Blog post #2 in the series, Lies We Tell Our Students
We all know what a run-on sentence is. We’ve known since our early school years that run-on sentences are not okay. As this Grammarly.com quote states, “Run-on sentences are strings of complete sentences without sufficient punctuation to make them readable.” Today I’d like to coin a new term, run-on equations. And here’s the description for this new term: Run-on equations are strings of multiple unequal expressions connected with equal signs, making the entire equation untrue.
You’ve seen it…I know you have. Your students are explaining their thinking, and it goes something like this: “I added 3+4 to get 7. Then I multiplied by 2 to get 14. Then I multiplied by 5 to get 70. And then I subtracted 12….” And as they say these words, here’s what they write:
3 + 4 = 7 x 2 = 14 x 5 = 70 – 12
I’ve been in countless classrooms where students and teachers, alike, record their thoughts like this, using the equal sign as an “action” symbol rather than a “relationship” symbol. Have you witnessed this?
The (Inadvertent) Lie
We examined a similar problem last week. They’re thinking of the equal sign as a symbol of action rather than relationship. They’re using the equal sign to show the action of finding an answer and then continuing their thinking from there. The problem is that in doing so, they create a series of unequal expressions, making the entire equation false.
Now here’s the catch. Writing equations with multiple equal signs is not problematic, as long as all expressions are equal. Strings of multiple expressions can be correctly connected with equal signs if the expressions are truly representing equal quantities. Here’s a case in point from a first-grade classroom where students have written pairs of numbers that total nine on post-it notes and then put equal signs between each:
1+8 = 2+7 = 3+6 = 4+5 = 5+4 = 6+3 = 7+2 = 8+1
As you can see, this looks a lot like the run-on equation as recorded in the story, above. However, there is a distinct difference. This string of equations is actually true! The equal sign appears between two equal expressions every single time, and every single expression is equal to every other expression. This equation uses the equal sign to represent equal relationships rather than actions, and it is, therefore, not a run-on equation.
So, how would one go about correcting the “equation” in the story, above, to truthfully describe the situation? S/he would need to separate each equation, rewriting the answer from the previous equation as the “start number” for the new equation.
3 + 4 = 7
7 x 2 = 14
14 x 5 = 70
70 – 12
To sum up this week’s Lie We Tell Our Students:
- The (inadvertent) lie: When explaining one’s reasoning, it’s okay to use run-on equations to show the action taking place (a corollary to the lie, “The equal sign indicates the action of finding the answer” from last week’s blog post).
- The truth: Multiple expressions may be connected by multiple equal signs, but every expression, in its entirety, must be equal to every other expression (a corollary to the truth, “The equal sign indicates relationship, not action” from last week’s blog post).
As always, we love hearing from you! Do you think your students fully understand the meaning of the equal sign? Have you caught your students (or yourself) using run-on equations? Please share your stories in the comments box, below.
Next Steps for Teachers: First of all, model good practice for recording equations to represent thinking. Never, ever, ever, ever, ever, ever let yourself take a short-cut that leads to run-on equations. Secondly, explicitly teach the meaning of the equal sign as a symbol that represents an equal relationship between two expression. You might use the equality flashcard idea from last week’s blog post. Or, if your students are writing equations with multiple expressions and multiple equal signs, have them get out their highlighters. They can highlight each expression a different color and then determine if every single expression in the equation is equal to every other expression. If not, they probably fell into the trap (again) of using the equal sign as an action symbol rather than as a relationship symbol.
Next Steps for Leaders: During grade-level meetings, PLC meetings, or staff meetings, provide time for teachers to discuss the meaning of the equal sign. You may want to begin by replicating the story told above, recording with run-on equations, and continuing until someone in the room stops you. If no one does, then at some point, you’ll need to interrupt yourself and check in to see what they’re thinking. I cannot emphasize this enough…do not let this practice infiltrate your classrooms. Improper understanding of the equal sign will impact student learning for years to come, especially when the students get into algebra.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.