Howdy howdy! After a brief hiatus, I wanted to start back up a discussion about the amazingness of children’s thinking. Tonight’s post is about subitizing, even and odd numbers, and division. I know that’s a lot in just one post, so I promise to connect them across!
In a post on 27 February, I tweeted out a picture of a child’s thinking (we will call him “Edgar” for now) during SEE Math program (website) who was working on a subitizing task inspired by the Intentional Talk book. But before we get to Edgar’s amazing thinking, we might need to first talk about, what in the world is “subitizing?”
Subitizing is to look at a collection of objects and to know how many are in that group (and in the total) without having to count individually or to estimate (Kaufman, Lord, Reese, Volkman, 1949).
(Disclaimer: the point of this post isn’t to go into super deep into the EXTENSIVE early childhood research about number discrimination, subitizing, and visualizing numbers. My research is not in early childhood numeracy so I would encourage you to explore the research of subitizing, which dates back to the 1940’s and 1950’s. This area is still extremely active today!) The research is clear that kids need to time and experience to learn how to recognize numbers that are “considerably more accurate, more rapid, and more confident process than estimating” (Kaufman, Lord, Reese, & Volkman, 1949, p. 520).
If that’s your end destination, then subitizing can be the Prius vehicle that gets you where you wanna go in the most efficient way.
So how does the notion of “subitizing” connect to the four operations? Well, when you learn to subitize, you can rely on your knowledge of repeated addition and multiplication (and sometimes the connection between the two processes). Kids can learn how to create “sub-groups” (to “SUBitize”, wink wink) and to efficiently determine the total value. You wanna add them up? Cool. You wanna find groups? SUPER COOL too. As kids learn how to recognize groups and practice seeing things in “groups of,” they can learn to connect complex mathematical ideas. Think that counting dots isn’t complex? Try this on for size: if I show you the following picture, how might you count the dots?
Some example responses might be from a child:
- I know this is 15 because I can count: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, (This isn’t subitizing, that’s just counting individually. But they are still counting so that’s cool.)
- I know this is 15 because 5+5+5 is 15” (This is like additive thinking or repeated addition).
- I know this is 15 because there are THREE GROUPS of FIVE (This is getting into the complex ideas of multiplication).
These justifications can connect to and across the operations of addition and multiplication. Note that the subitizing process gets more challenging as the total number increases and/or as the groups become less defined. Even with that being said, what if I gave you this picture below, could you count them just as a quickly?
It’s still 15 dots, but they aren’t grouped in a way that helps you to quickly count them. So the groupings helped, huh? Exactly the point of subitizing. If we wanted kids to strengthen more of their number sense to see things in “groups of,” (and to lay the foundation for multiplication), we can also lay the foundations for the notion of even and odd numbers (spoiler alert).
So let’s go back to Edgar and his dots. When Edgar saw the picture of 10 dots, there were a lot of different ways he could determine the total. I knew he would get the answer of 10, but I wasn’t sure how he would circle his dots.
Edgar looked at the picture, made two big circles around groups of five and said, “Oh, I know it is ten because I see two groups of 5.” I then asked him, “Is what you have circled altogether an even or an odd number of dots?”
First, let’s take a pausy-pause and think about what would you have said? Personally, as a student, I always remembered my teachers talking about even numbers being “pairs or partners.” Suppose I saw 11 dots. I should circle groups of two and if I found that there was a dot that didn’t have a partner or match, then it was “odd.” The “odd man out” scenario was how I always learned as a kid to determine the even or odd nature of a number. Looking back on my experiences, holy cow is that language problematic for many reasons (one of which it perpetuates a heteronormative stereotype of “you’re odd if you don’t have a partner.” Uhh yeah, no.) But the notion of odd numbers as “not having a partner” isn’t a new scenario. There is a history of thinking of even and odd numbers as a partnership (Zazkis (1998) talks about this in greater detail regarding the history of even and odd numbers in education).
The “odd man out/partner” justification can relate to the notion of measurement division (e.g., If you have 15 dots and you want to know if it is even, you can start “measuring” out 2s until you run out. If you can “measure out” all of your objects in pairs, then it should be even). This also connects to an idea of repeated subtraction (keep subtracting pairs from the total until you run out).
I assumed Edgar would use the pairing strategy to justify even and odd numbers because I still hear so many kids rely on this strategy in schools… but he didn’t. Edgar looked back at his picture and said confidently:
“I know it’s an even number because I just made TWO EQUAL groups.”
He just turned to me and smiled. He didn’t change anything about his picture when answering me. Everything about that picture communicated the total and the evenness of the total. And I loved it.
Note that Edgar’s justification wasn’t about measuring out partners or pairs. Instead, he used fairly divided the number of dots into TWO EQUAL GROUPS to determine the even/odd nature of the number. It didn’t matter that there was an ODD number in each group (five). What mattered is that Edgar created an even number of groups (two).
What I found so interesting about this whole interaction is how we expose kids to fair share ALL THE TIME when they divide whole numbers (I have four cookies and two friends, how many cookies does each child get?) … but then we expect them to rely on an idea of repeated subtraction/measurement division when learning about even/odd numbers (pair them up until you run out). I am excited to think about what else we as teachers can do with these subitizing tasks to helpkids see the complex nature of their own thinking!
Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The Discrimination of Visual Number. The American Journal of Psychology, 62(4), 498-525. doi:10.2307/1418556
Kazemi, E., & Hintz, A. (2014). Intentional talk: How to structure and lead productive mathematical discussions. Portland, ME: Stenhouse.
von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50(194), 191-218
Zazkis, R. (1998). Odds and ends of odds and evens: An inquiry into students’ understanding of even and odd numbers. Educational Studies in Mathematics, 36(1), 73-89.