4 + 8 = ? + 5), I was convinced that the majority of my students would ‘get’ this without any difficulty.  I certainly did not believe that many of them would write “12”.

“I asked students to think about the problem I wrote on the whiteboard, without talking to a neighbour or working out any math; just to think about it.  After a minute or so, I asked them to write what they thought would go in the box on their own whiteboards.  I was floored as I walked around the room and saw the overwhelming majority of responses being ‘12’!  I couldn’t understand it!”

After some discussion, some students offered that the answer must be 12 because 4 + 8 = 12.  They completely ignored the + 5 part of the equation."

Clearly, there is a trend: this misconception exists in several grades across multiple schools in this system. As a next step, we asked the students what the equal sign means to them:

“The equation before it needs to be equal to the one after it.”

“The first part before = has to equal second part.”

“The sum (or total) of 4 + 8.”

“Just a way of adding them up.”

“Equal to or the same as.”

“Equivalent equations that have to have the same answer.”

“The total of numbers you’re trying to find the answer to.”

“Equal up to ___.”

“The answer.

“The total.”

“The equal sign means they are equal to each other.”

“This sign means the sum of the question/equation.”

In his article A Balancing Act, Henry Borenson identifies two inherent meanings students can acquire for the equal sign - relational and “calculator”:

The use of the equal sign to indicate the unique numerical result of the sequence of computations that precede it (the “calculator use” of the equal sign) is a valid and necessary use of this sign...However, understanding the relational meaning of the equal sign also is essential for success in mathematics, and particularly in algebra. Whereas such equations as 3x + 5 = 26 can be understood knowing only the operational meaning of the equal sign (because 26 can be considered the result of the operations that appear to the left of the equal sign), examples such as 4 + 3 = __ + 2 and 4x + 2 = 2x + 6 cannot (Kieran 1992). In these examples, the equal sign indicates equivalence between two sets of expressions, each one of which includes one or more operations within it.

Thinking back to our initial question -- Do your students understand what the equal sign means? -- there is sufficient evidence to conclude that across the system students have a range of misconceptions about what the equal sign means in the relational context of the question 4 + 8 = __ + 7. The next question then naturally becomes: What can we do as teachers as a next step to immediately respond to this misconception?

How do you think your students will answer if you gave them the question above as a cold task? We would love to know what happens in your class - let us know in the comments below.

Thank you to guest blogger, Christopher St. Amand, SCCDSB Student Work Study Teacher. He can be found on Twitter at @MrStAmand 

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