This post is from Raquel Goya, a Kindergarten teacher at Hoover Elementary in Palo Alto, CA. Raquel noted that, “Although my kids are young, I want them to see themselves as mathematicians, capable of constructing arguments and thinking flexibly by understanding that multiple approaches exist to solving a problem.” She has seen how her commitment to number strings and related routines “invites enthusiasm in my classroom” and says that her students “take pride in seeing their thinking represented on the board and grow as they grapple with each type of problem.” We look forward to more contributions from Raquel.
In this number string, I wanted my kids to find a quick way to group and identify a certain amount (in this case, five). My students often feel satisfied once they determine how many and are able to articulate how they know. This exercise required my kids to step back and see if there were other, perhaps quicker or more interesting, ways to group quantities. It also encouraged the kids to make relationships among numbers.
When presenting each image, I asked “How many dots do you see?” and “How do you see them?” Although some students claimed they saw other quantities besides five, when pressed for how they saw them they indicated that the smaller quantity was only a part of the total. There was a surprising amount of enthusiasm to share their thinking and find their name on the board.
In the first image, Ethan and Jocelyn saw five in different combinations. As they shared their strategies, I represented their thinking so that the entire class would have two ways — hearing and seeing — to make sense of their ideas. I asked if there were any other ways — Jeffrey was certain he moved the dot from the center up and imagined an extra 6th dot, which he subsequently took away. It was almost as if he was making 2 even rows of 3 and then knew he needed to remove one. A few thumbs went up when I asked if anyone tried Jeffrey’s approach, and I thanked Jeffrey for sharing his thinking. It was surprising to have this approach suggested, but I think it helped open minds up for all the possibilities that exist for seeing and understanding a quantity.
[Click on images to enlarge]
The idea that you can re-arrange the dots and the quantity stays the same will support my kids to eventually think about fact families and the commutative property. And Jeffrey’s idea of adding one to the image and then taking one away is really a cool way of thinking about compensation, which will be a big idea for kids soon enough, when kids work on smart ways to do problems like 19 + 21.
As I hoped it would, image 1B solicited similar approaches to 1A. Two of the same number sentences came up (2 + 3 = 5 and 4 + 1 = 5). Audrey also volunteered a new way of seeing a quantity: by breaking it apart into multiple smaller quantities. She shared that she saw the dots as 2 and 2 and 1, which makes 5.
Jana followed this breaking apart strategy and explained her thinking when presented with the final image (1C). She envisioned the image as 2 and 2, which she knew was 4, plus one more, which makes five. As we listened to her strategy, I represented her thinking so that we all could see how Jana saw the 2, the 2 and the 1.
Amidst the controlled chaos of strategy sharing and discussing, Yu-Ming offered us the option that could might just realize it was five without counting or adding. He knew it was five, he said, “because it looks like a dice.” This piece of social knowledge, that not all children have, helped him here to recognize the quantity of five without counting. Typically, subitizing refers to our innate ability to see quantities of one, two, three and sometimes four without actually counting. But here he is using a familiar arrangement of five dots, and trusting it, to know it is five without counting.
Several students asked, “What about the number sentence?” I responded, “Sometimes, you look at an amount and you just know. If you didn’t add or subtract anything to get your answer, there’s no need for it!” With that, I concluded the string, leaving my kids to noodle on the variety of combinations and strategies they might try out next time.