The Relational Meaning of the Equal Sign

A quick review of the research literature will reveal that many researchers believe that students have difficulty in learning the relational meaning of the equal sign. However, I claim that it is all a matter of instruction.

With proper instruction, even kindergarten kids pick it up very quickly. With kids that age we begin with 5 chips. We then spread them out in different ways and write down what we see. For example, we may get 2 + 3 = 4 + 1 if the first partition has two chips together and three chips together and the second has four chips together and one chip by itself. Another partition might be 1 + 2 + 2 = 3 + 1 + 1. Whatever the kids come up with is fine. With this approach kids naturally see how the equal sign is used in this context. The problem comes when educators expect kids to intuit the relational meaning from knowing the operational meaning. My recent article A Balancing Act from the September 2013 issue of Teaching Children Mathematics addresses this issue. Here is a link to a review of that article by Kim Marshall:


That Pesky Equal Sign
In this insightful article in Teaching Children Mathematics, consultant/author Henry Borenson explains how we use the equal sign = in two quite different ways: 

• First, operational, indicating the unique numerical result of the sequence of computations that preceded it – for example, 10 + 15 = ___. This is the way the equal sign on a calculator works – you push it at the end of a computation and only one number (the answer) pops up. 
• Second, relational, indicating equivalence between two sets of expressions, each of which includes one or more operations – for example, 4 + 3 = ___ + 6. 

Because most students start with the operational use of the equal sign, they have trouble dealing with a relational problem like 8 + 4 = ___ + 5. In fact, a study of hundreds of grade 1-6 students conducted in 1999 found that only five percent solved that problem correctly. 

“We can therefore conclude that the relational meaning of the equal sign is not something that students find intuitive or self-evident,” says Borenson, “nor is it an understanding that naturally follows from knowing the operational meaning of the equal sign.” In an ideal world, there would be a different sign for relational equations – perhaps a third horizontal line above the two in the conventional equal sign, or an arrow pointing in both directions. But we’re stuck with the dual-purpose sign. 

The challenge for teachers is that an operational understanding of the equal sign can hinder learning its relational application. When students are given the problem 8 + 4 = ___ + 5 and asked to fill in the missing number and justify their answers, most will say that 12 belongs in the space because “the answer follows the equal sign” – in other words, the equal sign triggers the operational definition in their minds. If students are asked, “What about the plus five?”, they say, “Maybe they just put it there to confuse us” or “It’s there to see if we can tell what’s important and what isn’t” (as often happens with extraneous information in story problems). 

Very few students will get the correct answer (7), and once the majority decide the answer is 12 and see most of their classmates agreeing, it’s very difficult to dissuade them – they’ll cling tenaciously to the way they’ve been using the equal sign. “This instructional problem will be compounded,” says Borenson, “if the teacher, in trying to teach the relational meaning of the equal sign, says that ‘the equal sign does not mean that the answer comes next.’ What then are the students to think? They know how they have used the equal sign countless times. Is the teacher asking them to discard their prior understanding? Resistance sets in.” 

Teachers need to anticipate this crucial juncture in math instruction and have a strategy to avoid the scenario just described. They’ve got to avoid triggering the operational meaning of the equal sign – and yet validate students’ prior experience with that use of the sign. 
Borenson recommends introducing students (in second or third grade) to the idea of balanced equations using concrete objects rather than numbers and the equal sign – perhaps a see-saw with weights on each side or a diagram and a question like, “If you have 3 oranges and 2 apples on one side, what will balance it on the other side?” Once students get the idea, the equal sign can be introduced with the balancing explanation. Studies have shown that if the relational meaning of the equal sign is introduced this way, students will get it, becoming “bilingual” in their understanding and use of the equal sign.

“A Balancing Act” by Henry Borenson in Teaching Children Mathematics, September 2013 (Vol. 20, #2, p. 90-94), http://bit.ly/1fvrbKS; Borenson is at henry@borenson.com. 

From the Marshall Memo #503

In my Hands-On Equations program we take a similar approach. In three easy lessons kids learn to solve equations such as 4x + 3 = 3x + 9, and in the process, they learn the relational meaning of the equal sign.

This is why I say it is all a matter of instruction.  When kids are presented with the relational use of the equal sign in context, without the teacher making a big fuss about it, kids pick it up very easily.