I was at my friend’s house the other day describing this cool thing I saw a student do with 8+8, but before I went into my description, I asked her soon-to-be-third-grade son “how do you think about 8+8?”
He looked up toward to the ceiling and said, “8 and 8 is…8..12..16!”
I said, “How did you get that?”
He said, “I just knew it.”
I said, “I heard you whispering ‘8..12..16’…what were you doing there?”
“Oh! I thought about how 8 is two fours, so if I have 8 and add a four I get 12 and then add another 4, it’s 16!”
I turned to my friend and said, “See, right there, he used his understanding of the structure of numbers to figure out 8 and 8 without counting on by ones. That’s what we want our students to do once they understand that they can count on as a strategy.”
Her son wasn’t thinking about this image below when he was solving, rather, he was using groups of four and probably adding through ten. He didn’t verbalize adding through ten which would be “8 and 2 more is 10, I have 6 more from the 8 and 6+10=16” so I don’t know if he did it.
Once kids realize that they don’t have to count from one anymore when adding two collections, we want them to start to use their knowledge of the structure of numbers so they can do math mentally without counting by ones like “I saw 8 and 8 more is: 8–9,10,11,12,13,14,15,16” while tracking the 8 on their fingers. They know when to stop because they see a 5 and a 3 on their hands and they know that 5 and 3 are 8, so they are using some structuring to track their count.
If we flash just the left side of the rekenrek to second graders and have them talk about what they saw, we might hear, “I saw 5 reds on the top and 5 reds on the bottom, that’s 10. I saw 3 whites and 3 whites, that’s 6. So 10 +6=16.” I’d want them to also make sure to say that they saw 8 on the top and 8 on the bottom, so 8 and 8 is 16. And that’s the cool thing that I saw a student do with 8+8!
{Learning to Think Mathematically with the Rekenrek is an excellent resource to guide you if you’ve never used a rekenrek, which is a math tool that should be used with the guidelines. Using a the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics is one of my favorite resources also. This Blog has links to even more guides and has video examples.}
We don’t want to start using the Rekenrek too early! We should be working on making sure that our students know all finger patterns on their fingers, can recognize regular dot patterns, and know all dot dice combinations. More to come in more blog posts. 🙂
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Whose Voice in Math Class? 