Over the past year and a half, I have attended numerous CCSS trainings, read the standards and examined the CCSS learning trajectories. It is evident there is an emphasis placed on understanding of the properties of operations in the elementary grades. I don’t know about anyone else, but I remember it being taught to me as a lesson: Commutative Property is a+b=b+a… and such. No meaning behind it, simply some symbols, that if you could memorize and recite each, you were considered successful (as far as grades were concerned) in math class.
Fast forward to my second year as a K-5 math specialist. Having taught nothing below 5th grade in my previous 15 years in education, I am slowly wrapping my head around the depth of conceptual knowledge in grades K-1. I always knew K-1 was very “hands-on” but I have to admit, I really did not understand the complexity and beauty in the way kindergarteners “see” math until this year.
The other day I did a number talk with a class of kindergarten students. I displayed various dot images with anywhere from 5-10 dots arranged in different patterns. My goal was to have students subitizing the dot patterns and writing addition equations to match the groupings.
I flashed the first dot image on the smartboard for @ 2 seconds and the students wrote the number of dots they saw on their dry erase board. Students shared their answer with a partner and showed me their boards. I put the image back up and asked how they saw (visualized) the dots. We talked about different groupings, circled the dots for each, and practiced writing a couple equations together.
Feeling confident about the goals i had set for the number talk, i began to rethink them a bit after the following image:
Students quickly shared the answer of seven and then I asked, “How did you see the dots?”
The first student said,”I saw 2, 1 ,1,1, 2.” I had him circle the dots the way he saw them on the SMARTBoard and asked the students to write an equation for that grouping. Many successfully wrote a version (with some backwards 2s) of 2 + 1 +1+ 2+1=7. As I was looking around, I noticed one little girl had written all of the possible ways to arrange the 2s and 1s in the equation on her dry erase board. I realized at that moment, THIS is the commutative property in action! We shared all of the equations and I wrote them on the Smartboard. I posed the wondering to the class: How can these equations look different but still have the same answer? They talked to their neighbor and the common response was because no dots left the picture…not exactly what I was looking for, but good answer. I thought maybe it was too many numbers in the equation to see the commutative property or i just asked the question wrong, so i continued.
I asked for another way they saw it. Tons of thumbs went up (this is our sign for having a strategy) and the next student came to the board and circled 5 and 2. She knew it was a five, she explained because of a dice and she just knew two (there was the subitizing i wanted, but at this point we were going deeper). I asked students to write an equation for that grouping. They shared with their partner and we recorded 2+5=7 and 5+2=7. I was excited because two students had already written both equations on their boards before the share out. Now I posed the same type of question, worded differently, “What do you notice about the two equations we just wrote?”
I got responses like:
“The have the same numbers”
“Seven is at the end”
“Seven is the answer”
“He took my eraser” (all a part of the kindergarten learning curve)
“5,2,7 are there, mixed up”
I went with that comment and pressed further… “So how can the 5 and 2 be mixed up and still have the same answer?”
After a minute or two, one little girl said, “It’s just how you look at it. From that way (she pointed left) it is 2 then 5. If you look that way (she pointed right) it is 5 then 2.”
So there you have it teachers…the commutative property is “just the way you look at it.” Simple and beautiful.