Consider the problem “3 + 5 = ____ + 2”. Any elementary teacher will tell you instantly the most common WRONG answer is 8. The teacher will then lament that students just aren’t reading the entire problem. I’m not convinced this is the case. I believe students think that the equal sign is an operation.
An operation versus a comparison
In elementary mathematics students learn that there are 4 operations – addition, subtraction, multiplication, and division. An “operation” is an instruction to perform a task. So, for example, in the problem 3 + 5 we teach students to perform the addition of 3 and 5 and arrive at an answer of 8.
A comparison, on the other hand, has to do with determining if two numbers are the same or if one is larger than another. The common symbols for these comparisons of course are “=”, “<“, and “>”.
Unfortunately, it appears that many students treat the equal sign as an operation, meaning they are to perform a task. Think, for example, about a calculator.
When someone enters in “3 + 5” to the calculator, the operation is not performed. To perform the operation we must press the “=” button. Thus, we tend to start thinking about the equal sign as meaning an operation – performing a task – rather than a comparison.
Worksheets that treat the equal sign as an operation
But often we as teachers perpetuate the same issue, even if we don’t use calculators in our classroom. Consider this worksheet sample:
Notice how this is setup. The equal sign is subtly communicating the idea that it is OK to perform this operation now. Rather than giving the instructions “Compute the answer to the following problems” and then giving the problems as “1) 6 + 1”, we use the equal sign to communicate “do the operation.”
Thus, when a student sees “3 + 5 = ___ + 2”, they have been carefully trained when they see the equal sign to complete the operation before moving on, and so of course they come up with the incorrect answer of “8”.
So what do we do?
First, as teachers we need to recognize the difference between an operation and a comparison.
Second, we need to be extraordinarily careful in our use of calculators and how we present worksheets. Make sure that whenever we are using the equal sign it is to COMPARE two quantities and not to get an answer.
Finally, as early as possible embed problems such as “3 + 5 = ____ + 2” so students learn from an early age that the equal sign is to compare and not to compute.
Download a two page poster of Operations and Comparisons for free: https://www.teacherspayteachers.com/Product/Operations-and-Comparison-Poster-6607671
Explained this way, it makes so much sense that students make those mistakes that we call “linkage errors” when they get to AP Calculus!
Thank you so much for the detailed explanation to this visual-kinesthetic learner who has an aversion to mathematical operations (due to a very negative experience with several math teachers in my own schooling).
As a former Language Arts Teacher and Special Needs consultant, I’m always seeking resources to the Homeschool Whisperer FB group families I help with.
It was through a wonderful post your wife shared in our mutual special needs group that I found your article.