Blog post #1 in the series, Lies We Tell Our Students
A while back, I was working in a fifth-grade classroom, and as a warm-up, I put the following equation on the board:
8 + 4 = 1 + 11
I then asked the students, “Is this statement true or false?” After waiting several seconds, I asked them to raise their hands to indicate their votes. At that point, two students voted for true, and the other 22 students voted that the statement was false.
Being the good teacher that I am, I asked the students to discuss their thinking, believing that those two students who said it was true would quickly convince the others. However, when I brought the class back together and once again asked my questions, 24 hands went up stating that the answer was false.
“Yikes!” I said, “What’s going on here? This statement is actually true. Can you tell me what you’re thinking?”
“Mrs. Rimbey,” started one of those original two students, ”8 + 4 does not equal 1.”
I’ve since come to learn that this is not an isolated case. Many students across the grade levels believe that the equal sign is a symbol that indicates the action of finding the answer, when, in fact, it actually signals a relationship between the expressions on each side. In the vast majority of classrooms, students develop this misconception because virtually every equation they see in the early grades follows the same format: 3 + 3 = 6; 4 x 4 = 16; 20 – 13 = 7; and so forth. The “answer” appears last.
So, what do we do about this? We can be sure to provide students with multiple daily opportunities to see equations written in different formats, pointing out that the equal sign indicates an equal relationship, not the action of finding the answer. Provide examples with the “answer first” (e.g., 6 = 2 + 4), with “nothing to do” (e.g., 1000 = 1000), or with “no answer” (e.g., 32 + 42 = 20 + 5).
This blog post is the first of a series that addresses the lies we tell our students. It’s important to note that these lies are not deliberate untruths. Rather, they are rules, procedures, mnemonics, and tricks we teach kids in an effort to make math easier for them. Sometimes the lies come in the form of omission, such as the example of the equal sign, shared above, where we don’t provide enough varied examples. Sometimes they come in the form of half-truths, such as addition and multiplication always make bigger, which is only true with natural numbers (not including 0).
To sum up this week’s Lies We Tell Our Students:
- The inadvertent lie: The equal sign indicates the action of finding the answer.
- The truth: The equal sign indicates relationship, not action.
As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you tried any of these (or other) strategies with your students? Please share your thoughts in the comments box, below.
Next Steps for Teachers: Use a variety of equation structures with students on a daily basis (see examples above). Consider making a set of 4-6 “true-false cards” each day to show the class. Simply write one equation on each card, using a variety of formats, making some true and some false. Then ask the students whether each is true or false and to justify their answers.
Next Steps for Leaders: Discuss the meaning of the equal sign with grade levels, focusing on the notion that the equal sign indicates relationship, not the action of finding the answer. Although this concept typically appears the K-1 math standards, this misconception continues to exist in middle school and beyond. Discuss ways to avoid this misconception in each grade level, and then share their ideas across grade levels.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.