Blog post #3 in the series, Lies We Tell Our Students
The precision of our explanations matters! The words we choose and the way we put those words together can make all the difference between teaching children “tricks” and teaching for understanding. Imprecise language is another way we inadvertently lie to our students.
Today we turn our attention to the commutative property. Without thinking about it, we may very well say something similar to the quote in the illustration above, “With the commutative property, order doesn’t matter.” Hmmmmm…is that really the truth, or is there an inadvertent lie hiding in there somewhere?
Here’s the story I like to use to get things started. Let’s say my doctor tells me to take 3 pills per day for 9 days. So he gives me 27 pills. Now…is that the same thing as taking 9 pills per day for 3 days? The total is still 27 pills. In this context, however, 3 x 9 is certainly not the same thing as 9 x 3. In this context, order does matter! (Disclaimer, please do not use this context with children!)
The (Inadvertent) Lie
Without intending to do so, the teacher who says that “order doesn’t matter” with the commutative property is teaching an arithmetic trick that could very well lead students astray later on.
Instead, we want students to understand that the commutative property allows us to mentally add or multiply in either order to get to the same total (sum or product). When decontextualized, the numbers can be reversed and still provide the same result: the order can be changed to find the total of two addends, and the order can be changed to find the total of two factors. The language we use needs to be precise enough to indicate that the total is what stays the same, not necessarily the behavior.
Using the story from above, I like to represent 3 x 9 and 9 x 3 on a ten frame. In the pictures below, you can see 9 groups of 3 on the left and 3 groups of 9 on the right.
9 x 3 = 27
You can clearly figure out that the total is 27 for both 9×3 and 3×9. However, are they really the same thing? Is taking 3 pills per day for 9 days the same thing as taking 9 pills a day for 3 days? Of course not! The total is the same, but the context leads to entirely different behaviors!
To sum up this week’s Lie We Tell Our Students:
- The (inadvertent) lie: According to the commutative property, the order of the addends (or factors) doesn’t matter.
- The truth: According to the commutative property, you can add (or multiply) two addends (or factors) in any order and get the same total.
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 imprecise language about the commutative property? Please share your stories in the comments box, below.
Next Steps for Teachers: Take time to think through your mathematical explanations beforehand. During your planning time, think about the many concepts you will be teaching during the upcoming lesson. Mentally rehearse your explanations and listen for imprecise language that might get in the way of understanding.
Next Steps for Leaders: Initiate discussions during collaborative planning time. Ask groups of teachers to discuss concepts for which they might be using imprecise language. Here are a few ideas to get you started: rectangles have two long sides and two short sides, squares have four equal sides, perimeter is the outside, “top number” and “bottom number” for numerator and denominator, reducing fractions, borrowing & carrying, using the word “makes” for “equals.”
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.